HEAT TRANSFER IN BONE DURING DRILLING

Sean R.H. Davidson

A thesis submitted in conformity with the requirements for the degree of Master of Applied Science

Graduate Deparmient of Mechanical and industrial Engineering Institute of B iomatenals and Biomedical Engineering

University of Toronto

O Copyright by Sean R.H. Davidson 1999

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HEAT TRANSFER IN BONE DURING DRKLLING Sean RH. Davidson

Master of Applied Science, 1999 Graduate Department of Mechanical and Industrial Engineering

Institute of Biomaterials and Biomedicd Engineering University of Toronto

A numencd simulation was used to perform a parametric analysis of the thermal impact

of bone drilling operations The foliowing parameters were varied: dm rotational speed, feed

rate, helix angle, point angle, drill diameter, and the deasity, specific heat, and thennal

conductivity of bone. The maximum temperature and resultant thermal damage increase with

drill rotational speed. The temperature rises with feed rate kom 0.45 to 1.8 mm/s, and decreases

slightly up to the maximum feed rate of 4.5 mm/s. In the range of helix/point angie combinations

examined, the minimum temperature nse occm at 3 8 O helix angle and 1 30° point angle. The

maximum temperature increases wi th dri 11 diameter. The temperature and thermal damage

decrease with increasing thermal property values.

An experimental investigation of the thermal conductivity of bovine corticai bone was

also conducted. The conductivity was found to be 0.58 * 0.018 W/mK in the longitudinal

direction, 0.53 * 0.030 W/mK in the circumferential direction, and 0.54 S0.020 W/mK in the

radial direction. These results indicate that bone can be considered thermdly isotropie for

ACKNOWLEDGEMENTS

First and foremost, 1 wouid iike to thank Prot David F. James. His guidance was

invaluable. The respect that he showed me and the independence that he gave me to pumie

my own ideas (and make my own mistakes) led to a w m and niendly relationship that

made the research enjoyable.

Both Prof. C. A. Ward and Prof. S. Chandra were kind enough to provide advice at

various tirnes throughout the project.

Thanks to David Esdaile and the staff in the machine shop. Thanks also to Len

Roosman and Mike Smith for putting up with my constant requests for sharper saw blades

and better access to the student machine shop, and to Leo at Segovia Meat Market Ltd- for

granting my strange request for a regular supply of cow bone.

Financial support for the research was provided by the Natural Sciences and

Engineering Research Council of Canada (NSERC).

Finally, 1 wouid Iike to express my deepest appreciation to my fnends in the fnstitute

of Biomaterials and Biomedical Engineering and to the community of Massey College. They

helped to turn Toronto into a home. Without them, life would have far less meaning than it

does.

Table of Contents

Abstrac t

Acknowledgements

Table of Contents

List of Tables

List of Figures

1. Introduction

2. Literature Survey

2.1 Previous Investigations of the Thermal Properties of Cortical

Bone

2.1.1 Thermal Conductivity

2.1.2 Density

2.1.3 Specific Heat

2.2 Investigations of the Threshold for Thermal Necrosis

2.3 Previous Investigations of the Effect of Heat Created by Drilling

2.3.1 The Influence of Drill Speed, Applied Force, and Feed

Rate

2.3.2 The Influence of Irrigation

2.3.3 The Influence of Tool Wear

2.3.4 The Influence of Pre-Dnlling

2.3 -5 The Influence of Drill Geometry

2.4 Summary of Literature Survey and Objectives of Current

Research

3. Experimental Methods and Materials

3.1 Experimental Apparatus

3.2 Temperature Measurement and Calîbration

3.3 Preparation of the Bone Specimens

3.4 Experimental Procedure

3.5 Selection of a Material of Known Conductivity

3.6 Measurement of the 'Known' Themal Conductivity

3.7 Testing the Apparatuses

3.8 Evaporation h m the Specimens

4. Results and Discussion: Thermal Conductivity Experirnents

4.1 Thermal Conductivity Measurements of Bovine Cortical Bone

4.2 The Effect o f Evaporation

5. Cornputer Simulation of Bone Driliing

5.1 Description of the Problem

5.2 Application of the Finite Element Method

5.2.1 Mesh and Spatial Discretization

5.2.2 Temporal Discretization and Time Step

5.2.3 Implementation of the Boundary Conditions

5.2.4 Thermal Properties

5.2.5 Solution Method

5.3 Modelling the Heat Generated by Drilling

5.4 Modelling the Axnount of Thermal Damage Incurred

5.5 Testing the Computer Simulation

5.5.1 Analytical Solutions

5.5.2 Finite Element Code

5.5.3 Theoretical Drilling Model

5.6 Scope of the Computer Simulations

6. Results: Computer Simulation

6.1 Effects of Feed Rate and Drill Speed

6.2 Effects of Point Angle and Helix Angle

6.3 Effects of Drill Diameter

6.4 Effects of Bone Thermal Properties

7. Discussion of Simulation Results

7.1 Effects of Feed Rate and Drill Speed

7.2 Effects of Point Angle and Helix Angle

7.3 Effects of Drill Diameter

7.4 Effects of Bone Thermal Properties

8. Summary and Conclusions

8.1 Summary

8.2 Conclusions

8.3 Contributions

Appendix A - Raw Data: Thermal Conductivity Experiments

Appendix B - Raw Data: Testing the Apparatuses

Appendix C - Development of the FEM Numencal Model

C. 1 Introduction to the Finite EIement Method (FEM)

C.2 Spatial Discretization

C.3 Unsteady (r,z,t) FEM Solver

C.4 Boundary Conditions

CS Isoparametric Transformation

C.6 Gaussian htegration

C.7 Matrix Storage

C.8 Solvuig the System of Equations

C.9 Andyticd Solutions

C. 10 Flow Chart

Appendix D - Error Analysis

D. I Secondary Apparatus

D.2 Primary Apparatus

References

List of Tables

Table 2.1

Table 4. t

Table 5.1

Table 5.2

Table A. 1

Table A.2

Table A.3

TabIe B.1

Table B.2

Thermal conductivity of cortical bone.

Results of thermal conductivity experiments.

Comparison of heat values predicted by drilling theory vs. power

utikization in experïments,

Comparison of maximum temperatures: those predicted by the

simulation versus those measured experimentally.

Specimen dimensions.

Evaporation data,

Individual thermal conductivity measurements,

Data fiom control materiai conductivity measurements.

Results t?om testing the main apparatus.

viii

List of Figures

Figure 1.1

Figure 2.1

Figure 2.2

Figure 2.3

Figure 3.1

Figure 3.2

Figure 3.3

Figure 3.4

Figure 4.1

Figure 5.1

Figure 5.2

Figure 5.3

Figure 5.4

Figure 5.5

Anatomical directions of a Iong bone.

Tirne-temperature curve for thermal necrosis of epithelial cells.

Schematic drawing of a twist drill bit, iliustrating heik angle.

Schematic drawing of a twist drill bit, illustrating point angle.

Schematic drawing of the experimentai apparatus used to

measure the thermal conductivity of cortical bone.

CaIibration data for three thermocouples.

Schematic drawing of a cross-section of bovine fernur, with

typical specimen size and location highhghted.

Schematic drawing of the apparatus used to measure the thermal

conductivity of plastics.

Thermal conductivity in three anatornical directions.

Schematic drawing of a section of long bone with drill site

modeled.

Volume of bone modeled in the computer simulation.

Cornparison of temperature profiles for simulations run with three

different values of &.

Schematic drawing of the element mesh.

Temperature profiles for a point on the boundary r = Ri for

simulations run with two djfferent values of Np

Figure 5.6

Figure 5.7

Figure 5.8

Figure 5.9

Fiam 5.10

Figure 5.1 1

Figure 5.12

Figure 5.1 3

Figure 5.14

Figure 5.15

Figure 5.16

Figure 5.17

Figure 6.1

Figure 6.2

Figure 6.3

Figure 6.4

Temperature profiles for a point on the boundary r = Ri + 0.5 mm

for simulations run with two different values of Np

Variation of temperature profile with scaling factor.

Schematic of orthogonal cutting.

Zones of heat generation in orthogonal cutting.

Schematic drawing of a twist drill bit, illustrating point angle.

Schematic drawing of a twist driU bit, illustrating helix angle.

Hyperboiic streamline through the primary deformation zone.

Exarnple of a test of the steady FEM model.

Test of the convergence of the unsteady analyticai model.

Comparison of temperature histones from the unsteady FEM

model and the unsteady analytical solution.

Temperature versus time data for three points in a test of the

drilling simulation.

Comparison of heat generated by the drill and heat absorbed by

the bone.

Variation of heat generation with drill speed and feed rate.

Variation of energy absorbed with drill speed and feed rate.

Variation of maximum temperature with depth at a distance of 0.5

mm fiom the drill.

The average maximum temperature 0.5 mm fiom the drill, as a

fûnction of drill meed and feed rate.

Figure 6.5

Figure 6.6

Figure 6.7

Figure 6.8

Figure 6.9

Figure 6.10

Figure 6.1 1

Figure 6.12

Figure 7.1

Figure C. 1

Figure C.2

The average maximum temperature 0.5 mm h m the drill,

varying with feed rate at 100,000 rpm.

Changes in necrosis penetration distance with drill speed and feed 8 1

rate.

Average maximum temperature at 0.5 mm distance fkom the drill 82

as a function of helix angle and point angle.

Amount of thermal damage as a h c t i o n of heüx angle and point 83

angle.

Effect of drill diameter on average maximum temperature at 0.5 84

mm distance fiom the drill.

Changes in depth of necrosis penetration with increasing drill

diameter.

Average maximum temperature at 0.5 mm distance fiom the drill 85

as a function of bone thermal properties.

Average penetration distance of thermal necrosis, varying with

bone themal properties.

Plot of temperature versus distance fiom the drill in both the

longitudinal and the transverse (circumferential) directions, fiom

Abouzgia

Schematic drawing of volume modeiled by the cornputer

simulation.

Schematic drawing of the element mesh, with image of drill

placed for reference.

Figure C.3 Transformation to an isoparametric element.

Figure C.4 Isopararneû-ic element.

xii

Introduction

When mechanized cutting tools (saws, driils, etc) are used to eut a materiai, heat is

produced, which raises the temperature of both the tool and the matenal. In orthopaedic and

dental clinical practice, high-speed cutting tools are often applied to bone. The heat generated is

of particular concem in these operations because studies have shown that, if the temperature of

bone is raised above a threshold, the value of which depends on exposure t h e , thermal necrosis

will ensue 123,461. It is important to keep temperatures below this threshold since thermal

necrosis can have a negative impact on the outcome of surgical drillkg procedures. For example,

operations that require rigîd fixation of pins can fail because the implants become encapnitated

by soft tissue instead of new bone if the bone matrix around the pin is damageci [25]. Post-

operative complications can also result, as was demonstrated by Christie 1171; in four patients

treated for tibia1 hctures with skeletal traction, infections developed in the necrotic boue

surrounding the pin insertion site after the pins were removed. Furthemore, because the bone

was weakened, al1 four patients suffered subsequent fractures across the pin insertion defects.

The severity of the necrosis problem motivates the research in this thesis. Although the

problem has been investigated many times through experiment, conflicting results were

produced, leading to disagreement in the proper approach to surgical drilling. For example, holes

were drilled into bone specimens from a variet. of species and anatomical locations for the

purpose of determining the thermal impact of the drill rotational speed Cl, 12,48, 741. However,

there is no consensus on the optimal drill speed. Some recommend the use of low speeds, whiie

others claim that high drill speeds are more beneficial [28].

An aitemate approach to performing physical experiments is to evduate the temperature

distribution surroundhg the drill site with a mathematical model. The accuracy of such a model

depends on the reliability of input parameters, including the thermal properties. Although most of

the relevant physical properties for bone are weli established, a searcb of the literature reveals

that the thermal conductivity of cortical bone has been measured only a handfiil of times [8,33,

46,77,82], and that the hdings do not agree with each other. Furthemore, although it is weli

known that the mechanical properties of bone are anisotropic [20,59], o d y one investigator [82]

measured the thermal conductivity of cortical bone in different directions (shown in Figure 1.1).

radial

longitudinal + Figure 1.1: Anatomicai directions of a long bone.

An experiment was designed, therefore, to measure the thermal conductivity of cortical

bone in the longitudinal, circwnferential, and radial directions. The resuits of this experiment

then contributed to a compter mode1 that simulates the themal effect of drilling into bone.

CHAPTER 2

Literature Survey

2.1 Previous hvestigations of the Thermal Properties of Cortical Bone

2.1.1 Thermal Conductivity

Previous measurements of the

thermal conductivity of cortical bone are

summarized in Table 2.1. Surpnsingly,

the values span almost two orders of

magnitude, despite the fact that the

experimental apparatuses share the same

operating principle. The specimens were

subjected to a heat source, and the

thermal conductivity was calculated

fkom rneasurements of heat flow,

TABLE 2.1 : Thermal conductivity of cortical bone

Investigr tor(s) Coaduc tivity Notes WmKJ

Biyikii, Modest, 0.2 dry SarnPles Tarr 181 0 3 îÏesh samples

Zelenov [82] 12.8 axial 9.7 radial

1) Lundskog [46J ( 3.56 1 dry hvnvn lamples

Vachon et al. 0.601 dry 0x C771 2269 f iah ox

1 0.888 to 3.08 ( bovine specimens and caprine

Chato [33] 0.38 fiesh human samples

- t Conductivity was measured as a fhction of temperature.

temperature gradient, and specimen Values presented in the table are for T = 37OC-

dimensions.

In their study, Biyikli et al. [8] used a "cut-bar" apparatus. A 'sandwich' was formed

by placing the bone specirnen between two carbon steel bars. Heat was introduced at one end

of the sandwich by an electrically heated aluminum bar, and carried away at the other end by

a water-cooled heat sink. The apparatus was insulated to minimize transverse heat flow. The

heat flow through the apparatus was determined by measuring the temperature &op dong a

section of one of the steel bars, which had a h o w n thermal conductivity. Both the ârop in

temperature dong the bar and across the specimen were measured with thennocouples.

Zelenov's [82 ] apparatus also took the f o m of a sandwich, this time with the heating

elernent at the centre. Bone specimens were placed on either side of the heater. The

specimens and heater were then placed betsveen two large cylindrical containers such that the

exposed surface of each specimen was in contact with the base of a cylinder. The containers,

holding water maintained at a constant temperature, acted as heat sinks. The heat flowing

through each specimen was assumed to be equal to half the power supplied to the heating

eIement. The temperature drop across one of the specimens was measured with a differential

thermocouple.

Zeienov's experiment had two problerns. First, there is no mention of any insulation,

so heat was probably escaping into the environment, which Zelenov assumed was being

conducted through the bone specimens. Second, the heating element is descnbed as "a 0.1-

mm-diameter Constantan wire placed with a 1 -mm spacing and cernented between two layers

of vamished cambric." [82] The thickness of the heating element is not indicated. if it were

too thin, the temperature profile produced would not be uniform and the temperature drop

measured by the differential thermocouple would depend on the positions of its junctions.

While Zelenov's study is the only one to measure the thermal conductivity in different

directions (see Figure 1.1), the vdues differ so greatly fiom those of other investigators that

the accuracy of the results is questionable.

Instead of a sandwich design, Lundskog 1461 applied heat directly to the end of a bone

specimen with a resistance wire. The other end of the specimen was embedded in a copper

block, which acted as a heat sink. The temperature drop across the specimen was measured

by a differential thermocouple whose junctions were placed in holes drilled into the

specimen. The heat flow was assurned to be equai to the power fed through the resistance

wire.

The main problem with this experiment is that the only insulation mentioned is the

closed box into which the apparatus was placed. Although laboratory air currents were

elirninated, there was still a significant convective heat loss h m the specimen, and fiom the

resistance wire itself. That is, the assumption that al1 the power fed into the heater also

flowed through the spechnen is invalid. In addition, it is uncertain whether one-dimensional

heat flow was achieved, due to the holes in which the themocouples were placed, and to the

way in which the resistance wire was wrapped around the end of the specimen.

Vachon et al. 1771 used a completely different approach. The thermal conductivity of

bone was measured by a "thermal cornparator" technique. The conductivity was calculated

corn the difîerence in cooling rates of two copper spheres, one of which was in contact with

the specimen, the other held away fiom it. Vachon et al., however, question the reliability of

these measurements by indicating that m e r development of the technique is required.

As for the Iast two entries in Table 2.1, the results korn Chato's work are cited in

Huiskes [33], but a description of the measurernent method is not available. Similarly, the

results from Kirkland's work are cited in Krause et al. [40] and the original work is not

available.

It is unlikely that the natural variation in conductivity is as large as the range of values

listed in Table 2.1. The diversity in reported values is likely due to the problems discussed for

each apparatus. M e r comparing the different approaches, the apparatus used by Biyikli et al.

[8] seems to be the most accurate. In their apparatus, one-dimensional heat flow through the

specimen was achieved by placing the specimen between a heat source and a heat sink and

wrapping it in insulation. Furthemore, the heat flow was me& not assumed, as was

done by both Zelenov and Lundskog. These design elements reduce experimental eïror and

therefore an apparatus that incorporates these elements should provide accurate

measwements. The main drawback to Biyikli et al-% apparatus is the need to machine

specimens to a specific size. The design of the apparatus can be improved by making it

adaptable to specimens with different dimensions.

While Zelenov was the only researcher to measure the anisotropic nature of bone

thermal conductivity, other investigators have drawm conclusions about anisotropy fiom

indirect observations. From his drilling expenments, Abouzgia Cl] noted that the temperature

rise in the longitudinal direction was consistently greater than in the circumferential direction.

On that basis, he concluded that the thermal properties of cortical bone were anisotropic.

Lundskog [46] concluded the opposite when he observed circular isotherms surromding a

heated rod implanted in rabbit tibiae (the isothenns were obtained by thennography).

Given the disparity in conductivity data, and given that anisotropy has not been

sufficiently investigated, it is clear that M e r experiments are required. Furthemore, the

design of the apparatus used to measure the thermal conductivity of bone can be improved.

In addition to the themal conductivity, the other properties of bone relevant to the

curent research are the density and specific heat.

2.1.2 Density

Huiskes [33] summarized the results of three investigators who measured the density

of cortical bone. The range was 1 . 8 6 ~ 1 O-' to 2 . 9 ~ 1 o3 kg/m3, with an average of 2 . 2 ~ 1 O-'

kp/m3.

2.1.3 Specific Heat

In the same study, Huiskes summarized the published values for the specinc heat of

bone. While one researcher measured values in a range from 1 . 1 5 ~ 1 d to 1 . 7 3 ~ 1d J/k&

two researchers reported the same value, 1.26~ IO-' J/kgK

2.2 Investigations of the Threshold for Thermal Necrosis

Most investigations of the threshold temperahue above which ceil necrosis occurs

have been performed on epithelial and other soft tissues, as these are most ofien exposed to

elevated temperatures. One of the most thorough investigations of threshold temperature was

done by Moritz and Henriques [SOI. They measured the amount of time required to produce

damage to the dermal and epidermal layers of both hurnan and porcine skui over a large

temperature range (44°C to 1 00°C). They found that as the temperature increased, the amount

of time required to initiate thermal necrosis decreased, resulting in a time-temperature curve

simiIar to that shown in Figure 2.1.

l O 1 O0 200 300 400 500

exposure time [sec] 1

Figure 2.1 : The-temperature curve for thermal necrosis of epithelial celis. (The graph is based on data fiom Moritz and Henriques [50])

Few studies have looked at threshold temperature behaviour for bone, and the data

pertain to only a handfûl of discrete temperatures. Lundskog [46] measured a threshold

temperature of 5S°C at 30 seconds exposure. Eriksson and Albrektsson [23] established a

threshold temperature of 47°C at 1 minute exposure. The results fkom both studies are

consistent with those of Moritz and H e ~ q u e s .

2.3 Previous Investigations of the Effect of Heat Created by Drilling

The list of experiments presented in this section is not exhaustive. The survey is

limited to drilling experiments and to experiments which produced longitudinal cuts with

rotating tools. The scope was M e r limited to studies of drilling in bone, dentin and enamel.

Studies which examined the effect of cavity preparation on the pulp of teeth were not

included.

Since the present investigation continues the work of Abouzgia and James [2,3], their

results are presented hrst, followed by a brief history of the work that lead to their research

and to the present investigation.

Abouzgia and James [3] conducted drilling experiments on sections of bovine fmora,

examining the temperature nse in the region surroundhg the drill site under a range of loads

(1 -5 to 9.0 N) and with various rotational speeds (27,000 to 97,000 rpm, fiee-Nnning). They

did not use any irrigation. They found that the maximum temperature rise decreased with

increasing rotational speed, while the maximum temperature rise increased with force up to

3.6 N, then decreased as the force increased above 3.6 N. They explained this behaviour as

the result of a balance between two effects. As the force increases, the k a t generation rate

also increases. Conversely, higher force results in a shorter dribg time therefore, in total,

less heat is produced.

While Abouzgia and James were not the first to conduct such experiments, they made

a signi ficant contribution b y measuring the drilling speed throughout the course of the

operation, which had been done by only a srnaIl number of researchers before them.

Abouzgia and James found that the ro tational speed during the operation was lower than the

fiee-ninning speed, sometirnes by as much as 50% [2]. That there could be such a dramatic

decrease in the drill speed indicates that most of the work done by previous researchers, in

which only fiee-niMing drill speeds are reported, needs to be re-evaluated.

In the following bnef summary of previous bone drilling experiments, it will become

evident that the subject of bone drïlling is a complex one, and the debate continues on how

the results of such experiments should be reflected in clinical practice.

2.3.1 The Influence of Drill Speed, Applied Force, and Feed Rate

The drïlling parameters investigated most ofien have been the drill speed and the

applied load or feed rate. The influence of these parameters has been measured in two ways:

histological examination of the bone tissue, and measurement of the temperature rise at

various distances nom the drill site. Although temperature measurements are of greater

interest in the current research, histological studies provided the evidence that thermal

damage can occur and motivated the studies that measured temperatures Therefore, a

summary of the histological studies is presented fmt.

Investigations Based on Hiitological Examination of the Bone Tissue, Post-Operaton

Most of the early bone drilling studies examined the histological response of bone

tissue around the drill site to determine the effect of drill speed on thermal necrosis.

Both Thompson [74] and Pailan 1561 found necrotic tissue in the bone surrounding

pins after they were inserted, without imgation, into canine mandibles. Both researchers

measured the thickness of the necrotic band and found that it increased with drill speed fiom

125 to 2000 rpm.

In 1964, Moss [5 11 investigated the effect of higher rotational speeds, fiom 40,000

rpm to as high as 350,000 rpm. He drilled, without irrigation, into canine mandibles with a

number of different burs in three speed ranges and measued the aceliular zone around the

defects after a period of two weeks. He concluded that there was no disadvantage in using the

ultra-hi& drill speeds. While Moss' conclusion differs fkom that of both Thompson and

Pallan, the different drill speed ranges (125 to 2000 rpm vs. 40,000 to 350,000 rpm) may

explain the conflicting results. In that same year, Costich et al. 1191 performed a study similar

to that of Moss and obtained sirnilar results.

A year later, Spatz [69] published a cornparison of the early healing response (1,2,

and 7 days) of cuts made ui canine mandibles, without irrigation, by burs rotating at 12,000

rpm and 300,000 rpm. He showed that there was l e s trauma, and a faster healing response in

the cuts made at ultra-high rotational speeds. Spatz goes a step m e r than both Moss and

Costich et al. by stating that hîgher rotational speeds are preferable, rather than neutral-

The following year, Boyne [12] produced another study of the healing response of

cuts, this time made at 5000 rpm and 200,000 rpm, using a water spray at both speeds. He

concluded that although the short term healing response of cuts made at the higher rotational

speed was faster, there was no significant difference in the bng term healing response.

Boyne's results seem to consolidate the results of the previow three investigators. Spatz's

preference for higher drill speeds was based on short-term healing response while Moss and

Costich et al. looked at the bone after a longer penod of tirne, post-operation. However,

Boyne's results still confiict with those of Thampson and Pallan.

An investigation performed by Kramer [38] on extracted teeth indicated that use of

ultra-high ro tational speeds was detrimental. He found that heat damage was more likely at

higher drill speeds (320,000 rprn vs. 4000 to 8000 rpm), even when water spray was applied.

By cornparing the microscopic tooth structure after the drïlling experiments with dentine and

enarnel specimens exposed only to heat, Kramer was also able to establish that the damage

found at higher drill speeds was caused uniquely by heat.

Overall, histological studies seem to support the use of ultra-high rotational speeds

when drilling in bone. However, this hding is not compatible with the increase in thermal

darnage that was observed by Thompson and Pallan when drili speeds were increased within

the lower drill speed range (less than 2500 rpm).

Investigations Based on Temperature Rise During M i n g

In addition to his histological studies, mentioned at the begiming of the previous sub-

section, Thompson [74] aiso measured the temperature rise in bone in vivo during ske1eta.l pin

insertion. He found that the temperature rise at 2.5 mm and 5.0 mm fkom the drill site

increased with drill speed fkom 125 rpm to 2000 rpm. These results agree with those of his

histological study, and were later confimied by Pallan [56].

Four years afier Thompson, Rafel[60] published a similar study with opposite results.

Rafel drilled into human cadaveric mandibles with a variety of surgical drills without

irrigation and found that the maximum temperature nse occurred at the lower drill speed

(10,000 rpm vs. 350,000 rpm).

The study performed by Matthews and Hirsch [48] in 1972 was the most rigorous and

expansive at the time. They measured the temperature nse in human cadaveric femora over

ranges of loads (19.6 N to L 17.6 N) and drill speeds (345 rpm to 2900 rpm) using surgical

twist drills. The specimens were kept moist during the operation, but no imgation was

applied at the àrill site. They found that both the maximum temperature and temperature

dwation over 50°C decreased with increased loading. This was the first study to look at the

influence of applied load. The effect of increased drill speed was less pronounced but

followed a similar trend as the effect of increased force. These results contradict those

Thompson achieved 14 years earlier.

Vaughn and Peyton [78] driUed hto extracted molars without irrigation. They found

that the temperature nse increased both with increased drill speed (fiom 1 155 rprn to 1 1,300

rprn) and with increased ioad (fiom '/r to 2 lb.). These results agree with Thompson and

conflict with Maîthews and Hirsch, a fact that is particularly significant given that the speed

and load ranges overlap. Given that the different materials @ne, enamel, and dentin) &are

cornmon constituents and similar patterns of organization, it is doubtfid that the contradiction

in these findings is due solely to differences in materiai behaviour.

Sorenson et ai. [67] used a calometric approach. They drilled, without irrigation, into

blocks of dentine and found that the amount of heat tramferreci to the specimen increased

with increasing load fkom 0.3 N to 0.5 N (30 grarns to 50 grams), then decreased as the load

increased. The heat versus load behaviour is similar to the maximum temperature versus load

behaviour measured by Abouzgia and James [3]. Aithough Uie fkee-ninnuig speed was held

constant at 250,000 rprn throughout the experiment, the rotational speed was measured

during drilling and found to decrease dramatically as the load increased. This decrease in dm

speed was also noted by Abouzgia and James 121.

In 1982, Krause et al. [40] cut troughs, without irrigation, in cadaveric bovine femora

using rotating burs. The feed rate and depth of cut were varied. Two rotational speeds

(20,000 and 100,000 rpm) and two bur types were used. Although cutting troughs is not the

same as drilling, Krause et al. noted that the maximum temperature decreased with increasing

feed rate (fiom 1.80 to 6.35 d s ) . The effect of increasing the rotationai speed was

ambiguous. The temperature dropped signiticantly at lOO,Oûû rprn for one bur type, but not

for the other- Forces in the cutting direction were measured, but no correlation with

temperature nse was found. It should be noted however that a higher feed rate requires a

higher applied force, and thus the results of Krause et al. are similar to those obtained by

Matthews and Hirsch-

More than a decade afler Matthews and Hirsch examined temperatures during bone

driing, Matthews et al. 1471 measured temperatures durhg skeletal pin insertion into human

cadaveric long bones. No irrigation was used. Manual drilling, which produced rotational

speeds between 60 and 120 rpm, was compared to the use of electric drills at 300 rprn and

700 rpm. The maximum temperature nse was recorded at 700 rpm, foliowed by hand drilling,

followed by dnlling at 300 rpm. Conversely, the maximum temperature duration above 50°C

occured d u h g hand drilling, followed by drillhg at 300 rprn and then 700 rpm-

The results of the temperature measurement studies are divided into two groups: those

that indicate an increase in temperature with drill speed and applied load and those that

indicate the opposite. Furthemore, although most histological studies were performed at

ultra-high drill speeds, only Abouzgia and James [3] measured temperatures at these speeds.

2.3.2 The Influence of Irrigation

Several investigators [43,48,51, 73,791 studied the effect of irrigation on the

maximum temperature rise in bone and, not surprisingly, found significant decreases in

temperature were achieved when imgation was used. Other studies compared the effect of

interna1 irrigation, where the coolant is fed to the tip of the drill through channels in the drill

shaft, to that of extemal irrigation, where the coolant is applied to the surface of the drill at

the point of entty. Laveile and Wedgwood [43] compared the effectiveness of both irrigation

methods under low-speed (350 rpm) and moderate force (19 N), and found that intemal

irrigation decreased significantly the peak temperatures adjacent to the cavity.

Because irrigation has been proven to reduce the heat impact h m driilïng, one might

be tempted to ignore other aspects of drilling technique and rely solely on irrigation to

prevent thermal necrosis. There are still, however, good reasons to continue the investigation

of other dri1ling parameters, First, intemal irrigation is not dways posmile, especiaiiy for

srnaller tools. Furthemore, a histological cornparison of extemaily and internaliy cwled

implants, performed by Haider et al. [29], produced conflicting results. Near the surface of

the bone, extemal cooling proved more effective. There was less damage at deeper levels

when intemal irrigation was used. However, Haider et al. indicated that the reduced amount

of damage could be due to better thermal properties of the cancellous bone at those levels.

Second, external irrigation is not always effective, especially when the ratio of hole depth to

drill diameter becomes large. Eriksson et al. 1241 recorded temperatures as high as 96°C in

clinical drilling studies where irrigation was employed and Tetsch [73] noted that the

temperature in feline jaws increased above 100°C despite the use of cwlant. Third, when

skeletal pins are inserted, the tight fit required leaves little or no room for irrigation fluid to

flow into the hole.

2.3.3 The Influence of Tool Wear

Matthews and Hirsch [48] compared the performance of du11 (-200 uses) and

relatively sharp (C 40 uses) surgical twist drills. Not surprisingiy, the wom tmls produced

greater temperatures than new tools.

2.3.4 The Influence of Pre-Drilling

Matthews and Hirsch [48] determined that drilling a 2.2 mm hole and subsequently

enlarging it to 3.2 mm generated a much lower temperature rise than dnlling the 3.2 mm hole

directly. Matthews et al. 1471 also found that pre-dming signincantly decreased the

temperature nse during skeletal pin insertion.

2.3.5 The Influence of Drill Geometty

Food Oimtisn

I

Figure 23: Schematic dr&ing of a twist drill [modified fiom 641.

Figure 2.2: Scbematic drawing of a twist drill (modified fiom 1641).

Figures 2.2 and 2.3 contain schematic drawings of

a typical twist dm. The geometric features of interest in

the current research are the point angle (Iabelled 2p in

Figure 2.3), the helix angle (Figure 2.2), and the drill

diameter. Researchers [3 5,271 have investigated ways to optimize the design of the drill bit

without concem for the temperatures generated. Their cnteria were mechanical: faster

penetration rates, better chip clearance, and improved hole quality.

Saha et al. [64] sought to decrease the temperature by m o d i w g the design of the

standard surgicd driil bit. These new drill bits had a larger point angie (1 18') than the

standard bit and had helix angles fiom 34" to 36". The temperatures generated with the new

bits were 41 % lower than those created with standard sutgical drills of the sarne 118"

diameter. The temperatures generated by the new 1/8" bits were even lower than those

generated by the standard 3/16" bit. The geometry of the new bits is similar to those proposeci

by researchen who were concerned only with the mechanical aspects of the drilling

procedure.

2.4 Summary of Literature Survey and Objectives of Current Research

Even with the limitations on the scope of the survey, the variation amongst cirihg

experiments is large and thus it is difficult to summarize the results in a concise façhion.

With the exception of Kramer, the histological studies indicate that the use of ultra-

high drill speeds is no more damaging than the use of lower speeds, if not preferable.

Unfortunately, concIusions on the thermal impact of high drill speeds are limited to those

fiom histological studies because few temperature measurements have been made at ultra-

high speeds. There seems to be greater uncertainty in the studies conducted at lower speeds.

In some studies, temperature and resultant thermal damage increase with drill speed [74,78],

while in ouiers it decreases [48]. Matthews and Hirçch [48] found a decrease in temperature

with increasing applied load, both Sorenson et al. 1671 and Abouzgia [l] found a more

complex relation. Some investigations [46,38] had conflicting results within their own &ta.

The main problem in interpreting the results lies in the number of variables. The

range of drill speeds used by dental surgeons is different than that used by orthopaedic

surgeons, and it is difficult to compare the results of experiments that use different drill speed

ranges. The confusion over the role of drill speed might be aileviated if the entire range, fiom

very low to ultra-high, could be investigated under otherwise identical conditions. It is also

difficult to compare results fiom experiments with an applied force to those with an applied

feed rate, assuming these parameters are controlled, which occured oaly in a minority of the

studies examined in this chapter. Some studies used water irrigation, some used forced air,

some didn't use any irrigation. There was variation in the design of the différent twist drills;

b m , reamen, and diarnond tools were also used. It is not surprishg that results fiom these

studies are not in agreement.

The picture might be made clearer if an experiment were designed which used a

parametnc approach. In a parameaic analysis, one of the parameters - drill rotational speed,

feed rate, applied force, tool design, or tool geometry - is varïed while al1 others are held

constant. The importance of that parameter is deterxnined by measuring the change in thermal

impact caused by varyuig that parameter.

The number of permutations of the diflerent drillhg parameters makes the size of a

physical experiment daunting. Physical experiments also sufTer fiom other limitations. For

example, in many of the studies, precise positioning of temperature measurement devices

(usually thermocouples) was difficuit and oAen required drilling separate holes to

accommodate them. Furthemore, in al1 the experiments that measured temperature, it was

measured only at discrete locations or on the surface of the boue (using thermography). In

order to determine the full impact of the heat generated by drilling, the three-dimensional

temperature distribution is required-

The effect of parametric variation could be assessed if an accurate mathematical

mode1 of the drilling operation were developed. A mode1 would make the scope of the

problem more manageable and it would provide a more detailed picture of temperatures

generated and the amount of damage incurred. Hence it is proposed that a cornputer

simulation be used to nui the aforementioned parametric analysis.

A mode1 is only as good as its input data As was discussed in Section 2.1, the

physical properties of bone have been satisfactorily established, except for thermal

conductivity. The variation in published results for that property is large. Moreover, the

anisotropy of the conductivity has not been adequately established.

The study of the thermal effect of drilling in bone needs to continue on two hnts ,

both of which are tackled in the present investigation.

(1) The themai conductivity of bone needs to be measured in the longitudinal,

circumferential, and radial directions (see Figure 1.1) and the degree of anisotropy of the

conductivity needs to be determïned.

(2) The importance of individual driiling parameters and materiai propeaies on the

thermal impact of drilling in bone needs to be investigated.

The f k t objective was met by designing and building an apparatus capable of

measuring the thermal conductivity of bone in the three desired directions. Measurements

were taken for specirnens of bovine cortical bone. The second objective was accomplished by

performing a parametric analysis of the dnlling operation with a numericai simulation.

Experimental Methods and Materials

3.1 Experimental Apparatus

The main purpose of the apparatus was to create one-dimensional heat flow through a

bone specimen. Once this condition was established, the thermal conductivity was calculated

with the equation

where

kb = thermai conductivity of bone (in the direction being measured),

Q = heat flowing through the bone specimen,

tb = thiclcness of the bone specimen,

& = cross-sectional area of the bone specimen,

ATb = temperature cirop across the bone specimen.

Of the apparatuses discussed in Chapter 2, those that seemed to be the most

successfui in creating the required one-dimensional flow were of the sandwich variety [9,l8,

451, i.e., the specimen behg tested was placed behiveen a heat source and a heat sink and was

surrounded by insulation so that heat flowed through the specimen in only one direction. The

apparatus used in the present study to measure the thermal conductivity of cortical bone is

shown schematicaily in Figure 3.1.

Figure 3.1 : Schematic drawing of the wperimental apparatus useü to meâsuic the t h e d conductivity of cortical bone. A - specimeo, B - heated alumînum plate, C - cooled aïuminum pïate, D - heating element, E -

water bath, F - aquarium anll, G - indation, H - Plexiglas block, 1 - thennocouples, J - steel rods.

The bone specimen (A in Figure 3.1) was placed between two aluminum blocks (B

and C), each approximately 2" by 2" by W. A heating element (D) was made by embedding

coils of resistance wire in thermally conductive cement. The cernent 'glue& the element to

the upper aluminum block (B). The lower block (C) was partially submerged in water (E).

This was accomplished by holding the apparatus against the wall of an aquarium (F') with a c-

clamp (not shown). Heat l o s ~ fiom the specimen was rninimized by surrounding it with foam

insulation (G). The insulation also served to minimire evaporation fiom the specimen.

Power was fed into the heating element from a BK Precision DC power source

(mode1 1610A). In order to rneasure the heat flowing through the specimen, a block of

Plexiglas (H), whose thermal conductivity was measureü separately (see Sections 3.5 & 3.6),

was placed in series with the specimen. Each Plexiglas block was machineci such that its

cross-sectional area was close to that of the bone specimen being tested.

The heat flow Q flowing through the bone specimen was assumeci to be q u a i to that

flowing through the Plexiglas block and was calculated using equation (3. l), re-arrangeci and

applied to the Plexiglas block.

where

ko = thermal conductivity of the Plexiglas block, 0.228 W/mK,

& = cross-sectional area of the Plexiglas block,

AT0 = temperature drop across the Plexiglas block,

to = thickness of the Plexiglas block.

Substituting the expression for Q into the expression for kh

The temperature drops across both the Plexiglas block and the bone specimen were

measured by three thennocouples 0. The measurement junction of one of the thennocouples

was placed between the heated aluminum block (B) and the Plexiglas bIock (H), another

between the Plexiglas block CH) and the specimen (A), and the third between the specimen

(A) and the heat sink (C). Thermal contact between al1 elements in the senes was aided by

applying thermally conductive paste (OMEGATHERM 201) on the contact surfaces of both the

PIexigIas block and the bone specimen.

The second goal in the design of the apparatus was to minimize the amount of

machining required to prepare the bone specimens. In al1 previous experiments, specirnens

had to be cut to a specific size, requiring elaborate measures to keep the specimens moist and

to minimize thermal damage during specimen preparation. In contrast, the current apparatus

could be adapted to the dimensions of individuai boue specimens. Different specimen cross-

sectionai areas were accommodated by machuiing Plexiglas blocks with the required area.

Differences in thicbess were handled by slidùlg the heated block dong two guide rods (J in

Figure 3.1).

3.2 Temperature Measurement and Calibration

Temperatures were measured with thermocouples. The thermocouples were type K

and had a junction bead diameter of0.3 mm. The size of the bead was important because the

junctions were placed between two rigid objects, and this created a gap between the two

objects. Although the themally conductive paste eliminated the air in the gap, it was

desirable to rninimize the gap wvidth by using tbemocouples with srnail bead diameters. The

estimated temperature drop across the gap was very small (c 0.2OC) and was not expected to

affect significantly the conducfivity measurements. The test conditions were such that the use

of type K thermocouples was appropriate [SI. The signais fkom the three thennocouples were

fed into an amplifier built for previous research in this laboratory Cl]. The amplified signals

were then fed into a Fluke digital multimeter (mode1 8050A).

The thennocouples were calibrated against a laboratory themorneter in a heated water

bath and were checked periodically to assure linearity of the calibration curve. Sample

calibration data are provided in Figue 3.2.

O ' ,

O 0.1 0 -2 0.3 0.4 0.5 0.6

Voltage [V1

OTC #6

BTC #7 ATC #8

Figure 3.2: Caiiiration data for three thennocouples (TC).

3.3 Preparation of Bone Specimens

Bovine bone was selected because it is readily available and because it is stnichirally

similar to human bone [36]. Bovine bone has also been used in previous studies of themal

effects fiom drilling [1,40, 801.

Figure 33: Schematic drawing of a cross-section of bovine femur, with typical specimen sizc and location highlighted.

Slices of the mid-shaft of bovine femora were obtained fiom a local butcher, The

slices were stored in a fkeezer fkom the time they were cut fiom the bone to the time they

were sectioned to obtain specimens. Individual specimens were taken fiom sections of the

femur slices which had minimal curvature (see Figure 3.3). A band saw was used to cut the

fernur slices. When the blade was sharp, the cuts were made with minimal trauma to the

bone. Charring, smoke, or boiling around the blade was taken as evidence that the

temperature generated by the cutting was excessive. Whenever tbis occurred, the specimens

were rejected, and the blade was replaced.

After it was separated fiom the femur slice, the specimen was wrapped in gauze that

had been soaked in mammalian Ringer's solution. (Eünger's solution is used to rnimic

physio logical fluid. It is made b y dissolving sodium chloride, potassium chioride, glucose,

sodium bicarbonate and calcium chloride in water-) The specimen, thus wrapped, was placed

in a re-sealabfe plastic bag (ZipIoc). Individual specimens were stored in a fieezer until they

were used in tests.

3.4 Experimental Procedure

The specimens were oriented in the apparatus such that heat flowed in one of the three

directions in which the themal conductivity was desird: longitudinal, circumferential, or

radial (see Figure 1.1). At least four measurements were taken for each bone specimen.

Before each measurement, the specimen was weighed and then the apparatus - with

the specimen, the Plexiglas block, the insulation and the three thennocouples - was

assembled. The apparatus was secured against the wall of the aquarium with a c-clamp, and

the water level in the aquarium was raised so that the bottom alurninurn plate was partially

submerged. Power l a d s were then attached to the heating element, and the power source

turned on. Starting at l e s t two hours later, temperature readings were taken every 15 minutes

until the system reached steady state. It was assumed that this state was reached if three

thermal conductivity measurements in a row were identical to the three significant figures. A

set of three identical readings out of four consecutive measurernents was also accepted as

steady state.

After each test, the insulation was removed and the set-up was exarnined to determine

if alignment of the specbnen and the Plexiglas block, the thermal contact, and the

thermocouple positions were acceptable. Poor alignment, poor thermal contact, and an

Mproper thermocouple position were grounds for discarding the results.

The specimen was removed, wiped clean, and weighed again. The change in m a s of

each specirnen was recorded to detennine the effect of evaporation. This topic is dealt with in

detail in Section 3.8. The specimens were stored in re-sealable plastic bags between tests;

some were re-wrapped in the Ringer's-soaked gauze, some were not. The variation in the

conductivity measurernents of individual specimens was examinecl and there was no obvious

difference in the pattern of variation between specimens that were re-wrapped in the gauze

and those that weren't.

3.5 Selection of a Material of Known Conductivity

There were two requirements for the material used to measure the heat flow through

the specimen. It had to be easily machineci and it had to have a thema1 conductivity

comparable to that expected for bone. For these reasons, a number of plastics were

considered. A drawback with plastic is that the thennai conductivity for any given batch nom

a manufacturer is not known preciseIy. It was therefore necessary to measure the thermal

conductivity of the plastic using a separate apparatus-

3.6 Measurement of the 'Known' Thermal Conductivity

The apparatus used to measure the conductivity of the plastics is shown schematically

in Figure 3.4.

Figure 3.4: Schematic drawing of the apparatus used to measwe the t h e d conductivity of plastics. A - fiIm heater, BI& B2 - steel plates, C - specimens, I) - thermocouples, E - indation.

A sandwich was formed with a Kapton flexible heater (A) at the centre and, on each

side of the heater, a 2" square piece of steel (BI), a 2" square piece of the material being

measured (C), and another 2" square piece of steel (B2). The inner steel plates were used to

smooth out the temperature profile generated by the heater, and the outer plates were used as

support, and as heat sinks. A small depression had to be cut into one of the inner steel plates

to accommodate the junction where the power leads attached to the heater. Three

thermocouples 0) were placed on each side of one of the plastic specimens. Tbennally

conductive paste was applied between al1 layers. The entire sandwich was surrounded by

approximateIy 1 %" of polystyrene insulation (E - a ciiffereut materiai than that used in the

main apparatus), into which a groove was cut to accommodate the power leads and

thermocouple wires (not shown). It was estirnated that, with the indicated iosulation

thickness, the heat loss through the sides of the apparatus was iess than 10% of the power

input. The temperature &op across one of the specimens was calculated with the equation

A T = T ; ~ - ' ~ (3-4)

where

Tiavg = average of the three thermocouple measurements on the heated side of

the plastic specimen,

Travg = average of the three thermocouple measurements on the cwled side of

the plastic specimen.

The power fed into the heater was caiculated as the product of the voltage and current,

both of which were measured by the digital mdtimeter. It was assumed that half of the heat

generated passed through each specimen. It was thus possible to calculate the thermal

conductivity using equation (3.1). Four materials were tested: Plexiglas, PVC, Teflon, and

Delrin. The tests were run in a manner similar to those for the measurement of the thermal

conductivity of bone. Three hours after the heater was turned on, thermocouple readings were

taken at 15-minute intervais. If three consecutive conductivity values (or three out of four

consecutive values) were identicai to three significant figures, it was assumed that steady

state had been achieved and the conductivity was recorded. The data are presented in

Appendix B.

The measurements of al1 four materiais demonstrated good repeatability. The choice

of Plexiglas for use in the calculation of the heat flow was based on the results of tests of the

main apparatus, which are presented in Section 3.7.

3.7 Testing the Apparatuses

Since there were two apparatuses in this investigation, both for measuring thermal

conductivity, two separate terms will be used to reduce the possibility of confusion. The

apparatus used to measure the thermal conductivity of bone, described in Section 3.1, will be

referred to as the primary or main apparatus; the apparatus used to measure the thermal

conductivity of the plastics, described in Section 3.6, will be referred to as the secondary

app aratus.

After improvements, the secondary apparatus produced a set of thennal conductivity

measurements for Plexiglas wvith a standard deviation of less than 2%, which was considered

sufficiently accurate. The value obtained for Plexiglas, 0.228 W/mK, feii slightly above the

range of values published in engineering reference texts (0.2 W/mK [75], 0.187 W/mK [53]).

Once the thermal conductivity value for Plexiglas had been detennined, testing of the

main apparatus began. Tests were performed with two Plexiglas blocks in series, one used as

the test specimen, the other to calculate the heat flow. The fïrst set of tests resulted in thennal

conductivity values which were consistently higher than those expected for Plexiglas, and

repeatability was less than acceptable. The addition of more insulation led to a substantial

improvement. A set of 11 tests resulted in an average thermal conductivity which was 2.6%

higher than expected, with a standard deviation of 2.7%. Since the measured conductivity

was close to the expected value, it was assumed that the heat loss was minimal and that

therefore the amount of insulation was sufficient. The positive resuïts with Plexiglas led to

tests with other plastics. The themai conductivity of both PVC and Teflon were measured in

the same fashion as that of Plexiglas. The repeatability of these measurements was as go04 if

not better than for the Plexiglas measurements (see Appendix B).

The error in a series of tests using a combination of Plexiglas and Teflon specimens

was close to -20%. This large error was puzzling, given that subsequent tests with two

Plexiglas specimens still produced measurements within 5% of the expected value. It was

noted that the s m d Teflon blocks used in these tests were prone to melting when machined,

so Teflon was replaced with DeIrin (acetyl copolymer), which was more resistant to the

heating effects of the machining process.

The results corn a series of tests performed after the substitution of Delrin for Teflon

are presented in Table B.2. Steps were taken to reduce the error and improve the

repeatability. Thermocouples were re-calibrated or replaced. More care was taken in

removing excess conductive paste. The insulation was replaced. None of these changes

helped reduce the error to the point that it was consistently under the 5.4% predicted by error

analysis (presented in Appendix D), nor could any pattern be discerned which pointed to the

reason for the Iarger error.

It was decided, for several reasons, that the project should proceed. First, thne did not

permit a more thorough investigation of the problem. Furthemore, it was conceded that an

error above 5% could be accepted, given the expectations for a Master's level project. There

was also the expectation that the biological variation of the thermal conductivity of cortical

bone would be about 10%.

Plexiglas was chosen for use in the main apparatus because the tests that used

Plexiglas to measure the heat flow generally had lower experimental error.

3.8 Evaporation from the Specimens

Water is a constituent of bone tissue. The insulation in the experiment, in addition to

minimizing heat loss, also minimized evaporation. No attempts were made, however, to seal

completely the surface of the specimen. It was therefore necessary to detemillie the

importance of the evaporation which did occur.

In previous experiments, the amount of evaporation was generally measured by

weighing the specimen and determinhg the amount of mass loss over the course of a test. For

instance, Zelenov [82] found that as the temperature of the specimen was raised fiom 20°C to

9S°C, the moisture content decreased fiom approximately 1 1% by weight to approximately

4% by weight. Clattenburg et al. [18] attributed a "steady decline in conductivity" of

cancellous bone to moisture loss. in his measurements of thermal conductivity, Lundskog

[46] noted fluctuations in the temperature gradient, which he attributed to "evaporation nom

the surface of the specimen."

Hence there is suaicient evidence that evaporation fiom bone specimens occurs

during tests. The effect of evaporation on conductivity measurements seems to Vary with the

test conditions and it was not possible to know a priori whether the evaporation would be

significant in the present study. Therefore, the effect of evaporation was determineci by

measuring the mass of a specimen before and after each test and calculating the percentage

mass loss.

Results and Discussion: Thermal Conductivity Experiment s

4.1 Thermal Conductivity Measurements of Bovine Cortical Bone

Table 4.1 summarizes the measurements of cortical bone thermal conductivity, which

are also shown graphically in Figure 4.1.

T ~ L E 4.1 : Results of thermal conductivity experiments

No. No. No. k Stan. Dev. Direction Specimens Animais ExperUnents WhKI

Longitudinal 8 3 34 0.58 a0.018 (3.1%)

Circwnferential 7 3 29 0.53 0.030 (5.7%)

Radial 6 3 25 0.54 r 0.020 (3.7%)

The degree of thermal anisotropy in cortical bone does not appear to be large. The

largest difference, 0.5 W/mK between the circumferential and longitudinal directions, is less

than 10%. This hding is in iine with the observations made by Lundskog [46], but contrary

to the observations made b y Abouzgia [Il and to the experimental results of Zelenov [82]. In

Chapter 2, Zelenov's apparatus was discussed. The lack of insularion and the design of the

heating element may have led to an artificially high degree of anisotropy. Abouzgia observed

consistently higher temperatures in the longitudinal direction in his drilling experiments and

on that basis concluded that bone was thermally anisotropic. His results, however, s d e r

fiom a large amount of scatter, which may also have led to an artificially high directional

difference.

The average of the conductivity measurements in aii three directions is 0.56 W/mK,

which is closest to the value obtained by Vachon et al. [77] for dry ox bone. While this

similarity rnight lend some credibility to the thermal comparator device used by those

researchen, the large difference between dry and Eresh ox bone measured by Vachon et al.

continues to cast doubt on the viability of the method. The values obtained in the current

investigation are two to three times greater than those reported by Biyikli et al. [SI, who used

a cut-bar apparatus similar in design to the one used in the present investigation. The

measured values are comparable to the conductivity reported by Chato [33], as well as

published values for the thermal conductivity of cancellous bone [18,33]. Given the reliable

design of the cut-bar apparatus and given the fact that the difference between the themal

conductivity of cancellous and cortical bone should not be as large as one order of magnitude,

it is the author's opinion that a thermal conductivity in the range 0.3 to 0.6 WImK is more

accurate than one which is an order of magnitude higher, such as those reported by Luodskog

[46], and Zelenov [82].

1

i Iongitudinal circumferential radial

Figure 4.1: Thermal conductivity in three anatomical directions.

The repeatability of the experimental results proved to be quite go&. The average

standard deviation for measurements on an individual specimen was 2.2% (maximum: 4.5%.

minimum: OS%), while the average standard deviation for measurements on al1 specimens

fkom an individual animal was 3.1% (maximm = 4.8%' minimum = 2.1%). The raw data are

presented in Appendix A.

A statistical anaiysis was performed using Student's t-test. The clifferences in t h m a l

conductivity between the longitudinal and radial directions, and between the longitudinal and

circumferential directions are statisticdy signincant @ < 0.01). There is no significant

difference between the circumferential and the radial directions @ > 0.05). Statistical

significance does not necessarily translate to practical signincance, i.e., the difference

between the longitudinal direction and the other two directions may not signincautly &ect

the temperatures generated during dnlling. For the parametric analysis, the r d t s of which

are presented in Chapter 6, it was assumed that cortical bone can be treated as thermaily

isotropie. This assumption was tested by varying the thermal conductivity in the parametric

analysis and it will be seen that the assumption is valid.

4.2 The Effect of Evaporation

The measured mass losses are given for al1 but one specimen in Table A.2 (Appendix

A). The average percentage mass ioss was 2.8% (standard deviation of 1 .O%, maximum of

6.1%). The percentage loss was based on the initial mass, which ranged fiom 1-14 g to 2.57

g. The decrease in mass was assumed to be due to evaporation of water f?om the surface of

the specimen. Since the mass loss was, on average, less than 5% of the initial wet mass of the

specimen, it was concluded that the evaporation did not significantly affect the thermal

conductivity measurements of the bone.

Cornputer Simulation of Bone Drilling

A mode1 of a bone drilling operation was developed to perfom a parametric analysis

of surgical dnlling procedures. As noted in earlier chapters, there are mauy variables in the

drillhg process. To detennine the best method to decrease the thermal trauma caused by an

operation, the relative importance of these parameters must be laiown. This information can

be obtauied by performing an andysis of the drilling procedure in which one variable, e.g.

drill rotational speed, varies while the others are held constant The importance of that

variable is established by comparing the resdtant thermal damage, for example, with the

damage incurred when other parameters are varied.

The parametric analysis performed in the current study used a mathematical mode1 of

the drillhg procedure. An experimental mode1 was considered impractical due to the large

number of tests that need to be conducted. Furthermore, a parametric analysis requires that

al1 variables Save one remain constant, which is difficult with bone due to natural variation in

its properties.

5.1 Description of the Problem

The physical situation modelled in the analysis was the drilling of a hole in the mid-

diaphysis of a long bone pnor to pin insertion. The drill, tuming at a constant number of

revolutions per minute, starts at the outer surface of the bone and travels inward towards the

marrow. Heat is generated where the matenal is removed, along the cutting edge of the drill

bit. A portion of that heat is conducted into the surrounding bone, raising its temperature.

Depending on the degree and the duration of the temperature rise, thermal damage may

occw.

Themal damage depends on the history of the temperature distriiution around the

drill site. The temperature distribution was obtained by solving the Fourier heat conduction

equation (without heat generation), given in equation (5.1)

where

T = temperature,

t = tirne,

a = thermal diffisivity,

4 = forcing tenn used to mode1 heat input fiom the drill,

$2, dQo, dnN = computational domain, and the domain boundary, split into

portions over which Dirichlet

(D) and Neumann (N)

boundary conditions are

applied,

s = curvilinear CO-ordinate,

Ri, R,,, H = domain limits.

The computational domain was an

annulus with inner radius Ri, outer radius &,

Figure 5.1: Schematic drawing of a section of long bone with drill site modeiied.

and height H. The choice of cylindrical CO-ordinates to define the domain foîiowed naturally

fkom the cylindrical shape of the drill and the defect it creates. Figure 5.1 shows the location

of the computational domain with respect to the long bone and Figure 5.2 illustrates the

Figure 53: Volume of bone modelied in the cornputer

simulation,

domain in more detail. The inner radius, Ri, represented the

margin of the hole created by the drill and it is across this

boundary that the heat fiom the drilling operation was

transferred- The choice of the outer radius, %, will be

explained in Section 5.2.1. ih a mathematical model, it is

possible to have an outer radius at hfhify. It will mon becorne

evident, however, that a numerical approach was required to

solve the problem, and thus a finite & was necessary. The height of the annulus, H, was

chosen to represent the thickness of the cortical bone being drilled, with z = O representing

the surface of the bone exposed during the operation, and z = H the surface of the meduilary

canal.

The domain represented the drill site after the hole has been drilled. In a real

operation, the drill continuously removes matenal fiom the bone. A more realistic model

would incorporate material removal by continuously changuig the size and shape of the

domain. Modelling the removal of material was, however, considered beyond the scope of

the present study. The choice of domain was appropnate because it was the damage in the

material which remains f i er the operation that was of interest.

To solve equation (5.1), the boundary conditions must be dehed. The inner

boundary (r = Ri) was considered insulated, with the heat input fiom the drill modeiled as a

forcing term applied to a small portion of the boundary. The heat input was evedy

distributed around the circurnference of the inner boundary and travelied at a constant rate

Çom z = O to z = H. The heat input depended on the rotational speed of the drill, on the rate

at which the drill penetrated the bone, and on the geometry of the drill bit. In order to

calculate the heat created by the drill, relations designed for orthogonal cutting of metal were

adapted for drillhg in bone. Details of the forcing term are presented in Section 5.2.3, and a

description of the adaptation is presented in Section 5.3. The temperature at the boundary r =

I&, was assumed to be equal to nonnai body temperature, Le., 37°C.

The boundaries z = O and z = H were assumed to be insdated. The former boundary

represented the outer surface of the bone. Although, in reaîity, there is some convective heat

loss, the choice of a heat transfer coefncient would be arbitrary. Similariy, there is some

conductive loss where the bone is in contact with marrow, but once again, the amount of heat

loss was not known. In both cases, however, the heat loss was not expected to be large, and

thus the assurnption of an insulated boundary was reasonable.

Since this was a time-dependant problem, a set of initial conditions was also required.

The temperature within the domain was set to 37OC, normal body temperature, at t = 0-

A simplification of the mode1 was made by using the results of the experimental

investigation into bone thermal conductivity, presented in Chapter 4. There, it was concluded

that cortical bone could be treated as thermally isotropic. Due to the symmetry of the

goveming equation, the boundary conditions, and the thermal properties, only the

temperature in a two-dimensional (r,z) slice of the domain R needed to be calculated to

obtain temperature data for the whole volume. This two-dimensionai slice is presented in

Figure 5.2 as a darkened rectangle. With this simplification, equation (5.1) reduced to an

axisymmetric unsteady heat conduction problem in polar CO-ordinates.

where

r, z = cylindrical CO-ordinates,

k, p, c = themai conductivity, density, and specific heat of bone, respectively,

Equation (5.2) defines a boundary value problem that can be solved either

analyticaily or numerically. The heat input at the inner surface, representing the heat

generated during drilling, varied both in space and t h e . An anaiytical solution of equation

(5.2) was therefore difficult. The problem was amenable, however, to solution by numerical

approximation. Of the various numerical methods available, the Finite Element Method

(FEM) was chosen.

FEM is used to solve a wide range of boundary value problems, including stress

analysis, fluid flow, heat transfer, and magnetic fields. The Galerkin Finite Element Method

[26] was chosen over other numerical approaches for its known accuracy and ease of

application to the current problem.

Although several commercial FEM packages were available, it was decided to create

the simulation fkom scratch. The main reason for doing so was to enable the author to acquire

an in-depth understanding of the method. The use of a commercial package may have been

more convenient in some aspects, but it would have been a 'black box' into which input data

disappeared and fiom which temperature profiles appeared. The author was also unsue

whether the commercial packages could handle the shulated heat input. Finaily, the author

was restricted by financial and cornputer hardware Limitations. The simuiation was

programmed in C and run on a Personai iris SG workstation.

5.2 Application of the Finite Element Method

This section deals with the implernentation of the FEM to the drilling model. The

mathematical development is quite long and the reader should refer to Appendix C for the

complete derivation. The most important steps of the development are presented in the

folIowing sections.

5.2.1 Mesh and Spatial Discretization

The computational domain exarnined in the current study was the twoaimensional

slice shown in Figure 5.2. The line r = O coincided with the axis of rotation of the drill, thus

Ri was equal to the drill radius. Ri was varied to test the effect of drill diameter on the amount

of thermal trauma. The outer boundary, %, had to be sufficiently far away such that the

boundary condition of T = 37°C was vaiid. To determine an appropriate value for &, several

drilling simulations were m. The input parameters were identicai except for the value of E&,.

An arbitrary heat flux and feed rate were chosen. Figure 5.3 shows the temperature as a

fùnction of position in the r-direction, for the same location on the z-axis, at the same point in

time, for three values of &.

I

Figure 53: Comparison of temperature profiies for three values of & (Ri = 1.25 mm). N.B. The temperatures shown are not representative of driiüng in bone. Input parameters were chosen to create more extreme

conditions than were expected in bone drilling operations while testiug the simulation,

The simulations were run with an inner radius of 1.25 mm. It can be seen fiom Figure

5.3 that there is no difference between the curves obtained fiom simulations with the outer

boundary at distances of 4,s and 6 mm fiom the inner boundary. While 4 mm fiom the drili

(r = 5.25 mm) was sufficient to implement the Dirichlet boundary condition at r = &, the

pararnetric analysis was run with an outer boundary at r = 6.25 mm (% = Ri + 5 mm) to

insure that a l drill diameters could be handled.

The bone thickness, H, was set equal to 9 mm so that cornparisons could be made to

the results obtained by Abouzgia [Il. (The average specimen thickness in Abouzgia's

experiment was 9 mm.)

In the FEM, the domain of interest is divided into elements, each of whose position

and shape is determined (in 2-D space) by 3 or more nodes. The intercomected elements

together are referred to as the element rnesh. The rnesh used in the present simulation is

shown schematically in Figure 5.4, with an image of a drill bit located for reference.

Figure 5.4: Schematic drawing of the element mesh.

The elements used in the simulation were four-noded quacirilaterais, a choice based

on convenience. The temperature profile wi thin a four-noded quadrilateral element is

determined by iinear interpolation (in both the r- and z-directions) of the temperatures at the

four nodes defining the element. It was decided to start with this element and replace it with

an element with a higher order of interpolation, if necessary. It tumed out that the 4-noded

quadrilateral was sufficiently accurate (see Section 5.5).

The domain was divided into Nz elements in the z-direction and Nr elements in the r-

direction. While a large number of elements provides accuracy, the amount of cornputer

memory required and the duration of the simulation run also increase with the number of

elements. In choosing the h a 1 values of Nz and N,, a compromise had to be struck.

After perfoming preliminary tests, it was decided to use 40 elements in the r-

direction. Ddling simulations were then run to detemiii;~ L!e best value for N,, given Nr =

40. Figure 5.5 shows the variation of temperature with time for two points at the same

location on the boundary r = Ri. The only difference between the two curves is the value of

Nt.

Figure 5 3 : Temperature profiles for a point on the boundiuy r = for two different values of Nz (N, = 40). NB. The temperatures are not representative of boue drïüing operatiom.

There is a noticeable difference between the two curves. This was expected, however,

because the amount of heat applied at the boundary depended on the size of the element (this

will be explained in Section 5.2.3). To determine the importance of the value of N z on the

temperatures away fiom the imer boundary, data for a location at 0.5 mm distance from r =

Ri were plotted for the same simulations (Figure 5.6).

Figure 5.6: Temperature profiles for a location at r = Ri +- 0.5 mm for two different values of N, (N, = 40).

It is evident that there is no significant merence between the temperatures developed

a short distance away fiom the drill for the two values of Nz. Considering that the simulation

run times were prohibitively long at Nz = 40, it was decided that 20 elements in the z-

direction was adequate.

It can be seen in Figure 5.5, and to a lesser degree in Figure 5.6, that the temperature

dropped below the initial temperature of 37°C. This drop probably results h m the use of a

forcing term to model the drilling heat. An alternate approach to modelling the heat input is

discussed in Section C.4. The drop below the initial temperature was not serious, however,

because it lasted for only a bnef penod of time (less than 0.5 seconds), and the degree to

which the temperature was unredistic decreased rapidly with distance Erom the inner

boundary. Furthennore, it will be s h o ~ n in Section 5.5 that the overall heat balance for the

bone was maintained.

Whenever there are large temperature gradients, the density of the mesh must be

increased in the vicinity of the gradients to adequately describe the behaviour of the

temperature. This was done in the present model by increasïng the number of elements near

the imer boundary. An increase in the total number of elements, however, would aiso

increase the amount of memory and the nin t h e . Since large temperature gradients occurred

near the boundary r = Ri, and since the gradients near the boundary r = R,, were lower, (see

Figure 5.3), the number of elements in the region of large gradients was increased by

decreasing the length (in the r-direction) of the elements closest to r = Ri, while at the same

time increasing the length of the elements closest to r = &. This was accomplished by using

a scaling factor. The scaling factor was the ratio of the lengh (in the rdirection) of the

elements nearest the outer boundary to the length of the elements nearest the inner bomdary.

For a fixed ratio, the Iengths of the intervening elements increased with distance nom r = Ri.

Figure 5.7 shows the variation of temperature with time at the same location inside the

domain for simulations run with five different scaling factors.

1

time [sec]

Figure 5.7: Variation of temperature profile with scaling factor (r = R; + 0.5 mm, z = H.12).

There is Iittle dinerence between the temperature profiles in Figure 5.7, therefore a

scaling factor of 10: 1 was chosen for the parametric analysis.

In this section, the computational domain was dehned and discretized in trie spatial

dimensions. The next section describes how the domain was discretized with respect to t h e .

5.2.2 Temporal Discretization and Time Step

Equation (5.2) was discretized in the temporal domain using the method of central

differences, also known as the Crank-Nicolson method 1261. This method is a second order

time-stepping scheme. It was chosen because it is more accurate than methods such as the

Euler foward and backward schernes and because it is readily applicable to the problem at

hand.

When both the spatial and temporal discretizations are applied to the differential

equation (5.2), it is more easily expressed as a matrix equation,

where

[A], B] = the "stiffness" and "mass" matrices, the names borrowed fiom their

analogues in stress andysis, defined in Section 5.2.5 by equation (SS),

(8) = the vector of nodal temperatura,

(4) = vector of forcing terms

At = time step,

and where ' and '"w refer to the current and next t h e steps.

The Crank-Nicolson method is unconditionally stable, and so it does not place any

lirnits on the size of the t h e step. The time step had to be s m d enough, however, to capture

adequately the temperature behaviour and to calculate accurately the accumulated thennal

damage. A time step of 0.05 second was considered sufficiently small.

5.2.3 Implernentation of the Boundary Conditions

The insulated boundaries (homogeneous Neumann) at r = Ri, z = O and z = H required

no special irnplementation - such boundary conditions are assumed in the Galerkin finite

element fonnulation.

The Dirichlet boundary condition at r = % was appiied b y setting the corresponding

nodal temperatures to 37°C and then condensing the global matrices, Le. removing the rows

and columns corresponding to the boundary nodes. These boundary nodes were re-integrated

afier the condensed system of equations was solved.

Heat from the drilling operation was modelled by a forcing terni appiied to one

element at the boundary r = Ri. The heat flux created by applying the heat fiom the drill to a

single boundary element was calculated fiom equation (5.4).

where

q = heat flux appiied at boundary,

Qw = heat flowing into bone,

Ri = radius of drill bit,

h = Az of one element.

The forcing tenus {f$ were then calcuiated using this value of q (see Section CA).

The value of Qw was based on machining analysis, to be presented in Section 5.3.

The element at which the forcing terms were applied was that element whose position

in the z-direction corresponded with the position of the tip of the drill, which was caiculated

fiom the feed rate and the elapsed t h e . It was assumed that there was no heat transfer across

any other element on that boundary. Although the latter assumption may not reflect the true

dnliing conditions, there were insufficient data on heat transfer away ftom the point of

material removal, and thus any choice of heat transfer coefficient wouid have been ditrary.

It was assumed that the heat generated at the point of material removal was significantly

greater than the heat generated by friction at other points dong the drill surface.

The result was a travelling square wave, which started at z = O at t = O and stopped at

z = H at t = Wf, where f is the feed rate.

5.2.4 Thermal Properties

Based on values in the literature (Section 2.1), and on the present measurements of

thermal conductivity (Section 4.1), the following values were used in the computer

simulation for the properties of bone:

Density: 2200 kg/m3

Themai conductivity: 0.56 W/mK

Specific heat: 1300 UkgK

5.2.5 Solution Method

The simdation started by reading the input data h m a file. The drill speed, feed rate,

drill geometry, thermal properties, domain limits and mesh parameters were al1 included in

the input data. The element mesh was generated and the amount of driiling heat was

calculated. The next step was the generation of the global stiffness and mass matrices, [A]

and pl, which were constructed b y evahating, for each element,

Ni, Nj = the shape functions, which determine the behaviour of the

temperature field within the element (see Appendix C),

i, j = the local node numbers, which range fiom 1 to 4 for the four-noded

element used in the simulation,

n = the element number,

Q, = the portion of the domain over which the element n is defined.

The local node numbers, i and j, were mapped to their corresponding global node

numbers and the values of Ai and Biin added to the appropriate location in the global

matrices.

At t = 0, the initial conditions were set, then the simulation started marching through

tirne. At each tirne step, the temperature vector {O}' was filled with the nodal temperatures

fiom the previous t h e step, and the matrk multiplication on the right hand side of equation

(5.3) was perfonned. The location of the ciriII bit was then calcuiated fiom the feed rate and

the elapsed t h e , the forcing terms were added on the right hand side, and the Dirichlet

boundary condition was applied to the matrix equation. The system of equations was then

solved using a LowerNpper decomposition and back-substitution scheme [2q.

At each time step, the element temperatures were calculateci fiom the nodal

temperatures and rnonitored for potential thermal darnage (the monitoring wi11 be descn'bed

in detail in Section 5.4). The minimum duration of the simulation was the tirne it took for the

drill to travel the entire thickness of the bone. M e r the hole was completed, no more heat

was put into the system, and the temperatures began to &op. The simulation ended when ail

element temperatures dropped below the minimum temperature required to cause thetmal

damage (see Section 5.4).

A flow chart of the program is presented in Section C. IO.

5.3 Modeiiing the Heat Generated by Driiiing

In order to simulate the temperature nse that occurs in bone during d r i b g

operations, the amount of heat generated must be known. In the simulation, the heat

generated under various drilling conditions was estimated using machining theory that was

developed by Tay et al. [72] along with data on the shearing strength of bone published by

Saha [63].

The main disadvantage of using machining theory is that it was intended for metais

and other engineering matenals. Drilling, as weli as other machining operations Wce sawing

and tuming, separates chip matenal from the work piece through shearing, and bone's

shearing charactenstics are unlike those of metals. There were, however, two mitigating

factors. First, the shear strength of bone is known and was included in the model. Second, in

their analysis of orthogonal cutting of bone, Jacobs et al. [34] noted that, when examined

microscopically, the bone chips can resemble those produced when cuttïng metal. The latter

observation suggests that bone may behave like metal when machined and therefore that the

machining analysis developed for metals could be applied to bone.

Orthogonal Cutting

Machining theory has been

developed ptimarily for orthogonal

cutting, which is represented

sctiematically in Figure 5.9. The tool (at

right) moves with a cutting velocity v

Work piece ( with respect to the work piece (at

bottom), removing a layer of material

Figure 5.8: Schematic of orthogonal cutting, whose initial thickness is equal to the

depth of cut. The chip then travels with a velocity v, dong the tool face.

Originally, this change in the material velocity was modelled as a step change across

a "shear plane" [49]. The shear plane was defined as the surface that extends ftom the tip of

the cutting tool to the surface of the work material at an angle 4 (shear angle - see Figure

5.9) fkorn the cutting velocity.

Machinïng theory was refined after experirnents showed that a nnite zone of plastic

defonnation exists over which the chip velocity changes h m v to v, [IO]. This zone, labelled

A in Figure 5.10, is caIIed the primary deformation zone. The energy involveci in shearing the

material in this zone is converted to heat.

There are two other areas where heat is

generated. The secondary deformation zone,

labelled B in Figure 5.10, is a result of shearing of

the chip rnaterial due to a velocity gradient dong the

outward normal to the tool surface. The heat

generated in this secondary zone is shared between

Figure 5.9: Zones of heat generation in orthogonal cutting.

the chip and the tool. Since this analysis was concemed only with the amount of heat

entering the work piece, the heat generated in this secondary zone was neglected. The third

area where heat is generated is labeiled C in Figure 5.10. The heat produced here is a result

of friction between the tool and the new surface of the work piece. In the current model, it

was assurned that the tool was perfectly sharp, and therefore the amount of heat produced in

this third zone was negligible. The assumption that the use of sharp tools is the n o m in

clinical practice was based on investigations that examined the difference between worn and

new tools, which recomrnended the use of new tools to minimize thennal damage (see

Section 2.3.3). It was also assurned that the clearance angie between the back face of the tool

and the new work piece surface was greater than zero.

Although the cutting action in driUing is more complex than that in orthogonal

cutting, the basic mechanism of chip removal is the same, and the theory for orthogonal

cutting is still applicable.

Calculating the heat generated in the primary deformacion m e

Almost al1 of the energy used in matend removal is converted into heat [49]. The

work done, and therefore the amount of heat generated in material removal, was calculated

fiom

where

Q = heat,

Fs = force in the shear plane,

vs = shear velocity.

Calculating the shear velocity

The shear velocity, v,, was calculated fiom the cutting velocity and the shear angle

(refer to Figure 5.9).

where

v, = shear velocity,

v = cutting velocity,

4 = shear angle.

The shear angle, 4, was calculated b y Jacobs et al. [34] using Merchant's analysis,

24 +p-a=90° , where

a = rake angle of the cutting tool (angle between tool face and normal to the

cutting velocity),

B = fiction angle (37") [34&

The rake angle, a, actually changes along the cutting edges of a drill. The expression

for the rake angle at a distance i fiom the rotational axis was calculated by Saha et al. [64]

with equation (5.9). In the present analysis, the average rake angle over the length of the

cuttîng edge was used. The average was calculated by integrating numericaliy equation (5.9)

over the drill diameter and dividing the result by the diameter.

Figure 5.10: Schematic drawing of a twist drill (modifled fiom 1641)-

Figure 5.1 1: Schematic drawing of a twist drill (modified fiom [64]).

where

ai = orthogonal rake angie at distance i fiom the rotational axis,

d = drill diameter (Figure 5.1 l),

Q = chisel edge diameter (bottom of Figure 5.1 l),

di = diameter at distance i fÏom the rotational axis,

0 = helix angle (Figure 5-12},

p = half of point angle (bottom of Figure 5.1 1).

The cutting velocity, v, also changes dong the cutting edges of a drill.

v(r) = 2mN/60

where

r = distance fkom a ~ i s of drill rotation (refer to Figure 5.1 l),

N = drill rotational speed, in rpm.

The average cutting speed was calculated fiom equation (5.11).

Using the equation for the amount of heat generation (equation (5.6)), it is possible to

calculate an average shear velocity given the following parameters:

chisel edge diameter, &,

drili diameter, d,

drill rotational speed, N,

helix angle, 0,

half point angle, p.

In the parametric analysis, the drill diarnetcr, rotational speed, helix angle, and point

angle were al1 varied to determine their importance in causing thermal trauma The chisel

edge diameter was set to zero.

Calmlating the shear force

The shear force was calculated fiom equation (5.12).

F, = T, A, , where

r, = ukimate shear stress for boue,

As = area of the shear plane-

Bone is a viscoelastic material [66], and one consequence is that the value of the

ultimate shear stress, rs, varies with the shear rate. To determine t, both the shear rate and

the relation between the ultimate shear stress and the shear rate needed to be known.

Calculating the shear rate Y

To calculate the shear rate,

liyperbolic b%emline the behaviour of the material as it ywana - s) = a

passes through the primary

deformation zone needed to be x R

I u

known, or to be assumed. The

analysis by Tay et al. [72} assumes Figure 5.12: Hyperboiic streamline through the primary defonnation zone,

that the matenal being removed by

the tool travels through the deformation zone dong hyperbolic streamlines of the form (see

Figure 5.13):

where

x, y = Cartesian CO-ordinates,

a = rake angle,

a = constant which determines the curvature of the byperbola, which was

calculated by Tay et aI. [723 with equation (5.14).

a = 1 6 ~ ' sin' (+)[tan(a) + cot(t#~)] '

where

tl = undeformed chip thickness (= depth of cut pet revolution),

+ = shear angIe,

C = material constant, with units of lengtb.

The value of C is material-specific and is not known for bone, hence the value for

low-carbon steel (C = 5.9 1721) was used. Since C was used to calculate the curvature of the

material sireamlines, the use of C = 5.9 required the assumption that the rnanner in which

bone defoms under shear is simila. to that of steel. There is some experimental evidence that

this assumption was reasonable [34]. Furthemore, a prelimuiary investigation hdicated that

the heat generation predicted by the current anaiysis was not very sensitive to the value of C.

Thus the fact that the value of C for bone was not known should not have significantly

affected the accuracy of the analysis.

Once the value of a was calculated h m equation (5.141, the shear rate at any point in

the primary deformation zone could be cdculated. Recall that the shear rate was required to

detennine the ultimate shear stress. The shear rate and therefore the ultimate shear stress

varied throughout the defonnation zone. In order to simpli@ the analysis, the shear rate on

the line AB (see Figure 5.13) was used to detennine the ultimate shear stress. The line AB

corresponds to the shear plane in the original Merchant aaalysis [49]. From Tay et ai. [72],

the strain rate dong AB, +, , was calculated using equation (5.15).

The values of v, a, a, and + al1 depend on the drilling parameters, which were varied

in the parametric analysis. Thus the shear rate on the line AB depends in a complex fashion

on the drilling parameters.

Determining the ultimate shear sîress for bone

The ultimate shear stress for cortical bone and its variation with shear rate was

determined by combining the results of several studies of bone properties.

Saha 1631 measured the ultimate shear stress of bone as 50.4 + 14.1 MPa The tests

were done at a very low shear rate (approximately 0.00042 s").

Carter and Hayes 1151 found that the compressive strength of cortical bone is

approximately proportional to the strain rate raised to the 0.060 power, while Carter and

Caler [14] f~und that the tensile strength of cortical bone is proportionai to the strain rate

raised to the 0.055 power.

From the latter two studies, it was assurned that the ultimate shear strength behaves in

a similar fashion with respect to shear rate, i.e.,

f , o y - . (5.W

Using the single shear rate value measured by Saha, the constant of proportionality

was calculated as 80 MPa.

so,

For cornparison, the constant of proportionality for tensile strength is 147 MPa [20].

The expression for the amount of heat generation was re-written as

Q = A , w S , where

A, = shear plane area,

r, = ultimate shear stress,

v, = shear velocity.

Cairnlaring the shear plane area

The final parameter in the calculation of the amount of heat generated is the shear

plane area,

The depth of cut, 11, was calculated by Saha et al. [64] with equation (5.20),

where

f = feed rate,

N = rotational speed,

The heat that is generated is distnbuted among the tool, the chip, and the work piece.

For the current andysis, the hc t ion of the heat that enters the work piece needed to be

determined. This fiaction, q, was measured empirïcally by Boothroyd [IO] for orthogonal

cutting of steel. Since the value of q depends on the geometry of the problem, Boothroyd's

relations could not be used. An appropriate value of q for the current problem was

determined by comparing the temperatures predicted by the current analysis with temperature

measurements obtained by Abouzgia [l]. This process is discussed in Section 55.2.

The final expression for the rate of heat entering the work piece is given by equation

(5.21)

Q w = W U s ( ~ r n ) v s 9 (5.21)

where the brackets denote a fûnctional relationship.

5.4 Modeliing the Amount of Thermal Damage Incurred

In their work on the threshoid for thermal damage in epitheiial tissue, Moritz and

Henriques [SOI detemiined that the temperature required to cause darnage to both porcine and

hurnan skin varied with exposure tirne. The result is a tirne-temperature cuve iike the one

presented in Figure 2.1.

Given that the study by Moritz and H e ~ q u e s is the most extensive, and given that

studies on bone [23,46] agree closely with their results, their curve was used in this study- In

order to simpli@ the analysis, the the-temperature c w e in Figure 2.1 was divided into five

damage categories:

52°C to 54°C for 60 seconds

54°C to 57°C for 30 seconds

57°C to 60°C for 15 seconds

60°C to 65°C for 5 seconds

65°C or above for any length of time

At each time step in the simulation, the temperature of each element was calculated as

the average of its respective nodal ternperatures. The amount of time spent above the five

threshold temperatures (52"C, 54"C, etc.) was stored in arrays. At the end of the simulation,

an element whose temperature exceeded any one of the threshold temperatures for a penod of

t h e longer than the corresponding threshold duration was tagged as 'damaged'. For

exarnple, if the temperature in element 21 stayed above 60°C for more than 5 seconds, it was

assumed that the bone in that element would become necrotic.

The amount of necrotic bone was recorded in two ways. The volume of al1 of the

'damaged' elements was calculated and printed to a file. In addition, the penetration of

thermal necrosis into the bone in the r-direction, as a fûnction of position in the z-direction,

was detemined by calculating, for each element row in the mesh, the centre of the

'darnaged' elernent fùrthest fiom the inner boundary r = Ri.

5.5 Testing the Cornputer Simulation

5.5.1 Analytical Solutions

The complexity of the heat input at the inner boundary required numerical

approximation to solve equation (5 -3) when drilling conditions were simulated. However,

analytical solutions could be obtained if the boundary condition at r = Ri was simplifieci. The

analytical solutions to simplified problems were used in benchmark tests of the finite element

code in its early stages of development. Solutions for two differential equations were

developed. The hrst was the 2-D (r,z) steady-state heat conduction equation- The boundary

conditions were identicai to those used in drilling simulations, except at the inner boundary (r

= Ri), where either a heat flux or a temperature boundary condition was prescribed dong the

entire boundary. The h e r boundary conditions appiied to the 2-D equation were described

by continuous functions of z. The second equation was the 1-D (r) unsteady heat conduction

equation with a constant temperature boundary condition at r = R,,, and either a constant heat

flux or a constant temperature boundary condition at r = Ri. The solutions were obtained by

standard mathematical analysis [54,55] and are given in Appendix C. Predictions fiom the

solutions were evaluated by cornputer programs written by the author.

5.5.2 Finite Element Code

The first set of tests performed on the FEM code were designed to verify 2-D (r,z)

steady-state behaviour by comparing the output fiom the FEM mode1 to that fiom the 2-D

analytical solutions. Both steady temperature and heat flux boundary conditions, defhed over

the entire surface of the inner boundary (r = Ri), were applied to the analyticai solution. The

forcing ternis, {fq): were used in the FEM code in tests where a heat flux boundary condition

was applied to the analytical solutions- For each boundary condition type, linear, shusoidal,

and exponential fùnctions were tested. The conditions on the remaining boundaries were

identical to those used in drilling simulations. Figure 5.14 contains both the numerical and

analytical output for a temperature boundary condition at the inner boundary, which is

described by an exponential hc t ion.

Figure 5.13: Example of a test of the steady FEM model. h this case, the FEM d e l was compared to an analytical mode1 with the B-C. at r = Ft,: T(z) = 37 + 1 0 e ' ~ . [r/R = r/&, - R;)]

The excellent agreement between the numencal and analytical solutions exhibited in

Figure 5.14 was evident in al1 tests of the steady-state behaviour.

A two-stage test was performed to veri& the unsteady behaviour of the simulation.

First, calculations were carried out to ensure that the unsteady analyticd solution converged

to the expected steady-state temperatures. These tests were perfomed with constant heat flux

and then with constant temperature applied at r = Ri. The steady-state values were obtained

by solving the 1-D (r) steady-state equation. Figure 5.15 contains the output fkom one such

test, and it can be seen that the unsteady analytical model converged properly.

dR = 0-875, unsteady - dR = 0.125, steady - - - - - - dR = 0375, steady

dR = 0-625, steady - - - - dR = 0.875, steady

35

O 5 1 O 15

time [secl

Figure 5.14: Test of the convergence of the unsteady anaiyticd mode1 (used in subsequent tests to verify the unsteady FEM modei). [constant kat flux at r = Rr; r/R = r/(% - Ri)]

Next, the output fiom the unsteady 2-D FEM model was compared to that from the

unsteady 1-D analytical solution. A constant boundâry condition (or forcing term) was

applied at the inner boundary of both tt.le analytical solution and the numerical model. The 2-

D unsteady analytical solution was not used because the computer time required for

calculation was too great. Figure 5-16 demonstrates that both analytical and numencal

temperature profiIes match closely.

- r/R = 0,125, analytical

1 - - - r/R = 0.375, analyticai

r/R= 0.375, FEM 1 ; r/R = 0.875, FEM

I time [secl ! i 1

Figure 5.15: Cornparison of temperature histories fiom the unsteady FEM model and the unsteady analytical solution. [constant heat flux at r = Ri; r/R = r/(% - Ri)]

When it came to testing the unsteady behaviour of the model under simulated drillhg

conditions, cornparisons to analytical models could not be made because the analytical

solution could not be obtained. Nevertheless, an examination of temperature data fiom one

such simulation (Figure 5.1 7) indicates that, quaiitatively, at least, the behaviour of the model

is correct. The c w e s in Figure 5.17 represent data for three points, dl the same distance

from the inner boundary, but each at a different distance dong the z-axis. As the distance

fiom the boundary z = O increases, the tirne required to reach the maximum temperature also

increases, which makes sense.

36.5

O 5 10 15 20

time [secl

Figure 5.16: Temperature-time data for three points in a test of the dnlling simulation

A ngorous test of the numerical model under simulated dnlling conditions

investigated heat balance in the model. The to ta1 amount of heat applied at the boundary at

time t was calculated fiom equation (5.22),

Q , = q - A - t , O<t<tdding, where

Q, = heat applied,

q = applied heat flux,

A = area over which heat flux was applied,

t = time,

bri,ling = dnlling tirne, i.e. time during which heat flux was applied.

The heat accrued in the model during each time step i was estimated accordhg to

equation (5.23),

where

the superscnpt " refers to the element nurnber, n = l...N

the subscripis i, i-1 refer to the current and previous t h e steps,

N = total number of elements in the model,

p = density,

c = specific heat,

V = volume,

T = temperature.

Note that the elernent temperatures were calculated as the average of the temperatures

of the nodes that define the element. The results fiom one test are given in Figure 5.18.

2 4 6

time [sec]

j- heat applied at I

f bounciary (5.22)

Figure 5.17: Cornparison of heat generated and heat absorbed The heat generation was 1.49W for a drilling M i e of 2 seconds.

It is evident that the energy is bdanced, therefore it was concluded that the numericd

simulation likely provides reliable results.

5.5.3 Theoretical Drilling Mode1

In addition to ensuring that the finite element code worked pmperly, it was also

necessary to check the theory used to calculate the drilling heat. Predictions h m the

simulation were compared to experimental results obtained by Abouzgia [l]. in his bone

dnlling tests, Abouzgia measured the power used by the dm. The m e d power values

are compared in Table 5.1 to the heat predicted by the theoretical drilling analysis developed

in Section 5.2 for four different cases.

JTABLE 5.1 : Cornparison of heat values predicted by 1

power CI drilling theory vs. power utilization in

It is evident that the values do not agree. It was noted, however, that the trends are the

drill time [secl

7.5 2

3 -5 2.5

same. For example, when the measured power increases, so does the heat predicted by the

curent model. Although it is not possible to determine the mechanicd losses in the drill used

drill speed kpml

70000 45000 58000 49000

by Abouzgia, these losses and other factors, including fiction between the drill and the sides

heat predicted

rw (Section 5.2)

5.9 20 12 16

of the hole, and fiction between the drill and the chips as they fiowed up through the dm

flutes, used some fhction of the power consumed by the drill. The similarity in trends was

sufficiently encouraging to proceed with a cornparison of the maximum temperatures

generated by the simulation to those measured by Abouzgia, the results of which are

presented in Table 5.2.

TABLE 5.2: Cornparison o f maximum temperatures: those predicted by the simulation

drill speed [wml 70000

45000

28000

49000

heat predicted b! curreat mode1 rw (ri = 0.9

2.95

distance from drill

mas. temp predicted

["Cl 82 60 45

dm. w.r.t. Abouzgia

["Cl +10 -7 +1 +32 +8 +6 -32 +4 +6

Recall that q represents the fiaction of heat generated which enters the bone.

Calculations for different values for q indicated that q = 0.5 produced the best agreement

with Abouzgia's experiments, as can be seen in the last column of Table 5.2. The majonty of

predicted values are within l+lO°C of those rneasured by Abouzgia. Given the considerable

amount of scatter in Abouzgia's measurements, the agreement obtained was considered

sufficient to proceed with the simulations.

5.6 Scope of the Computer Simulations

The following parameters were varied in the parametric anaiysis: drill rotationai

speed, feed rate, helix angle, point angle, drill diameter, and the specific heat, thermal

conductivity, and density of bone. This section discusses the choice of an appropriate range

for each of these variables.

Drifl Speed [rpm]

The literature review in Chapter 2 revealed that a wide range of drill speeds was used

in past experiments. At the lower Limit, hand drillhg produced rotational speeds as low as 60

rprn [74]. The highest reported rotational speed was 350,000 rprn in the study conducteci by

Rafel[60]. Other researchers [38, 5 11 used rotational speeds above 200,000 rpm, but these

were fkee-running speeds. As mentioned earlier, Abouzgia and James 121 demonstrateci that

drilling speed decreases by as much as 50% during a dnlling operation. The upper bound for

these simulations was therefore chosen as 200,000 rpm.

To cover the range tkom 100 to 200,000 rpm, simulations were nui at the following

rotational speeds:

100 rpm 1000 rpm 10 000 rpm 100 000 rpm 250 rpm 2500 rpm 25 000 rpm 150 000 rpm 500 rpm 5000 rpm 50 000 rpm 200 000 rpm 750 rpm 7500 rpm 75 000 rpm

Feed Rate [ id . ]

The range of feed rates used in the current study was based on the work of Abouzgia

[ l ] in order to make cornparisons with his results. He recorded drilling times between 2 and

16 seconds for the normalized sample ttiiclaiess of 9 mm, and consequently the selected

drilling times for the present study were 2,5 , 10, 15, and 20 seconds. For a thickuess of 9

mm, the corresponding feed rates ranged from 0.45 to 4.5 d s , which overlaps with those

applied b y prior investigators.

The experiments doue by Jacobs et al. [35] were performed at 1,2, and 5 in/&

(0.423,0.847, and 2.12 mm/sec). Saha et al. [64] applied a feed rate of 2 mds. In a separate

experiment, they applied a constant load and measured feed rates in a range of 4 to 14 mds.

Famworth and Burton [27] rneasured feed rates between 0.00476 mm/s and 0.1 67 mmh. The

upper limit of the range measured by Saha et al. is significantly higher than the upper limit

used in the current study; however, feed rates greater than 4.5 d s are unredistic in a

clinical setting. Similady, the feed rates measured by Famworth and Burton would result in

impracticably long driUing times.

The combination of 15 drill speeds and 5 feed rates produced a set of 75 simulations.

The diameter of the drill bit in these simulations, 2.5 mm, was chosen so that cornparisons

could be made to the results of Abouzgia A helix angle of 27" and a point angle of 118"

were chosen for the same reason. The chisel edge diameter was set to zero because it

primarily affects the temperature in the material directly below the drill (643, which is not

modelled in the numerical simulation.

A note on simulations driven by appfied force

Ideally, the simulations would have been canied out with a constant applied force

since a constant force is more relevant in a clinical settïng, i.e., most clinicians apply a

constant load rather than a constant feed rate. Udortunately, there is no theory available to

determine feed rate from applied force for an arbitrary drill bit. Experiments done by Jacobs

et al. [35] and Famworth and Burton [27] have measured feed rate as a fuoction of force, but

their results are specific to the dnll used, and could not be applied to this study.

Drill Bit Geornetty

Three parameters of the drill bit geometry were varied in order to detennine their

effects on thermal damage: drill diameter, helix angle, and point angle.

Diameter: A set of simulations was run with drill diameters ftom 1.0 to 3.5 mm in 0.5

mm increments, corresponding to the range of diameters reportecl for the various drills, burs,

and pins used in past experiments. These simulations were run with a helix angle of 27O and

a point angle of 11 8 O . The dnll speed and feed rate were 7500 rpm and 0.9 d s

respectively. The choice of drill speed and feed rate corresponds to the mid-range values for

both parameters.

Helix Anele and Point Anele: Few papers reported the values of the helix and point

angles of the drill bits used [27,35, 64,801. Published values for helix angle fa11 in the range

of 10" to 36O, while point angles have ranged fiom 30" to 180". Based on the scope of

previous experiments, five values for the point angle were chosen: 40°, 70°, 1 ûOO, 130°, and

MO0, as well as five values for the helix angle: IO0, L P , 24O, 3 Io, and 389 Twenty-five

simulations were nui to determine the combined effects of the two angles.

These simulations were run with a sbulated drill diameter of 2.5 mm- The drilling

speed and feed rate were 7500 rpm and 0.9 mmk respectively.

Physical Properties of Bone

The three physicai properties which play a role in heat transfer are the density, the

specific heat, and the thermal conductivity. Since these propertîes vary fiom person to

person, it would be usefiil to know the sensitivity of the temperature distribution to changes

in these heat transfer properties. This was determined in three separate tests in which each of

the density, thermal conductivity, and specific heat were vacîed by -IO%, -5% O%, +5% and

+IO% of their nominal vaiues. Hence the test values were:

Specific heat [J/kgK]: 1 170, 1235, 1300, 1365, 1430.

Density [kg/rn": 1980,2090,2200,23 10,2420.

Thermal conductivity p/mK]: 0.504,0.532,0.560,0.588, 0.616.

These simulations were run with a drill diameter of 2.5 mm, a helix angle of 27" and

a point angle of 118". The dnlling speed and feed rate were 7500 rpm and 0.9 mmls

respectively.

Results : Computer Simulation

In this chapter, results fiom the parameiric analysis of drillhg operations are

presented. The analysis and discussion of these results is presented in Chapter 7.

6.1 Effects of Feed Rate and Drill Speed

The temperature and thermal darnage depend on the rate at which heat is

generated by the drill. As shown in Figure 6.1, the rate of heat generation predicted by

the mode1 varies considerably. There is a moderate increase in heat generation with

increasing drill rotational speed, and a significant increase with feed rate.

driti speed [rpm j = gj

Figure 6.1: Variation of heat generation with drill speed and feed rate.

In the graph presented in Figure 6.1, and in other surface graphs presented in this

chapter, the different shading patterns represent sub-ranges of the independent variable.

For example, in Figure 6.1, there are five shading patterns, representing the sub-ranges O

to5 W,5to 10W, Loto 15W, LSto20Wand20to25W.

In addition to the heat generation rate, the temperature and thmal damage also

depend on the total time the heat is applied. The variation of the total heat absorbed,

tirne, is shown in Figure 6.2.

which is obtained by multiplying the heat transfer rate to the bone by the total drilling

-

energy

n t e

O

drill speeâ [rpml

Figure 6.2: Variation of energy absorbed with drill speed and feed rate.

It is evident fiom Figure 6.2 that the decreased drilling time at higher feed rates

acts to counter-balance the increased heat generation. The result is that the energy

absorbed increases with increasing drill speed and decreases slightly with increasing feed

rate.

The cornputer simulation was written to keep track of the maximum temperature

at each node. The maximum temperature at a distance of 0.5 mm ftom the drill site was

interpolated from the temperature at the nodes which straddle the 0.5 mm point. The

distance of 0.5 mm was chosen because it was a nominal distance at which temperatures

were measured in many of the previous experiments. It was found that the maximum

temperature varied as a h c t i o n of the depth, z, as shown in Figure 6.3.

t * 50 70 90 110 130 1

temperature [ O C ) ,

Figure 63: Variation of maximum temperature witb depth (2-direction) at a distance of 0.5 mm fiom the drill. [drill speed: 7500 rpm, feed rate: 0.9 mm/s]-

Note that there is a large increase in maximum temperature at the boundary z = H

(9 mm). This increase is related to the fact that the lower boundary is insulated and

therefore traps the heat generated by the drill near the end of the operation, while the heat

near the boundary z = O does not have tirne to build up because the heat source travels

away fkom it. While this increase is probably not realistic, it did not signincantly affect

the results of the analysis because the maximum temperatures were averaged over the

thiclmess of the bone. A plot of the average maximum temperature at a distance of 0.5

mm fiom the drill site is presented in Figure 6.4 as a fimction of feed rate and drill speed.

average , temperature l0Cl

drill specd [rpml

Figure 6.4: The average maximum temperature at 0.5 mm distance fiom drill, varying with drill speed and feed rate.

Figure 6.4 indicates that the average maximum temperahire increaseç with

increasing drill speed. There is also an increase in temperature with feed rate up to 1.8

mds, after which the temperahire plaieaus, or decreases slightly. Since this behaviour is

difficult to see in Figure 6.4, the temperature versus feed rate data for 100,000 rpm are

presented in Figure 6.5. There, the decrease in temperature at feed rates above 1.8 mm/s

c m be seen more easily.

O 1 2 3 4 5

feed rate [mmls]

Figure 6.5: The average maximum temperature at 0.5 mm distance fiom the drill, varyïng with feed rate at 100,000 rpm.

The penetration distance of necrosis (Figure 6.6) follows a pattern sunilar to that

o f maximum temperature.

penetration of necrosis [mm]

drill speed (rpm]

Figure 6.6: Changes in average necrosis penetration distance with driU speed and feed rate.

Recall that the penetration distance of necrosis at any depth z is equal to the

centre of the 'damaged' element furthest fiom the boundary r = Ri at that depth. As with

maximum temperature, the amount of damaged bone increased in the z-direction, with

great penetration near the boundary z = H. The penetration distance plotted in Figure 6.6

is the average across the thickness of the bone.

6.2 Effects of Point Angle and Helix Angle

Figure 6.7 contains a plot of the average maximum temperature, at a distance of

0.5 mm fiom the drill site, as a function of both helix angle and point angle.

1

l

tempemture IOCI 92

i

point angle [O1 160

Figure 6.7: Average maximum temperature at 0.5 mm distance fiom the drill as a firriction of helix angle and point angle.

With the exception of the data close to 40° point angle, the temperature decreases

monotonically with increasing helix angle. The effect of point angle is small - the

temperature varies less than 2°C for any given helix angle in the range tested.

Necrosis penetration, shown in Figure 6.8, follows a similar pattern.

penetration of necrosis [mm]

point angle Io] 130 '

160

Figure 6.8: Arnount of t h e m l damage as a hct ion of helix angle and point angle.

As with drill speed and feed rate, the temperature and damage penetration

behaviour follows the same pattern as the amount of heat absorbed.

6.3 Effects of Drill Diameter

Both the temperature and the amount of damage increase with drill diameter, as

shown in Figures 6.9 and 6.10 respectively.

O

1 1.5 2 2.5 3 3.5 4

diameter [mm] - - --

Figure 6.9: Effect of drill diameter on average maximum temperature at 0.5 mm distance fiom the drill-

1 O , 4

1 1 -5 2 2.5 3 3 -5 4

diametcr [mm]

Figure 6.10: Changes in depth of necrosis penetration with increasing dnll diameter.

6.4 Effects of Bone Thermal Properties

The effects o f density, specific heat, and thermal conductivity can be seen in

Figures 6.1 1 and 6.12.

86 '

-10 -5 O 5 10

% change

I

i + density, specific i j ! heat

conductivity ! j

,

Figure 6-1 1: Average maximum temperature at 0.5 mm distance fiom the drill as a îünction of bone thermal properties.

, + density, specific I heat

conductivity j

-5 O 5

% change l

Figure 6.12: Average penetration distance of thenaal necrosis, va-g with bone thermal properties.

Both the temperature and the thermal damage decrease with increasing thermal

properties, with the importance of the thermal conductivity much less than that of either

density or specific heat-

The results presented in this chapter are discussed in the following chapter.

Discussion of Simulation Results

In the previous chapter, the results of the parametric analysis were presented. In this

chapter, these results are analyzed in the context of previous bone drillhg investigations.

7.1 Effects of Feed Rate and Driii Speed

The graphs in Figures 6.4 and 6.6 indicate that both the average maximum

temperature at 0.5 mm distance fiom the drill and the resdtant amount of thermal damage

increase with drill rotational speed over the entire range fkom 100 to 200,000 rpm. These

graphs also indicate that both temperature and thennai damage increase with feed rate fiom

0.45 to 1.8 mmfs and then decrease siightly up to a feed rate of 4.5 mmk It is clear fiom

these two graphs, and fiom the data presented in Figure 6.2, that the temperatures produced

in the work piece, the resultant thermal damage, and the amount of heat energy absorbed by

the bone al1 follow the same pattern. This makes sense because the temperature depends on

the heat input and the thermal damage depends on the temperature. Since the total energy is

calculated independently fkom the maximum temperature, the similarity in their behaviour

provides a f o m of intemal verification.

The surface graphs representing the amount of thermal damage (Figures 6.6 and 6.8)

are no t as smoo th as the energy and temperature graphs, a result of discretization. First, there

is an effect fkom the spatial discretization. The depths of penetration were caiculated fiom the

centres of elements labelled as 'damaged' by the simuIation, therefore the penetration depths

were limited to a set of discrete values. Secondly, the continuous the-temperature curve for

the tissue damage threshold was divided into discrete categories. The result of using this

method for calculating damage is that a slightly higher temperature or a slightly longer

duration at a given temperature can result in a large change in the amount of damage recorded

by the simulation, Despite the unevenness of the damage graphs, it is evident that the trends

for al1 three parameters (energy, temperature, and thermal damage) are the same and M e r

rehements of the simuiation could lead to smoother behaviour of the damage hdicator.

The average maximum temperature values at 0.5 mm distance fiom the drill fa11 in the

range of approximately 60°C to 120°C- It was shown in Section 5.52 that, once a proper

value of q had been chosen, the computer simulation was able to reproduce the maximum

temperatures recorded in Abouzgia's experiments with reasonable accuracy. Cornparisons

with other studies are more difficult because it is rare that al1 the drilling parameters are

reported. Temperatures in the range of 60°C to 120°C are, however, certainly not unusud.

The equipment used by Thompson [74] could not record temperatures above 65.S°C, but this

maximum was reached, and assumed to be exceeded, at rotational speeds of 1000 rpm at

distances of 2.5 mm and 5.0 mm fkom the pin insertion site. The temperatures at 0.5 mm

were much higher, and so were consistent with the present temperatures of 60°C to 120°C.

Similarly, Matthews and HUsch [48] recorded temperatures over 100°C at 0.5 mm with low

drill speeds and with a drill diameter of 3.2 mm. Vaughn and Peyton [78], while drilling Uito

dentine and enamel, measured temperatures upwards of 160°F (7 1 OC).

The variation of temperature with feed rate is rerniniscent of the temperature versus

force data reported by Abouzgia Cl] and of the specific cutting energy versus force data

presented by Sorenson et al. [67] Both studies show an increasing trend in the dependent

variable with force, reaching a maximum and then decreasing. In the current work, feed rate

is not equal to force but increasing the appiied force increases the feed rate. Thus, if the

results of Abouzgia and Sorenson et al. were plotted against feed rate, they would be

89

qualitatively similar to those presented in Figure 6.5. The results fiom the simulation are also

consistent with the study by Krause et al. [40], which shows a decrease in specific cuttuig

energy with feed rate fiom 1.80 mmls to 6.35 d s while making longitudinal cuts in bone

with dental burs. In the variation of temperature with feed rate, there is a balancing act

between an increase in heat due to increasing rates of heat generation and a decrease in heat

due to decreasing drillhg times.

An increase in maximum temperature with drill speed was reported by al1

investigators that used low to moderate drill speeds except Matthews and Hirsch 1481, who

did not find any significant change in the maximum temperature for drill speeds ftom 345 to

2900 rpm. That the simulation results show a similar upward trend with drill speed provides

assurance that the simulation models correctly the physical situation at these rotationai

speeds.

A discrepancy between previous experiments and the current model exists at higher

drill speeds. Most histological studies indicate that the use of higher rotational speeds is

either benign or beneficiai. Furthemore, Abouzgia [l], who was the only researcher to

measure temperatures at high drill speeds*, found a decrease in peak temperature with

increasing drill speed. The nurnencai model, however, suggests that the use of higher drill

rotational speeds is detrimental because the temperatures are significantly higher.

There are two possible expianations for this discrepancy. Either the computer

simulation is correct and the observations made by previous researchers are in some way

-

Rafel[60] rneasured temperatures at 350,000 rpm, fiee-running, but did not mesure the speed whiie drilluig. 90

aawed, or the simulation does not adequately represent the drilling operation under ail

conditions.

There are several arguments for disregarding the results of previous histological

studies. For instance, statements made by previous researchers indicate that t h e was no

agreement on what parameter should be used as a rneasure of thermal damage in bone tissue.

Boyne [12] stated that the acellular zone is not a good indicator of thermal damage. Spatz

[69] could not con- the Iink behveen the depth of basopbilic staining and the amount of

themal damage. Both the acellular zone and the amount of basophilie staining have been

used as a measure of thermal impact. The lack of a standard measure for bone damage could

leave the results of most histological studies in doubt. It should also be remembered that

Abouzgia and James [2] demonstrated that the drill speed during an operation can be

substantially lower than the fiee-running speed. Most of the previous experiments reported

only the fiee-ntnning speed, which means that, in almost al1 experiments that reported either

temperature or thermal damage as a fhction of drill speed, the actual speed was not known.

Lfall such experiments fiom those discussed in Chapter 2 are eliminated, only three remain.

Thompson [74] used a stroboscopie method to control the drill speed, but he perfomed

experiments only at low speeds. Sorenson et al. [67] looked at the change in rotational speed

with applied load, but did not tuid the relation between heat absorbed and rotational speed.

That leaves the experiment performed by Abouzgia and James [3], which indicates a decrease

in maximum temperature with increasing drill speed. Thek results contain a great deal of

scatter (see Figure 7. l), and this puts the accuracy of the experirnental technique in question.

Aithough these arguments do not establish the accuracy of the cornputer simulation, they

should raise sufficient doubt in the resdts of previous experiments that the current work

cannot be dismissed entirely.

- Longitudinal direction -+- Transverse direction

2 -1 2.3 2.5 2 -7 2-9 3.1 3.3 3.5 Rodial distance (R) mm

Figure 7.1 : Plot of temperature versus distance tÏom the drill in both the longitudinal and the transverse (circumferential) directions, fiom Abouzgia [l]. Note the large amount of scatter about both curves.

The altemate interpretation of the discrepancy is that the mode1 is misshg a physical

factor which occurs only at higher drill speeds. It is the author's opinion that this is the more

plausible interpretation. First, while both Boyne and Spatz may have been correct in stating

that there is no standard way of measuring bone damage, al1 histological experiments on bone

indicate that there is no significant disadvantage to using ultra-high drill speeds, regardles of

the measure of themal damage. Ifmeasuring the acellular zone, determining the amount of

basophilie staining, and examining the healing response al1 indicate the same trend in bone

damage with respect to drill speed, then the trend shouid not be an artefact of the

measurement technique and the conclusions drawn on the effect of drill speed should be

valid. Second, although there was scatter in Abowgia's temperature measurements, he

measured the power consumed by the drill and found that it also decreased with both applied

force and drill speed. The power was measured independentiy of the temperature. Since the

temperature depends on the amount of heat absorbed by the bone, which in tum c m be

related to the power consurned by the drill, it is likely that the drop in temperature recorded

by Abouzgia is real, and not a result of experimental inaccuracies. Given that each argument

against the accuracy of previous investigations has a stronger counter-argument in support of

their accuracy, it was decided that the discrepancy was due to a flaw in the compter

simulation.

That the drilling model is likely incomplete leads to the question of how the model

can be improved in order to make it more consistent with observations. The most likely cause

of the discrepancy is the assumption made about the failure mode of bone. It was assumed

that bone undergoes shear failure throughout the entire range of shear rates. Wiggins and

Mallcin [80], however, exarnined bone chips produced by driliing and determined that most

chips were separated by fracture. Not only c m bone fail through fhcture, but there is

evidence that bone has two types of fracture mechanisms. Krause [39] perfomed

expenments on orthogonal cutting of bone and discovered that the fracture energy decreases

at higher feed rates. Furthemore, in Jacobs et al. [34], evidence is cited which points to a

decrease in hcture energy with increasing shear rate. These studies indicate that the failure

mode of bone passes through three regimes. Bone undergoes shear f a i lw at low shear rates,

with a transition to fiacture failure as the shear rate increases, and a second transition at even

higher shear rates to a fracture mode that releases less energy- Reduced energy wouid result

in lower temperatures and less thermal damage. Incorporating this more complex model of

bone failure may bring the results of the cornputer simulation in line with previous

experimental results.

It was assumed that the use of rnachining anaiysis originally developed for metals and

other engineering materials was appropriate for bone (Tay 1721). It is possible that this

assumption is not valid and is therefore another factor in the discrepancy between the

simulation results and experimental observations. The conclusion that bone can fail through

fracture rnakes it even more likely that the adaptation of Tay's analysis is inappropriate.

Fortunately, there is experimental evidence, presented by Krause 1391, that suggests the use

of Tay's analysis is nonetheless correct. Krause performed orthogonal cutting experiments on

bone and observed an excellent match with Merchant's [49] minimum energy cutting

condition* despite his evidence that bone chips are separated by hcture.

In addition to modifying failure mechanics, fiuther refinements could be made to the

model. Bonfield and Li [9] showed that the stress-strain behaviour of bone changes

considerably when its temperature rises above 50°C. The hcture behaviour might likewise

change with temperature, but no data appear to be available. Wiggins and Malkin [80]

showed that clogging of the flutes at greater depths of penetration increases significantly the

torque and specific cutting energy. The friction between the chips and the drill hole surface

* 2rp + .t - a = 89.4g0 vs. equation (5.8) (Tay's analysis - adapted in the present work - is based Merchant's). 94

also increases the temperature at the surface. The effect of clogging could WEely be modelied

if the traction stresses created by the chips could be determined-

7.2 Effects of Point Angle and Heür Angle

The importance of the drill geometry can be seen in Figures 6.7 and 6.8. Except at

low point angles, there is a decrease in the temperature with increasing helix angle. At the

larger helix angles, the optimum point angle seems to be between 100" and 130°. These

resuits are interesting because they agree with recommendations which were based only on

mechanical considerations [27,64]. For example, Famworth and Burton [27] recommended

point angles between 120" and 140°, and a helix angle of S7O. The temperature drop across

the range of angles in Figure 6.7 is not as large as the 41% drop achieved by the drill design

proposed by Saha et al [64], indicating that the improvement in thermal performance they

achieved was probably a result of improved peuetration times.

7.3 Effects of Drill Diameter

The average maximum temperature (Figure 6.9) and the resultant thermal damage

(Figure 6.1 0) increase almost iinearly with drill diameter. Although the increase in

temperature with drill diameter is hardly surprising, the data may be usefid in determining

optimal drill sizes for pre-drilling.

7.4 Effects of Bone Thermal Properties

It is evident fiom Figures 6.1 1 and 6.12 that the effect of changing the themai

conductivity is much less pronounced than changing either the specific heat or the density.

For a 1 0 % change in thema1 conductivity, there was less than a +1.3% change in either the

average maximum temperature or the arnount of thermal damage. That the change in thermal

95

effect is so small confirms the assumption that cortical bone couid be treated as thermaily

isotropie. The corresponding changes were approximately fi% in the temperature and

approxirnately k!J% in the amount of thermal damage when the density and specific heat were

changed withh + 10% of their nominal values.

In his numerical model of the heat generated by curing cernent in hip arthropIasty,

Huiskes [33] arrived at the opposite conclusion with respect to the importance of the different

thexmai properties. He found that changing the conductivity of bone had a greater influence

than changing the heat capacity of bone. The geometry of his model was different, however,

and may explain the difference in conclusions. Furthemore, Huiskes based his conclusion on

a much larger range of conductivity than that covered in the current study.

The change in thermal damage with density and specific heat is notably differ~nt than

the correspondhg change in maximum temperature. In general, the thermal damage and

maximum temperature follow the same pattern. The difference in this case is that, not only is

the temperature in the bone afTected by the change in thermal properties, but so is the rate at

which the heat dissipates fiom the drill site. Since the heat clearance rate affects not only the

temperature rise, but also the time spent at elevated temperatures, and since thermal damage

is sensitive to both these parameters, the change in clearance rate is likely the reason for the

non-linear behaviour of the curve in Figure 6.12.

CHAPTER 8

Surnmary and Conclusions

8.1 Summary

The purpose of the present research is to investigate the thennal impact of drilling

into bone. Since a physical experiment would not add significantly to the body of knowledge

that has been accumulated to date, it was decided that using a numerical simulation to

perform a parametnc analysis would elicit a better understanding of the importance of

individual drilling parameters.

A search of the iiterature for the thermal properties of bone, which were required for

the simulation, revealed that it was necessary to measure the thermal conductivity of cortical

bone. In particdar, it was necessary to determine whether the thermal conductivity varied

with the direction of heat flow. An apparatus to measure the thermal conductivity was

designed, built, and tested. The apparatus created one-âimensionai heat flow through bone

specimens cut fkom the diaphysis of bovine femora and used the temperature drop across a

material of known conductivity, placed in series with the specimen, to calculate the heat flow

through the specimen. The apparatus was capable of handling different specimen dimensions.

The thermal conductivity of cortical bone was measured as 0.58 * 0.018 W/mK in the

longitudinal direction, 0.53 * 0.030 W/mK in the circderential direction, and 0.54 a0.020

W/mK in the radiai direction. The differences in conductivity between the longitudinal

direction and the other two directions were statistically significant, whiIe the difference

between the radial and circumferential directions was not.

A computer model was constmcted, based the finite element method, to calculate the

time-varying temperature distribution around a drill site in cortical bone. The heat input inta

the FEM model was cakulated using an analysis adapted fkom machining theory for metals.

A parametric analysis of the drillhg operation was performed to determine the importance of

severai drilling parameters on the temperatures generated in bone and on the resultant thermal

damage, which results f?om the temperature remaining above the time-temperature threshold

curve for thermal necrosis. The following drilling parameters were varied: drill rotational

speed, feed rate, point angle, heiix angle, and drill diameter. The density, specific heat, and

thermal conductivity of bone were also varied in order to detennine the importance of the

natural variance in these properties. The force applied to the drill could not be used as an

input parameter because the relation between applied force and feed rate is specific to the

drill geometry and cannot be known a priori.

The results of the computer simulation indicate that the maximum temperature and

the resultant thermal damage increase with drill rotational speed within the range modelled

(100 to 200,000 rpm). The maximum temperature also increases with feed rate from 0.45

mm/s to 1.8 d s . In the range of 1.8 mm/s to 4.5 d s , the maximum temperature either

remains the same or drops slightly.

The drill point angle and helix angle have small, but not negligible, effects on the

temperatures generated. The lowest maximum temperature occurs at a point angle of 130"

and a helix angle of 38". The maximum temperature and thermal damage increase in an

almost linear fashion with diameter (1 .O to 3.5 mm).

The average maximum temperature and corresponding amount of thermal damage

decrease with increasing density, thermal conductivity, and specific heat of bone in the range

o f f 10% of their nominal values. The importance of thermal conductivity is much Iess than

that of the other two themal properties.

The results fiom the simulation were compared with previous experimentai resuits

and they are consistent with those investigations which used lower drill speeds (up to 10,000

rpm). At higher drill speeds, results fkom the simulation disagree with the experimental

evidence that the temperature and thermal damage decrease as the rotational speed inmeases.

8.2 Conclusions

This study concludes corn the thermal conductivity measurements that bovine

cortical bone can be considered thermally isotropic, at least for purposes of rnodelling. Since

the structure and composition of cortical bone is sunilar for different species, it is likely that

the thermal conductivity of human cortical bone can aiso be considered isotropic.

The drilling simulation was found to be realistic at low drill speeds, but a physical

factor appears to be missing fiom the simulation because there is a discrepancy with

experhental evidence at high drill speeds. It was decided that the most probable cause for

the discrepancy is an incorrect modelling of bone's failure mode at high drill speeds. Despite

the discrepancy, the results of the simulation indicate that it is possible to model driiiing

operations numencally. The results also indicate that machining analysis for orthogonal

cutting of metal can be adapted for drilling in bone, given a proper model of bone failure.

Because of the discrepancy, the computer simulation was not considered sufkiently

refined to produce clinically relevant conclusions on drilling technique. Nevertheless, it

should be possible to improve the simulation to the point where it can be w d for planning

orthopaedic or dental operations. Necessary modifications to the simulation include a switch

fiom shear failure to Eracture failure, with a change in the fhcture mode at higher shear rates

that reduces the energy required to separate chips nom the work piece.

Any recomrnendations stemming fiom the model must, however, be bahceci with

other requirements. The ideal driliing operation, in addition to minimi;inng thermal trauma to

the bone surrounding the osteotomy, would minirnize cutting time, thus reducing the time

spent under anaesthesia, would create a hole with an accurate diameter and a smooth surface,

and would rninimize mechanical trauma to the surrounding tissue. Law drill speeds ( l e s than

500 rpm) rninimize the temperature rise, yet produce rough, hgmented and irregular

surfaces [74]. WhiIe higher drill speeds tend to lessen the effort to penetrate the bone, very

fast penetration is not desirable because of potential damage to the soft tissue on the opposite

side. Although the weights attached to thermal considerations, patient comfort, effort

required by the surgeon, and dimensional accuracy Vary with each situation, an informeci

decision is aiways preferable and simulations such as the one constnxcted for this thesis help

tu provide as much information as possible to the practising surgeon.

8.3 Contributions

The research presented in this thesis has made the following contributions to the field

of biomechanics:

Measurements of the thermal conductivity of cortical bone fiom the bovine femoral

diaphysis in the longitudinal, radial, and circumferential directions, which show that cortical

bone can be treated as thermally isotropie.

Development of a numerical model of the thermal impact of drilling into bone,

which uses the finite element method combined with a heat generation model based on

machining analysis.

A parametric analysis of the drillhg operation that examines the importance in the

thermal impact of bone dnlling of the following parameters: drill ro tationd speed, feed rate,

point angle, helix angle, drill diameter, and the thermal properties o f bone (density, specific

heat, and thermal conductivity).

Raw Data: Thermal Conductivity Experiments

1 TABLE A. 1 : Specimen dimensions.

1 Specimen 1 Area [cm2] 1 Thickness [cm]

II TABLE A.2: Evaporation data.

Specimen ti Initial Mass [g] Final Mass [g) 1 Mass Loss [g]

Mass loss data for sample I l not available.

Average mass loss: 2.8%.

[TABLE A.3: Individual thermal conductivity rneasurements.

:ircumferential

specimen k WfmKJ

longitudinal

specimen k WImK]

radial

specimen k [WImK]

TABLE A-3: Individuai thermal conductivity me-ernents.

llspecirnen k w/ml(l (specimen k w/mKj radiai

specimen k [W/mK)

APPENDK B

Raw Data: Testing the Apparatuses

II Table B. 1: Data fiom control material conductivity measurements.

/ Erperimeot 1 Specimen

Plexiglas

Table B. 1 : Data fiom control material conductivity measurements.

Experiment Specimen k w m w

JANSA Dekh 1 PVC 1 0.244 0.226 7.96%

JAN3B Plexiglas 1 Delrin 1 0.426 0.439 -2.96%

JAN4A Plexiglas 1 Delrin 1 0.408 0.439 -7.06%

JAN4B Plexiglas 1 Delrin 1 0.420 0.439 -4.52%

JAN3A Plexiglas 1 Delrin 1 0.415 0.439 -5.47% --

JAN8B Plexiglas 1 Plexiglas III 0.240 0.228 5.26%

JANSA Plexiglas 1 Plexiglas III 0.240 0.228 5.26%

DEC14B Plexiglas 1 Plexiglas ïII 0.248 0.228 8.77%

DEClOA Plexiglas 1 Plexiglas III 0.254 0.228 1 1 -4%

DEC11A Plexiglas 1 Plexiglas IiI 0.247 0.228 8.33%

DEC14A Plexiglas I Plexiglas III 0.243 0.228 6.58% "

DEC 15A Plexiglas 1 PVC 1 0.226 0.226 0%

1 DECISB 1 Plexiglas 1 ( PVC 1 1 0.223 ( 0.226 1 -1.33%

TABLE B.2: Resuits corn testing the main apparatus.

1 knorm k 1 specimen 1 measured 1 expected 1 difIerence

DEC16B Plexiglas III PVC 1 0.227 0.226 0.442%

DEC16A Plexiglas III PVC 1 0.235 0.226 3.98%

JAN7A PVC 1 Plexiglas 1 0.241 0.228 5.70% - 1 i i 1 0.239 1 0.228 4.60%

J A N 7 8 PVCI 1 Plexiglas1 , DEC20A PVC II PVC 1 0.248 0.226 9.73%

DEC17A 1 PVC II 1 PVC 1 1 0.242 1 0.226 1 7.08%

Development of the FEM Numerical Mode1

C.1 Introduction to the Finite Element Method (FEM)

The Finite Element Method is a subset of Weighted Residual Methods (WRMs),

whose purpose is to provide a numerical method for s o h g boundary value problems of the

L0 = q + Boundary Conditions + Initial Conditions, (C-1)

where L is an operator. WRMs solve for 8 by imposing the requirement that (in 1-D

prob f ems)

where (Wi(x)} is a set of weighting fiinctions. Furthemore, WRMs express 8 as

where {Oj} is a set of constants, and (Nj(x)} is a set of ba i s functions. In the Galerlrin

formulation, used in the present analysis,

Nj(x) Wj(x) -

In the FEM approach, the domain R is sub-divided into N elements, each of whose

position and geometry are defined (in 2-D space) by at least three nodes, for a total of M

nodes over the entire domain.

There are three main steps in creating and solving a Finite Element Model:

Generate element mesh

Build system of equations and

solve for temperatures

Post-process the results

C.2 Spatial Discretization

The cornputahonal domain, Q, was modelled as an

annulus with inner radius Ri and outer radius &. The inner

radius represented the margin of the hole created by the drill

and it is across this boundary that the heat fkom the drillhg

operation was applied. The height of the annulus, H, was

chosen to represent the thickness of the corticd bone being

drined, with z = O representing the surface of the bone exposed

Figure C.l: Schematic drawing of volume modeiied by the computer simulation

during the operation, and z = H the surface of the medullary canal.

The axisymmetry of the mode1 (goveming equation, boundary conditions, and thermal

properties) allowed modelling of on1 y a two-dimensional (r,z) slice (darkened rectangle in

Figure C. 1).

Figure C2: Schematic drawing of the eIement mesh, with image of drill placed for reference.

The element mesh, shown schematically in Figure C.2, was fonned with 4-noded

quadrilateral elements. That element type was chosen because dekition of the mesh with

quaclrilaterai elements was simple, given the geometry of the domain. The boundaries of the

domain, as well as the nurnber of elements in the r- and z-directions were read fkom a data

file. A restriction was placed on the inner radius; it had to be greater than zero. The outer

radius was chosen so that the boundary condition imposed on it (T = 37OC) was valid.

The number of elements in the mesh was chosen so that a balance was struck between

the size of the model, which affects both the memory and the run tirne, and the accuracy of

the results. In general, the greater the number of elements, the higher the accuracy. This is an

important consideration in areas of large temperature gradients. In thîs model, large spatial

and temporal gradients were expected near the inner radius, where the heat fkom drilling was

applied, and the temperature gradients near the outer radius were expected to be low. It was

possible to take advantage of this by using a scaling factor to adjust the ratio of the length (in

the direction) of the elernents nearest the outer radius to the length of the elements nearest

the inner radius. The scaling factor was used to increase the number of elements nearest the

imer radius without increasing the number of elements overall. The result was an increased

accuracy in a region with large temperature gradients without increasing the amount of

memory or the duration of the simulation run. (The scaling factor in Figure C.2 is 1 : 1 .)

C.3 Unsteady (r,z,t) FEM Solver

The differential equation solved by the current mode1 was the two-dimensional

unsteady heat conduction equation, with no heat generation, in polar CO-ordinates.

t(aZe8+1de +- a2e) = P C - + ~ , de ûr2 r û r aZz a t

The termf, is a forcing term used ta simulate the heat input fkom the drill. Since it is

applied only at one portion of the h e r boundary, its implementation in the FEM formulation

will be discussed in Section (2.4.

The first step was to apply the weighted residual requirement, Le.,

The domain was then split into N elements.

(Ca When the Galerkin formuIation was combined with the FEM approach, the

requirement imposed on the differentiai equation became

K.8)

After perforrning the differentiation and integrating by parts, the Galerkin fomulation

became

The set {N,(r,z)) are the weighting hctions, also cailed shape functions, which are

defined locaily. Each fùnction is equal zero at al1 nodes Save one, and each node has only one

shape function with a non-zero value. For example, N14 has a non-zero value at only node 14,

and al1 other shape functions in the set {N,(r,z)) are zero at node 14. When this property of

the shape hc t ions was taken into account, the Galericin formulation became

(C. 10)

The constant n~ is the number of local nodes, Le., the number of nodes required to

d e h e a single element. Equation (C. 10) represents a system of equations, which were solved

to obtain the temperahue at each of the nodes, (0,).

The above equation is discretized in space, but not in time, which is required for an

unstead y anal ysis. The mode1 used Crank-Nicolson tune-stepping. The fully discretized

equation c m be more easily expressed by converthg to a matrix equation.

[A] and [BI were constmcted by evaluating, for each element,

(C. 12)

The subscripts i j refer to the local node numbers. In constnrcting the global matrices

[A] and @3], the local node numbers were mapped to the global node numbers. The mapping

information was contained in the element comectivity table.

C.4 Boundary Conditions

The boundaries in the z-direction, i-e., representing the outer surface of the bone and

the surface of the medullary canal, were insulated Homogeneous Neumann boundary

conditions are asswned in the finite element formulation, and thus no special implementation

was required.

The boundary condition prescribed on the outer boundary (r = &) was a constant

temperature of 37°C (normal body temperature). This Dirichlet boundary condition was

implemented by condensing the system of equations to remove the rows and columns that

corresponded to the boundary nodes at r = &. The basic procedure involved, for each node

on the boundary,

subtracting the column of the global matrix corresponding to the boundary node,

multipiied by the applied boundary temperature, fiom the nodal temperature

vector,

eliminating the boundary node fiom the nodal temperature vector, {ej), and

re-sizing the global matrix, eliminating the row and column associated with the

boundary node.

The condensed values were re-integrated d e r the condensed system of equations was

solved.

The heat input nom the drill was modelled as a set of forcing terms (4) applied at the

boundary. Otherwise, the inner boundary \vas considered insulated. In order to mode1 the

localized nature of the heat input, forcing terms were calculated only for the boundary nodes

of the element whose location corresponded to the location of the drill bit.

where

n = element number,

i j = local nodes (mapped to global nodes before adding to the system),

Mi, M, = one-dimensional shape fùnctions (integral evaluated in the z-

direction only),

z,, z,+~ = boundaries of the element in the z-direction.

q = heat flux, calculated fkom equation C. 14.

(C. 13)

(C. 14)

where

Qw = heat flowing into bone,

Ri = radius of dnll bit,

h = Az of one element.

The forcing terms were then added to the right hand side of equation (C. 1 1) after the

rnatrix multiplication was performed.

An altemate approach would have been to treat the heat input as a heat flux boundary

condition, which would have involved re-writhg the matrix equation as

The elements of the matrix [f l would be calculated in a manner similar to the forcing

tems (4}, Le.

(C. 16)

The local node numbers (1,2) would be mapped to the appropnate global nurnbers

before the heat flux terms were added to the system.

C S Isoparametric Transformation

Evaluating the double integrals for Aijn and Bi/ can be problematic. The task was

simplified greatly by transforming each element to an isoparametric element. An example of

this transformation is given in the figure below.

Figure 42.3: Transformation to an isoparameh-ïc element.

The transformation (r,z) + (6,q) was accompiished via the Jacobian and its inverse.

Using the fact that

L a ? as.

the integrals to be evaluated were re-written.

(C. 'L 9 )

If the shape functions {Ni) are dehed in tems o f (c,~), and are the same for ail

elements, then the shape fûnctions can be defmed globaily, not locally. There were therefore

only as many shape functions as there were bcai nodes per element. The shape functions

used with the elements in this model, dong with a schematic of an isoparametric element

showing the locations of the local nodes, are given below.

Figure C.4: Isoparametric element

C.6 Gaussian Integration

The numerical integration routine that was used to evaluate Aidn and Bijn was the

Gaussian integration scheme, which involves taking a weighted s w n of the integrand

evaluated at key, or Gaussian, points. For the isoparametric element with boundaries 5 = *1

and q = "1, the Gaussian integration scheme is denned as

C.7 Matrix Storage

The global matrices were banded. The bandwidth depended on the maximum

difference between node aumbers in each element. The banded nature of the matrix was

exploited to reduce the amount of memory required. The matrices [A] and p ] were stored in

1-D arrays. The size of the arrays was defined as

size([A]) = size(p]) = (m x (2 x N - m - 1) + N + 2) (C.23)

where,

m = bandwidth,

N = # nodes.

Elements of the matrix were accessed through a subroutine point (i , j ) which

returned the position in the 1-D array corresponding to the indices of the 2-D matrix.

C.8 Solnng the System of Equations

Once the boundary conditions were applied, the system was solved using LU-

decomposition and back substitution.

C.9 Analytical Solutions

Analytical solutions to the heat transfer equation were used in benchmark tests of the

Finite Element Code. Four solutions were obtained, two to the 2-D (r,z) steady-state heat

equation, and two to the 1-D (r) unsteady heat equation. Two solutions were obtained for

each equation because two types of boundary conditions were applied at the inner boundary

of each problem. For the 2-D (r,z) steady equation, both a temperature boundary condition

and a heat flux boundary condition were applied. In both cases, the boundary condition was

described by a continuous function of z. For the 1-D (r) equation, both a constant temperature

and a constant heat flux were applied at r = Ri. It was aecessary that these boundary

conditions were simpler than that which represented the heat nom drilling; analytical

solutions would not be available otherwise.

For the 2-D problem descnbed by

d2T ln aZT dr2 r d r az-

=O; T(Ro,z)=To; T(Ri,z)-To =e i (z )

where

T, = temperature at the bouudary r = %,

Bi(z) = hinction defining behaviour of temperature at r = Ri,

the solution is

For the 2-D problem described by

where

q(z) = function denning behaviour of heat flux boundary condition at r = Ri,

the solution is

For the 1-D probEem descnbed by

where

a = thexmai diffisivity,

Ti = temperature at the boundary r = Ri.

T, = temperature at the boundary r = &, and also the initiai temperature,

the soIution is

For the 1-D problem described by

B.C.:T(R,, t) = T,, = qi ; I.C. : T(r,O) = T,

where

qi = heat flux at the boundary r = Ri,

the solution is

C.10 Flow Chart

Read input data 1 f i o y file. 1

Calculate drilling

1

' Build global stifhess [A] 1

Re-write [A] to hold LHS 1 of equation 5.2, i.e.

2 i (%+EL) ,

1

Set temperatures to 1 initial values. 1

1 I

1 on RHS of equation5.3, Le. 1

r

/ Calculate position o f drill point / & apply ail boundary conditions.

- -

J

Track the time spent over threshuld temperatures for darnage.

1 Track maximum temperatures. / Calculate and print heat 1 energy inthesystem.

Ptint nodal temperatures '1 to file.

1 Detennine which elements / are 'damaged'.

I

Print maximum temperatures 1 1 and d a m y S o to file 1

Error Analysis

In order to perform an error analysis of the prirnary apparatus, an error analysis of the

secondary apparatus (the apparatus used to measure the thermal conductivity of the plastics)

was necessary because its measurements were used to calculate the thermal conductivity of

bone.

D.l Secondary Apparatus

The thexmal conductivity in the secondary apparatus is calculated according to the

equation

where

Q = heat flow,

Ax = specimen thickness,

A = specïmen cross-sectional area,

AT = temperature drop across the sample.

In order to h d the error 4, each of the terms in the above equation is evaluated in

order to determine their individual errors, i.e. e ~ , e b e ~ , and em.

N.B. The values in this analysis are taken fiom test W C .

Heat Flow: Q

The heat flow is a product of the voltage, V, and the current, I, both of which

were measured using a mukirneter.

V = 29.l3V * 0.005V I = 4 . 8 0 7 x l 0 - ~ ~ * 5 x 1 0 ~ ~

The thickness of the materid was measured with Vernier c d

AX = 8 . 4 5 ~ 1 0 - ~ m * ~ x 1 0 ~ r n

eh = 5x 104m

Cross-sectional Area: A

The cross-sectional area is the product of two dimensions (length, 1, and

width, w) both of which were measured using Vernier calipers.

1 = 5.078x10-~m * 5x1o4rn

w = 5.104~10-*m * 5x10~rn

A = l - w

= ( 5 . 1 0 4 ~ 10-~)'(5x lod)' + (5.078 x 10-')'(5 x 1 0 ~ ) ~

-7 ' e ~ = 3 . 6 0 x 1 0 m'

-Temperature dmp: AT

The temperature drop across the specimen is the difference between the

averages of three temperatures measured on each side of the sample, Le.,

AT = Tht - Tcold

Tbt = (Tq + T5 f T6)/3

Tcold = (T7 + Tg + T9)/3

Ta = 49.6S°C * O. 174°C T7 = 40.17OC * 0.174"C

TS = 49.68"C f 0.174"C T8 = 39.8 1 OC * 0.174"C

Tg = 49.3 1 OC f O. 174°C Tg = 39.59"C * 0.174"C

The error in the individual temperatures is equal to the standard deviation of

data points around the thennocouple calibration c w e .

e a ~ = (0.174OC + 0.174OC + 0.174OC)/3 + (0.174OC + 0.174OC + 0.174"C)/3

= 0.34g°C

The error in the thermal conductivity measured by the secondary apparatus can now

be calculated,

D.2 Primary Apparatus

The thermal conductivity of bone, kz, is calculated fiom the equation:

where,

Q = heat flowing through the bone specimen,

Axz = thickness of the bone specimen,

At = cross-sectionai area of the bone specimen,

AT2 = temperature drop across the bone specimen.

As before, the error in the individual variables in the equation above needs to be

evaluated in order to determine m.

N.B. Values in this analysis were taken fiom test MAR16C.

-Heatflow: Q

The heat flow is calculated fkom the equation

where

kl = thermal conductivity of Plexiglas bIock, measued using

secondary apparatus,

Al = cross-sectional area of Plexiglas block,

ATI = temperature drop across the PlexigIas block,

Axi = thickness of the Plexiglas block.

To calculate ep, qi, e ~ i , e ~ ~ i and e h , must be determïned.

Thermal conductivitv of Plexiglas: kl

The value of the themal conductivity of Plexiglas (0.228 W/mK) was

taken as the average of the results fiom tests JUN13A to JUN18B. The

expected error for the thermal conductivity of the Plexiglas specimen was

determined in section D. 1.

ekl = 0.0085 W/mK

Cross-sectional area: Al -

The cross-sectional area was the product of two dimensions, length, II,

and width, wl, measured by Vernier caiipers.

I I = 8.94xloJrn * 5x104m

w1 = 9 . 5 2 ~ 1 oJrn * 5x lo4m

= (9.52 x 10")~(5 x 1 ~ ~ ) ~ + (8.94 x ~ o - ~ ) ~ ( s x 10-~)'

-a 2 e ~ l = 6 . 5 3 ~ 10 m

Temperature drop: ATI

The temperature drop across the Plexiglas block was the difference of

the temperature measurements nom two thermocouples (#6 and #7).

T7 = 55.52OC f 0.174OC

Tg = 34.57OC f 0.174°C

Once again, the errors in the individual temperatures are equal to the

standard deviation of data points about the thermocouple caiibration curve.

ATi = T7 - Ts = (55.52OC f 0.174OC) - (34.57OC * 0.174OC)

e ~ ~ i = 0.348"C

Thickness: Ax

The thickness of the Plexiglas specimen was measured using Vernier

caiipers:

hi = 8.44x10"m 1 5 x 1 0 ~ r n

e & ~ = 5x1odrn

The error in the heat flow, e ~ , can now be calculated:

ep = 1 . 9 7 x 1 0 - ~ ~

Thickness: A-:

The thickness of the bone sample was measured using Vernier calipers.

e m = 5x 10%

Cross-sec&ional: A2

The cross-sectional area is the product of two dimensions, the length, 12, and

the width, WZ, both measured with Vernier calipers.

12 = 8 . 8 0 ~ 1 F3m * sx 1 04m

wz = 9-44x loJm + sx iodm

Az = 1 2 . ~ 2

- &, - wiet+iie2 w2 =(9.44~10-')~(5~l0~)~+(8.80~10'~)~(5~10~)'

-8 2 e u = 6.45~10 m

Temperature dmp: ATr

The temperature drop across the specimen was the difference between

readings fÏom two different thermocouples (#6 and #8).

Tg = 34.57OC * 0.174OC Tg = 24.7S°C * O.174OC AT2 = Tg - Tg = (34.57OC f 0.174OC) - (24.75OC f O. l74OC)

e ~ m = 0.348"C

e~ = 0.029 W/mK

For a nominal thermal conductivity of 0.538 W/mK,

eu = *5.4%.

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