Monte Carlo Simulation of X-ray Scattering for Quantitative Characterization of Breast Cancer

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Monte Carlo simulation of x-ray scattering for quantitative characterization of breast cancer

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2009 Phys. Med. Biol. 54 3773

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IOP PUBLISHING PHYSICS INMEDICINE ANDBIOLOGY

Phys. Med. Biol.54(2009) 37733784 doi:10.1088/0031-9155/54/12/011

Monte Carlo simulation of x-ray scattering forquantitative characterization of breast cancer

Wael M Elshemey and Wafaa B Elsharkawy

Biophysics Department, Faculty of Science, Cairo University, Egypt

E-mail: [email protected]

Received 15 February 2009, in final form 7 April 2009

Published 28 May 2009

Online atstacks.iop.org/PMB/54/3773

Abstract

In the last decade there has been growing interest in the possibility of

characterizing breast cancer using differences in the coherent x-ray-scattering

profiles of normal and malignant tissues. To a great extent, characterization has

depended on the differences in the peak positions of both tissues in addition to

the overall profile which exhibits a distinctive sharp adipose peak in the case of

a normal breast. In many excised tissue samples, breast cancer samples may

be mixed with a variable percentage of other tissues which affect the shape

of the x-ray-scattering profile and consequently the ability to characterize the

tissue. Moreover, fibroglandular tissue produces a scattering profile showing

an extent of similarity to breast cancer. The present study introduces a Monte

Carlo simulation code capable of tracing photon transport inside a mixedtwo-component sample. The code is utilized to simulate and best fit x-ray-

scattering profiles of the measured samples. This provides reliable breast

tissue characterization in addition to a quantitative estimate of the percentage

of each component in a given sample. It is expected that the present simulation

would potentially enhance the characterization of breast cancer using the x-

ray-scattering technique.

1. Introduction

Biological samples have distinctive x-ray-scattering profiles with one or more peaks in theforward direction of scattering. These peaks arise from the interference of coherently scattered

photons and are found to hold characteristic features of the investigated samples.

Several authors have presented x-ray-scattering measurements on a variety of important

biological samples such as muscle, liver, fat, blood, bone, brain white and gray matter, breast

tissue, hair, bioequivalent materials, in addition to a number of lyophilized proteins and

biological samples (Kosanetzkyet al1987, Evans et al1991, Royle and Speller1991,1995,

Farquharson and Speller1997, Kidaneet al 1999, Elshemeyet al 2001, 2003, Desoukyet al

0031-9155/09/123773+12$30.00 2009 Institute of Physics and Engineering in Medicine Printed in the UK 3773
http://dx.doi.org/10.1088/0031-9155/54/12/011mailto:[email protected]://stacks.iop.org/PMB/54/3773http://stacks.iop.org/PMB/54/3773mailto:[email protected]://dx.doi.org/10.1088/0031-9155/54/12/011


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3774 W M Elshemey and W B Elsharkawy

2001, Polettiet al2002a, and James2006). Their measurements supported the expectation of

Speller and Horrocks (1991) about the x-ray-scattering technique becoming a new source of

information in medicine and biology.

While most of these measurements were performed using the angle dispersive method of

conventional x-ray diffractometers, some authors used an energy dispersive x-ray-scattering(EDXRS) method employing synchrotron radiation to obtain similar information (Ryan and

Farquharson2004and Castroet al2004).

The dependence of the shape of the x-ray-scattering profile on the interference of

coherently scattered photons directed the efforts of some authors toward obtaining measured

molecular form factors which accounted for molecular interference effects (Tartari et al1997,

2002, Peplow and Verghese 1998 and Poletti et al 2002b). Johns and Wismayer (2004)

reported that a degree of variation existed in the form factor values obtained using different

diffractometers.

Elshemey etal (1999) used a Monte Carlo simulation code employingmeasured molecular

form factors by Peplow and verghese (1998) in order to examine the possibility of tissue

characterization of several biological samples.

A number of significant medical applications of the x-ray-scattering technique have beenpresented by many authors. These applications included the determination of bone mineral

density (Royle and Speller 1991, 1995), the detection of liver cirrhosis and hepatocellular

carcinoma (Elshemey et al 2003), the development of new imaging techniques such as

coherent x-ray scatter imaging and diffraction enhanced imaging (Harding and Schreiber

1999, Bohndieket al 2008 and Griffiths et al 2008) and the measurement of x-ray scatter

signatures for normal and neoplastic breast tissue based on the differences in fat content of

breast in both cases. The last application was first introduced by Kidane et al(1999) followed

by valuable contributions by a number of research groups (Ryan and Farquharson2004,2007,

Castroet al2004, Ryanet al2005, Bohndieket al2008and Griffithset al2008).

Theodorakou and Farquharson (2008) reviewed the x-ray techniques used for human

soft tissue analysis. The reviewed techniques included x-ray diffraction, x-ray fluorescence,

Compton scattering, Compton to coherent scattering ratio and attenuation measurements. The

classified soft tissues included brain, breast, colon, fat, kidney, liver, lung, muscle, prostate,skin, thyroid and uterus.

In this work, a Monte Carlo simulation code is modified so that it becomes capable of

simulating x-ray scattering from mixed two-component samples. The code is compared with

the old version (Elshemey et al 1999) for x-ray scattering from single-component samples.

It is also compared with practical measurements for single and mixed biological samples

with predefined components percentages by volume. Mixed biological samples included

fibroglandular, normal and malignant breast tissues. Quantitative estimation of the percentage

of each component in mixed samples is also assessed.

2. Theoretical background

Low-angle scattering of low-energy x-ray photons (e.g., 8.047 keV) is dominated by the elastic

(coherent) scattering process. The energy transferred to the struck electron is small compared

with the binding energy of the electron. The atom is neither ionized nor excited and the recoil

momentum is absorbed by the entire atom. Under these conditions, the wavelength of the

radiation scattered by the bound electrons of an atom is essentially the same as that of the

incident photon. There is a fixed phase relationship among the scattered radiations, which are

thus capable of producing constructive interference. The differential coherent scattering cross


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Monte Carlo simulation of x-ray scattering for quantitative characterization of breast cancer 3775

section per atom for unpolarized radiation determines the new direction of the photon after

coherent scattering and can be expressed approximately as

dacoh

d

=

r2o

21 + cos

2 . [F(x,Z)]2 , (1)

whererois the classical electron radius,is the photon scattering angle and

r2o

2

1 + cos2

(2)

is the energy-independent Thomson differential cross section per electron, (deT/d). The

variable F(x, Z)is called the atomic form factor which is the sum of the electronic form factors

and represents the ratio of the amplitude of the coherently scattered radiation by an entire

atom to that by a single free electron. The square of this form factor is the probability that

Zelectrons of the atom take up a recoil momentum, (q), without absorbing any energy. The

form factor depends on the combined variable x= sin( /2)/, whereis the wavelength of

the incident photon.

When considering a molecule, the coherent differential scattering cross section is given

bydmcoh

d=

r 2o

2

1 + cos2

.F2m(x), (3)

whereF2m(x) is the square of molecular form factor.

While coherent scattering is the dominant interaction process at low photon energies,

incoherent scattering still takes place but with lower probability. The differential cross section

of incoherent scattering including electron binding effects can be given as the product of

KleinNishina differential cross section dKN,e/d(for Compton collision between a photon

and a free electron) and the incoherent scattering function of an atom S(x, Z). The latter

represents the probability that an atom will be raised to any excited or ionized state when

a photon imparts a recoil momentum to an atomic electron. The quantityx is the same as

that previously defined for the form factor. The differential cross section of a molecule for

incoherent scattering determines the new direction of photon after incoherent scattering andcan be written as

dincoh,m

d=

dKN,e

dSm(x) =

r 2o

2

E/

E

2 E/

E+

E

E/+ cos2 1

Sm(x), (4)

whereSm(x) is the incoherent scattering function of a molecule, considering that atomic cross

sections for incoherent scattering combine independently. EandE are the energies of incident

and scattered photons, respectively.

At low photon energies, the third possible interaction of an incident photon with a

biological sample is photoelectric absorption, where an incident photon is totally absorbed at

the interaction site. The attenuation of an x-ray beam in the diagnostic energy range is due to

all three interaction processes.

For a biological sample of known composition, its mass attenuation coefficient / can

be approximately evaluated from the tabulated coefficientsi/ifor the constituent elements

according to the weighted average wi of each element, where

=

i

wi i

ior = .

i

wi i

i(5)

iand iare, respectively, the attenuation coefficient and density of element i, while and

are, respectively, the linear attenuation coefficient and density of the biological sample

(Hubbell1969).


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3. Monte Carlo simulation

The present Monte Carlo simulation is a modification of the Monte Carlo simulation by

Elshemey et al (1999). The latter is a step-by-step tracing of the simulation algorithm and

sampling procedures described in detail by Chan and Doi (1983) which was capable ofsimulating the photon transport in a single sample at a time.

For each photon incident on the sample, the first step is to calculate its free path t in

order to predict the first interaction site. This is carried out by sampling from the exponential

probability density function:p(t)= et, suchthatt=1/( ln r), where ris a randomnumber

uniformly distributed in the interval [0, 1] and is the total linear attenuation coefficient of

the scattering medium at the energy of the incident photon (Chan and Doi 1983).

In this work, in order to simulate the photon transport in a two-component mixture, first

the percentage by volume of each component in the mixture is identified. The first photon

free path length is then calculated as the sum of two displacements. The first displacement is

given byd1 = t1v1wheret1is the simulated free path length of photon in the first component

given by t1 = 1/(1 ln r1) and v1 is the percentage by volume of the first component in the

mixture. Similarly, the second displacement is equal to the simulated free path length of the

photon in the second component multiplied by its percentage by volume in the mixture, d2 =

t2v2. In this way, the effective free path length, d, is determined based on the relative volume

of each component in the mixture where, d= d1+ d2.

If the free path length of the photon falls within the sample dimension, the component

with which the photon will interact is selected by generating a uniform random number (from

0 up to 1). If this random number is less than or equal to the percentage by volume of the

first component in the mixture (note that, for example, 20% is equal to 0.2), then the photon

will interact with the first component, otherwise, the photon will interact with the second

component.

The type of interaction mechanism of the photon in the selected component is determined

by the relative probability of interaction at the given photon energy. A random number is

drawn, and according to its value an interaction mechanism is selected (Chan and Doi1983).

The photon is either absorbed (photoelectric effect) and consequently the program willgenerate a new photon, or it will be coherently scattered thus the program will continue tracing

the photon by calculating the new photon coordinates, new free path length and interaction

site taking into consideration the photon scattering angle () and simulated azimuthal angle

(). If the photon is incoherently scattered, the program will follow the same steps as coherent

scattering except that it will take into consideration the change in scattered photon energy.

If the free path length of photon at any step falls outside the sample all photon information

including energy and scattering angle will be saved in a file for the development of the x-ray

photon-scattering profile. At this end, the program will generate another photon up to a pre-

defined maximum number of photons. In the present simulation, 9 106 photons are generated

in each run of the program. The values of the incoherent scattering function are obtained from

Hubbellet al(1975) and the values of measured coherent scattering form factors accounting

for molecular interference effects are obtained from Peplow and Verghese(1998), whereas the

values of photon attenuation coefficients are obtained from Hubbell(1977). These tabulations

were previously proven to be reliable and used by many authors. The block diagram in

figure1summarizes the main simulation steps of the present Monte Carlo code.

4. Materials and methods

Fresh beef muscle and adipose tissues are purchased from a local market, and are mixed at

different percentages by volume using a mixer until acceptable homogeneity is reached. The


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Programinputs

Energy

1st sample

percentageSample

dimension

Number ofphotons

2nd samplepercentage

Uploading files containing data for incoherent, coherent,& photoelectric cross-sections for selected samples

Calculate free path length of photon in mixed sampleaccording to percentage of each sample in the mixture

Free path length >sample dimension

Transmitted

photon

Select the sample

with which photonwill interact

according to itspercentage (relativeprobability) in the

mixture

Determine interaction type according to relative interactionprobabilities for selected sample at present photon energy

New photoncoordinates

Coherent Incoherent Photoelectric

Generate New , ,keep E unchanged Generate new, and E Absorbed photon

Energy < threshold Generate newphoton

No

No

Yes

Yes

Figure 1. A block diagram of the Monte Carlo codes main simulation steps.

samples are then measured using a Philips (X-Pert) x-ray diffractometer operating at 40 kV

and 10 mA producing collimated monoenergetic Cu Kx-rays of 8.047 keV. Measurements

are done in the step mode at a step equal to 0.25. The water sample is measured using

the same diffractometer. Breast samples are obtained from women undergoing mastectomyand are preserved in formalin until measured (Peplow and Verghese 1998). The samples

are characterized by a histopathologist for being normal or malignant. Normal breast tissue

samples are usually obtained from the safety margin of a tumor. Breast samples are measured

using a Shimadzu x-ray diffractometer working at 40 kV and 30 mA. The device employs a Cu

target to produce a monoenergetic, 8.047 keV highly collimated x-ray beam. Measurements

are done in step mode at a step equal to 0.5. Diffraction data are collected using a scintillation

detector employing a sodium iodide crystal and a graphite monochromator. Measured and


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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Momentum transfer (nm -1)

Normalizedcounts

simulated (new) simulated (old)

Water

(b)

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated (new) simulated (old)

Beef muscle

(a)

Figure 2. Simulation of x-ray scattering from (a) beef muscle and (b) water using the present codeand the code used by Elshemey et al(1999).

simulated data are subject to three-point smoothing. In the present simulation, pork muscle

is used to simulate breast cancer (Peplow and Verghese 1998and Griffiths et al 2007) while

fibroglandular tissue is simulated as a mixture of adipose tissue and stroma (Kidane et al

1999). Water is used to simulate stroma.

5. Results and discussion

The current simulation is made to pass over several steps of validation. First, the new

simulation (current version) is compared with the old simulation (Elshemey et al 1999) for

x-ray scattering from samples composed of a single component. In order to simulate scattering

from a single component in the modified version, the percentage of the second component is

set to zero. Figures2(a) and (b) present simulated x-ray-scattering profiles from beef muscle

and water, respectively, using the old and new simulation codes. The good agreement between

the old and new simulated x-ray-scattering profiles for both samples indicates that the new

version is still capable of producing scattering profiles similar to that produced by the validated

old program.

The current simulation is further validated by comparing x-ray-scattering profiles from

samples composed of a single component using the present code with measured profiles of

water, beef muscle and beef adipose (figures3(a), (b) and (c), respectively). The results show

that simulated data fit well with the measured samples.

The ability of the present code to simulate x-ray scattering from mixed two-component

samples is examined by comparison with measured profiles of samples mixed at well-definedpercentages by volume. For such purpose, three mixtures having different percentages

(27:73%, 53:47% and 77:23%) of beef adipose and muscle respectively are measured and

compared to simulated profiles at the same percentages (figures3(d), (e) and (f), respectively).

An apparent agreement can be seen between simulated and measured samples. Measuring the

correlation coefficient between simulated and measured data further reinforces this agreement.

The correlation coefficient values for simulated and measured data sets in figure 3 are as

follows: water (0.9793), beef muscle (0.9757), beef adipose (0.9095) and for the three mixtures


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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

(b)

Muscle

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

(c)

Adipose tissue

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

(d)

77% adipose

+ 23% muscle

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Momentum transfer(nm-1)

Normalizedcounts

simulated measured

(e)

53% adipose

+ 47% muscle

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

(f)

27% adipose

+ 73% muscle

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

(a)

Water

Figure 3. Simulated and measured x-ray-scattering profiles of (a) water, (b) beef muscle, (c) beefadipose and (d)(f) mixtures of beef adipose and muscle.

of beef adipose and muscle (27:73%, 53:47% and 77:23%), the values are 0.9308, 0.9631 and

0.9587, respectively.

In this work, the correlation coefficient is measured starting at a momentum transfer value

of 0.9 nm1 (equivalent to a scattering angle of 16 for 8.047 incident x-ray photons) in order


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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

measured (breast cancer)

measured (normal breast)

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated(breast cancer)

simulated (normal breast)

(b)(a)

Figure 4. Measured (a) and simulated (b) x-ray-scattering profiles of normal breast and breast

cancer samples.

to avoid the disagreement between measured and simulated data at low momentum transfer

values. This disagreement arises mainly from excess measured primary x-rays reaching the

detector at low momentum transfers. For this reason, Peplow and Verghese (1998) used

extrapolated molecular form factor values in their tabulations at low momentum transfers.

Recent work on the characterization of normal and malignant breast tissues using coherent

x-ray scattering has potentially depended on the differences in the percentage of fat and muscle

in normal and malignant breast tissues. Normal breast tissue has higher fat content compared

to malignant tissue. Thus, the scattering profile of normal breast shows a sharp adipose

peak at 1.1 nm1 and a much shorter shoulder at 1.6 nm1. On the other hand, malignant

breast tissue is characterized by reduced intensity at 1.1 nm1 and increased intensity at

1.6 nm1 (kidaneet al1999, Ryan and Farquharson2004, Castroet al2004and Theodorakou

and Farquharson2008). Figure4(a) shows the measured x-ray-scattering profiles of normal

and malignant breast tissues. The difference in the peak position in both cases is obvious.

Figure4(b) illustrates these differences using the present Monte Carlo simulation.

Figure5presents simulated and measured x-ray-scattering profiles of (a) breast adipose,

(b) normal breast, (c)(e) three breast cancer samples and (f) mixed sample. A remarkable

agreement is shown between simulated and measured profiles with a correlation coefficient of

about 0.9955 for all samples except the mixed sample (figure5(f)), which shows a correlation

coefficient of 0.9807. The differences in the scattering profiles at momentum transfer values

less than 0.9 nm1 are discussed earlier in this section.

It has been possible using the current simulation to predict a quantitative estimate of

the composition of the mixed breast tissue sample presented in figure5(f). Despite beingcharacterized as a breast cancer by the histopathologist, the x-ray-scattering profile of this

sample looked remarkably different from the profiles of other breast cancer samples, except

that it still retains the same peak position. The comparison of such sample with simulated x-ray

scattering from breast cancer also yielded low correlation. This suggested the presence of a

component other than breast cancer tissue that has contributed to the x-ray-scattering profile in

figure5(f). After performing a number of iterative simulations and measuring the correlation

coefficient in each step, simulated and measured data yielded best correlation (0.9807) at


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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcount

s

simulated measured

Breast adipose

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

Normal breast

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

Breast cancer

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

Breast cancer

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

Breast cancer

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

simulated measured

Mixed sample

(b)

(c) (d)

(e) (f)

(a)

Figure 5.Simulatedand measured x-ray-scattering profiles of (a) breast adipose, (b) normal breast,(c)(e) three breast cancer samples and (f) mixed sample.

80% breast cancer and 20% normal tissue. The selection of breast tissue components to fit a

measured profile is not random and depends on the available knowledge of the characteristics

of x-ray-scattering profiles of different breast tissue compositions. This knowledge makes the

fitting process faster, knowing that a single simulation takes less than a minute. The ability


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0

1000

2000

3000

4000

5000

6000

0 1 2 3 4


Counts

100%a 75%a : 25%c50%a : 50%c 25%a : 75%c100%c

0

400

800

1200

1600

2000

2400

2800

3200

0 1 2 3 4


Counts

100%n 75%n : 25%c50%n :50%c 25%n : 75%c100%c

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

25%a:75%s normal breastbreast cancer

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Momentum transfer(nm -1)

Normalizedcounts

Kidane et al 1999 23%a : 77%s

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4


Normalizedcounts

30%a : 70%s 25%a : 75%s20%a : 80%s 15%a : 85%s

(b)

(c) (d)

(e)

(a)

Figure 6. Simulated x-ray-scattering profiles of (a) mixtures of adipose and breast cancer, (b)mixtures of normal breast and breast cancer, (c) different compositions of fibroglandular tissue, (d)fibroglandular, normal breast and breast cancer and (e) fitting of fibroglandular tissue from Kidaneet al(1999) using present simulation (23% adipose:77% stroma).

of the present code to fit mixed samples having well-predefined percentages with remarkable

success (figures3(d), (e) and (f)) would probably add confidence to the predicted component

types and percentages.


12/13


Figure 6 presents simulated x-ray-scattering profiles of mixtures that are possibly

encountered when investigating an excised breast tissue sample. Figure 6(a) shows

mixed samples of different percentages of breast adipose and breast cancer revealing some

characteristic features differentiating each sample composition. As the percentage of breast

cancer increases, the intensity of the adipose peak at 1.1 nm1 decreases while the intensityat 1.6 nm1 increases. Similar behavior is seen in figure6(b) for different percentages of

normal and cancerous breast tissue. Figure6(c) presents possible x-ray-scattering profiles

of fibroglandular tissue composed of different percentages of breast adipose and stroma.

The characteristic profile exhibited by each composition shows how much a Monte Carlo

simulation would be useful in characterizing and predicting the compositions of such tissues.

This remark also applies to the tissue compositions presented in figures 6(a) and (b).

Figure6(d) shows how fibroglandular tissue (25% adipose and 75% stroma), normal breast

and breast cancer can easily be distinguished from their x-ray-scattering profiles. The present

simulation has also been used to fit the fibroglandular x-ray-scattering profile presented by

Kidaneet al1999(figure6(e)). The program estimates that the fitted fibroglandular tissue is

composed of 23% breast adipose and 77% stroma based on a correlation coefficient of 0.9649

between measured and simulated data.

6. Conclusion

It has been possible to construct and validate a Monte Carlo simulation capable of simulating

x-ray scattering from breast tissue samples. The code has shown remarkable reliability in

simulating the measured x-ray-scattering profiles of single and mixed breast tissue samples

with a high degree of correlation between simulated and measured sets of data. The

program has been able to efficiently simulate x-ray scattering from two-component mixtures

of known percentages by volume and producing quantitative estimates of the percentage of

each component in measured breast tissue samples. It is expected that the present simulation

would be very supportive in providing a quantitative approach for the characterization of breast

tissues. A fully automated version of the program using Computer Aided Diagnosis would

make it possible to provide an on-spot quantitative characterization of breast tissue.

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