Calculation algorithms in radiation therapy treatment planning systems

Colleen DesRosiers, Ph.D.AAMD Region III annual meeting, Indianapolis, Indiana

April 12, 2013

Learning Objectives1. The learner will be able to describe the different

types of algorithms used in treatment planning systems

2. The learner will be able to identify strengths and weakness of algorithms used in treatment planning systems

3. The learner will be able to identify reasons for disagreement between monitor unit calculations generated by treatment planning systems and by monitor unit check programs.

Image from: http://oeaeuprrp.blogspot.com/2011/09/workshop-how-to-write-and-assess.html

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I am not the expert …

1. Khan, FM Gerbi, BJ Treatment Planning in Radiation Oncology, 3rd

edition, 20122. Khan, The Physics of Radiation Therapy, 4th edition, 20103. Murlidhar, KR, Murthy, NP, Raju, AK, Sresty, NVNM. Comparative study

of convolution, superposition, and fast superposition algorithms in conventional radiotherapy, three-dimensional conformal radiotherapy, and intensity modulated radiotherapy techniques for various sites, done on CMS XIO planning system J Med Phys [serial online] 2009 [cited 2013 Apr 1];34:12-22.

4. Ahnesjö , Anders, Basic modeling concepts in treatment planning dose dose calculations, fluence, raytracing, kernels, etc- Uppsala University, Sweden, 2013

5. Ahnesjö , Anders Patient dose calculation models in TPS.

More references…6. Wiesmeyer MD, Miften MM. A multigrid approach for accelerating three-

dimensional photon dose calculation Med Phys 1999; 26 :1149 (Abstract)7. Mackie TR, Bielajew AF, Rogers DWO, Battista JJ. Generation of photon energy

deposition kernels using the EGS Monte Carlo code. Phys Med Biol 1988;33:1-20. 8. Sharpe MB, Battista JJ. Dose calculations using convolution and superposition

principles: The orientation of dose spread kernels in divergent X-ray beams. Med Phys 1993;20:1685-94.

9. Mackie TR, Scrimger JW, Battista JJ. A convolution method of calculating dose for 15 MV X-rays. Med Phys 1985;12:188-96

10. Gagné, I, Zavgorodni, S. Evaluation of the analytical anisotropic algorithm in an extreme water–lung interface phantom using Monte Carlo dose calculations. Journal of Applied Clinical Medical Physics, vol. 8, (1), Winter 2007

11. Das, I, Cheng, C.W., Srivastava, S, et al. Variability of Low-Z Inhomogeneity Correction in IMRT/SBRT: A Multi-Institutional Collaborative Study AAPM annual meeting 2008

Acknowledgments

Vadim Moskvin, Ph.DIndra Das, Ph.D.

al·go·rithm \ˈal-gə-ˌri-thəm\

1. a procedure for solving a mathematical problem in a finite number of steps that frequently involves repetition of an operation;

2. a step-by-step procedure for solving a problem or accomplishing some end especially by a computer

From: http://www.merriam-webster.com/dictionary/

Where, in the planning system, do we find algorithms?

• MU calculations• Isodose distributions• DVH generation• IMRT optimization• DRR generation• Brachytherapy calculations• Any process that occurs when the user does not

dictate each step (e.g. generating a 3D image from a series of slices, placing a margin around a structure, etc.)

In the beginning …• Calculations were performed

strictly based on empirical (directly measured) data in tabular format.

• Isodose curves were generated based on PDD and profile data

• Corrections based on patient were very simplistic, depth corrections only (attenuation)

Most advanced …Monte Carlo method

Histories of millions of photons and secondary electrons are traced to calculate dose deposition based on physics interactions in matter

Monte Carlo method is the most accurate method for dose calculation but requires the greatest processing time. Most calculation algorithms use pre-calculated MC kernels.

4. Ahnesjö , Anders

Monte Carlo terms

• Random number generator (RNG)– the random number generator selects a number between 0 and 1 to determine the path of the particle (photon)

• History – the tracking of a single particle (how is the photon losing energy as it passes through the medium)

• Phase space – characterizes position in 6D

• Events – PE, CE, PP, electron interactions

More Monte Carlo terms…• Sampling – the draw of the parameters of events from

the probability distributions using RNG• Scoring – acquiring the value of the parameter of

interest during the simulation• Estimator – mathematical and algorithmical description

of the scoring method• Kernel – Pencil or point; a Monte Carlo simulation that

has been “scored” of a small “pencil” beam which incorporates the events in that path, or that occur at a “point”

Photon interactions… or events in Monte Carlo terms

1. Photoelectric effect – photon transfers all energy to electron, ejected, increases with Z3 and decreases with E3

2. Compton effect – dominant at therapy energies, results in scattered photons and secondary electrons, decreases slowly with E and is independent of Z

3. Pair Production – results in the creation of electron/positron

pair, annihilation, dependent on Z2 and E2.Which lead to Electron interactions – ionization, excitation,

bremsstrahlung, ultimately dose deposition(There are other effects that contribute to dose, such as neutron

production, which may or may not be modeled)

Monte Carlo method• MC method uses known probabilities and probability

distributions in sampling to predict results of interactions (events) (e.g. Compton: Klein-Nishina coefficients)

• MC utilizes the Law of Large Numbers (LLN)Theorem: The average of the results obtained from a large number of trials should be close to the expected value, and will tend to become closer as more trials are performed.

• The convergence of the Monte Carlo method follows the Central Limit Theorem (CLT)

Example: 2 Gy dose in 3x3x3 mm voxel could be delivered by 1011 photons crossed this voxel. However, simulation of 1011 photons multiplied on field size will take infinite amount of time. According to LLN, we can simulate 107 photons for whole field and get results according to CLT with 1% accuracy.

Monte Carlo method• Direct Monte Carlo method

o Particle trajectory is simulated in details on the even-by-even basis with the maximum accuracy in details.

o Binary estimator is used. Example: General-purpose code algorithms, PENELOPE, EGSnrc, FLUKA

• Specialized Monte Carlo methodo Variance reduction based simulation or certain

simplifications in the transport descriptiono Weighted estimator is used.

Example: FLUKA and MCNPX neutron transport module, MMC used in TPS, DPM electron-photon code.

Monte Carlo code uses programming subroutines and interaction

probabilities to calculate dose

Why does MC method work?• As the number of photons required for dose delivery

decreases, dose delivery uncertainty increases• Example: a quarter landed on “heads” or “tails”

Probability = 50%If the quarter is flipped 10 timesIf the quarter is flipped 100 timesIf the quarter is flipped 1011 times

• If only a few photons were needed to deliver dose, the MC method would not be accurate

So, if MC is the most accurate method for calculation, why develop other algorithms?

1. Time! Processing time makes MC calculations impractical for clinic as the TPS engine, but may serve well for treatment plan verification.

2. Since the cost associated with higher performance computing has decreased over the years, more sophisticated algorithmsemploying MC calculations are clinically available.Khan, 2010 (1)

Two atoms are sitting in a field of ionizing radiation.One atom says, "I think I lost an electron."

The other says, "Are you sure?"The first atom says, "I'm positive!"

On the light side …

Phase Space• Phase space is a 6 dimensional

characterization of the particles

),,,,,(6),,(

EzyxDzyxr

Monte Carlo accuracy - Modeling• Highly dependent on

the modeling of the components of the accelerator head

• Where can secondary electrons be produced? Where can scatter occur? Where does the calculation start?

http://health-7.com/imgs/20/7507.jpg

Monte Carlo input • Finite photon source size Open fluence distribution Fluence modulation Head scatter sources flattening filter collimators wedges Monitor back scatter Collimator leakage, including MLC interleaf leakage

shape of MLC leaf ends • Beam spectra • Spectral changes• Electron contamination

Monte Carlo Codes• EGS4• EGSnrc• GEANT• PENELOPE• MCNP• MCNPx• FLUKA

Different types of calculation algorithms

• Semi-empirically based• Model based• Direct Monte Carlo• Hybrid

Give me 30 minutes and I can confuse anyone. I don’t need to prepare.

Lech Papiez, Ph.D.Indiana University

Dr. Papiez’ response to a request to give a talk on algorithms to medical residents on the same day of the request.

Why do we need calculation algorithms?

Measurements are performed under specific conditions:- Fixed square fields- Fixed depths- Homogenous medium (water)- Flat surface

Monte Carlo based algorithms and the simplest of empirically based algorithms reasonably agree in homogeneous media

Physical DensityMaterial Density

(g/cm3)Relative to

waterImpact

Air 0.0012 1/800 Attenuation and scattering

Lung 0.2-0.3 1/5 Attenuation and scattering

Water(soft tissue)

1.000 1.000

Bone 1.600 1.6X (Attenuation and scattering)

Titanium 4.5 4.5X Attenuation

Steel 7.5 7.5X Attenuation

Semi Empirical (also called Correction or Factor based)

• Based on measured dataPDDProfile

• TAR• ETAR• Batho• Power Law• Clarkson

Photon dose calculation methods “Dose engines”

Method characteristics

Remarks

Monte Carlo Explicit particle transport simulation + Accurate - Noisy distributions

Standard research tool, clinical use under development

Point kernel methods Convolution/superposition ,Collapsed Cone

Implicit particle transport + Accurate - Minor systematic errors

Current workhorse for accurate calculations in lung.

Pencil Kernel Methods Heterogeneity impact through corrections

The workhorse for many applications

Scatter dose estimations ”Semi” pencil kernel metods

Often used for factor based calculation schemes

1D heterogeneity corrections Models what happen along the incident beam direction only

Can be used to correct dose calculated with any method for heterogeneities

Model Based

Factor Based

Ahnesjö, 2013 (4)

Calculation algorithms for TPS (photons)Elekta XiO• Clarkson• FFT Convolution• Multigrid Convolution• Superposition• Fast SuperpositionPhillips Pinnacle• Collapsed Cone Convolution• Pencil BeamVarian ECLIPSE• AAA Collapsed Cone Convolution• Pencil Beam

Model based algorithms• Empirically based algorithms rely on measurement,

corrections performed based on patient characteristics

• Model based algorithms rely less on measured data, more on predictions of dose distribution (equations, probabilities)

• No clear distinction!Empirical based algorithms use models for

corrections and model based algorithms use some measured data.

Pencil Beam algorithm (Convolution)• Monte Carlo “kernels”• Spatially invariant (non-divergent)• Scatter not modeled well

- Lateral scatter not considered- Heterogeneity correction largely attenuation

correction only (generally convolved with Batho or other correction)

- Generally a “hybrid” algorithm (modeled with semi-empirical correction)

Monte Carlo kernel Summation of kernels

In the pencil beam algorithm, kernels are spatially invariant, i.e., parallel to surface, non-divergent,

source of some inaccuracy

Homogeneous media Inhomogeneous media

Pencil beam algorithm predictions Changes in side scatter not modeled

Attenuation correction only

High density heterogeneity

Low density heterogeneity

Inhomogeneous media

High density heterogeneity

Low density heterogeneity

Increased areas of scatter not modeled

Will result in higher MUs than is needed

Decreased areas of scatter not modeled

Will result in lower MUs

Convolution känvəˈlo͞oSHən• A coil or twist, esp. one of many• A thing that is complex and difficult to follow• a mathematical operation on two functions f

and g, producing a third function that is typically viewed as a modified version of one of the original functions.

• It is based on theory of Laplace transformation of functions.

http://en.wikipedia.org/wiki/Convolution

Convolution-Superposition method (Collapsed Cone, Superposition)

• Most commonly used and widely accepted algorithm class in radiotherapy planning systems.

• Primary photons are treated separately from scattered photons and electrons set in motion

Convolution-Superposition algorithms• Point kernels (finer resolution than pencil kernels)• Consideration of divergence• Consideration of lateral scatter• Consideration of energy spectrum• Consideration of primary/secondary interactions with

inhomogeneous media• Effects of collimator, flattening filter• Less averaging than pencil beam

Equations

scatter for ally simplistic accounts'

mass in the releasedenergy totalTERMA,

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rT

rdrrArT

rdrrArrD

p

p

sitephoton the tosource thefrom distanceradiologic theis and sitephoton primary the tosite deposition dose thefrom distance radiologic theis' where

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rrr

rdrrArT

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Convolution Equation

Convolution-Superposition

Equation

Data and Clinical Examples

Experimental geometry for the treatment planning and measurement. The phantom consists of 14.4 cm of lung equivalent material (Cork, =0.25g/cm3) sandwiched between slabs of solid water. Measurements were made at various depth with micro-chamber

Treatment Planning and Measurements

Das et al., 2008 (11)

medium

Charged particle equilibrium (CPE)

Pencil beam algorithm does not accurately account for

secondary electron production, calculations at

the tissue/lung interface are not accurate.

Region of non-equilibrium

From Khan, 2010 (2)

Plot = Di/Dh

CMS-XiO, 6 MV

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18 20

Depth (cm)

Cor

rect

ion

Fact

or (D

i/Dh)

1x12x23x34x45x56x68x810x10

Pencil Beam Convolution

Pencil Beam

CMS-XiO, 6 MV

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18 20

Depth (cm)

Cor

rect

ion

Fact

or (D

i/Dh)

1x12x23x34x45x56x68x810x10

Superposition

Monte Carlo, 6 MV

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0 2 4 6 8 10 12 14 16 18 20

Depth (cm)

Cor

rect

ion

Fact

or (D

i/Dh)

1x12x23x34x43x36x6

PENELOPE, MC

Superposition Monte Carlo

Das et al., 2008 (11)

Lung Case

1. SBRT case –10 fields

2. 1 cm volume

3. 6 MV

4. AAA and Pencil beam

5. Heterogeneity corrections “on”

AAA100% White

95%

90%

80%

Calculation verification program predicts 11.3%

higher dose than AAA with same MUs

Pencil Beam100% White

95%

90%

80%

Calculation verification program agrees well with

Pencil beam (.01%)

Comparison of isodose curvesAAA Pencil Beam

Max dose = 100.8%

80% volume = 42.4 cc

Max dose = 102.2%

80% volume = 57.7 cc

The 80% volume generated in PBC is 36% greater than AAA

Spine Case – High ZAPPA spine

6 MV beams

High density implanted devices, 3000+ HU

Bone measured density values = 900-1500 HU

AAA and Pencil Beam

Heterogeneity corrections on

AAA MU sum = 206

Pencil Beam MU

sum = 208

AAA – edge of heterogeneity PBC– edge of heterogeneity

PBC results in 10% higher dose at edge of high Z heterogeneity

AAA – inside heterogeneity PBC – inside heterogeneity

PBC results in 10% higher dose inside the high Z heterogeneity

Spine case

No implant

APPA

16 MV photons

200 cGy anterior to vertebral body

No difference in MUs between AAA and PBC

Heterogeneity corrections on

AAA PBC

Less than 1% difference in dose calculated to cord

AAA PBC

Less than 1% difference in calculated dose in bone

Summary• Algorithms are step by step processes which are used

in planning systems (and otherwise) to complete specific tasks.

• The simplest of algorithms perform as accurately as the most sophisticated algorithms for ideal conditions.

• Time is a critical factor in the development of treatment planning algorithms.

• Heterogeneities pose the greatest challenge to predicting accurate dose distributions in patients

Summary (cont.)• The Monte Carlo method is the most accurate method

for calculating dose in heterogeneities. The most accurate currently available algorithms incorporate Monte Carlo kernels.

• Discrepancies in calculations more likely to arise from low density media than from high density media

• Safer to rely on your “convolution-superposition” algorithm than your verification calculation, since your verification calculation uses a simpler, less accurate algorithm

www.pythian.com

The academic portion of the lecture is now

complete.

An Irish blessing for the Medical Dosimetrist …

May your first optimization meet your dose constraintIf not, may you meet with a little dose paintMay your calc’d and your plan MUs always agreeMay you need not replan for a re-drawn GTVMay your MD not change his mind post approvalAnd your user rights never suffer removalMay your IMRT QA turn out rightAnd you get to go home while there is still light

Questions?

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