1 Computational Modeling in Quantitative Cancer Imaging Biomedical Science and Engineering...

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Computational Modeling in Quantitative Cancer Imaging

Biomedical Science and Engineering Conference

18 March 2009

Tom Yankeelov, Nkiruka Atuegwu, John C. Gore

Institute of Imaging Science, Departments of Radiology, Physics, Biomedical Engineering, and Cancer Biology

Vanderbilt University

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Outline

2) Basic Idea: Use Imaging to Drive Math Models

1) Mathematical modeling of tumors

3) What can imaging provide?

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4) An example

33

Outline




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4) An example

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Math Modeling of Tumors, 1/4Math Modeling of Tumors, 1/4

• Over last 10-15 years, many math models of tumor growth have appeared

• Much experimental data on the growth kinetics of avascular tumors have been integrated into growth models

Quaranta et al. Clinica Chimica Acta 2005;357:173-9.

• Deterministic reaction-diffusion equations used to model spatial spread of tumors

• Continuum/solid mechanics, mechano-chemical modeling, including physical pressure and forces between cells and matrix

Let’s look at an example…

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Quaranta et al. Clinica Chimica Acta 2005;357:173-9.

• Simulation of spatial distribution of tumor cells and tumor invasion

• What do the equations look like that generate these kind of results?

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Anderson et al. Cell 2006;127:905-15.

Avascular Model Example

MDE = matrix degrading enzyme

MM = matrix molecule

• These are “conservation of matter” equations… and they work quite well; but…

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… their application to in vivo physiological events has been extremely limited

• Current models of tumor growth rely on knowledge of data that is quite difficult to measure in an intact living system:

metrics of chemotaxis, haptotaxis, growth factor gradients, MDEs, etc.

• Thus, general limitation of these models is that they are driven by parameters that can be measured only by highly invasive methods or in idealized systems

• In addition to difficulty of measuring these parameters in animal systems, it is extremely difficult to measure in clinical setting

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Outline




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4) An example

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Basic Idea: Use Imaging to Drive Math ModelsBasic Idea: Use Imaging to Drive Math Models

• Thus, models driven by imaging data will make predictions based on individual tumor characteristics that can be tested during longitudinal studies

• We propose to construct math models of tumor growth that can be parameterized by data obtained from noninvasive imaging experiments

• Approach is fundamentally different as models would be driven by parameters obtained noninvasively and in 3D

Can be measured repeatedly to update and refine predictions

Allows for in vivo hypothesis testing

Existing models driven by parameters obtained invasively; not 3D; clinically?

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Outline




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4) An example

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• Fundamental characteristic of cancer is unchecked cell proliferation

can be quantified by “diffusion weighted MRI” (DW-MRI)

What can imaging provide, 1/2?What can imaging provide, 1/2?

• Water molecules wander about randomly in tissue (Brownian Motion)

• In free solution, after time t, molecules travel (on average) a distance L

• But in cellular tissue, compartment effects hinder movement = restricted diffusion

• Thus, the Apparent Diffusion Coefficient (ADC) is lowered ~√t

Distancefrom

originalposition

Free

Restricted

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Diffusion weighted MRI

• Increasing cell density (cellularity); more cell membranes per cm to hinder diffusion lower ADC

0 0.2 0.4 0.6 0.8 10.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Volume fraction

AD

C

• ADC depends on cell volume fraction

• Tumor cellularity may be monitored by DW-MRI

Hall et al. Clin Canc Res 2004;10:7852Anderson et al. Magn Reson Imaging. 2000;18:689-95

ADC (MRI)

17 days 24 days 31 days

What can imaging provide, 2/2?What can imaging provide, 2/2?

tumor

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Outline


4) An example



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01 , ( 0)dN N

kN with N t Ndt

•The logistic growth model incorporates exponential growth of tumor cells early; asymptotically approaches cellular carrying capacity

N0 = number of cells initially present

k = cells proliferative rate

= carrying capacity of the population

An example, 1/5 An example, 1/5

Byrne. "Modelling Avascular Tumor Growth," in Cancer Modeling and Simulation

• Solution given by:

0

0 0

( )( ) e kt

NN t

N N

Goal is to apply this in the imaging setting

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An example, 2/5 An example, 2/5 0

0 0

( )( ) e kt

NN t

N N

First application: obtain maps of the proliferation rate, k

• Assign the parameters from available imaging data

• Carrying capacity:

= (voxel volume)/(cell volume)

• Need to measure cell number, N(t), or at least relative cell number, Nrel(t)

• ADC is related to cell number:

ADC(t) = ADCw – N(t),

ADCw = ADC of free waterN(t) = cell number

= proportionality constant

0 0.2 0.4 0.6 0.8 10.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Volume fraction

AD

C

Anderson et al. Magn Reson Imaging. 2000;18:689-95. /21/21

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An example, 3/5 An example, 3/5 0

0 0

( )( ) e kt

NN t

N N


• Then we can write the relative number of cells at time t :

Nrel(t) = N(t)/N0 = [ADC(t) – ADCw]/[ADC(0) – ADCw]

• Thus, we have converted from measured ADC at time t to a relative cell number at time t given by Nrel(t)

• Rewrite the solution to logistic equation as:

(0)( )

(0) ( (0))erel

rel ktrel rel

NN t

N N

Every term in above relation is known, except proliferation rate, k

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(0)( )

(0) ( (0))erel

rel ktrel rel

NN t

N N

• Consider, a rat brain tumor model where multiple imaging sessions are planned in both treated and control animals

• Tumors are allowed to grow and ADC(t) is measured to estimate Nrel(t)

• Since is fixed, and Nrel(t) and Nrel(0) are measured, can fit data to extract k for each voxel thereby yielding a proliferation rate map

• Testable hypothesis: rats from treated and untreated groups would display different k distributions

could be used to separate responders from non-responders

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• Can also use this approach to simulate growth by combining with other methods

Nkiruka Atuegwu, Ph.D.

• Taking data from ADC (MRI), proliferation (FLT-PET), hypoxia (FMISO-PET)

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SummarySummary

• We have presented an approach whereby imaging data can drive a (simple) mathematical model of tumor growth

• The example provided here is quite simple, represents only a first step

• Models driven by imaging data will make predictions based on individual tumor characteristics that can be tested during longitudinal studies

• Approach is fundamentally different as our models are driven by parameters obtained noninvasively in 3D

Can be measured repeatedly to update and refine predictions

Allows for in vivo hypothesis testing

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2020

AcknowledgementsAcknowledgements

• VUIIS Director

John C. Gore, Ph.D.

• Collaborators

Jim Nutaro, Ph.D. Nkiruka Atuegwu, Ph.D.

Mike Miga, Ph.D. Shelby Wyatt, Ph.D.

Tuhin Sinha, Ph.D.

• FundingNIBIB 1K25 EB005936 (Career Development Award)

NCI 1R01CA129961

NIBIB R01 EB000214/21/21

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Thank you very much for your time Thank you very much for your time and attention.and attention.

Vanderbilt University Institute of Imaging Science /21/21

1 Computational Modeling in Quantitative Cancer Imaging Biomedical Science and Engineering...

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