M. Garbey and G. Zouridakis Modeling Tumor Growth: from Differential Deformable Models to Growth Prediction of Tumors Detected in PET Images Modeling Tumor

Modeling Tumor Growth: from Differential Deformable Models to Growth Prediction of Tumors Detected in PET Images

M. Garbey and G. Zouridakis

Modeling Tumor Growth:Modeling Tumor Growth: from Differential Deformable Models to from Differential Deformable Models to Growth Prediction of Tumors Detected in Growth Prediction of Tumors Detected in

PET ImagesPET Images

M. Garbey1 and G. Zouridakis1,2

1Department of Computer Science and 2Department of Electrical and Computer Engineering

University of Houston, TX, USA

Thanks: A.Deutsch & N.Mullani



Abstract Abstract

• Modeling of a growing tumor over time is extremely difficult, because of the complex biological phenomena underlying cancer proliferation.

• Existing mathematical models can mostly describe in vitro experiments of spherically-shaped avascular tumors, but they cannot match the highly heterogeneous and complex-shaped tumors seen in cancer patients, except may be for the gliomas growth.

• We propose a new time-dependent geometric deformable model that may match tumors of complex shape, such as vascular tumors, but are connected in a simplified manner to the dynamic of tumor growth.



I. INTRODUCTIONI. INTRODUCTION• The development of cancer is a very complex biological

phenomenon but, in general, it can be separated into two phases: an initial relatively benign phase of avascular growth, which is then followed by a potentially life-threatening vascular growth.

• The first (and second) phase is usually modeled by the bio-mathematical community as a system of reaction-diffusion equations or, alternatively, as a cellular automaton.

• Typically, PDE models consist of a system of differential equations that describe the growth rate of a population of cells as a function of cell mobility, nutrient consumption, cell proliferation, and apoptosis, and often include one or more competing colonies of both normal and cancer cells coupled to fast time scale biochemistry processes (glucose, endostatin…).



PET image to track Glucose MetabolismPET image to track Glucose MetabolismData are for day one, 28 and 56.Data are for day one, 28 and 56.



Tool: reconstruction of the 3d contourTool: reconstruction of the 3d contour,,with for example the snake algorithm.with for example the snake algorithm.



Further tool: combining several modalitiesFurther tool: combining several modalities

on the same pictureon the same picture..



Our goal

• an approach based on a time-dependent geometric deformable model that “fills the gap” between plain image analysis and direct numerical simulation of tumor growth.

• a global model simple enough to fit medical data, and rich enough to reproduce the complex geometry of tumors.

• A model that can improve the understanding of the medical image.



Open questions Ref. P.Tracqui et Al. (MMMAS 99).

• do we have enough clinical data to retrieve the model parameters?

• can we get from this modeling a dynamic grading of the tumor?

• Our Difficulties:

– matching in space of several images taken at different period of time,

– tumor is under medical treatment, – not enough data in the time series …. – time invariance of metabolism during the day?, – Etc…



II. METHODOLOGYII. METHODOLOGY

We build a PDE model using the analogy of reactive flow in porous media. In this model, the tumor growth is induced by the pressure H, and the basic equation can be written as

in,)graddiv( 0FFHKt

H (1)

• the permeability constant K may correspond to the properties of cell mobility and tissue vascularization and morphology

• the source term F may correspond to the metabolic activity of cells and the extent of vasculature that supplies nutrients.

• Remark: H is not a pressure in the physical sense, but rather a commodity…



Hypothesis: Most of the metabolic activity that drives tumor expansion is due to cancer cells near the boundary of the tumor,

we assume that the gradient of F is localized close to the boundary of the tumor:

where dist(M, S(t)) is the Euclidian distance of M to the front location of the tumor S(t).

,1,

)(,distexp),(

2

Fo

tSMyxF



• Pressure equation is nonlinear, because F depends on the front location.

• F must be positive and, therefore, F0 is a threshold such that when

F < F0, the pressure decreases, the tumor shrinks, and the cancer cells die.

• Analogy: The function H accounts for the dynamics of the tumor:

– in order to grow, the tumor applies some “pressure” on its environment.

– Depending on the vascularization and mobility of cells, which are given by the function K, this pressure will be more or less effective.

– The source of pressure is the metabolic activity of cells supported by the supply of nutrients represented by the source term F.



We track the growth of the tumor using Darcy’s law,

HKV (2)which gives the speed of expansion or shrinking of the tumor.

Finally, we use the following nonlinear convection-reaction equation

)1()( 0 CCCCCVt

C

(3)

to monitor a set level C = C0 which represents the boundary location S(t) of the tumor. The set level value is somewhat arbitrary, and in practice we choose C0 = 0.5.



• Let us denote with SM(ti), t0<t1<…<tnT, a sequence of tumor contours provided by segmenting a series of medical images from a given patient.

• The mathematical problem is to control K and possibly F in order to

minimize some norm of S(ti)-SM(ti), i=1..n.

• The objective function is the L2 norm of the difference between the two numerical representations of the front locations.

• To solve this problem, we require that the function spaces for the two unknown scalar functions K and F are well-defined and dependent only on a limited number of parameters.



In the one dimensional case, the model writes :

x

HKV

x

CV

t

C

The confined constraint gives the following boundary conditions for all time

02

;02

L

x

HL

x

H

FoxFx

HxK

xt

H

)()( ),0(,)2

,2

( TtLL

x



L

LxywithymayaaxK m

2/,)cos(...)cos()( 10

Fo

xxxxxF rightleft

22

expexp2

)(

The inverse problem is to optimize the set of parameters

,,...,,, 0 maain order to get the front location xleft(t) and xright(t) that fit some prescribed positions

xM,left/right(ti) for the set of time values t0,…,tn

The tumor is bounded by two interfaces that will move in opposit directions

Let us represent K as follows





Details of the numerical implementation in one and two space dimensions:

• The pressure equation is approximated with a finite-volume scheme that is second order in space, and an implicit first order Euler scheme for time integration. We need a time step of the same order as the space step, since T is large.

• The gradient of pressure is computed with a central formula, and should be smooth in space and slowly varying in the time scale under consideration.

• The convection of the contour C is integrated using the method of characteristics, which is an implicit solver. This method satisfies a maximum principle, since we use second order linear interpolation in space to compute the roots of the characteristics.

• The solution of CC0, which gives the location of the interface where metabolic activity is higher, is obtained by second order interpolation.



III. RESULTSIII. RESULTSOne Dimensional Example: Solution at final time T = 20. The curve with ‘+’ represents H, the curve with ‘o’ the function C, the two peaks in solid line represent the source term F, and the plateau represents the extent of the tumor.



Propagation of the boundaries of the tumor (upper row) and the corresponding speeds (lower row) for the symmetric case. after a small initial time interval, the rate of tumor growth remains almost constant. the smaller the K, the larger the transient.



Two Dimensional Example

• the model is a straightforward generalization of the previous case. The domain of computation is the square (L/2, L/2)2.

• We use a regular grid of dimension N × N for the discretization in space with constant step in both space directions x and y.

• The numerical difficulty is to define the source term F that is attached to the front location, and to minimize the effect of grid orientation.

• We have investigated the pattern formation of a tumor for a non-constant function K.

– The initial condition is a very small radially symmetric tumor.

– K is five time larger in a narrow strip Q, to simulate a blood vessel

– we observed that a growing tumor has a tendency to follow the vessel.



Impact of a pseudo-vessel on a growing tumor: domain of computation is (-5, 5)2, T = 7, = 5 ,and = 0.03. K reaches its maximum value of 0.1 in (-3.6, 3.6)×(0.8, 1.2), whereas K = 0.02 elsewhere.



C. Solution of the inverse problem

• We look first at tumors that correspond to a star-shaped domain centered on the initial position of the tumor. Based on our hypothesis, S(t) can be described in polar coordinates by its radius R(, t) :

(4)

(4) is the Fourier expansion of R(, ,t) of order m .• We choose as an initial guess for (K, ) to be a set of

constants in the level set given by a surface response for the radial symmetric case that matches the least square circle approximation of R(, ,t) .

2

2

)(),(m

mk

jkk etrtR



We keep K fixed and look for the optimum “metabolism”

(5)

• We process our inverse problem using an iterative optimization procedure.

• To facilitate the search, we iterate with increasing degree n of the Fourier expansion in (5) with 1 n m. We use the optimum solution obtained with n-1 as an initial guess for the search of the optimum solution corresponding to n, and increase n until the matching between the front location of the tumor at time T and the target given by (4) is satisfactory.

• n can be less than m depending on the accuracy of segmentation of the PET images that provide the contour S(T).

2

2

)(n

nk

jkke



We have preliminary successful experimental results with this procedure, provided that

• a constraint optimization method, such as sequential quadratic programming, is used to limit the amplitude of variation of the unknown Fourier coefficients k.

• In general, this condition is not a problem, since the decay of k as |k| increases reflects the smoothness of the Fourier expansion of the target function given by (4).

Similar work is currently under way to reconstruct the 2-D function K with fixed which, however, requires the identification of more unknowns parameters.



IV. DISCUSSIONIV. DISCUSSION

It seems that the proposed model has a number of interesting features that are in qualitative agreement with observations of real tumors, including the following:

• the larger the value of K, the faster the tumor growth. This also agrees with the fact that higher mobility of cells or higher vascular density result in increased rates of cell growth.

• the larger the value of the source term factor ¸ the faster the tumor growth. This is coherent if represents the metabolic rate of cancer cells.

• the larger the thickness of the source term, the faster the tumor growth: a larger layer of active cancer cells may result in a faster expanding tumor.

• After a fast decay of K and/or , the tumor may start shrinking only after some delay.



• One objective of this line of research is to correlate morphological data and metabolic activity obtained from CT and PET images, respectively, at discrete time points ti with the solution of the inverse problem.

• there is no unique solution to the inverse problem.

• However, it is expected that the use of additional information obtained from more elaborate analyses of the medical imaging data, such as a rough estimate of the scalar field of metabolic activity from PET metabolic images and vascularization from blood flow PET images, will provide enough additional information to allow for a unique solution of the inverse problem, which, in turn, may lead to some prediction of tumor growth.



V. CONCLUSIONV. CONCLUSION

• Our preliminary results suggest that the proposed porous-media-like model may adequately describe the growth of complex tumors that give rise to irregular patterns with rough surfaces while keeping the number of unknowns to a minimum.

• The all computational machinery coupling image processing and direct simulation via an optimization loop might be reused for a number of models such reaction-diffusion.

• This tool might be a practical way to integrate medical experience stored in data basis, knowledge in mathematical modeling and image analysis that is routinely used by the doctors.



ACKNOWLEDGMENTACKNOWLEDGMENT

The authors would like to thank The Texas Learning and Computation Center for the support of this project.



REFERENCESREFERENCES

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M. Garbey and G. Zouridakis Modeling Tumor Growth: from Differential Deformable Models to Growth Prediction of Tumors Detected in PET Images Modeling Tumor