Inverse problems in tumor growth modelling

Transcription

1 MC2 Damiano Lombardi Angelo Iollo Thierry Colin Olivier Saut Marseille,

2 Outline 1) Introduction: phenomenological model for tumor growth models 2) Models based on mixture theory: a simplified model 3) Inverse problems: formulation and numerical procedure 4) Realistic Applications: Prognosis tool for lung tumors 5) Conclusions: analysis of results and perspectives

3 Introduction Tumor growth is an intrinsic multi-scale phenomenon: Microscopic scale: description of genetic mutations and molecular pathways; sub cellular level. Mesoscopic scale: behavior of a cell in its social context; biochemical signals and transport phenomena; cellular level. Macroscopic scale: interactions between cellular phenotypes in a tissue, mechanical propagation; tissue and organ level. It is impossible to model tumor growth using first principles

4 Introduction The role of mathematical modeling: Understand quantitatively the complex relationships involved Use models in realistic applications: prognosis tools, treatment efficacy evaluations, therapy impact Different type of models: Individual based: discrete models, agent based models and cellular automata. PDE models: continuous type models, based on reaction diffusion equations, transport equations.

5 Continuous type models Population dynamics: mass conservation P i t + (J) =Γ(P j,c k ) N(P j,c k ) Variation of the cell density for the species i: J is the flux Production and Reduction, depending on other species and nutrients The choice of J determines the model nature: J depending on the gradient of P: reaction-diffusion model J depending on P and a vector field v: transport model P i t + (v ip i )=Γ(P j,c k ) N(P j,c k ),

6 Continuous type models The meaning of v: v(x; t) = X(x; t) t Evolution of the Lagrangian Coordinate v is unknown: conservation of momentum (P i v i ) t + (P i v i v i )= σ i + j U i,j (P i,p j ; v i, v j ), Variation of momentum in a given point, at time t Divergence of Stress Tensor: Interaction between species: The stress tensor is linked to v through constitutive equations

7 Continuous type models Nutrients evolution: a reaction diffusion model C k t (D(P i ) C k )= ( ) S j (P j ) M j (P j ) j C k Time variation of the nutrients concentration; Diffusion: in general it is non-uniform and non-isotropic; Production and Consumption. The nutrient equation is coupled with the population dynamics Coefficients depend on tissue properties

8 Simplified 2D Darcy model 2 cellular species: P t Q t + (vp ) = (2γ 1)P, + (vq) = (1 γ)p. P is the proliferating cell density Q is the dead cell density Healthy tissue is denoted by S, and P+Q+S=1: ( ) v = γp. v = k(p, Q) Π. { } Equation for the divergence Darcy-Law for the velocity The concentration of oxygen: C t (D(Y ) C) =α S Ω 1 Y αpc λc 1 Y dω

9 Simplified 2D Darcy model Equation for the porosity: k = k 1 +(k 2 k 1 )(P + Q) Equation for the diffusivity: D = D max K(P + Q) Hypoxia threshold: γ = 1 + tanh(r(c C hyp)) 2, Non-uniform, isotropic, linear Non-uniform, isotropic, linear Regularization of unit step Hypoxia function realizes the coupling between oxygen diffusion and population dynamics.

10 The need of inverse problems: Inverse Problems; The model is a system of coupled non-linear parametric PDE. Microscopic scales are not modeled: their effects is lumped into parameters that are not known a priori In order to apply models, parameters have to be determined via inverse problems, the data coming from medical imagery

11 Inverse problems: Inverse Problems; Observables have to be defined: Y = P + Q With Scan different tumor phenotypes may not be distinguished: the sum is the only observable. The time evolution of the observable is derived: Y t + (vy )=γp The other equations become: v = γp k(y ) v = k(y ) v C t (D(Y ) C) = αpc λc,

12 Inverse problems: Inverse Problems; Unknowns: k 2 /k 1,D max, K, α, λ,c hyp v, P, C Parameters Fields that are not observables From a medical standpoint P and C are of the highest importance Normally only two-three scans are available: the problem is greatly under-determined Infinite solutions are possible and fit the images in an optimal sense.

13 Inverse problems: Inverse Problems; In order to find one solution a regularization approach is defined. It consists in choosing a priori a particular functional space, and looking for the solution that belongs to this space. The base of the space is constructed by means of semiempirical eigenfunctions, defined using the results of numerical simulations. It is equivalent to add information imposing a functional constrain to the unknown fields.

14 Inverse problems: Inverse Problems; P = a (P ) i φ (P ) i,i=1,..., N P, C = a (C) i φ (C) i,i=1,..., N C, v = a (v) i φ (v) i,i=1,..., N v, γp = a (γp ) i φ (γp ) i,i=1,..., N γp, Eigenfunctions are extracted from a database of simulations by means of Proper Orthogonal Decomposition (POD). POD given the dimension N maximizes the energy representation of a given signal (P) φ i = j bi jp j λ 1/2 i. λ and b are the eigenvalues and the eigenvectors of the Autocorrelation matrix A (P ) ij =< P k (t h ),P m (t n ) > h, n =1,..., N; k, m =1,..., M

15 Inverse problems: Inverse Problems; The POD expansions are substituted into the equations written for the observable Y Ẏ + a v i (Y φv i )=aγp j φ (γp ) j a v i (φv i )=aγp j φ γp j k(y )a v i φ v i = k(y ) a v i φ v i a C i φc i a C i (K(Y ) φ C i )= αa p i ac i φ C i φ P i λa C i φ C i 2a γp i φ γp i = 1 + tanh(r(a C i φc i C hyp )) Unknowns: k 2 /k 1,D max, K, α, λ,c hyp a P i,a C i,a v i,a γp i Parameters Expansion Coefficients

16 Inverse problems: procedure Direct equations written at certain time instants Are data sufficient to perform a complete identification? yes Inverse Problems; no We build a data base varying all the parameters We need to add information We extract an eigenfunction basis We identify the system using a non-linear Newton solver

17 Realistic cases: Tumour in lung: 2D scan (Institut Bergonié) first scan: second scan: third scan: Starting from the first and the second scan we try to reconstruct the third one

18 Realistic cases: 1) Identification: all the parameters and the unknown quantities are identified in such a way that we are able to simulate the entire tumour growth; 2) Simulation: we take the first image as initial condition and we simulate the tumour (125 simulations in the database); we observe the differences between the third image and the tumour simulated at the same time. The simulation is rather realistic: 1) Tumour grows up in the same way; 2) It moves towards the boundary; 3) The dynamics seem to be well captured.

19 Realistic cases: Simulations of the second and third scans second and third scans

20 Realistic cases: Volume curve: - Simulation (continuous line); - scan (circle) Errors: Simulations reach the volume of the third scan a little bit after; Volume T(days) Why? 1) We have only 2D partial information; 2) The model is approximated, not explicitly designed for a lung; 3) We do not have considered angiogenesis.

21 Realistic cases: Another case, from the other side of the biological spectrum: slow, quasi-stable lung nodules The amount of information is higher, we try to solve the same problem

22 Realistic cases: Simuation databases: 125 and 800 simulations. The offline stage is parallelized. Given the first two images, we tried to reconstruct all the others. Initial condition is the first image, for all the simulations. The simulation is realistic since a slow tumor growth is represented. The shape is not so good. Irregularity in the boundary are introduced mainly from initial condition

23 Realistic cases: Simulations of the second and third scans second and third scans

24 Realistic cases: 0.09 Volume curve: what about non-logistic behavior? 0.08 Predictions Vol Data months There is an intrinsic non-linearity in the model

25 Conclusions: An efficient numerical procedure have been set up: The numerical cost is concentrated in the construction of the database and the eigenmodes, but this stage is parallelized Prognose is realiable on the time scale of medical exams Errors and open questions: Insufficient representation of tissue anisotropies: the shape of the tumor is not yet well captured. High dimensional database imply more difficult optimization with local methods: study the sensitivity to initial guess. The reconstruction of the initial conditions is a critical stage for several reasons. Regularization process and time scale of the observed phenomena.

26 Perspectives: It is difficult to improve the accuracy, but we have to improve the reliability. Study the sensitivity to initial conditions, the variance of the results. Identification in several cases: a population approach. Models: Model for lungs: tissue anisotropies and particular geometry Angiogenesis and therapy model Inverse problems: General database construction, 3D cases Optimal trasport theory to approximate evolution of observables using semiempirical eigenfunctions

27 Thank you for your attention!