An inverse problem formulation for parameter estimation of a reaction–diffusion model of low grade gliomas

Abstract

We present a numerical scheme for solving a parameter estimation problem for a model of low-grade glioma growth. Our goal is to estimate the spatial distribution of tumor concentration, as well as the magnitude of anisotropic tumor diffusion. We use a constrained optimization formulation with a reaction–diffusion model that results in a system of nonlinear partial differential equations. In our formulation, we estimate the parameters using partially observed, noisy tumor concentration data at two different time instances, along with white matter fiber directions derived from diffusion tensor imaging. The optimization problem is solved with a Gauss–Newton reduced space algorithm. We present the formulation and outline the numerical algorithms for solving the resulting equations. We test the method using a synthetic dataset and compute the reconstruction error for different noise levels and detection thresholds for monofocal and multifocal test cases.

Appendices

Appendix A: Operator definitions

The definitions of operators in Eqs. 34, 35, 36, and 37 are as follows:

$$\begin{aligned}&J^T\tilde{\alpha }:=-{\frac{\partial \tilde{\alpha }}{\partial t}}-D\tilde{\alpha } -{\frac{\partial R}{\partial c}}\bigg |_{c^0} \tilde{\alpha }+\int _0^1\delta (t-T)\tilde{\alpha }dt \end{aligned}$$

(47)

$$\begin{aligned}&N\tilde{c}:=-{\frac{\partial ^2 R}{\partial c^2}}\bigg |_{c^0}\tilde{c}\alpha ^0 +O^TO\int _0^1\delta (t-T)\tilde{c}dt \end{aligned}$$

(48)

$$\begin{aligned}&Z^T\tilde{k_f}:=-\nabla \cdot (\tilde{k_f}\mathbf{T}\nabla \alpha ^0) \end{aligned}$$

(49)

$$\begin{aligned}&g_p:=Bu-\varPhi ^T\alpha _0^0 \end{aligned}$$

(50)

$$\begin{aligned}&Bu:=(\beta _p + O_0^TO_0\varPhi )p \end{aligned}$$

(51)

$$\begin{aligned}&Z\tilde{c}:=\int _0^1\int _{\varOmega }(\mathbf{T}\nabla \tilde{c})\cdot (\nabla \alpha ^0)d\varOmega dt \end{aligned}$$

(52)

$$\begin{aligned}&W\tilde{\alpha }:=\int _0^1\int _{\varOmega }(\mathbf{T}\nabla c^0)\cdot (\nabla \tilde{\alpha })d\varOmega dt \end{aligned}$$

(53)

$$\begin{aligned}&g_{k}:=\int _0^1\int _{\varOmega }(\mathbf{T}\nabla c^0)\cdot (\nabla \alpha ^0)d\varOmega dt \end{aligned}$$

(54)

$$\begin{aligned}&J\tilde{c}:={\frac{\partial \tilde{c}}{\partial t}}-D\tilde{c}-{\frac{\partial R}{\partial c}}\bigg |_{c^0}\tilde{c}+\int _0^1\delta (t)\tilde{c}dt \end{aligned}$$

(55)

$$\begin{aligned}&W^T\tilde{k_f}:=-\nabla \cdot (\tilde{k_f}\mathbf{T}\nabla c^0) \end{aligned}$$

(56)

The reduced Hessians of Eq. 39 are defined as follows:

$$\begin{aligned}&H_{pp}=B + \varPhi ^TJ^{-T}NJ^{-1}\varPhi , \end{aligned}$$

(57)

$$\begin{aligned}&H_{pk}=-\varPhi ^TJ^{-T}(-Z^T+NJ^{-1}W^T), \end{aligned}$$

(58)

$$\begin{aligned}&H_{kp}=(Z-WJ^{-T}N)J^{-1}\varPhi , \end{aligned}$$

(59)

$$\begin{aligned}&H_{kk}=- ZJ^{-1}W^T+WJ^{-T}(-Z^T+NJ^{-1}W^T). \end{aligned}$$

(60)

To compute the matvec of $H_{pp}\tilde{p}$ one needs to take the following steps:

• 1. $J^{-1}\varPhi$:

  Solve Eq. 31 with $\tilde{k_f}=0$ and initial condition of $\tilde{c}_0=\varPhi \tilde{p}$ to get $\tilde{c}$

• 2. $-J^{-T}NJ^{-1}\varPhi$:

  Solve Eq. 28 with $\tilde{k_f}=0$ and initial condition of $\tilde{\alpha }_1=-O^TO\tilde{c}_1$ to get $\tilde{\alpha }$

• 3. $B+\varPhi ^TJ^{-T}NJ^{-1}\varPhi$:

  Compute $-\varPhi ^T\alpha _0$ and add $Bu$

To compute the matvec of $H_{pk}\tilde{k_f}$ one needs to take the following steps:

• 1. $J^{-1}W^T$:

  Solve Eq. 31 with zero initial condition to get $\tilde{c}$

• 2. $J^{-T}(-Z^T+NJ^{-1}W^T)$:

  Solve Eq. 28 with initial condition of $\tilde{\alpha }_1=-O^TO\tilde{c}_1$ to get $\tilde{\alpha }$

• 3. $-\varPhi ^TJ^{-T}(-Z^T+NJ^{-1}W^T)$:

  Compute $-\varPhi ^T\alpha _0$

To compute the matvec of $H_{ku}\tilde{p}$ one needs to take the following steps:

• 1. $J^{-1}\varPhi$:

  Solve Eq. 31 with $\tilde{k_f}=0$ and initial condition of $\tilde{c}_0=\varPhi \tilde{p}$ to get $\tilde{c}$

• 2. $ZJ^{-1}\varPhi$:

  Compute $\int _0^1\int _{\varOmega }(\mathbf{T}\nabla \tilde{c})\cdot (\nabla \alpha ^0)d\varOmega dt$

• 3. $-J^{-T}NJ^{-1}\varPhi$:

  Solve Eq. 28 with $\tilde{k_f}=0$ and initial condition of $\tilde{\alpha }_1=-O^TO\tilde{c}_1$ to get $\tilde{\alpha }$

• 4. $-WJ^{-T}NJ^{-1}\varPhi$:

  Compute $\int _0^1\int _{\varOmega }(\mathbf{T}\nabla c^0)\cdot (\nabla \tilde{\alpha })d\varOmega dt$

• 5. Add 2 and 4

To compute the matvec of $H_{kk}\tilde{k_f}$ one needs to take the following steps:

• 1. $J^{-1}W^T$:

  Solve Eq. 31 with zero initial condition to get $\tilde{c}$

• 2. $ZJ^{-1}W^T$:

  Compute $\int _0^1\int _{\varOmega }(\mathbf{T}\nabla \tilde{c})\cdot (\nabla \alpha ^0)d\varOmega dt$

• 3. $J^{-T}(-Z^T+NJ^{-1}W^T)$:

  Solve Eq. 28 with initial condition of $\tilde{\alpha }_1=-O^TO\tilde{c}_1$ to get $\tilde{\alpha }$

• 4. $WJ^{-T}(-Z^T+NJ^{-1}W^T)$:

  Compute $\int _0^1\int _{\varOmega }(\mathbf{T}\nabla c^0)\cdot (\nabla \tilde{\alpha })d\varOmega dt$

• 5. Add 2 and 4.

Appendix B: Fictitious domain method

We use a fictitious domain method in which the original brain domain, $\mathcal {B}$ , is extended to a cubic box, denoted by $\varOmega$ (Hogea et al. 2008b; Mang et al. 2012; Tracqui et al. 1995). The original homogeneous boundary conditions imposed on $\varGamma$ can be satisfied using a penalty method (Del Pino and Pironneau 2003). To do so we define a new diffusion coefficient $\mathbf K_\epsilon (x)$, $\mathbf{x}\in \varOmega$ as follows:

where the penalty parameter $\epsilon$, is a small positive number. The actual boundary condition on $\varGamma$ will be satisfied in the limit of $\epsilon \rightarrow 0$ (Del Pino and Pironneau 2003). The original boundary conditions can be re-imposed on the extended cubic domain, $\varOmega$, for both the forward Eq. 1 and adjoint equation Eq. 20.