Tensorial Reduced-Order Models for Parametric Coupled Reaction-Diffusion Systems: Application to Brain Tumor Growth Modeling

Abstract

We construct efficient surrogate models for parametric forward operators arising in brain tumor growth simulations, governed by coupled semilinear parabolic reaction-diffusion systems on heterogeneous two- and three-dimensional domains. We consider two models of increasing complexity: a scalar single-species formulation and a six-state, nine-parameter multi-species go-or-grow model. The governing equations are discretized using a finite volume method and integrated in time via an operator-splitting strategy. We develop tensorial reduced-order model (TROM) surrogates based on the Higher-Order Singular Value Decomposition in Tucker format and the Tensor Train decomposition, each in intrusive and non-intrusive variants. The models are compared against a classical proper orthogonal decomposition (POD) ROM baseline. Numerical experiments with up to $m=9$ model parameters demonstrate speedups of $85\times$-$120\times$ relative to the full-order solver while maintaining excellent accuracy, establishing tensorial surrogates as a rigorous and efficient computational foundation for many-query workflows.

Figures (10)

• Figure 1: Exemplary simulation results for a single-species model in three dimensions (ambient space). We solve the FOM. We show (from left to right) an axial, coronal and sagittal view of the 3D volume. The native resolution (number of mesh points) of this data set is $193 \times 229 \times 193$ at pseudo-time $t=1$. High tumor cell densities are shown in red and low densities are shown in green.
• Figure 2: Simulation results for the multi-species model in two dimensions (ambient space). We solve the FOM. We show (from left to right) axial slices for the white matter density $u_{\textit{w}}$, the gray matter density $u_{\textit{g}}$, and the densities for the infiltrative tumor cells $u_{\textit{i}}$, proliferative tumor cells $u_{\textit{p}}$, and necrotic tumor cells $u_{\textit{n}}$, as well as the oxygen concentration $u_{\textit{o}}$ at pseudo-time $t=1$. High cell densities are shown in red and low densities are shown in green.
• Figure 3: Parameter sensitivities for the multi-species model. We plot the $\ell^2$-norm of the residual for each species ($u_{\textit{p}}$, $u_{\textit{i}}$, $u_{\textit{n}}$, $u_{\textit{o}}$, $u_{\textit{g}}$, $u_{\textit{w}}$) with respect to a reference solution located at the center of each parameter interval $\Theta_i = [\theta_i^{\mathit{min}}, \theta_i^{\mathit{max}}] \subset \mathbb{R}$ as a function of the perturbation from this reference parameter. The reference solution is computed at the center of the hypercube $\Theta$ defined by the parameter ranges $\Theta_i$. We report results for all nine model parameters $\theta = (\rho, \alpha, \gamma_p, \gamma_i,\lambda_d, \kappa_c, \kappa_s, u_o^\textit{hyp}, u_o^\textit{inv})$. The error is zero at the center of each parameter domain. As we move away from the center we expect the error to increase if the model is sensitive with respect to the considered parameter. Each plot provides the norm of the residuals for these perturbations for each individual parameter for all species (states). Averages are reported in \ref{['t:sensitivity-analysis-summary']}.
• Figure 4: Illustration of the refinement of the parameter samples in $\Theta_i$. We gradually increase the number of parameter samples $n_{\theta_i}$ from 3 to 17. At each refinement step, we add new samples midway between the existing ones, while keeping all previously selected samples.
• Figure 5: Relative error with respect to the number of samples $n_{\theta_i}$ drawn to construct the HOSVD-TROM for each individual model parameter $\theta_i$. We consider the intrusive and non-intrusive variant of the HOSVD-TROM. We report the relative error compared to the FOM solution for an increasing number of samples $n_{\theta_i}$ drawn. The samples are varied per parameter $\theta_i$, $i = 1,\ldots, 9$, with the remaining parameters $\theta_j$, $j \not=i$ fixed. Each plot shows the trend of the mean error averaged across 100 random off-grid samples $\theta_i$. The envelopes correspond to the max and min error across all 100 trials. These results are for $n_t = 16$ using the tolerances reported in \ref{['t:rom-hyperparameters']}.