Integrated Biophysical Modeling and Image Analysis: Application to Neuro-Oncology.

Central nervous system (CNS) tumors come with vastly heterogeneous histologic, molecular, and radiographic landscapes, rendering their precise characterization challenging. The rapidly growing fields of biophysical modeling and radiomics have shown promise in better characterizing the molecular, spatial, and temporal heterogeneity of tumors. Integrative analysis of CNS tumors, including clinically acquired multi-parametric magnetic resonance imaging (mpMRI) and the inverse problem of calibrating biophysical models to mpMRI data, assists in identifying macroscopic quantifiable tumor patterns of invasion and proliferation, potentially leading to improved (a) detection/segmentation of tumor subregions and (b) computer-aided diagnostic/prognostic/predictive modeling. This article presents a summary of (a) biophysical growth modeling and simulation,(b) inverse problems for model calibration, (c) these models' integration with imaging workflows, and (d) their application to clinically relevant studies. We anticipate that such quantitative integrative analysis may even be beneficial in a future revision of the World Health Organization (WHO) classification for CNS tumors, ultimately improving patient survival prospects.

WHERE DID THE TUMOR START? AN INVERSE SOLVER WITH SPARSE LOCALIZATION FOR TUMOR GROWTH MODELS.

We present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction-diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of Magnetic Resonance Imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.

Simulation of glioblastoma growth using a 3D multispecies tumor model with mass effect.

In this article, we present a multispecies reaction-advection-diffusion partial differential equation coupled with linear elasticity for modeling tumor growth. The model aims to capture the phenomenological features of glioblastoma multiforme observed in magnetic resonance imaging (MRI) scans. These include enhancing and necrotic tumor structures, brain edema and the so-called "mass effect", a term-of-art that refers to the deformation of brain tissue due to the presence of the tumor. The multispecies model accounts for proliferating, invasive and necrotic tumor cells as well as a simple model for nutrition consumption and tumor-induced brain edema. The coupling of the model with linear elasticity equations with variable coefficients allows us to capture the mechanical deformations due to the tumor growth on surrounding tissues. We present the overall formulation along with a novel operator-splitting scheme with components that include linearly-implicit preconditioned elliptic solvers, and a semi-Lagrangian method for advection. We also present results showing simulated MRI images which highlight the capability of our method to capture the overall structure of glioblastomas in MRIs.

Ensemble Inversion for Brain Tumor Growth Models With Mass Effect.

We propose a method for extracting physics-based biomarkers from a single multiparametric Magnetic Resonance Imaging (mpMRI) scan bearing a glioma tumor. We account for mass effect, the deformation of brain parenchyma due to the growing tumor, which on its own is an important radiographic feature but its automatic quantification remains an open problem. In particular, we calibrate a partial differential equation (PDE) tumor growth model that captures mass effect, parameterized by a single scalar parameter, tumor proliferation, migration, while localizing the tumor initiation site. The single-scan calibration problem is severely ill-posed because the precancerous, healthy, brain anatomy is unknown. To address the ill-posedness, we introduce an ensemble inversion scheme that uses a number of normal subject brain templates as proxies for the healthy precancer subject anatomy. We verify our solver on a synthetic dataset and perform a retrospective analysis on a clinical dataset of 216 glioblastoma (GBM) patients. We analyze the reconstructions using our calibrated biophysical model and demonstrate that our solver provides both global and local quantitative measures of tumor biophysics and mass effect. We further highlight the improved performance in model calibration through the inclusion of mass effect in tumor growth models-including mass effect in the model leads to 10% increase in average dice coefficients for patients with significant mass effect. We further evaluate our model by introducing novel biophysics-based features and using them for survival analysis. Our preliminary analysis suggests that including such features can improve patient stratification and survival prediction.

Fully Automatic Calibration of Tumor-Growth Models Using a Single mpMRI Scan.

Our objective is the calibration of mathematical tumor growth models from a single multiparametric scan. The target problem is the analysis of preoperative Glioblastoma (GBM) scans. To this end, we present a fully automatic tumor-growth calibration methodology that integrates a single-species reaction-diffusion partial differential equation (PDE) model for tumor progression with multiparametric Magnetic Resonance Imaging (mpMRI) scans to robustly extract patient specific biomarkers i.e., estimates for (i) the tumor cell proliferation rate, (ii) the tumor cell migration rate, and (iii) the original, localized site(s) of tumor initiation. Our method is based on a sparse reconstruction algorithm for the tumor initial location (TIL). This problem is particularly challenging due to nonlinearity, ill-posedeness, and ill conditioning. We propose a coarse-to-fine multi-resolution continuation scheme with parameter decomposition to stabilize the inversion. We demonstrate robustness and practicality of our method by applying the proposed method to clinical data of 206 GBM patients. We analyze the extracted biomarkers and relate tumor origin with patient overall survival by mapping the former into a common atlas space. We present preliminary results that suggest improved accuracy for prediction of patient overall survival when a set of imaging features is augmented with estimated biophysical parameters. All extracted features, tumor initial positions, and biophysical growth parameters are made publicly available for further analysis. To our knowledge, this is the first fully automatic scheme that can handle multifocal tumors and can localize the TIL to a few millimeters.

Estimating Glioblastoma Biophysical Growth Parameters Using Deep Learning Regression.

Glioblastoma ( GBM ) is arguably the most aggressive, infiltrative, and heterogeneous type of adult brain tumor. Biophysical modeling of GBM growth has contributed to more informed clinical decision-making. However, deploying a biophysical model to a clinical environment is challenging since underlying computations are quite expensive and can take several hours using existing technologies. Here we present a scheme to accelerate the computation. In particular, we present a deep learning ( DL )-based logistic regression model to estimate the GBM's biophysical growth in seconds. This growth is defined by three tumor-specific parameters: 1) a diffusion coefficient in white matter ( Dw ), which prescribes the rate of infiltration of tumor cells in white matter, 2) a mass-effect parameter ( Mp ), which defines the average tumor expansion, and 3) the estimated time ( T ) in number of days that the tumor has been growing. Preoperative structural multi-parametric MRI ( mpMRI ) scans from n = 135 subjects of the TCGA-GBM imaging collection are used to quantitatively evaluate our approach. We consider the mpMRI intensities within the region defined by the abnormal FLAIR signal envelope for training one DL model for each of the tumor-specific growth parameters. We train and validate the DL-based predictions against parameters derived from biophysical inversion models. The average Pearson correlation coefficients between our DL-based estimations and the biophysical parameters are 0.85 for Dw, 0.90 for Mp, and 0.94 for T, respectively. This study unlocks the power of tumor-specific parameters from biophysical tumor growth estimation. It paves the way towards their clinical translation and opens the door for leveraging advanced radiomic descriptors in future studies by means of a significantly faster parameter reconstruction compared to biophysical growth modeling approaches.

Multiatlas Calibration of Biophysical Brain Tumor Growth Models with Mass Effect.

We present a 3D fully-automatic method for the calibration of partial differential equation (PDE) models of glioblastoma (GBM) growth with "mass effect", the deformation of brain tissue due to the tumor. We quantify the mass effect, tumor proliferation, tumor migration, and the localized tumor initial condition from a single multiparameteric Magnetic Resonance Imaging (mpMRI) patient scan. The PDE is a reaction-advection-diffusion partial differential equation coupled with linear elasticity equations to capture mass effect. The single-scan calibration model is notoriously difficult because the precancerous (healthy) brain anatomy is unknown. To solve this inherently ill-posed and illconditioned optimization problem, we introduce a novel inversion scheme that uses multiple brain atlases as proxies for the healthy precancer patient brain resulting in robust and reliable parameter estimation. We apply our method on both synthetic and clinical datasets representative of the heterogeneous spatial landscape typically observed in glioblastomas to demonstrate the validity and performance of our methods. In the synthetic data, we report calibration errors (due to the ill-posedness and our solution scheme) in the 10%-20% range. In the clinical data, we report good quantitative agreement with the observed tumor and qualitative agreement with the mass effect (for which we do not have a ground truth). Our method uses a minimal set of parameters and provides both global and local quantitative measures of tumor infiltration and mass effect.

IMAGE-DRIVEN BIOPHYSICAL TUMOR GROWTH MODEL CALIBRATION.

We present a novel formulation for the calibration of a biophysical tumor growth model from a single-time snapshot, multiparametric magnetic resonance imaging (MRI) scan of a glioblastoma patient. Tumor growth models are typically nonlinear parabolic partial differential equations (PDEs). Thus, we have to generate a second snapshot to be able to extract significant information from a single patient snapshot. We create this two-snapshot scenario as follows. We use an atlas (an average of several scans of healthy individuals) as a substitute for an earlier, pretumor, MRI scan of the patient. Then, using the patient scan and the atlas, we combine image-registration algorithms and parameter estimation algorithms to achieve a better estimate of the healthy patient scan and the tumor growth parameters that are consistent with the data. Our scheme is based on our recent work (Scheufele et al., Comput. Methods Appl. Mech. Engrg., to appear), but we apply a different and novel scheme where the tumor growth simulation in contrast to the previous work is executed in the patient brain domain and not in the atlas domain yielding more meaningful patient-specific results. As a basis, we use a PDE-constrained optimization framework. We derive a modified Picard-iteration-type solution strategy in which we alternate between registration and tumor parameter estimation in a new way. In addition, we consider an ℓ 1 sparsity constraint on the initial condition for the tumor and integrate it with the new joint inversion scheme. We solve the sub-problems with a reduced space, inexact Gauss-Newton-Krylov/quasi-Newton method. We present results using real brain data with synthetic tumor data that show that the new scheme reconstructs the tumor parameters in a more accurate and reliable way compared to our earlier scheme.