Tina A. Schuetz

Biophysical modeling of brain tumor progression: From unconditionally stable explicit time integration to an inverse problem with parabolic PDE constraints for model calibration

PURPOSE A novel unconditionally stable, explicit numerical method is introduced to the field of modeling brain cancer progression on a tissue level together with an inverse problem (IP) based on optimal control theory that allows for automated model calibration with respect to observations in clinical imaging data. METHODS Biophysical models of cancer progression on a tissue level are in general based on the assumption that the spatiotemporal spread of cancerous cells is determined by cell division and net migration. These processes are typically described in terms of a parabolic partial differential equation (PDE). In the present work a parallelized implementation of an unconditionally stable, explicit Euler (EE(⋆)) time integration method for the solution of this PDE is detailed. The key idea of the discussed EE(⋆) method is to relax the strong stability requirement on the spectral radius of the coefficient matrix by introducing a subdivision regime for a given outer time step. The performance is related to common implicit numerical methods. To quantify the numerical error, a simplified model that has a closed form solution is considered. To allow for a systematic, phenomenological validation a novel approach for automated model calibration on the basis of observations in medical imaging data is developed. The resulting IP is based on optimal control theory and manifests as a large scale, PDE constrained optimization problem. RESULTS The numerical error of the EE(⋆) method is at the order of standard implicit numerical methods. The computing times are well below those obtained for implicit methods and by that demonstrate efficiency. Qualitative and quantitative analysis in 12 patients demonstrates that the obtained results are in strong agreement with observations in medical imaging data. Rating simulation success in terms of the mean overlap between model predictions and manual expert segmentations yields a success rate of 75% (9 out of 12 patients). CONCLUSIONS The discussed EE(⋆) method provides desirable features for image-based model calibration or hybrid image registration algorithms in which the model serves as a biophysical prior. This is due to (i) ease of implementation, (ii) low memory requirements, (iii) efficiency, (iv) a straightforward interface for parameter updates, and (v) the fact that the method is inherently matrix-free. The explicit time integration method is confirmed via experiments for automated model calibration. Qualitative and quantitative analysis demonstrates that the proposed framework allows for recovering observations in medical imaging data and by that phenomenological model validity.PURPOSE A novel unconditionally stable, explicit numerical method is introduced to the field of modeling brain cancer progression on a tissue level together with an inverse problem (IP) based on optimal control theory that allows for automated model calibration with respect to observations in clinical imaging data. METHODS Biophysical models of cancer progression on a tissue level are in general based on the assumption that the spatiotemporal spread of cancerous cells is determined by cell division and net migration. These processes are typically described in terms of a parabolic partial differential equation (PDE). In the present work a parallelized implementation of an unconditionally stable, explicit Euler (EE⋆ ) time integration method for the solution of this PDE is detailed. The key idea of the discussed EE⋆ method is to relax the strong stability requirement on the spectral radius of the coefficient matrix by introducing a subdivision regime for a given outer time step. The performance is related to common implicit numerical methods. To quantify the numerical error, a simplified model that has a closed form solution is considered. To allow for a systematic, phenomenological validation a novel approach for automated model calibration on the basis of observations in medical imaging data is developed. The resulting IP is based on optimal control theory and manifests as a large scale, PDE constrained optimization problem. RESULTS The numerical error of the EE⋆ method is at the order of standard implicit numerical methods. The computing times are well below those obtained for implicit methods and by that demonstrate efficiency. Qualitative and quantitative analysis in 12 patients demonstrates that the obtained results are in strong agreement with observations in medical imaging data. Rating simulation success in terms of the mean overlap between model predictions and manual expert segmentations yields a success rate of 75% (9 out of 12 patients). CONCLUSIONS The discussed EE⋆ method provides desirable features for image-based model calibration or hybrid image registration algorithms in which the model serves as a biophysical prior. This is due to (i) ease of implementation, (ii) low memory requirements, (iii) efficiency, (iv) a straightforward interface for parameter updates, and (v) the fact that the method is inherently matrix-free. The explicit time integration method is confirmed via experiments for automated model calibration. Qualitative and quantitative analysis demonstrates that the proposed framework allows for recovering observations in medical imaging data and by that phenomenological model validity.

A novel method for simulating the extracellular matrix in models of tumour growth.

A novel hybrid continuum-discrete model to simulate tumour growth on a cellular scale is proposed. The lattice-based spatiotemporal model consists of reaction-diffusion equations that describe interactions between cancer cells and their microenvironment. The fundamental ingredients that are typically considered are the nutrient concentration, the extracellular matrix (ECM), and matrix degrading enzymes (MDEs). The in vivo processes are very complex and occur on different levels. This in turn leads to huge computational costs. The main contribution of the present work is therefore to describe the processes on the basis of simplified mathematical approaches, which, at the same time, depict realistic results to understand the biological processes. In this work, we discuss if we have to simulate the MDE or if the degraded matrix can be estimated directly with respect to the cancer cell distribution. Additionally, we compare the results for modelling tumour growth using the common and our simplified approach, thereby demonstrating the advantages of the proposed method. Therefore, we introduce variations of the positioning of the nutrient delivering blood vessels and use different initializations of the ECM. We conclude that the novel method, which does not explicitly model the matrix degrading enzymes, provides means for a straightforward and fast implementation for modelling tumour growth.

A Generic Framework for Modeling Brain Deformation as a Constrained Parametric Optimization Problem to Aid Non-diffeomorphic Image Registration in Brain Tumor Imaging

OBJECTIVES In the present paper a novel computational framework for modeling tumor induced brain deformation as a biophysical prior for non-rigid image registration is described. More precisely, we aim at providing a generic building block for non-rigid image registration that can be used to resolve inherent irregularities in non-diffeomorphic registration problems that naturally arise in serial and cross-population brain tumor imaging studies due to the presence (or progression) of pathology. METHODS The model for the description of brain cancer dynamics on a tissue level is based on an initial boundary value problem (IBVP). The IBVP follows the accepted assumption that the progression of primary brain tumors on a tissue level is governed by proliferation and migration of cancerous cells into surrounding healthy tissue. The model of tumor induced brain deformation is phrased as a parametric, constrained optimization problem. As a basis of comparison and to demonstrate generalizability additional soft constraints (penalties) are considered. A back-tracking line search is implemented in conjunction with a limited memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) method in order to handle the numerically delicate log-barrier strategy for confining volume change. RESULTS Numerical experiments are performed to test the flexible control of the computed deformation patterns in terms of varying model parameters. The results are qualitatively and quantitatively related to patterns in patient individual magnetic resonance imaging data. CONCLUSIONS Numerical experiments demonstrate the flexible control of the computed deformation patterns. This in turn strongly suggests that the model can be adapted to patient individual imaging patterns of brain tumors. Qualitative and quantitative comparison of the computed cancer profiles to patterns in medical imaging data of an exemplary patient demonstrates plausibility. The designed optimization problem is based on computational tools widely used in non-rigid image registration, which in turn makes the model generally applicable for integration into non-rigid image registration algorithms.

Modelling of glioblastoma growth by linking a molecular interaction network with an agent-based model

In this work, a mathematical model of malignant brain tumour growth is presented. In particular, the growth of glioblastoma is investigated on the intracellular and intercellular scale. The Go or Grow principle of tumour cells states that tumour cells either migrate or proliferate. For glioblastoma, microRNA-451 has been shown to be an energy dependent key regulator of the LKB1 (liver kinase B1) and AMPK (AMP-activated protein kinase) pathway that influences the signalling for migration or cell division. We introduce a mathematical model that reproduces these biological processes. The intracellular molecular interaction network is represented by a system of nine ordinary differential equations. This is put into a multiscale context by applying an agent-based approach: each cell is equipped with this interaction network and additional rules to determine its new phenotype as either migrating, proliferating or quiescent. The evaluation of the proposed model by comparison of the results with in vitro experiments indicates its validity.

A computational multiscale model of glioblastoma growth: Regulation of cell migration and proliferation via microRNA-451, LKB1 and AMPK

A new computational multiscale model of glioblastoma growth is introduced. This model combines an agent-based model for representing processes on the cellular level with a molecular interaction network for each cell on the subcellular scale. The network is based on recently published work on the interaction of microRNA-451, LKB1 and AMPK in the regulation of glioblastoma cell migration and proliferation. We translated this network into a mathematical description by the use of 17 ordinary differential equations. In our model, we furthermore establish a link from the molecular interaction network of a single cell to cellular actions (e.g. chemotactic movement) on the microscopic level. First results demonstrate that the computational model reproduces a tumor cell development comparable to that observed in in vitro experiments.

Cyclic Numerical Time Integration in Variational Non-Rigid Image Registration based on Quadratic Regularisation

In the present work, a novel computational framework for variational non-rigid image registration is discussed. The fundamental aim is to provide an alternative to approximate approaches based on successive convolution, which have gained great popularity in recent years, due to their linear complexity and ease of implementation. An optimise-then-discretise framework is considered. The corresponding Euler-Lagrange equations (ELEs), which arise from calculus of variation, constitute a necessary condition for a minimiser of the variational optimisation problem. The conventional, semi-implicit (SI) time integration for the solution of the ELEs is replaced by an explicit approach rendering the implementation straightforward. Since explicit methods are subject to a restrictive stability requirement on the maximal admissible time step size, they are in general inefficient and prone to get stuck in local minima. As a remedy, we take advantage of methods based on cyclic explicit numerical time integration. With this the strong stability requirement on each individual time step can be replaced by a relaxed stability requirement. This in turn results in an unconditionally stable method, which is as efficient as SI approaches. As a basis of comparison, SI methods are considered. Generalisability is demonstrated within a generic variational framework based on quadratic regularisation. Qualitative and quantitative analysis of numerical experiments based on synthetic test data demonstrates accuracy and efficiency.

Identification of Crucial Parameters in a Mathematical Multiscale Model of Glioblastoma Growth

Glioblastomas are highly malignant brain tumours. Mathematical models and their analysis provide a tool to support the understanding of the development of these tumours as well as the design of more effective treatment strategies. We have previously developed a multiscale model of glioblastoma progression that covers processes on the cellular and molecular scale. Here, we present a novel nutrient-dependent multiscale sensitivity analysis of this model that helps to identify those reaction parameters of the molecular interaction network that influence the tumour progression on the cellular scale the most. In particular, those parameters are identified that essentially determine tumour expansion and could be therefore used as potential therapy targets. As indicators for the success of a potential therapy target, a deceleration of the tumour expansion and a reduction of the tumour volume are employed. From the results, it can be concluded that no single parameter variation results in a less aggressive tumour. However, it can be shown that a few combined perturbations of two systematically selected parameters cause a slow-down of the tumour expansion velocity accompanied with a decrease of the tumour volume. Those parameters are primarily linked to the reactions that involve the microRNA-451 and the thereof regulated protein MO25.

A Cross-scale Model of Tumor Growth: Do We Need to Model Molecular Interactions in Separate Artificial Compartments within a Cell?

Abstract We modified an existing cross-scale model of avascular tumor growth that couples a gene-protein interaction network with an agent based model. In the original model each biological cell is artificially subdivided into four geographic compartments to account for a spatial polarity of the molecules. Since these artificial compartments result in eight ordinary differential equations (ODEs) — instead of one — that need to be solved for each molecular species we tried to reduce the related computational burden. We renounced to include these artificial compartments and assume a homogeneous distribution of molecular concentrations within each cell. By adapting the conditions for nutrient uptake and the neighborhood options for cell migration and division we achieve results that are comparable to the original model. However, for this modified model the number of ODEs that need to be evaluated for each molecular species within each cell and thus the computing time could be significantly reduced.

Personalisierte Modellierung der Progression primärer Hirntumoren als Optimierungsproblem mit Differentialgleichungsnebenbedingung

Die vorliegende Arbeit liefert einen neuartigen Ansatz fur die Individualisierung bildbasierter, biophysikalischer Modelle der Progression primarer Hirntumoren. Das verwendete mathematische Modell ist etabliert. Es basiert auf einer parabolischen, partiellen Differentialgleichung (PDG). Die Modellierung der Migration von Tumorzellen entlang der Nervenbahnen der weisen Substanz wird durch eine Integration von Diffusionstensordaten realisiert. Die Modellindividualisierung basiert auf der Losung eines Parameteridentifikationsproblems. Der verwendete Ansatz fuhrt auf ein Optimierungsproblem mit Differentialgleichungsnebenbedingung. Eine qualitative und quantitative Analyse fur patientenindividuelle Bildgebungsdaten demonstriert die phanomenologische Validitat des verwendeten Modells. Die gute Ubereinstimmung zwischen der geschatzten Zustandsfunktion (Losung des direkten Problems) und der Observable (gewonnen aus den Bildgebungsdaten) bestatigt die Methodik.

In-silico Modelling of Tumour-Immune System Interactions for Glioblastomas

Abstract In the present work, a new mathematical approach for modelling the influence of the immune system, more precisely of microglial cells, on the progression of malignant primary brain tumours is presented. A hybrid approach is used to model the cellular tumour progression, the development of the local nutrient concentration and of the density of the extracellular matrix (ECM). The resting microglia in primary brain tumours are activated and attracted by signals emitted by tumour cells, which are described by a partial differential equation. The secretion of matrix degrading enzymes from amoeboid immune cells can be modelled with the help of an additional term for the degradation of the ECM. This supports a more invasive migration of tumour cells. To our knowledge, we present for the first time a model of microglial cells in the context of tumour growth. The qualitative results are identical to the cell arrangements described in the literature. In addition, the comparison with in-vitro data matches in a qualitative manner. The proposed model, thus, represents a promising approach for modelling brain tumour growth at the cellular level in the light of the innate immune system.