Alina Toma

In-silico oncology: an approximate model of brain tumor mass effect based on directly manipulated free form deformation

PurposeThe present work introduces a novel method for approximating mass effect of primary brain tumors.MethodsThe spatio-temporal dynamics of cancerous cells are modeled by means of a deterministic reaction-diffusion equation. Diffusion tensor information obtained from a probabilistic diffusion tensor imaging atlas is incorporated into the model to simulate anisotropic diffusion of cancerous cells. To account for the expansive nature of the tumor, the computed net cell density of malignant cells is linked to a parametric deformation model. This mass effect model is based on the so-called directly manipulated free form deformation. Spatial correspondence between two successive simulation steps is established by tracking landmarks, which are attached to the boundary of the gross tumor volume. The movement of these landmarks is used to compute the new configuration of the control points and, hence, determines the resulting deformation. To prevent a deformation of rigid structures (i.e. the skull), fixed shielding landmarks are introduced. In a refinement step, an adaptive landmark scheme ensures a dense sampling of the tumor isosurface, which in turn allows for an appropriate representation of the tumor shape.ResultsThe influence of different parameters on the model is demonstrated by a set of simulations. Additionally, simulation results are qualitatively compared to an exemplary set of clinical magnetic resonance images of patients diagnosed with high-grade glioma.ConclusionsCareful visual inspection of the results demonstrates the potential of the implemented model and provides first evidence that the computed approximation of tumor mass effect is sensible. The shape of diffusive brain tumors (glioblastoma multiforme) can be recovered and approximately matches the observations in real clinical data.

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.

Coupling tumor growth with brain deformation: a constrained parametric non-rigid registration problem

A novel approach for coupling brain tumor mass effect with a continuous model of cancer progression is proposed. The purpose of the present work is to devise an efficient approximate model for the mechanical interaction of the tumor with its surroundings in order to aid registration of brain tumor images with statistical atlases as well as the generation of atlases of brain tumor disease. To model tumor progression a deterministic reaction-diffusion formalism, which describes the spatio-temporal dynamics of a coarse-grained population density of cancerous cells, is discretized on a regular grid. Tensor information obtained from a probabilistic atlas is used to model the anisotropy of the diffusion of malignant cells within white matter. To account for the expansive nature of the tumor a parametric deformation model is linked to the computed net cell density of cancerous cells. To this end, we formulate a constrained optimization problem using an inhomogeneous regularization that in turn allows for approximating physical properties of brain tissue. The described coupling model can in general be applied to estimate mass effect of non-convex, diffusive as well as multifocal tumors so that no simplification of the growth model has to be stipulated. The present work has to be considered as a proof-of-concept. Visual assessment of the computed results demonstrates the potential of the described method. We conclude that the analogy to the problem formulation in image registration potentially allows for a sensible integration of the described approach into a unified framework of image registration and tumor modeling.

A model of tumour induced brain deformation as bio-physical prior for non-rigid image registration

The present paper introduces extensions to a novel model of tumour induced brain deformation in order to aid non-rigid registration of images displaying brain tumour pathology to a standard reference atlas. The model serves as a bio-physical prior and by that resolves the inherent irregularities that naturally arise in the considered registration problem. The proposed model is formulated in terms of a constrained optimisation problem. At this, the data term is modelled on the basis of the population density of cancerous cells obtained from the solution of an initial boundary value problem. A soft constraint allows for approximating bio-mechanical properties of brain tissue. It is demonstrated that introducing a non-linear weighting functional with respect to the computed density of cancerous cells into both - the data term and the soft constraint - allows for an adaptive control of the deformation pattern. Additionally, we explicitly penalise deformations of rigid structures and extend the numerical scheme by exploiting analytical derivatives as well as the compact support of the employed parametric deformation model during optimisation. Further, we have made available a strategy for re-orientation of diffusion tensors subject to spatial deformation.

Modellierung tumorinduzierter Gewebedeformation als Optimierungsproblem mit weicher Nebenbedingung

Ein ungelostes Problem in der nicht-rigiden Bildregistrierung ist die Behandlung von pathologie-bedingten, morphologischen Unterschieden, wie sie beispielsweise bei der raumlichen Normalisierung neuroradiologischer Datensatze, die Tumorpathologie abbilden, auftreten. Mit der vorliegenden Arbeit liefern wir einen Baustein fur einen modellbasierten Losungsansatz. Wir schlagen vor, die entstehenden Irregularitaten durch eine explizite Modellierung der Pathologie zu umgehen. Im Detail stellen wir die Erweiterung eines auf der Formulierung eines Optimierungsproblems basierenden Ansatzes zur Modellierung tumorinduzierter Gewebedeformation vor. Dieser bietet potentiell die Moglichkeit einer direkten Integration in Verfahren der nicht-rigiden Bildregistrierung. Neben einer Darstellung des theoretischen Zusammenhangs mit existierenden Verfahren, zeigen wir experimentell, dass die Hinzunahme einer nicht-linearen Wichtung der Terme des Zielfunktionals eine adaptivere Steuerung der resultierenden Deformationsmuster erlaubt.

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.