Andreas Mang

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.

An Inexact Newton--Krylov Algorithm for Constrained Diffeomorphic Image Registration

We propose numerical algorithms for solving large deformation diffeomorphic image registration problems. We formulate the nonrigid image registration problem as a problem of optimal control. This leads to an infinite-dimensional partial differential equation (PDE) constrained optimization problem. The PDE constraint consists, in its simplest form, of a hyperbolic transport equation for the evolution of the image intensity. The control variable is the velocity field. Tikhonov regularization on the control ensures well-posedness. We consider standard smoothness regularization based on H1- or H2-seminorms. We augment this regularization scheme with a constraint on the divergence of the velocity field (control variable) rendering the deformation incompressible (Stokes regularization scheme) and thus ensuring that the determinant of the deformation gradient is equal to one, up to the numerical error. We use a Fourier pseudospectral discretization in space and a Chebyshev pseudospectral discretization in time. The latter allows us to reduce the number of unknowns and enables the time-adaptive inversion for nonstationary velocity fields. We use a preconditioned, globalized, matrix-free, inexact Newton-Krylov method for numerical optimization. A parameter continuation is designed to estimate an optimal regularization parameter. Regularity is ensured by controlling the geometric properties of the deformation field. Overall, we arrive at a black-box solver that exploits computational tools that are precisely tailored for solving the optimality system. We study spectral properties of the Hessian, grid convergence, numerical accuracy, computational efficiency, and deformation regularity of our scheme. We compare the designed Newton-Krylov methods with a globalized Picard method (preconditioned gradient descent). We study the influence of a varying number of unknowns in time. The reported results demonstrate excellent numerical accuracy, guaranteed local deformation regularity, and computational efficiency with an optional control on local mass conservation. The Newton-Krylov methods clearly outperform the Picard method if high accuracy of the inversion is required. Our method provides equally good results for stationary and nonstationary velocity fields for two-image registration problems.

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.

Consistency of parametric registration in serial MRI studies of brain tumor progression

ObjectThe consistency of parametric registration in multi-temporal magnetic resonance (MR) imaging studies was evaluated.Materials and methodsSerial MRI scans of adult patients with a brain tumor (glioma) were aligned by parametric registration. The performance of low-order spatial alignment (6/9/12 degrees of freedom) of different 3D serial MR-weighted images is evaluated. A registration protocol for the alignment of all images to one reference coordinate system at baseline is presented. Registration results were evaluated for both, multimodal intra-timepoint and mono-modal multi-temporal registration. The latter case might present a challenge to automatic intensity-based registration algorithms due to ill-defined correspondences. The performance of our algorithm was assessed by testing the inverse registration consistency. Four different similarity measures were evaluated to assess consistency.ResultsCareful visual inspection suggests that images are well aligned, but their consistency may be imperfect. Sub-voxel inconsistency within the brain was found for allsimilarity measures used for parametric multi-temporal registration. T1-weighted images were most reliable for establishing spatial correspondence between different timepoints.ConclusionsThe parametric registration algorithm is feasible for use in this application. The sub-voxel resolution mean displacement error of registration transformations demonstrates that the algorithm converges to an almost identical solution for forward and reverse registration.

Constrained

We propose regularization schemes for deformable registration and efficient algorithms for their numerical approximation. We treat image registration as a variational optimal control problem. The deformation map is parametrized by its velocity. Tikhonov regularization ensures well-posedness. Our scheme augments standard smoothness regularization operators based on H1- and H2-seminorms with a constraint on the divergence of the velocity field, which resembles variational formulations for Stokes incompressible flows. In our formulation, we invert for a stationary velocity field and a mass source map. This allows us to explicitly control the compressibility of the deformation map and by that the determinant of the deformation gradient. We also introduce a new regularization scheme that allows us to control shear. We use a globalized, preconditioned, matrix-free, reduced space (Gauss-)Newton-Krylov scheme for numerical optimization. We exploit variable elimination techniques to reduce the number of unknowns of our system; we only iterate on the reduced space of the velocity field. Our current implementation is limited to the two-dimensional case. The numerical experiments demonstrate that we can control the determinant of the deformation gradient without compromising registration quality. This additional control allows us to avoid oversmoothing of the deformation map. We also demonstrate that we can promote or penalize shear whilst controlling the determinant of the deformation gradient.We propose regularization schemes for deformable registration and efficient algorithms for its numerical approximation. We treat image registration as a variational optimal control problem. The deformation map is parametrized by a velocity field. Quadratic Tikhonov regularization ensures well-posedness of the problem. Our scheme augments standard smoothness vectorial operators based on

H^1

H^1

-Regularization Schemes for Diffeomorphic Image Registration

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

H^2

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

-seminorms with a constraint on the divergence of the velocity field. Our formulation is motivated from Stokes flows in fluid mechanics. We invert for a stationary velocity field as well as a mass source map. This allows us to explicitly control the compressibility of the deformation map and by that the determinant of the deformation gradient. In addition, we design a novel regularization model that allows us to control shear. We use a globalized, preconditioned, matrix-free (Gauss-)Newton-Krylov scheme. We exploit variable elimination techniques to reduce the number of unknowns of our system: we only iterate on the reduced space of the velocity field. Our scheme can be used for problems in which the deformation map is expected to be nearly incompressible, as is often the case in medical imaging. Numerical experiments demonstrate that we can explicitly control the determinant of the deformation gradient without compromising registration quality. This additional control allows us to avoid over-smoothing of the deformation map. We demonstrate that our new formulation allows us to promote or penalize shear whilst controlling the determinant of the deformation gradient.