Extrapolating glioma invasion margin in brain magnetic resonance images: suggesting new irradiation margins

A metric for quantitative evaluation of glioma margin changes in magnetic resonance imaging

BACKGROUND

Gliomas differ from meningiomas in their margins, most of which are not separated from the surrounding tissue by a distinct interface.

PURPOSE

To characterize the margins of gliomas quantitatively based on the margin sharpness coefficient (MSC) is significant for clinical judgment and invasive analysis of gliomas.

MATERIAL AND METHODS

The data for this study used magnetic resonance image (MRI) data from 67 local patients and 15 open patients to quantify the intensity of changes in the glioma margins of the brain using MSC. The accuracy of MSC was assessed by consistency analysis and Bland-Altman test analysis, as well as invasive correlations using receiver operating characteristic (ROC) and Spearman correlation coefficients for subjects.

RESULTS

In grading the tumors, the mean MSC values were significantly lower for high-grade gliomas (HGG) than for low-grade gliomas (LGG). The concordance correlation between the measured gradient and the actual gradient was high (HGG: 0.981; LGG: 0.993), and the Bland-Altman mean difference at the 95% confidence interval (HGG: -0.576; LGG: 0.254) and the limits of concordance (HGG: 5.580; LGG: 5.436) indicated no statistical difference. The correlation between MSC and invasion based on the margins of gliomas showed an AUC of 0.903 and 0.911 for HGG and LGG, respectively. The mean Spearman correlation coefficient of the MSC versus the actual distance of invasion was -0.631 in gliomas.

CONCLUSION

The relatively low MSC on the blurred margins and irregular shape of gliomas may help in benign-malignant differentiation and invasion prediction of gliomas and has potential application for clinical judgment.

The influence of anisotropy on the clinical target volume of brain tumor patients

Objective.Current radiotherapy guidelines for glioma target volume definition recommend a uniform margin expansion from the gross tumor volume (GTV) to the clinical target volume (CTV), assuming uniform infiltration in the invaded brain tissue. However, glioma cells migrate preferentially along white matter tracts, suggesting that white matter directionality should be considered in an anisotropic CTV expansion. We investigate two models of anisotropic CTV expansion and evaluate their clinical feasibility.Approach.To incorporate white matter directionality into the CTV, a diffusion tensor imaging (DTI) atlas is used. The DTI atlas consists of water diffusion tensors that are first spatially transformed into local tumor resistance tensors, also known as metric tensors, and secondly fed to a CTV expansion algorithm to generate anisotropic CTVs. Two models of spatial transformation are considered in the first step. The first model assumes that tumor cells experience reduced resistance parallel to the white matter fibers. The second model assumes that the anisotropy of tumor cell resistance is proportional to the anisotropy observed in DTI, with an 'anisotropy weighting parameter' controlling the proportionality. The models are evaluated in a cohort of ten brain tumor patients.Main results.To evaluate the sensitivity of the model, a library of model-generated CTVs was computed by varying the resistance and anisotropy parameters. Our results indicate that the resistance coefficient had the most significant effect on the global shape of the CTV expansion by redistributing the target volume from potentially less involved gray matter to white matter tissue. In addition, the anisotropy weighting parameter proved useful in locally increasing CTV expansion in regions characterized by strong tissue directionality, such as near the corpus callosum.Significance.By incorporating anisotropy into the CTV expansion, this study is a step toward an interactive CTV definition that can assist physicians in incorporating neuroanatomy into a clinically optimized CTV.

Toward image-based personalization of glioblastoma therapy: A clinical and biological validation study of a novel, deep learning-driven tumor growth model

Background

The diffuse growth pattern of glioblastoma is one of the main challenges for accurate treatment. Computational tumor growth modeling has emerged as a promising tool to guide personalized therapy. Here, we performed clinical and biological validation of a novel growth model, aiming to close the gap between the experimental state and clinical implementation.

Methods

One hundred and twenty-four patients from The Cancer Genome Archive (TCGA) and 397 patients from the UCSF Glioma Dataset were assessed for significant correlations between clinical data, genetic pathway activation maps (generated with PARADIGM; TCGA only), and infiltration (Dw) as well as proliferation (ρ) parameters stemming from a Fisher-Kolmogorov growth model. To further evaluate clinical potential, we performed the same growth modeling on preoperative magnetic resonance imaging data from 30 patients of our institution and compared model-derived tumor volume and recurrence coverage with standard radiotherapy plans.

Results

The parameter ratio Dw/ρ (P < .05 in TCGA) as well as the simulated tumor volume (P < .05 in TCGA/UCSF) were significantly inversely correlated with overall survival. Interestingly, we found a significant correlation between 11 proliferation pathways and the estimated proliferation parameter. Depending on the cutoff value for tumor cell density, we observed a significant improvement in recurrence coverage without significantly increased radiation volume utilizing model-derived target volumes instead of standard radiation plans.

Conclusions

Identifying a significant correlation between computed growth parameters and clinical and biological data, we highlight the potential of tumor growth modeling for individualized therapy of glioblastoma. This might improve the accuracy of radiation planning in the near future.

Data driven modeling of pseudopalisade pattern formation

Pseudopalisading is an interesting phenomenon where cancer cells arrange themselves to form a dense garland-like pattern. Unlike the palisade structure, a similar type of pattern first observed in schwannomas by pathologist J.J. Verocay (Wippold et al. in AJNR Am J Neuroradiol 27(10):2037-2041, 2006), pseudopalisades are less organized and associated with a necrotic region at their core. These structures are mainly found in glioblastoma (GBM), a grade IV brain tumor, and provide a way to assess the aggressiveness of the tumor. Identification of the exact bio-mechanism responsible for the formation of pseudopalisades is a difficult task, mainly because pseudopalisades seem to be a consequence of complex nonlinear dynamics within the tumor. In this paper we propose a data-driven methodology to gain insight into the formation of different types of pseudopalisade structures. To this end, we start from a state of the art macroscopic model for the dynamics of GBM, that is coupled with the dynamics of extracellular pH, and formulate a terminal value optimal control problem. Thus, given a specific, observed pseudopalisade pattern, we determine the evolution of parameters (bio-mechanisms) that are responsible for its emergence. Random histological images exhibiting pseudopalisade-like structures are chosen to serve as target pattern. Having identified the optimal model parameters that generate the specified target pattern, we then formulate two different types of pattern counteracting ansatzes in order to determine possible ways to impair or obstruct the process of pseudopalisade formation. This provides the basis for designing active or live control of malignant GBM. Furthermore, we also provide a simple, yet insightful, mechanism to synthesize new pseudopalisade patterns by linearly combining the optimal model parameters responsible for generating different known target patterns. This particularly provides a hint that complex pseudopalisade patterns could be synthesized by a linear combination of parameters responsible for generating simple patterns. Going even further, we ask ourselves if complex therapy approaches can be conceived, such that some linear combination thereof is able to reverse or disrupt simple pseudopalisade patterns; this is investigated with the help of numerical simulations.

A stochastic hierarchical model for low grade glioma evolution

A stochastic hierarchical model for the evolution of low grade gliomas is proposed. Starting with the description of cell motion using a piecewise diffusion Markov process (PDifMP) at the cellular level, we derive an equation for the density of the transition probability of this Markov process based on the generalised Fokker-Planck equation. Then, a macroscopic model is derived via parabolic limit and Hilbert expansions in the moment equations. After setting up the model, we perform several numerical tests to study the role of the local characteristics and the extended generator of the PDifMP in the process of tumour progression. The main aim focuses on understanding how the variations of the jump rate function of this process at the microscopic scale and the diffusion coefficient at the macroscopic scale are related to the diffusive behaviour of the glioma cells and to the onset of malignancy, i.e., the transition from low-grade to high-grade gliomas.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Modeling the propagation of tumor fronts with shortest path and diffusion models—implications for the definition of the clinical target volume

Objective. The overarching objective is to make the definition of the clinical target volume (CTV) in radiation oncology less subjective and more scientifically based. The specific objective of this study is to investigate similarities and differences between two methods that model tumor spread beyond the visible gross tumor volume (GTV): (1) the shortest path model, which is the standard method of adding a geometric GTV-CTV margin, and (2) the reaction-diffusion model. Approach. These two models to capture the invisible tumor ‘fire front’ are defined and compared in mathematical terms. The models are applied to example cases that represent tumor spread in non-uniform and anisotropic media with anatomical barriers. Main results. The two seemingly disparate models bring forth traveling waves that can be associated with the front of tumor growth outward from the GTV. The shape of the fronts is similar for both models. Differences are seen in cases where the diffusive flow is reduced due to anatomical barriers, and in complex spatially non-uniform cases. The diffusion model generally leads to smoother fronts. The smoothness can be controlled with a parameter defined by the ratio of the diffusion coefficient and the proliferation rate. Significance. Defining the CTV has been described as the weakest link of the radiotherapy chain. There are many similarities in the mathematical description and the behavior of the common geometric GTV-CTV expansion method, and the definition of the CTV tumor front via the reaction-diffusion model. Its mechanistic basis and the controllable smoothness make the diffusion model an attractive alternative to the standard GTV-CTV margin model.

Deep Learning for Reaction-Diffusion Glioma Growth Modeling: Towards a Fully Personalized Model?

Simple Summary

Mathematical tumor growth models have been proposed for decades to capture the growth of gliomas, an aggressive form of brain tumor. However, the estimation of the tumor cell-density distribution at diagnosis and model parameters from partial observations provided by magnetic resonance imaging are ill-posed problems. In this work, we propose a deep learning-based approach to address these problems. 1200 synthetic tumors are first generated using the mathematical model over brain geometries of 6 volunteers. Two deep convolutional neural networks are then trained to (i) reconstruct a whole tumor cell-density distribution and (ii) evaluate the model parameters from partial observations provided in the form of threshold-like imaging contours, with state-of-the-art results. From the estimated cell-density distribution and parameter values, the spatio-temporal evolution of the tumor can ultimately be accurately captured by the mathematical model. Such an approach could be of great interest for glioma characterization and therapy planning.

Abstract

Reaction-diffusion models have been proposed for decades to capture the growth of gliomas, the most common primary brain tumors. However, ill-posedness of the initialization at diagnosis time and parameter estimation of such models have restrained their clinical use as a personalized predictive tool. In this work, we investigate the ability of deep convolutional neural networks (DCNNs) to address commonly encountered pitfalls in the field. Based on 1200 synthetic tumors grown over real brain geometries derived from magnetic resonance (MR) data of six healthy subjects, we demonstrate the ability of DCNNs to reconstruct a whole tumor cell-density distribution from only two imaging contours at a single time point. With an additional imaging contour extracted at a prior time point, we also demonstrate the ability of DCNNs to accurately estimate the individual diffusivity and proliferation parameters of the model. From this knowledge, the spatio-temporal evolution of the tumor cell-density distribution at later time points can ultimately be precisely captured using the model. We finally show the applicability of our approach to MR data of a real glioblastoma patient. This approach may open the perspective of a clinical application of reaction-diffusion growth models for tumor prognosis and treatment planning.

Carbonized paramagnetic complexes of Mn (II) as contrast agents for precise magnetic resonance imaging of sub-millimeter-sized orthotopic tumors

Paramagnetic complexes containing gadolinium ions have been widely used for magnetic resonance imaging (MRI) in clinic. However, these paramagnetic complexes pose some safety concerns. There is still a demand for the development of stable MRI contrast agents that exhibit higher sensitivity and superior functionality to existing contrast agents. Here, we develop carbonized paramagnetic complexes of manganese (II) (Mn@CCs) to encapsulate Mn²⁺ in sealed carbonized shells with superhigh r₁ relaxivity. Compared to the most common clinical contrast agent Magnevist, investigations in vivo demonstrate that the Mn@CCs cross the intact blood-brain barrier of normal health mice with minor metal deposition; preferentially target the glioma tissues distribute homogeneously with high penetration in an intracranial mouse model; delineate clear tumor margins in MRIs of ultrasmall single-nodule brain tumors, and multi-nodular liver tumors. The sensitivity, accuracy and low toxicity offer by Mn@CCs provides new opportunities for early molecular diagnostics and imaging-guided biomedical applications.

Improving the imaging of cancer may enhance the treatment of patients, Here, the authors develop a Mn(II) based nanoparticle contrast agent for MRI imaging and show that the nanoparticles can cross the brain barrier and image glioma cells.

Effect of energy requirements in the growth of brain tumor: a theoretical approach

A malignant tumor is an uncontrolled growth of tissues receiving energy in form of the nutrients provided by the microvascular networks. It is proposed that the supplied energy to a tumor is used for three purposes: the creation of new cells, maintenance of tumor cells, and tumor volume expansion by overcoming external pressure. A mathematical model studying the effects of energy required for maintenance and overcoming external pressure, the energy required creating a single cell, death rate, and tumor cell density on tumor development has been formulated. Including a term, residual energy for tumor growth in the tumor growth equation, the well-known logistic equation has been re-derived for tumors. Analytical solutions have been developed, and numerical analysis for the growth in brain tumors with the variation of parameters related to energy supply, the energy required for maintenance, and expansion of tumor has been performed. Expressions for the tumor growth rate(r) and carrying capacity(C) of the tumor are formulated in terms of the parameters used in the model. The range of 'r', estimated using our model is found within the ranges of tumor growth rates in gliomas reported by the other researchers. Selecting the model parameters precisely for a particular individual, the tumor growth rate and carrying capacity could be estimated accurately. Our study indicates that the actual growth rate and carrying capacity of a tumor reduce and tumor saturation time increases with the increase of death rate, the energy required for a single cell division, and energy requirement for the tumor cell maintenance.

Initial Condition Assessment for Reaction-Diffusion Glioma Growth Models: A Translational MRI-Histology (In)Validation Study

Reaction-diffusion models have been proposed for decades to capture the growth of gliomas. Nevertheless, these models require an initial condition: the tumor cell density distribution over the whole brain at diagnosis time. Several works have proposed to relate this distribution to abnormalities visible on magnetic resonance imaging (MRI). In this work, we verify these hypotheses by stereotactic histological analysis of a non-operated brain with glioblastoma using a 3D-printed slicer. Cell density maps are computed from histological slides using a deep learning approach. The density maps are then registered to a postmortem MR image and related to an MR-derived geodesic distance map to the tumor core. The relation between the edema outlines visible on T2-FLAIR MRI and the distance to the core is also investigated. Our results suggest that (i) the previously proposed exponential decrease of the tumor cell density with the distance to the core is reasonable but (ii) the edema outlines would not correspond to a cell density iso-contour and (iii) the suggested tumor cell density at these outlines is likely overestimated. These findings highlight the limitations of conventional MRI to derive glioma cell density maps and the need for other initialization methods for reaction-diffusion models to be used in clinical practice.

Evaluating potential of multi-parametric MRI using co-registered histology: Application to a mouse model of glioblastoma

BACKGROUND

Conventional MRI fails to detect regions of glioblastoma cell infiltration beyond the contrast-enhanced T1 solid tumor region, with infiltrating tumor cells often migrating along host blood vessels.

PURPOSE

MRI is capable of generating a range of image contrasts which are commonly assessed individually by qualitative visual inspection. It has long been hypothesized that better diagnoses could be achieved by combining these multiple images, so called multi-parametric or multi-spectral MRI. However, the lack of clinical histology and the difficulties of co-registration, has meant this hypothesis has never been rigorously tested. Here we test this hypothesis, using a previously published multi-dimensional dataset consisting of registered MR images and histology.

STUDY TYPE

Animal Model.

SUBJECTS

Mice bearing orthotopic glioblastoma xenografts generated from a patient-derived glioblastoma cell line.

FIELD STRENGTH/SEQUENCES

7 Tesla, T₁/T₂ weighted, T₂ mapping, contrast enhance T_1, diffusion-weighted, diffusion tensor imaging.

ASSESSMENT

Immunohistochemistry sections were stained for Human Leukocyte Antigen (probing human-derived tumor cells). To achieve quantitative MRI-tissue comparison, multiple histological slices cut in the MRI plane were stacked to produce tumor cell density maps acting as 'ground truth'.

STATISTICAL TESTS

Sensitivity, specificity, accuracy and Dice similarity indices were calculated. ANOVA, t-test, Bonferroni correction and Pearson coefficients were used for statistical analysis.

RESULTS

Correlation coefficient analysis with co-registered 'ground truth' histology showed interactive regression maps had higher correlation coefficients and sensitivity values than T2W, ADC, FA, and T2map. Further, the interaction regression maps showed statistical improved detection of tumor volume.

DATA CONCLUSION

Voxel-by-voxel analysis provided quantitative evidence confirming the hypothesis that mpMRI can, potentially, better distinguish between the tumor region and normal tissue.

CANet: Context Aware Network for Brain Glioma Segmentation

Automated segmentation of brain glioma plays an active role in diagnosis decision, progression monitoring and surgery planning. Based on deep neural networks, previous studies have shown promising technologies for brain glioma segmentation. However, these approaches lack powerful strategies to incorporate contextual information of tumor cells and their surrounding, which has been proven as a fundamental cue to deal with local ambiguity. In this work, we propose a novel approach named Context-Aware Network (CANet) for brain glioma segmentation. CANet captures high dimensional and discriminative features with contexts from both the convolutional space and feature interaction graphs. We further propose context guided attentive conditional random fields which can selectively aggregate features. We evaluate our method using publicly accessible brain glioma segmentation datasets BRATS2017, BRATS2018 and BRATS2019. The experimental results show that the proposed algorithm has better or competitive performance against several State-of-The-Art approaches under different segmentation metrics on the training and validation sets.

Visualizing Glioma Infiltration by the Combination of Multimodality Imaging and Artificial Intelligence, a Systematic Review of the Literature

The aim of this study was to systematically review the literature concerning the integration of multimodality imaging with artificial intelligence methods for visualization of tumor cell infiltration in glioma patients. The review was performed in accordance with the preferred reporting items for systematic reviews and meta-analysis (PRISMA) guidelines. The literature search was conducted in PubMed, Embase, The Cochrane Library and Web of Science and yielded 1304 results. 14 studies were included in the qualitative analysis. The reference standard for tumor infiltration was either histopathology or recurrence on image follow-up. Critical assessment was performed according to the Quality Assessment of Diagnostic Accuracy Studies (QUADAS2). All studies concluded their findings to be of significant value for future clinical practice. Diagnostic test accuracy reached an area under the curve of 0.74–0.91 reported in six studies. There was no consensus with regard to included image modalities, models or training and test strategies. The integration of artificial intelligence with multiparametric imaging shows promise for visualizing tumor cell infiltration in glioma patients. This approach can possibly optimize surgical resection margins and help provide personalized radiotherapy planning.

The role of computational methods for automating and improving clinical target volume definition

Treatment planning in radiotherapy distinguishes three target volume concepts: the gross tumor volume (GTV), the clinical target volume (CTV), and the planning target volume (PTV). Over time, GTV definition and PTV margins have improved through the development of novel imaging techniques and better image guidance, respectively. CTV definition is sometimes considered the weakest element in the planning process. CTV definition is particularly complex since the extension of microscopic disease cannot be seen using currently available in-vivo imaging techniques. Instead, CTV definition has to incorporate knowledge of the patterns of tumor progression. While CTV delineation has largely been considered the domain of radiation oncologists, this paper, arising from a 2019 ESTRO Physics research workshop, discusses the contributions that medical physics and computer science can make by developing computational methods to support CTV definition. First, we overview the role of image segmentation algorithms, which may in part automate CTV delineation through segmentation of lymph node stations or normal tissues representing anatomical boundaries of microscopic tumor progression. The recent success of deep convolutional neural networks has also enabled learning entire CTV delineations from examples. Second, we discuss the use of mathematical models of tumor progression for CTV definition, using as example the application of glioma growth models to facilitate GTV-to-CTV expansion for glioblastoma that is consistent with neuroanatomy. We further consider statistical machine learning models to quantify lymphatic metastatic progression of tumors, which may eventually improve elective CTV definition. Lastly, we discuss approaches to incorporate uncertainty in CTV definition into treatment plan optimization as well as general limitations of the CTV concept in the case of infiltrating tumors without natural boundaries.

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.

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.

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.

Radiation protraction schedules for low-grade gliomas: a comparison between different mathematical models

We optimize radiotherapy (RT) administration strategies for treating low-grade gliomas. Specifically, we consider different tumour growth laws, both with and without spatial effects. In each scenario, we find the optimal treatment in the sense of maximizing the overall survival time of a virtual low-grade glioma patient, whose tumour progresses according to the examined growth laws. We discover that an extreme protraction therapeutic strategy, which amounts to substantially extending the time interval between RT sessions, may lead to better tumour control. The clinical implications of our results are also presented.

Precision diagnostics based on machine learning-derived imaging signatures

The complexity of modern multi-parametric MRI has increasingly challenged conventional interpretations of such images. Machine learning has emerged as a powerful approach to integrating diverse and complex imaging data into signatures of diagnostic and predictive value. It has also allowed us to progress from group comparisons to imaging biomarkers that offer value on an individual basis. We review several directions of research around this topic, emphasizing the use of machine learning in personalized predictions of clinical outcome, in breaking down broad umbrella diagnostic categories into more detailed and precise subtypes, and in non-invasively estimating cancer molecular characteristics. These methods and studies contribute to the field of precision medicine, by introducing more specific diagnostic and predictive biomarkers of clinical outcome, therefore pointing to better matching of treatments to patients.

Molecular and Clinical Insights into the Invasive Capacity of Glioblastoma Cells

The invasive capacity of GBM is one of the key tumoral features associated with treatment resistance, recurrence, and poor overall survival. The molecular machinery underlying GBM invasiveness comprises an intricate network of signaling pathways and interactions with the extracellular matrix and host cells. Among them, PI3k/Akt, Wnt, Hedgehog, and NFkB play a crucial role in the cellular processes related to invasion. A better understanding of these pathways could potentially help in developing new therapeutic approaches with better outcomes. Nevertheless, despite significant advances made over the last decade on these molecular and cellular mechanisms, they have not been translated into the clinical practice. Moreover, targeting the infiltrative tumor and its significance regarding outcome is still a major clinical challenge. For instance, the pre- and intraoperative methods used to identify the infiltrative tumor are limited when trying to accurately define the tumor boundaries and the burden of tumor cells in the infiltrated parenchyma. Besides, the impact of treating the infiltrative tumor remains unclear. Here we aim to highlight the molecular and clinical hallmarks of invasion in GBM.

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.

Coupling brain-tumor biophysical models and diffeomorphic image registration

We present SIBIA (Scalable Integrated Biophysics-based Image Analysis), a framework for joint image registration and biophysical inversion and we apply it to analyze MR images of glioblastomas (primary brain tumors). We have two applications in mind. The first one is normal-to-abnormal image registration in the presence of tumor-induced topology differences. The second one is biophysical inversion based on single-time patient data. The underlying optimization problem is highly non-linear and non-convex and has not been solved before with a gradient-based approach. Given the segmentation of a normal brain MRI and the segmentation of a cancer patient MRI, we determine tumor growth parameters and a registration map so that if we "grow a tumor" (using our tumor model) in the normal brain and then register it to the patient image, then the registration mismatch is as small as possible. This "coupled problem" two-way couples the biophysical inversion and the registration problem. In the image registration step we solve a large-deformation diffeomorphic registration problem parameterized by an Eulerian velocity field. In the biophysical inversion step we estimate parameters in a reaction-diffusion tumor growth model that is formulated as a partial differential equation (PDE). In SIBIA, we couple these two sub-components in an iterative manner. We first presented the components of SIBIA in "Gholami et al., Framework for Scalable Biophysics-based Image Analysis, IEEE/ACM Proceedings of the SC2017", in which we derived parallel distributed memory algorithms and software modules for the decoupled registration and biophysical inverse problems. In this paper, our contributions are the introduction of a PDE-constrained optimization formulation of the coupled problem, and the derivation of a Picard iterative solution scheme. We perform extensive tests to experimentally assess the performance of our method on synthetic and clinical datasets. We demonstrate the convergence of the SIBIA optimization solver in different usage scenarios. We demonstrate that using SIBIA, we can accurately solve the coupled problem in three dimensions (256³ resolution) in a few minutes using 11 dual-x86 nodes.

Differentiation of Glioblastoma and Solitary Brain Metastasis by Gradient of Relative Cerebral Blood Volume in the Peritumoral Brain Zone Derived from Dynamic Susceptibility Contrast Perfusion Magnetic Resonance Imaging

OBJECTIVE

The purpose of our study was to evaluate the efficacy of the relative cerebral blood volume (rCBV) gradient in the peritumoral brain zone (PBZ)-the difference in the rCBV values from the area closest to the enhancing lesion to the area closest to the healthy white matter-in differentiating glioblastoma (GB) from solitary brain metastasis (MET).

METHODS

A 3.0-T magnetic resonance imaging (MRI) machine was used to perform dynamic susceptibility contrast perfusion MRI (DSC-MRI) on 43 patients with a solitary brain tumor (24 GB, 19 MET). The rCBV ratios were acquired by DSC-MRI data in 3 regions of the PBZ (near the enhancing tumor, G1; intermediate distance from the enhancing tumor, G2; far from the enhancing tumor, G3). The maximum rCBV ratios in the PBZ (rCBVp) and the enhancing tumor were also calculated, respectively. The perfusion parameters were evaluated using the nonparametric Mann-Whitney test. The sensitivity, specificity, accuracy, and the area under the receiver operating characteristic curve were identified.

RESULTS

The rCBVp ratios and rCBV gradient in the PBZ were significantly higher in GB compared with MET (P < 0.05 for both rCBVp ratios and rCBV gradient). The threshold values of 0.50 or greater for rCBVp ratios provide sensitivity and specificity of 57.69% and 79.17%, respectively, for differentiation of GB from MET. Compared with rCBVp ratios, rCBV gradient had higher sensitivity (94.44%) and specificity (91.67%) using the threshold value of greater than 0.06.

CONCLUSIONS

The parameter of rCBV gradient derived from DSC-MRI in the PBZ seems to be the most efficient parameter to differentiate GB from METs.

MR Imaging Features of Anaplastic Pleomorphic Xanthoastrocytoma Mimicking High-Grade Astrocytoma

BACKGROUND AND PURPOSE

Anaplastic pleomorphic xanthoastrocytoma, which has been recently defined as a distinct entity in the 2016 World Health Organization classification, may exhibit aggressive clinical behavior and relatively worse prognosis than pleomorphic xanthoastrocytoma. This study aimed to investigate whether there were any differences in MR imaging characteristics between these 2 tumors.

MATERIALS AND METHODS

This retrospective study included 9 patients with anaplastic pleomorphic xanthoastrocytoma and 10 patients with pleomorphic xanthoastrocytoma who underwent MR imaging before an operation. DWI was performed in 17 patients (8 with anaplastic pleomorphic xanthoastrocytoma, 9 with pleomorphic xanthoastrocytoma); and DSC-PWI, in 9 patients (5 with anaplastic pleomorphic xanthoastrocytoma, 4 with pleomorphic xanthoastrocytoma). Demographics, conventional imaging characteristics (location, size, cystic degeneration, enhancement, peritumoral edema, and leptomeningeal contact), minimum relative ADC ratio, and maximum relative CBV ratio were evaluated between the anaplastic pleomorphic xanthoastrocytoma and pleomorphic xanthoastrocytoma groups.

RESULTS

Anaplastic pleomorphic xanthoastrocytoma was more likely to demonstrate high-grade features than pleomorphic xanthoastrocytoma, including greater maximum tumor diameter (4.7 ± 0.6 cm versus 3.1 ± 1.1 cm, P = .001), more frequent heterogeneous contrast enhancement of solid portions (88.9% versus 20.0%, P = .01), more obvious peritumoral edema (2.3 ± 0.9 cm versus 1.0 ± 0.9 cm, P = .008), lower minimum relative ADC on DWI (1.0 ± 0.2 versus 1.5 ± 0.4, P = .008), and higher maximum relative CBV on DSC-PWI (2.6 ± 0.8 versus 1.6 ± 0.2, P = .036).

CONCLUSIONS

Anaplastic pleomorphic xanthoastrocytomas often have more aggressive MR imaging features mimicking high-grade astrocytomas than pleomorphic xanthoastrocytomas. DWI and DSC-PWI might be useful in the characterization and differentiation of anaplastic pleomorphic xanthoastrocytoma and pleomorphic xanthoastrocytoma.

Subject-specific Brain Tumor Growth Modelling via An Efficient Bayesian Inference Framework

An accurate prediction of brain tumor progression is crucial for optimized treatment of the tumors. Gliomas are primarily treated by combining surgery, external beam radiotherapy, and chemotherapy. Among them, radiotherapy is a non-invasive and effective therapy, and an understanding of tumor growth will allow better therapy planning. In particular, estimating parameters associated with tumor growth, such as the diffusion coefficient and proliferation rate, is crucial to accurately characterize physiology of tumor growth and to develop predictive models of tumor infiltration and recurrence. Accurate parameter estimation, however, is a challenging task due to inaccurate tumor boundaries and the approximation of the tumor growth model. Here, we introduce a Bayesian framework for a subject-specific tumor growth model that estimates the tumor parameters effectively. This is achieved by using an improved elliptical slice sampling method based on an adaptive sample region. Experimental results on clinical data demonstrate that the proposed method provides a higher acceptance rate, while preserving the parameter estimation accuracy, compared with other state-of-the-art methods such as Metropolis-Hastings and elliptical slice sampling without any modification. Our approach has the potential to provide a method to individualize therapy, thereby offering an optimized treatment.

Radiomic signature of infiltration in peritumoral edema predicts subsequent recurrence in glioblastoma: implications for personalized radiotherapy planning

Standard surgical resection of glioblastoma, mainly guided by the enhancement on postcontrast T1-weighted magnetic resonance imaging (MRI), disregards infiltrating tumor within the peritumoral edema region (ED). Subsequent radiotherapy typically delivers uniform radiation to peritumoral FLAIR-hyperintense regions, without attempting to target areas likely to be infiltrated more heavily. Noninvasive in vivo delineation of the areas of tumor infiltration and prediction of early recurrence in peritumoral ED could assist in targeted intensification of local therapies, thereby potentially delaying recurrence and prolonging survival. This paper presents a method for estimating peritumoral edema infiltration using radiomic signatures determined via machine learning methods, and tests it on 90 patients with de novo glioblastoma. The generalizability of the proposed predictive model was evaluated via cross-validation in a discovery cohort ([Formula: see text]) and was subsequently evaluated in a replication cohort ([Formula: see text]). Spatial maps representing the likelihood of tumor infiltration and future early recurrence were compared with regions of recurrence on postresection follow-up studies with pathology confirmation. The cross-validated accuracy of our predictive infiltration model on the discovery and replication cohorts was 87.51% (odds ratio = 10.22, sensitivity = 80.65, and specificity = 87.63) and 89.54% (odds ratio = 13.66, sensitivity = 97.06, and specificity = 76.73), respectively. The radiomic signature of the recurrent tumor region revealed higher vascularity and cellularity when compared with the nonrecurrent region. The proposed model shows evidence that multiparametric pattern analysis from clinical MRI sequences can assist in in vivo estimation of the spatial extent and pattern of tumor recurrence in peritumoral edema, which may guide supratotal resection and/or intensification of postoperative radiation therapy.

Comparative study between spatio-temporal models for brain tumor growth

Modeling of brain tumor growth simulator can estimate life expectancy for individual patients, estimate future effect of brain damages toward human senses and attitude and help in evaluating the efficiency of applied treatments. Brain tumor growth can be calculated based on Spatio-Temporal mathematical models namely the isotropic reaction-diffusion model and the anisotropic reaction-diffusion model where the second model produces more realistic results. Tumor normally grows in White Matter (WM) five times faster than in Gray Matter (GM) which makes brain tissues modeled as inhomogeneous-anisotropic material to assign different parameters to each tissue. In this research a comparative study between several tumor growth models has been achieved to clarify the effect of different algorithms on modeling tumor grow.

Diffusion tensor magnetic resonance imaging driven growth modeling for radiotherapy target definition in glioblastoma

BACKGROUND

The clinical target volume (CTV) in radiotherapy is routinely based on gadolinium contrast enhanced T1 weighted (T1w + Gd) and T2 weighted fluid attenuated inversion recovery (T2w FLAIR) magnetic resonance imaging (MRI) sequences which have been shown to over- or underestimate the microscopic tumor cell spread. Gliomas favor spread along the white matter fiber tracts. Tumor growth models incorporating the MRI diffusion tensors (DTI) allow to account more consistently for the glioma growth. The aim of the study was to investigate the potential of a DTI driven growth model to improve target definition in glioblastoma (GBM).

MATERIAL AND METHODS

Eleven GBM patients were scanned using T1w, T2w FLAIR, T1w + Gd and DTI. The brain was segmented into white matter, gray matter and cerebrospinal fluid. The Fisher-Kolmogorov growth model was used assuming uniform proliferation and a difference in white and gray matter diffusion of a ratio of 10. The tensor directionality was tested using an anisotropy weighting parameter set to zero (γ0) and twenty (γ20). The volumetric comparison was performed using Hausdorff distance, Dice similarity coefficient (DSC) and surface area.

RESULTS

The median of the standard CTV (CTVstandard) was 180 cm³. The median surface area of CTVstandard was 211 cm². The median surface area of respective CTV_γ0 and CTV_γ20 significantly increased to 338 and 376 cm², respectively. The Hausdorff distance was greater than zero and significantly increased for both CTV_γ0 and CTV_γ20 with respective median of 18.7 and 25.2 mm. The DSC for both CTV_γ0 and CTV_γ20 were significantly below one with respective median of 0.74 and 0.72, which means that 74 and 72% of CTVstandard were included in CTV_γ0 and CTV_γ20, respectively.

CONCLUSIONS

DTI driven growth models result in CTVs with a significantly increased surface area, a significantly increased Hausdorff distance and decreased overlap between the standard and model derived volume.

Moments of von Mises and Fisher distributions and applications

The von Mises and Fisher distributions are spherical analogues to the Normal distribution on the unit circle and unit sphere, respectively. The computation of their moments, and in particular the second moment, usually involves solving tedious trigonometric integrals. Here we present a new method to compute the moments of spherical distributions, based on the divergence theorem. This method allows a clear derivation of the second moments and can be easily generalized to higher dimensions. In particular we note that, to our knowledge, the variance-covariance matrix of the three dimensional Fisher distribution has not previously been explicitly computed. While the emphasis of this paper lies in calculating the moments of spherical distributions, their usefulness is motivated by their relationship to population statistics in animal/cell movement models and demonstrated in applications to the modelling of sea turtle navigation, wolf movement and brain tumour growth.

Non-standard radiotherapy fractionations delay the time to malignant transformation of low-grade gliomas

Grade II gliomas are slowly growing primary brain tumors that affect mostly young patients. Cytotoxic therapies (radiotherapy and/or chemotherapy) are used initially only for patients having a bad prognosis. These therapies are planned following the “maximum dose in minimum time” principle, i. e. the same schedule used for high-grade brain tumors in spite of their very different behavior. These tumors transform after a variable time into high-grade gliomas, which significantly decreases the patient’s life expectancy. In this paper we study mathematical models describing the growth of grade II gliomas in response to radiotherapy. We find that protracted metronomic fractionations, i.e. therapeutical schedules enlarging the time interval between low-dose radiotherapy fractions, may lead to a better tumor control without an increase in toxicity. Other non-standard fractionations such as protracted or hypoprotracted schemes may also be beneficial. The potential survival improvement depends on the tumor’s proliferation rate and can be even of the order of years. A conservative metronomic scheme, still being a suboptimal treatment, delays the time to malignant progression by at least one year when compared to the standard scheme.

Mathematical Analysis of Glioma Growth in a Murine Model

Five immunocompetent C57BL/6-cBrd/cBrd/Cr (albino C57BL/6) mice were injected with GL261-luc2 cells, a cell line sharing characteristics of human glioblastoma multiforme (GBM). The mice were imaged using magnetic resonance (MR) at five separate time points to characterize growth and development of the tumor. After 25 days, the final tumor volumes of the mice varied from 12 mm³ to 62 mm³, even though mice were inoculated from the same tumor cell line under carefully controlled conditions. We generated hypotheses to explore large variances in final tumor size and tested them with our simple reaction-diffusion model in both a 3-dimensional (3D) finite difference method and a 2-dimensional (2D) level set method. The parameters obtained from a best-fit procedure, designed to yield simulated tumors as close as possible to the observed ones, vary by an order of magnitude between the three mice analyzed in detail. These differences may reflect morphological and biological variability in tumor growth, as well as errors in the mathematical model, perhaps from an oversimplification of the tumor dynamics or nonidentifiability of parameters. Our results generate parameters that match other experimental in vitro and in vivo measurements. Additionally, we calculate wave speed, which matches with other rat and human measurements.

Personalized Radiotherapy Planning Based on a Computational Tumor Growth Model

In this article, we propose a proof of concept for the automatic planning of personalized radiotherapy for brain tumors. A computational model of glioblastoma growth is combined with an exponential cell survival model to describe the effect of radiotherapy. The model is personalized to the magnetic resonance images (MRIs) of a given patient. It takes into account the uncertainty in the model parameters, together with the uncertainty in the MRI segmentations. The computed probability distribution over tumor cell densities, together with the cell survival model, is used to define the prescription dose distribution, which is the basis for subsequent Intensity Modulated Radiation Therapy (IMRT) planning. Depending on the clinical data available, we compare three different scenarios to personalize the model. First, we consider a single MRI acquisition before therapy, as it would usually be the case in clinical routine. Second, we use two MRI acquisitions at two distinct time points in order to personalize the model and plan radiotherapy. Third, we include the uncertainty in the segmentation process. We present the application of our approach on two patients diagnosed with high grade glioma. We introduce two methods to derive the radiotherapy prescription dose distribution, which are based on minimizing integral tumor cell survival using the maximum a posteriori or the expected tumor cell density. We show how our method allows the user to compute a patient specific radiotherapy planning conformal to the tumor infiltration. We further present extensions of the method in order to spare adjacent organs at risk by re-distributing the dose. The presented approach and its proof of concept may help in the future to better target the tumor and spare organs at risk.

Imaging Surrogates of Infiltration Obtained Via Multiparametric Imaging Pattern Analysis Predict Subsequent Location of Recurrence of Glioblastoma

BACKGROUND

Glioblastoma is an aggressive and highly infiltrative brain cancer. Standard surgical resection is guided by enhancement on postcontrast T1-weighted (T1) magnetic resonance imaging, which is insufficient for delineating surrounding infiltrating tumor.

OBJECTIVE

To develop imaging biomarkers that delineate areas of tumor infiltration and predict early recurrence in peritumoral tissue. Such markers would enable intensive, yet targeted, surgery and radiotherapy, thereby potentially delaying recurrence and prolonging survival.

METHODS

Preoperative multiparametric magnetic resonance images (T1, T1-gadolinium, T2-weighted, T2-weighted fluid-attenuated inversion recovery, diffusion tensor imaging, and dynamic susceptibility contrast-enhanced magnetic resonance images) from 31 patients were combined using machine learning methods, thereby creating predictive spatial maps of infiltrated peritumoral tissue. Cross-validation was used in the retrospective cohort to achieve generalizable biomarkers. Subsequently, the imaging signatures learned from the retrospective study were used in a replication cohort of 34 new patients. Spatial maps representing the likelihood of tumor infiltration and future early recurrence were compared with regions of recurrence on postresection follow-up studies with pathology confirmation.

RESULTS

This technique produced predictions of early recurrence with a mean area under the curve of 0.84, sensitivity of 91%, specificity of 93%, and odds ratio estimates of 9.29 (99% confidence interval: 8.95-9.65) for tissue predicted to be heavily infiltrated in the replication study. Regions of tumor recurrence were found to have subtle, yet fairly distinctive multiparametric imaging signatures when analyzed quantitatively by pattern analysis and machine learning.

CONCLUSION

Visually imperceptible imaging patterns discovered via multiparametric pattern analysis methods were found to estimate the extent of infiltration and location of future tumor recurrence, paving the way for improved targeted treatment.

MRI Based Bayesian Personalization of a Tumor Growth Model

The mathematical modeling of brain tumor growth has been the topic of numerous research studies. Most of this work focuses on the reaction-diffusion model, which suggests that the diffusion coefficient and the proliferation rate can be related to clinically relevant information. However, estimating the parameters of the reaction-diffusion model is difficult because of the lack of identifiability of the parameters, the uncertainty in the tumor segmentations, and the model approximation, which cannot perfectly capture the complex dynamics of the tumor evolution. Our approach aims at analyzing the uncertainty in the patient specific parameters of a tumor growth model, by sampling from the posterior probability of the parameters knowing the magnetic resonance images of a given patient. The estimation of the posterior probability is based on: 1) a highly parallelized implementation of the reaction-diffusion equation using the Lattice Boltzmann Method (LBM), and 2) a high acceptance rate Monte Carlo technique called Gaussian Process Hamiltonian Monte Carlo (GPHMC). We compare this personalization approach with two commonly used methods based on the spherical asymptotic analysis of the reaction-diffusion model, and on a derivative-free optimization algorithm. We demonstrate the performance of the method on synthetic data, and on seven patients with a glioblastoma, the most aggressive primary brain tumor. This Bayesian personalization produces more informative results. In particular, it provides samples from the regions of interest and highlights the presence of several modes for some patients. In contrast, previous approaches based on optimization strategies fail to reveal the presence of different modes, and correlation between parameters.

Multi-scale Modeling in Clinical Oncology: Opportunities and Barriers to Success

Hierarchical processes spanning several orders of magnitude of both space and time underlie nearly all cancers. Multi-scale statistical, mathematical, and computational modeling methods are central to designing, implementing and assessing treatment strategies that account for these hierarchies. The basic science underlying these modeling efforts is maturing into a new discipline that is close to influencing and facilitating clinical successes. The purpose of this review is to capture the state-of-the-art as well as the key barriers to success for multi-scale modeling in clinical oncology. We begin with a summary of the long-envisioned promise of multi-scale modeling in clinical oncology, including the synthesis of disparate data types into models that reveal underlying mechanisms and allow for experimental testing of hypotheses. We then evaluate the mathematical techniques employed most widely and present several examples illustrating their application as well as the current gap between pre-clinical and clinical applications. We conclude with a discussion of what we view to be the key challenges and opportunities for multi-scale modeling in clinical oncology.

Sampling image segmentations for uncertainty quantification

In this paper, we introduce a method to automatically produce plausible image segmentation samples from a single expert segmentation. A probability distribution of image segmentation boundaries is defined as a Gaussian process, which leads to segmentations which are spatially coherent and consistent with the presence of salient borders in the image. The proposed approach is computationally efficient, and generates visually plausible samples. The variability between the samples is mainly governed by a parameter which may be correlated with a simple Dice's coefficient, or easily set by the user from the definition of probable regions of interest. The method is extended to the case of several neighboring structures, but also to account for under or over segmentation, and the presence of excluded regions. We also detail a method to sample segmentations with more general non-stationary covariance functions which relies on supervoxels. Furthermore, we compare the generated segmentation samples with several manual clinical segmentations of a brain tumor. Finally, we show how this approach can have useful applications in the field of uncertainty quantification, and an illustration is provided in radiotherapy planning, where segmentation sampling is applied to both the clinical target volume and the organs at risk.

Computational Trials: Unraveling Motility Phenotypes, Progression Patterns, and Treatment Options for Glioblastoma Multiforme

Glioblastoma multiforme is a malignant brain tumor with poor prognosis and high morbidity due to its invasiveness. Hypoxia-driven motility and concentration-driven motility are two mechanisms of glioblastoma multiforme invasion in the brain. The use of anti-angiogenic drugs has uncovered new progression patterns of glioblastoma multiforme associated with significant differences in overall survival. Here, we apply a mathematical model of glioblastoma multiforme growth and invasion in humans and design computational trials using agents that target angiogenesis, tumor replication rates, or motility. The findings link highly-dispersive, moderately-dispersive, and hypoxia-driven tumors to the patterns observed in glioblastoma multiforme treated by anti-angiogenesis, consisting of progression by Expanding FLAIR, Expanding FLAIR + Necrosis, and Expanding Necrosis, respectively. Furthermore, replication rate-reducing strategies (e.g. Tumor Treating Fields) appear to be effective in highly-dispersive and moderately-dispersive tumors but not in hypoxia-driven tumors. The latter may respond to motility-reducing agents. In a population computational trial, with all three phenotypes, a correlation was observed between the efficacy of the rate-reducing agent and the prolongation of overall survival times. This research highlights the potential applications of computational trials and supports new hypotheses on glioblastoma multiforme phenotypes and treatment options.

Optimal radiation fractionation for low-grade gliomas: Insights from a mathematical model

We study optimal radiotherapy fractionations for low-grade glioma using mathematical models. Both space-independent and space-dependent models are studied. Two different optimization criteria have been developed, the first one accounting for the global effect of the tumor mass on the disease symptoms and the second one related to the delay of the malignant transformation of the tumor. The models are studied theoretically and numerically using the method of feasible directions. We have searched for optimal distributions of the daily doses dj in the standard protocol of 30 fractions using both models and the two different optimization criteria. The optimal results found in all cases are minor deviations from the standard protocol and provide only marginal potential gains. Thus, our results support the optimality of current radiation fractionations over the standard 6 week treatment period. This is also in agreement with the observation that minor variations of the fractionation have failed to provide measurable gains in survival or progression free survival, pointing out to a certain optimality of the current approach.

An inverse problem formulation for parameter estimation of a reaction-diffusion model of low grade gliomas

We present a numerical scheme for solving a parameter estimation problem for a model of low-grade glioma growth. Our goal is to estimate the spatial distribution of tumor concentration, as well as the magnitude of anisotropic tumor diffusion. We use a constrained optimization formulation with a reaction-diffusion model that results in a system of nonlinear partial differential equations. In our formulation, we estimate the parameters using partially observed, noisy tumor concentration data at two different time instances, along with white matter fiber directions derived from diffusion tensor imaging. The optimization problem is solved with a Gauss-Newton reduced space algorithm. We present the formulation and outline the numerical algorithms for solving the resulting equations. We test the method using a synthetic dataset and compute the reconstruction error for different noise levels and detection thresholds for monofocal and multifocal test cases.

Extreme protraction for low-grade gliomas: theoretical proof of concept of a novel therapeutical strategy

Grade II gliomas are slowly growing primary brain tumours that affect mostly young patients and become fatal after a variable time period. Current clinical handling includes surgery as first-line treatment. Cytotoxic therapies (radiotherapy RT or chemotherapy QT) are used initially only for patients having a bad prognosis. Therapies are administered following the 'maximum dose in minimum time' principle, which is the same schedule used for high-grade brain tumours. Using mathematical models describing the growth of these tumours in response to radiotherapy, we find that an extreme protraction therapeutical strategy, i.e. enlarging substantially the time interval between RT fractions, may lead to better tumour control. Explicit formulas are found providing the optimal spacing between doses in a very good agreement with the simulations of the full 3D mathematical model approximating the tumour spatiotemporal dynamics. This idea, although breaking the well-established paradigm, has biological meaning since, in these slowly growing tumours, it may be more favourable to treat the tumour as the tumour cells leave the quiescent compartment and move into the cell cycle.

Effects of anti-angiogenesis on glioblastoma growth and migration: model to clinical predictions

Glioblastoma multiforme (GBM) causes significant neurological morbidity and short survival times. Brain invasion by GBM is associated with poor prognosis. Recent clinical trials of bevacizumab in newly-diagnosed GBM found no beneficial effects on overall survival times; however, the baseline health-related quality of life and performance status were maintained longer in the bevacizumab group and the glucocorticoid requirement was lower. Here, we construct a clinical-scale model of GBM whose predictions uncover a new pattern of recurrence in 11/70 bevacizumab-treated patients. The findings support an exception to the Folkman hypothesis: GBM grows in the absence of angiogenesis by a cycle of proliferation and brain invasion that expands necrosis. Furthermore, necrosis is positively correlated with brain invasion in 26 newly-diagnosed GBM. The unintuitive results explain the unusual clinical effects of bevacizumab and suggest new hypotheses on the dynamic clinical effects of migration by active transport, a mechanism of hypoxia-driven brain invasion.

Expert-validated CSF segmentation of MNI atlas enhances accuracy of virtual glioma growth patterns

Biomathematical modeling of glioma growth has been developed to optimize treatments delivery and to evaluate their efficacy. Simulations currently make use of anatomical knowledge from standard MRI atlases. For example, cerebrospinal fluid (CSF) spaces are obtained by automatic thresholding of the MNI atlas, leading to an approximate representation of real anatomy. To correct such inaccuracies, an expert-revised CSF segmentation map of the MNI atlas was built. Several virtual glioma growth patterns of different locations were generated, with and without using the expert-revised version of the MNI atlas. The adequacy between virtual and radiologically observed growth patterns was clearly higher when simulations were based on the expert-revised atlas. This work emphasizes the need for close collaboration between clinicians and researchers in the field of brain tumor modeling.

Glioma follow white matter tracts: a multiscale DTI-based model

Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25-39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma.

A multilayer grow-or-go model for GBM: effects of invasive cells and anti-angiogenesis on growth

The recent use of anti-angiogenesis (AA) drugs for the treatment of glioblastoma multiforme (GBM) has uncovered unusual tumor responses. Here, we derive a new mathematical model that takes into account the ability of proliferative cells to become invasive under hypoxic conditions; model simulations generate the multilayer structure of GBM, namely proliferation, brain invasion, and necrosis. The model is able to replicate and justify the clinical observation of rebound growth when AA therapy is discontinued in some patients. The model is interrogated to derive fundamental insights int cancer biology and on the clinical and biological effects of AA drugs. Invasive cells promote tumor growth, which in the long run exceeds the effects of angiogenesis alone. Furthermore, AA drugs increase the fraction of invasive cells in the tumor, which explain progression by fluid-attenuated inversion recovery (FLAIR) signal and the rebound tumor growth when AA is discontinued.