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

Modelling non-local cell-cell adhesion: a multiscale approach

Cell-cell adhesion plays a vital role in the development and maintenance of multicellular organisms. One of its functions is regulation of cell migration, such as occurs, e.g. during embryogenesis or in cancer. In this work, we develop a versatile multiscale approach to modelling a moving self-adhesive cell population that combines a careful microscopic description of a deterministic adhesion-driven motion component with an efficient mesoscopic representation of a stochastic velocity-jump process. This approach gives rise to mesoscopic models in the form of kinetic transport equations featuring multiple non-localities. Subsequent parabolic and hyperbolic scalings produce general classes of equations with non-local adhesion and myopic diffusion, a special case being the classical macroscopic model proposed in Armstrong et al. (J Theoret Biol 243(1): 98-113, 2006). Our simulations show how the combination of the two motion effects can unfold. Cell-cell adhesion relies on the subcellular cell adhesion molecule binding. Our approach lends itself conveniently to capturing this microscopic effect. On the macroscale, this results in an additional non-linear integral equation of a novel type that is coupled to the cell density equation.

Quantification of perineural satellitosis in pretreatment glioblastoma with structural MRI and a diffusion tensor imaging template

Background

Survival outcomes for glioblastoma (GBM) patients remain unfavorable, and tumor recurrence is often observed. Understanding the radiological growth patterns of GBM could aid in improving outcomes. This study aimed to examine the relationship between contrast-enhancing tumor growth direction and white matter, using an image registration and deformation strategy.

Methods

In GBM patients 2 pretreatment scans (diagnostic and neuronavigation) were gathered retrospectively, and coregistered to a template and diffusion tensor imaging (DTI) atlas. The GBM lesions were segmented and coregistered to the same space. Growth vectors were derived and divided into vector populations parallel (Φ = 0-20°) and perpendicular (Φ = 70-90°) to white matter. To test for statistical significance between parallel and perpendicular groups, a paired samples Student's t-test was performed. O6-methylguanine-DNA methyltransferase (MGMT) methylation status and its correlation to growth rate were also tested using a one-way ANOVA test.

Results

For 78 GBM patients (mean age 61 years ± 13 SD, 32 men), the included GBM lesions showed a predominant preference for perineural satellitosis (P < .001), with a mean percentile growth of 30.8% (95% CI: 29.6-32.0%) parallel (0° < |Φ| < 20°) to white matter. Perpendicular tumor growth with respect to white matter microstructure (70° < |Φ| < 90°) showed to be 22.7% (95% CI: 21.3-24.1%) of total tumor growth direction.

Conclusions

The presented strategy showed that tumor growth direction in pretreatment GBM patients correlated with white matter architecture. Future studies with patient-specific DTI data are required to verify the accuracy of this method prospectively to identify its usefulness as a clinical metric in pre and posttreatment settings.

Modelling microtube driven invasion of glioma

Malignant gliomas are notoriously invasive, a major impediment against their successful treatment. This invasive growth has motivated the use of predictive partial differential equation models, formulated at varying levels of detail, and including (i) "proliferation-infiltration" models, (ii) "go-or-grow" models, and (iii) anisotropic diffusion models. Often, these models use macroscopic observations of a diffuse tumour interface to motivate a phenomenological description of invasion, rather than performing a detailed and mechanistic modelling of glioma cell invasion processes. Here we close this gap. Based on experiments that support an important role played by long cellular protrusions, termed tumour microtubes, we formulate a new model for microtube-driven glioma invasion. In particular, we model a population of tumour cells that extend tissue-infiltrating microtubes. Mitosis leads to new nuclei that migrate along the microtubes and settle elsewhere. A combination of steady state analysis and numerical simulation is employed to show that the model can predict an expanding tumour, with travelling wave solutions led by microtube dynamics. A sequence of scaling arguments allows us reduce the detailed model into simpler formulations, including models falling into each of the general classes (i), (ii), and (iii) above. This analysis allows us to clearly identify the assumptions under which these various models can be a posteriori justified in the context of microtube-driven glioma invasion. Numerical simulations are used to compare the various model classes and we discuss their advantages and disadvantages.

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.

Application of Magnetic Resonance DTI Technique in Evaluating the Effect of Postoperative Exercise Rehabilitation

Magnetic resonance diffusion tensor imaging (DTI) is a new kind of magnetic resonance imaging technology. Its imaging principle is to distinguish different pathological tissues according to the movement of water molecules, which is higher than regular magnetic resonance diffusion-weighted imaging. Magnetic resonance diffusion tensor imaging has exact utility price in medical analysis and sickness evaluation. However, there are few researches on the utility of diffusion tensor imaging in the rehabilitation comparison of patients. This paper explores the utility of magnetic resonance DTI science in evaluating the impact of postoperative patients' exercising rehabilitation. Taking stroke patients as an example, through giving patients rehabilitation training method, using magnetic resonance DTI technology, the motor function rehabilitation of patients was evaluated, and FA changes of the affected side and healthy side and Fugl–Meyer score of two groups of patients before and after rehabilitation were observed. The software outcomes exhibit that, in the contrast of rehabilitation therapy impact of motor feature in sufferers with cerebral infarction, the use of magnetic resonance DTI technological know-how gives a foundation for clinicians to deeply apprehend the CST involvement of patients, which helps to scientifically evaluate the effect and quality of limb motor rehabilitation training of patients and provides a basis for disease treatment.

Modelling glioma progression, mass effect and intracranial pressure in patient anatomy

Increased intracranial pressure is the source of most critical symptoms in patients with glioma, and often the main cause of death. Clinical interventions could benefit from non-invasive estimates of the pressure distribution in the patient's parenchyma provided by computational models. However, existing glioma models do not simulate the pressure distribution and they rely on a large number of model parameters, which complicates their calibration from available patient data. Here we present a novel model for glioma growth, pressure distribution and corresponding brain deformation. The distinct feature of our approach is that the pressure is directly derived from tumour dynamics and patient-specific anatomy, providing non-invasive insights into the patient's state. The model predictions allow estimation of critical conditions such as intracranial hypertension, brain midline shift or neurological and cognitive impairments. A diffuse-domain formalism is employed to allow for efficient numerical implementation of the model in the patient-specific brain anatomy. The model is tested on synthetic and clinical cases. To facilitate clinical deployment, a high-performance computing implementation of the model has been publicly released.

Multi-Cue Kinetic Model with Non-Local Sensing for Cell Migration on a Fiber Network with Chemotaxis

Cells perform directed motion in response to external stimuli that they detect by sensing the environment with their membrane protrusions. Precisely, several biochemical and biophysical cues give rise to tactic migration in the direction of their specific targets. Thus, this defines a multi-cue environment in which cells have to sort and combine different, and potentially competitive, stimuli. We propose a non-local kinetic model for cell migration in which cell polarization is influenced simultaneously by two external factors: contact guidance and chemotaxis. We propose two different sensing strategies, and we analyze the two resulting transport kinetic models by recovering the appropriate macroscopic limit in different regimes, in order to observe how the cell size, with respect to the variation of both external fields, influences the overall behavior. This analysis shows the importance of dealing with hyperbolic models, rather than drift-diffusion ones. Moreover, we numerically integrate the kinetic transport equations in a two-dimensional setting in order to investigate qualitatively various scenarios. Finally, we show how our setting is able to reproduce some experimental results concerning the influence of topographical and chemical cues in directing cell motility.

Is Diffusion Tensor Imaging-Guided Radiotherapy the New State-of-the-Art? A Review of the Current Literature and Technical Insights

Despite the increasing precision of radiotherapy delivery, it is still frequently associated with neurological complications. This is in part due to damage to eloquent white matter (WM) tracts, which is made more likely by the fact they cannot be visualised on standard structural imaging. WM is additionally more vulnerable than grey matter to radiation damage. Primary brain malignancies also are known to spread along the WM. Diffusion tensor imaging (DTI) is the only in vivo method of delineating WM tracts. DTI is an imaging technique that models the direction of diffusion and therefore can infer the orientation of WM fibres. This review article evaluates the current evidence for using DTI to guide intracranial radiotherapy and whether it constitutes a new state-of-the-art technique. We provide a basic overview of DTI and its known applications in radiotherapy, which include using tractography to reduce the radiation dose to eloquent WM tracts and using DTI to detect or predict tumoural spread. We evaluate the evidence for DTI-guided radiotherapy in gliomas, metastatic disease, and benign conditions, finding that the strongest evidence is for its use in arteriovenous malformations. However, the evidence is weak in other conditions due to a lack of case-controlled trials.

Mathematical modeling of therapeutic neural stem cell migration in mouse brain with and without brain tumors

Neural stem cells (NSCs) offer a potential solution to treating brain tumors. This is because NSCs can circumvent the blood-brain barrier and migrate to areas of damage in the central nervous system, including tumors, stroke, and wound injuries. However, for successful clinical application of NSC treatment, a sufficient number of viable cells must reach the diseased or damaged area(s) in the brain, and evidence suggests that it may be affected by the paths the NSCs take through the brain, as well as the locations of tumors. To study the NSC migration in brain, we develop a mathematical model of therapeutic NSC migration towards brain tumor, that provides a low cost platform to investigate NSC treatment efficacy. Our model is an extension of the model developed in Rockne et al. (PLoS ONE 13, e0199967, 2018) that considers NSC migration in non-tumor bearing naive mouse brain. Here we modify the model in Rockne et al. in three ways: (i) we consider three-dimensional mouse brain geometry, (ii) we add chemotaxis to model the tumor-tropic nature of NSCs into tumor sites, and (iii) we model stochasticity of migration speed and chemosensitivity. The proposed model is used to study migration patterns of NSCs to sites of tumors for different injection strategies, in particular, intranasal and intracerebral delivery. We observe that intracerebral injection results in more NSCs arriving at the tumor site(s), but the relative fraction of NSCs depends on the location of injection relative to the target site(s). On the other hand, intranasal injection results in fewer NSCs at the tumor site, but yields a more even distribution of NSCs within and around the target tumor site(s).

Correlation of anatomical involvement patterns of insular gliomas with subnetworks of the limbic system

OBJECTIVE

Gliomas frequently involve the insula both primarily and secondarily by invasion. Despite the high connectivity of the human insula, gliomas do not spread randomly to or from the insula but follow stereotypical anatomical involvement patterns. In the majority of cases, these patterns correspond to the intrinsic connectivity of the limbic system, except for tumors with aggressive biology. On the basis of these observations, the authors hypothesized that these different involvement patterns may be correlated with distinct outcomes and analyzed these correlations in an institutional cohort.

METHODS

Fifty-nine patients who had undergone surgery for insular diffuse gliomas and had complete demographic, pre- and postoperative imaging, pathology, molecular genetics, and clinical follow-up data were included in the analysis (median age 37 years, range 21-71 years, M/F ratio 1.68). Patients with gliomatosis and those with only minor involvement of the insula were excluded. The presence of T2-hyperintense tumor infiltration was evaluated in 12 anatomical structures. Hierarchical biclustering was used to identify co-involved structures, and the findings were correlated with established functional anatomy knowledge. Overall survival was evaluated using Kaplan-Meier and Cox proportional hazards regression analysis (17 parameters).

RESULTS

The tumors involved the anterior insula (98.3%), posterior insula (67.8%), temporal operculum (47.5%), amygdala (42.4%), frontal operculum (40.7%), temporal pole (39%), parolfactory area (35.6%), hypothalamus (23.7%), hippocampus (16.9%), thalamus (6.8%), striatum (5.1%), and cingulate gyrus (3.4%). A mean 4.2 ± 2.6 structures were involved. On the basis of hierarchical biclustering, 7 involvement patterns were identified and correlated with cortical functional anatomy (pure insular [11.9%], olfactocentric [15.3%], olfactoopercular [33.9%], operculoinsular [15.3%], striatoinsular [3.4%], translimbic [11.9%], and multifocal [8.5%] patterns). Cox regression identified hippocampal involvement (p = 0.006) and postoperative tumor volume (p = 0.027) as significant negative independent prognosticators of overall survival and extent of resection (p = 0.015) as a significant positive independent prognosticator.

CONCLUSIONS

The study findings indicate that insular gliomas primarily involve the olfactocentric limbic girdle and that involvement in the hippocampocentric limbic girdle is associated with a worse prognosis.

Cortical and Subcortical Anatomy of the Orbitofrontal Cortex: A White Matter Microfiberdissection Study and Case Series

BACKGROUND

The literature on white matter anatomy underlying the human orbitofrontal cortex (OFC) is scarce in spite of its relevance for glioma surgery.

OBJECTIVE

To describe the anatomy of the OFC and of the underlying white matter fiber anatomy, with a particular focus on the surgical structures relevant for a safe and efficient orbitofrontal glioma resection. Based on anatomical and radiological data, the secondary objective was to describe the growth pattern of OFC gliomas.

METHODS

The study was performed on 10 brain specimens prepared according to Klingler's protocol and dissected using the fiber microdissection technique modified according to U.T., under the microscope at high magnification.

RESULTS

A detailed stratigraphy of the OFC was performed, from the cortex up to the frontal horn of the lateral ventricle. The interposed neural structures are described together with relevant neighboring topographic areas and nuclei. Combining anatomical and radiological data, it appears that the anatomical boundaries delimiting and guiding the macroscopical growth of OFC gliomas are as follows: the corpus callosum superiorly, the external capsule laterally, the basal forebrain and lentiform nucleus posteriorly, and the gyrus rectus medially. Thus, OFC gliomas seem to grow ventriculopetally, avoiding the laterally located neocortex.

CONCLUSION

The findings in our study supplement available anatomical knowledge of the OFC, providing reliable landmarks for a precise topographical diagnosis of OFC lesions and for perioperative orientation. The relationships between deep anatomic structures and glioma formations described in this study are relevant for surgery in this highly interconnected area.

The link between gliomas infiltration and white matter architecture investigated with electron microscopy and diffusion tensor imaging

•

Possible favorable factors for glioma infiltration were investigated with MRI, TEM and DTI analysis.

•

The infiltration of white matter bundles (WMB) displayed regional differences in three gliomaś subgroups.

•

Regional differences within the same WMB were detected by morphological (TEM) and DTI analysis.

•

HIF regions, common to all gliomas subgroups, displayed a smaller fiber diameter, lower FA and higher RD.

•

Morphological features and diffusion parameters of the VMB may be linked to preferential locations of gliomas.

Diffuse low-grade gliomas (DLGG) display different preferential locations in eloquent and secondary associative brain areas. The reason for this tendency is still unknown. We hypothesized that the intrinsic architecture and water diffusion properties of the white matter bundles in these regions may facilitate gliomas infiltration. Magnetic resonance imaging of sixty-seven diffuse low-grade gliomas patients were normalized to/and segmented in MNI space to create three probabilistic infiltration weighted gradient maps according to the molecular status of each tumor group (IDH mutated, IDH wild-type and IDH mutated/1p19q co-deleted). Diffusion tensor imaging (DTI)- based parameters were derived for five major white matter bundles, displaying regional differences in the grade of infiltration, averaged over 20 healthy individuals acquired from the Human connectome project (HCP) database. Transmission electron microscopy (TEM) was used to analyze fiber density, fiber diameter and g-ratio in 100 human white matter regions, sampled from cadaver specimens, reflecting areas with different gliomas infiltration in each white matter bundle. Histological results and DTI-based parameters were compared in anatomical regions of high- and low grade of infiltration (HIF and LIF) respectively. We detected differences in the white matter infiltration of five major white matter bundles in three groups. Astrocytomas IDHm infiltrated left fronto-temporal subcortical areas. Astrocytomas IDHwt were detected in the posterior-temporal and temporo-parietal regions bilaterally. Oligodendrogliomas IDHm/1p19q infiltrated anterior subcortical regions of the frontal lobes bilaterally. Regional differences within the same white matter bundles were detected by both TEM- and DTI analysis linked to different topographical variables. Our multimodal analysis showed that HIF regions, common to all the groups, displayed a smaller fiber diameter, lower FA and higher RD compared with LIF regions. Our results suggest that the both morphological features and diffusion parameters of the white matter may be different in regions linked to the preferential location of DLGG.

Feasibility and clinical usefulness of modelling glioblastoma migration in adjuvant radiotherapy

Glioblastoma (GBM) is one of the most common primary brain tumours in adults, with a dismal prognosis despite aggressive multimodality treatment by a combination of surgery and adjuvant radiochemotherapy. A detailed knowledge of the spreading of glioma cells in the brain might allow for more targeted escalated radiotherapy, aiming to reduce locoregional relapse. Recent years have seen the development of a large variety of mathematical modelling approaches to predict glioma migration. The aim of this study is hence to evaluate the clinical applicability of a detailed micro- and meso-scale mathematical model in radiotherapy. First and foremost, a clinical workflow is established, in which the tumour is automatically segmented as input data and then followed in time mathematically based on the diffusion tensor imaging data. The influence of several free model parameters is individually evaluated, then the full model is retrospectively validated for a collective of 3 GBM patients treated at our institution by varying the most important model parameters to achieve optimum agreement with the tumour development during follow-up. Agreement of the model predictions with the real tumour growth as defined by manual contouring based on the follow-up MRI images is analyzed using the dice coefficient. The tumour evolution over 103-212 days follow-up could be predicted by the model with a dice coefficient better than 60% for all three patients. In all cases, the final tumour volume was overestimated by the model by a factor between 1.05 and 1.47. To evaluate the quality of the agreement between the model predictions and the ground truth, we must keep in mind that our gold standard relies on a single observer's (CB) manually-delineated tumour contours. We therefore decided to add a short validation of the stability and reliability of these contours by an inter-observer analysis including three other experienced radiation oncologists from our department. In total, a dice coefficient between 63% and 89% is achieved between the four different observers. Compared with this value, the model predictions (62-66%) perform reasonably well, given the fact that these tumour volumes were created based on the pre-operative segmentation and DTI.

Multiscale modeling of glioma pseudopalisades: contributions from the tumor microenvironment

Gliomas are primary brain tumors with a high invasive potential and infiltrative spread. Among them, glioblastoma multiforme (GBM) exhibits microvascular hyperplasia and pronounced necrosis triggered by hypoxia. Histological samples showing garland-like hypercellular structures (so-called pseudopalisades) centered around the occlusion site of a capillary are typical for GBM and hint on poor prognosis of patient survival. We propose a multiscale modeling approach in the kinetic theory of active particles framework and deduce by an upscaling process a reaction-diffusion model with repellent pH-taxis. We prove existence of a unique global bounded classical solution for a version of the obtained macroscopic system and investigate the asymptotic behavior of the solution. Moreover, we study two different types of scaling and compare the behavior of the obtained macroscopic PDEs by way of simulations. These show that patterns (not necessarily of Turing type), including pseudopalisades, can be formed for some parameter ranges, in accordance with the tumor grade. This is true when the PDEs are obtained via parabolic scaling (undirected tissue), while no such patterns are observed for the PDEs arising by a hyperbolic limit (directed tissue). This suggests that brain tissue might be undirected - at least as far as glioma migration is concerned. We also investigate two different ways of including cell level descriptions of response to hypoxia and the way they are related .

Estimating the extent of glioblastoma invasion : Approximate stationalization of anisotropic advection-diffusion-reaction equations in the context of glioblastoma invasion

Glioblastoma Multiforme is a malignant brain tumor with poor prognosis. There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection–diffusion–reaction equations. The patient-wise parametrization of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are mostly missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate. Therefore we propose a stationalized approach to estimate the extent of glioblastoma (GBM) invasion at the time of a given MRI/DTI scan. The underlying dynamics can be derived from an instationary GBM model, falling into the wide class of advection-diffusion-reaction equations. The stationalization is introduced via an analytic solution of the Fisher-KPP equation, the simplest model in the considered model class. We investigate the applicability in 1D and 2D, in the presence of inhomogeneous diffusion coefficients and on a real 3D DTI-dataset.

A Flux-Limited Model for Glioma Patterning with Hypoxia-Induced Angiogenesis

We propose a model for glioma patterns in a microlocal tumor environment under the influence of acidity, angiogenesis, and tissue anisotropy. The bottom-up model deduction eventually leads to a system of reaction–diffusion–taxis equations for glioma and endothelial cell population densities, of which the former infers flux limitation both in the self-diffusion and taxis terms. The model extends a recently introduced (Kumar, Li and Surulescu, 2020) description of glioma pseudopalisade formation with the aim of studying the effect of hypoxia-induced tumor vascularization on the establishment and maintenance of these histological patterns which are typical for high-grade brain cancer. Numerical simulations of the population level dynamics are performed to investigate several model scenarios containing this and further effects.

A Stochastic Modelling Framework for Single Cell Migration: Coupling Contractility and Focal Adhesions

The interaction of the actin cytoskeleton with cell–substrate adhesions is necessary for cell migration. While the trajectories of motile cells have a stochastic character, investigations of cell motility mechanisms rarely elaborate on the origins of the observed randomness. Here, guided by a few fundamental attributes of cell motility, I construct a minimal stochastic cell migration model from ground-up. The resulting model couples a deterministic actomyosin contractility mechanism with stochastic cell–substrate adhesion kinetics, and yields a well-defined piecewise deterministic process. Numerical simulations reproduce several experimentally observed results, including anomalous diffusion, tactic migration and contact guidance. This work provides a basis for the development of cell–cell collision and population migration models.

Computational models to explore the complexity of the epithelial to mesenchymal transition in cancer

Epithelial to mesenchymal transition (EMT) is a complex biological process that plays a key role in cancer progression and metastasis formation. Its activation results in epithelial cells losing adhesion and polarity and becoming capable of migrating from their site of origin. At this step the disease is generally considered incurable. As EMT execution involves several individual molecular components, connected by nontrivial relations, in vitro techniques are often inadequate to capture its complexity. Computational models can be used to complement experiments and provide additional knowledge difficult to build up in a wetlab. Indeed in silico analysis gives the user total control on the system, allowing to identify the contribution of each independent element. In the following, two kinds of approaches to the computational study of EMT will be presented. The first relies on signal transduction networks description and details how changes in gene expression could influence this process, both focusing on specific aspects of the EMT and providing a general frame for this phenomenon easily comparable with experimental data. The second integrates single cell and population level descriptions in a multiscale model that can be considered a more accurate representation of the EMT. The advantages and disadvantages of each approach will be highlighted, together with the importance of coupling computational and experimental results. Finally, the main challenges that need to be addressed to improve our knowledge of the role of EMT in the neoplastic disease and the scientific and translational value of computational models in this respect will be presented. This article is categorized under: Analytical and Computational Methods > Computational Methods.

The Brain Imaging Collaboration Suite (BrICS): A Cloud Platform for Integrating Whole-Brain Spectroscopic MRI into the Radiation Therapy Planning Workflow

Glioblastoma has poor prognosis with inevitable local recurrence despite aggressive treatment with surgery and chemoradiation. Radiation therapy (RT) is typically guided by contrast-enhanced T1-weighted magnetic resonance imaging (MRI) for defining the high-dose target and T2-weighted fluid-attenuation inversion recovery MRI for defining the moderate-dose target. There is an urgent need for improved imaging methods to better delineate tumors for focal RT. Spectroscopic MRI (sMRI) is a quantitative imaging technique that enables whole-brain analysis of endogenous metabolite levels, such as the ratio of choline-to-N-acetylaspartate. Previous work has shown that choline-to-N-acetylaspartate ratio accurately identifies tissue with high tumor burden beyond what is seen on standard imaging and can predict regions of metabolic abnormality that are at high risk for recurrence. To facilitate efficient clinical implementation of sMRI for RT planning, we developed the Brain Imaging Collaboration Suite (BrICS; https://brainimaging.emory.edu/brics-demo), a cloud platform that integrates sMRI with standard imaging and enables team members from multiple departments and institutions to work together in delineating RT targets. BrICS is being used in a multisite pilot study to assess feasibility and safety of dose-escalated RT based on metabolic abnormalities in patients with glioblastoma (Clinicaltrials.gov NCT03137888). The workflow of analyzing sMRI volumes and preparing RT plans is described. The pipeline achieved rapid turnaround time by enabling team members to perform their delegated tasks independently in BrICS when their clinical schedules allowed. To date, 18 patients have been treated using targets created in BrICS and no severe toxicities have been observed.

Multiscale dynamics of a heterotypic cancer cell population within a fibrous extracellular matrix

Local cancer cell invasion is a complex process involving many cellular and tissue interactions and is an important prerequisite for metastatic spread, the main cause of cancer related deaths. As a tumour increases in malignancy, the cancer cells adopt the ability to mutate into secondary cell subpopulations giving rise to a heterogeneous tumour. This new cell subpopulation often carries higher invasive abilities and permits a quicker spread of the tumour. Building upon the recent multiscale modelling framework for cancer invasion within a fibrous ECM introduced in Shuttleworth and Trucu, (2019), in this paper we consider the process of local invasion by a heterotypic tumour consisting of two cancer cell populations mixed with a two-phase ECM. To that end, we address the double feedback link between the tissue-scale cancer dynamics and the cell-scale molecular processes through the development of a two-part modelling framework that crucially incorporates the multiscale dynamic redistribution of oriented fibres occurring within a two-phase extra-cellular matrix and combines this with the multiscale leading edge dynamics exploring key matrix-degrading enzymes molecular processes along the tumour interface that drive the movement of the cancer boundary. The modelling framework will be accompanied by computational results that explore the effects of the underlying fibre network on the overall pattern of cancer invasion.

Kinetic models with non-local sensing determining cell polarization and speed according to independent cues

Cells move by run and tumble, a kind of dynamics in which the cell alternates runs over straight lines and re-orientations. This erratic motion may be influenced by external factors, like chemicals, nutrients, the extra-cellular matrix, in the sense that the cell measures the external field and elaborates the signal eventually adapting its dynamics. We propose a kinetic transport equation implementing a velocity-jump process in which the transition probability takes into account a double bias, which acts, respectively, on the choice of the direction of motion and of the speed. The double bias depends on two different non-local sensing cues coming from the external environment. We analyze how the size of the cell and the way of sensing the environment with respect to the variation of the external fields affect the cell population dynamics by recovering an appropriate macroscopic limit and directly integrating the kinetic transport equation. A comparison between the solutions of the transport equation and of the proper macroscopic limit is also performed.

Human organotypic brain slice culture: a novel framework for environmental research in neuro-oncology

When it comes to the human brain, models that closely mimic in vivo conditions are lacking. Living neuronal tissue is the closest representation of the in vivo human brain outside of a living person. Here, we present a method that can be used to maintain therapeutically resected healthy neuronal tissue for prolonged periods without any discernible changes in tissue vitality, evidenced by immunohistochemistry, genetic expression, and electrophysiology. This method was then used to assess glioblastoma (GBM) progression in its natural environment by microinjection of patient-derived tumor cells into cultured sections. The result closely resembles the pattern of de novo tumor growth and invasion, drug therapy response, and cytokine environment. Reactive transformation of astrocytes, as an example of the cellular nonmalignant tumor environment, can be accurately simulated with transcriptional differences similar to those of astrocytes isolated from acute GBM specimens. In a nutshell, we present a simple method to study GBM in its physiological environment, from which valuable insights can be gained. This technique can lead to further advancements in neuroscience, neuro-oncology, and pharmacotherapy.

A novel radiological classification system for cerebral gliomas: The Brain-Grid

Purpose

Standard radiological/topographical classifications of gliomas often do not reflect the real extension of the tumor within the lobar-cortical anatomy. Furthermore, these systems do not provide information on the relationship between tumor growth and the subcortical white matter architecture. We propose the use of an anatomically standardized grid system (the Brain-Grid) to merge serial morphological magnetic resonance imaging (MRI) scans with a representative tractographic atlas. Two illustrative cases are presented to show the potential advantages of this classification system.

Methods

MRI scans of 39 patients (WHO grade II and III gliomas) were analyzed with a standardized grid created by intersecting longitudinal lines on the axial, sagittal, and coronal planes. The anatomical landmarks were chosen from an average brain, spatially normalized to the Montreal Neurological Institute (MNI) space and the Talairach space. Major white matter pathways were reconstructed with a deterministic tracking algorithm on a reference atlas and analyzed using the Brain-Grid system.

Results

In all, 48 brain grid voxels (areas defined by 3 coordinates, axial (A), coronal (C), sagittal (S) and numbers from 1 to 4) were delineated in each MRI sequence and on the tractographic atlas. The number of grid voxels infiltrated was consistent, also in the MNI space. The sub-cortical insula/basal ganglia (A3-C2-S2) and the fronto-insular region (A3-C2-S1) were most frequently involved. The inferior fronto-occipital fasciculus, anterior thalamic radiation, uncinate fasciculus, and external capsule were the most frequently associated pathways in both hemispheres.

Conclusions

The Brain-Grid based classification system provides an accurate observational tool in all patients with suspected gliomas, based on the comparison of grid voxels on a morphological MRI and segmented white matter atlas. Important biological information on tumor kinetics including extension, speed, and preferential direction of progression can be observed and even predicted with this system. This novel classification can easily be applied to both prospective and retrospective cohorts of patients and increase our comprehension of glioma behavior.

The Direction of Tumour Growth in Glioblastoma Patients

Generating MR-derived growth pattern models for glioblastoma multiforme (GBM) has been an attractive approach in neuro-oncology, suggesting a distinct pattern of lesion spread with a tendency in growing along the white matter (WM) fibre direction for the invasive component. However, the direction of growth is not much studied in vivo. In this study, we sought to study the dominant directions of tumour expansion/shrinkage pre-treatment. We examined fifty-six GBMs at two time-points: at radiological diagnosis and as part of the pre-operative planning, both with contrast-enhanced T1-weighted MRIs. The tumour volumes were semi-automatically segmented. A non-linear registration resulting in a deformation field characterizing the changes between the two time points was used together with the segmented tumours to determine the dominant directions of tumour change. To compute the degree of alignment between tumour growth vectors and WM fibres, an angle map was calculated. Our results demonstrate that tumours tend to grow predominantly along the WM, as evidenced by the dominant vector population with the maximum alignments. Our findings represent a step forward in investigating the hypothesis that tumour cells tend to migrate preferentially along the WM.

Cell adhesion heterogeneity reinforces tumour cell dissemination: novel insights from a mathematical model

BACKGROUND

Cancer cell invasion, dissemination, and metastasis have been linked to an epithelial-mesenchymal transition (EMT) of individual tumour cells. During EMT, adhesion molecules like E-cadherin are downregulated and the decrease of cell-cell adhesion allows tumour cells to dissociate from the primary tumour mass. This complex process depends on intracellular cues that are subject to genetic and epigenetic variability, as well as extrinsic cues from the local environment resulting in a spatial heterogeneity in the adhesive phenotype of individual tumour cells. Here, we use a novel mathematical model to study how adhesion heterogeneity, influenced by intrinsic and extrinsic factors, affects the dissemination of tumour cells from an epithelial cell population. The model is a multiscale cellular automaton that couples intracellular adhesion receptor regulation with cell-cell adhesion.

RESULTS

Simulations of our mathematical model indicate profound effects of adhesion heterogeneity on tumour cell dissemination. In particular, we show that a large variation of intracellular adhesion receptor concentrations in a cell population reinforces cell dissemination, regardless of extrinsic cues mediated through the local cell density. However, additional control of adhesion receptor concentration through the local cell density, which can be assumed in healthy cells, weakens the effect. Furthermore, we provide evidence that adhesion heterogeneity can explain the remarkable differences in adhesion receptor concentrations of epithelial and mesenchymal phenotypes observed during EMT and might drive early dissemination of tumour cells.

CONCLUSIONS

Our results suggest that adhesion heterogeneity may be a universal trigger to reinforce cell dissemination in epithelial cell populations. This effect can be at least partially compensated by a control of adhesion receptor regulation through neighbouring cells. Accordingly, our findings explain how both an increase in intra-tumour adhesion heterogeneity and the loss of control through the local environment can promote tumour cell dissemination.

REVIEWERS

This article was reviewed by Hanspeter Herzel, Thomas Dandekar and Marek Kimmel.

Aggregation of biological particles under radial directional guidance

Many biological environments display an almost radially-symmetric structure, allowing proteins, cells or animals to move in an oriented fashion. Motivated by specific examples of cell movement in tissues, pigment protein movement in pigment cells and animal movement near watering holes, we consider a class of radially-symmetric anisotropic diffusion problems, which we call the star problem. The corresponding diffusion tensor D(x) is radially symmetric with isotropic diffusion at the origin. We show that the anisotropic geometry of the environment can lead to strong aggregations and blow-up at the origin. We classify the nature of aggregation and blow-up solutions and provide corresponding numerical simulations. A surprising element of this strong aggregation mechanism is that it is entirely based on geometry and does not derive from chemotaxis, adhesion or other well known aggregating mechanisms. We use these aggregate solutions to discuss the process of pigmentation changes in animals, cancer invasion in an oriented fibrous habitat (such as collagen fibres), and sheep distributions around watering holes.

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.

3D Mathematical Modeling of Glioblastoma Suggests That Transdifferentiated Vascular Endothelial Cells Mediate Resistance to Current Standard-of-Care Therapy

Glioblastoma (GBM), the most aggressive brain tumor in human patients, is decidedly heterogeneous and highly vascularized. Glioma stem/initiating cells (GSC) are found to play a crucial role by increasing cancer aggressiveness and promoting resistance to therapy. Recently, cross-talk between GSC and vascular endothelial cells has been shown to significantly promote GSC self-renewal and tumor progression. Furthermore, GSC also transdifferentiate into bona fide vascular endothelial cells (GEC), which inherit mutations present in GSC and are resistant to traditional antiangiogenic therapies. Here we use three-dimensional mathematical modeling to investigate GBM progression and response to therapy. The model predicted that GSCs drive invasive fingering and that GEC spontaneously form a network within the hypoxic core, consistent with published experimental findings. Standard-of-care treatments using DNA-targeted therapy (radiation/chemo) together with antiangiogenic therapies reduced GBM tumor size but increased invasiveness. Anti-GEC treatments blocked the GEC support of GSCs and reduced tumor size but led to increased invasiveness. Anti-GSC therapies that promote differentiation or disturb the stem cell niche effectively reduced tumor invasiveness and size, but were ultimately limited in reducing tumor size because GECs maintain GSCs. Our study suggests that a combinatorial regimen targeting the vasculature, GSCs, and GECs, using drugs already approved by the FDA, can reduce both tumor size and invasiveness and could lead to tumor eradication. Cancer Res; 77(15); 4171-84. ©2017 AACR.

Structured models of cell migration incorporating molecular binding processes

The dynamic interplay between collective cell movement and the various molecules involved in the accompanying cell signalling mechanisms plays a crucial role in many biological processes including normal tissue development and pathological scenarios such as wound healing and cancer. Information about the various structures embedded within these processes allows a detailed exploration of the binding of molecular species to cell-surface receptors within the evolving cell population. In this paper we establish a general spatio-temporal-structural framework that enables the description of molecular binding to cell membranes coupled with the cell population dynamics. We first provide a general theoretical description for this approach and then illustrate it with three examples arising from cancer invasion.

A multiscale model for glioma spread including cell-tissue interactions and proliferation

Glioma is a broad class of brain and spinal cord tumors arising from glia cells, which are the main brain cells that can develop into neoplasms. They are highly invasive and lead to irregular tumor margins which are not precisely identifiable by medical imaging, thus rendering a precise enough resection very difficult. The understanding of glioma spread patterns is hence essential for both radiological therapy as well as surgical treatment. In this paper we propose a multiscale model for glioma growth including interactions of the cells with the underlying tissue network, along with proliferative effects. Our current accounting for two subpopulations of cells to accomodate proliferation according to the go-or-grow dichtomoty is an extension of the setting in [16]. As in that paper, we assume that cancer cells use neuronal fiber tracts as invasive pathways. Hence, the individual structure of brain tissue seems to be decisive for the tumor spread. Diffusion tensor imaging (DTI) is able to provide such information, thus opening the way for patient specific modeling of glioma invasion. Starting from a multiscale model involving subcellular (microscopic) and individual (mesoscale) cell dynamics, we perform a parabolic scaling to obtain an approximating reaction-diffusion-transport equation on the macroscale of the tumor cell population. Numerical simulations based on DTI data are carried out in order to assess the performance of our modeling approach.

Towards the Personalized Treatment of Glioblastoma: Integrating Patient-Specific Clinical Data in a Continuous Mechanical Model

Glioblastoma multiforme (GBM) is the most aggressive and malignant among brain tumors. In addition to uncontrolled proliferation and genetic instability, GBM is characterized by a diffuse infiltration, developing long protrusions that penetrate deeply along the fibers of the white matter. These features, combined with the underestimation of the invading GBM area by available imaging techniques, make a definitive treatment of GBM particularly difficult. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of GBM evolution in every single patient throughout his/her oncological history, in order to target therapeutic weapons in a patient-specific manner. In this work, we propose a continuous mechanical model and we perform numerical simulations of GBM invasion combining the main mechano-biological characteristics of GBM with the micro-structural information extracted from radiological images, i.e. by elaborating patient-specific Diffusion Tensor Imaging (DTI) data. The numerical simulations highlight the influence of the different biological parameters on tumor progression and they demonstrate the fundamental importance of including anisotropic and heterogeneous patient-specific DTI data in order to obtain a more accurate prediction of GBM evolution. The results of the proposed mathematical model have the potential to provide a relevant benefit for clinicians involved in the treatment of this particularly aggressive disease and, more importantly, they might drive progress towards improving tumor control and patient’s prognosis.

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.