Simulated brain tumor growth dynamics using a three-dimensional cellular automaton

The self-organized structure of glioma oncostreams and the disruptive role of passive cells

Oncostreams are self-organized structures formed by spindle-like, elongated, self-propelled cells recently described in glioblastomas and especially in gliosarcomas. Cells within these structures either move as large clusters in one main direction, flocks, or as linear, intermingling collections of cells advancing in opposite directions, streams. Round, passive cells are also observed, either inside or segregated from the oncostreams. Here we generalize a recently formulated particle-field approach to investigate the genesis and evolution of these structures, first showing that, in systems consisting only of identical self-propelled cells, both flocks and streams emerge as self-organized dynamic configurations. Flocks are the more stable configurations, while streams are transient and usually originate in collisions between flocks. Stream degradation is easier at low self-propulsion speeds. In systems consisting of both motile and passive cells, the latter block stream formation and accelerate their degradation and flock stabilization. Since the flock appears to be the most effective invasive structure, we thus argue that a phenotype mixture (motile and passive cells) may favor glioblastoma invasion. hlBy relating cellular properties to the observed outcome, our model shows that oncostreams are self-organized structures that result from the interplay between speed, shape, and steric repulsion.

"False friends" in Language Subcortical Mapping: A Systematic Literature Review

BACKGROUND

Subcortical brain mapping in awake glioma surgery might optimize the extent of resection while minimizing neurological morbidity, but it requires a correct interpretation of responses evoked during surgery. To define, with a systematic review: 1) a comprehensive 'map' of the principal white matter bundles involved in awake surgery on language-related networks, describing the most employed tests and the expected responses; 2) In linguistics, a false friend is a word in a different language that looks or sounds like a word in given language but differs significantly in meaning. Similarly, our aim is to give the surgeons a comprehensive review of potentially misleading responses, namely "false friends", in subcortical language mapping.

METHODS

Preferred Reporting Items for Systematic Reviews and Meta-Analyses guidelines were followed. Standardized data extraction was conducted.

RESULTS

Out of a total of 224 initial papers, 67 were included for analysis. Expected responses, common tests, and potential "false friends" were recorded for each of the following white matter bundles: frontal aslant tract, superior and inferior longitudinal fascicles, arcuate fascicle, inferior fronto-occipital fascicle, uncinate fascicle. Practical examples are discussed to underline the risk of intraoperative fallouts ("false friends") that might lead to an early interruption (false positive) or a risky surgical removal (false negative).

CONCLUSIONS

This paper represents a critical review of the present status of subcortical awake mapping and underlines practical "false-friend" in mapping critical crossroads in language-related networks.

Calibration of agent based models for monophasic and biphasic tumour growth using approximate Bayesian computation

Agent-based models (ABMs) are readily used to capture the stochasticity in tumour evolution; however, these models are often challenging to validate with experimental measurements due to model complexity. The Voronoi cell-based model (VCBM) is an off-lattice agent-based model that captures individual cell shapes using a Voronoi tessellation and mimics the evolution of cancer cell proliferation and movement. Evidence suggests tumours can exhibit biphasic growth in vivo. To account for this phenomena, we extend the VCBM to capture the existence of two distinct growth phases. Prior work primarily focused on point estimation for the parameters without consideration of estimating uncertainty. In this paper, approximate Bayesian computation is employed to calibrate the model to in vivo measurements of breast, ovarian and pancreatic cancer. Our approach involves estimating the distribution of parameters that govern cancer cell proliferation and recovering outputs that match the experimental data. Our results show that the VCBM, and its biphasic extension, provides insight into tumour growth and quantifies uncertainty in the switching time between the two phases of the biphasic growth model. We find this approach enables precise estimates for the time taken for a daughter cell to become a mature cell. This allows us to propose future refinements to the model to improve accuracy, whilst also making conclusions about the differences in cancer cell characteristics.

A Bayesian method with nonlinear noise model to calibrate constitutive parameters of soft tissue

The measured mechanical responses of soft tissue exhibit large variability and errors, especially for the softest brain tissue, while calibrating its constitutive parameters in a deterministic way remains a common practice. Here we implement a Bayesian method considering the nonlinear noise model to calibrate constitutive parameters of brain tissue. A probability model is first developed based on the measured experimental data, likelihood function, and prior function, from which the posterior distributions of model parameters are formulated. The likelihood function considers the nonlinear behaviors of the constitutive response and noise distribution of the experimentally measured data. Meanwhile, the prior predictive distribution is computed to check the probability model preliminarily. Secondly, the Markov Chain Monte Carlo (MCMC) method is used to compute the posterior distributions of model parameters, enabling assessment of parameter uncertainty, correlation, and model calibration error. Finally, the posterior predictive distributions of the overall response, constitutive response, and noise response are computed to validate the probabilistic model, all of which are consistent with the corresponding data. Furthermore, the effect of the prior distribution, experimental data, and noise model on posterior distribution is studied. Our study provides a general approach to calibrating constitutive parameters of soft tissue despite errors and large variability in experimental data.

Development of a coupled modeling for tumor growth, angiogenesis, oxygen delivery, and phenotypic heterogeneity

Analysis of the evolution and growth dynamics of tumors is crucial for understanding cancer and the development of individually optimized therapies. During tumor growth, a hypoxic microenvironment around cancer cells caused by excessive non-vascular tumor growth induces tumor angiogenesis that plays a key role in the ensuing tumor growth and its progression into higher stages. Various mathematical simulation models have been introduced to simulate these biologically and physically complex hallmarks of cancer. Here, we developed a hybrid two-dimensional computational model that integrates spatiotemporally different components of the tumor system to investigate both angiogenesis and tumor growth/proliferation. This spatiotemporal evolution is based on partial diffusion equations, the cellular automation method, transition and probabilistic rules, and biological assumptions. The new vascular network provided by angiogenesis affects tumor microenvironmental conditions and drives individual cells to adapt themselves to spatiotemporal conditions. Furthermore, some stochastic rules are involved besides microenvironmental conditions. Overall, the conditions promote some commonly observed cellular states, i.e., proliferative, migrative, quiescent, and cell death, depending on the condition of each cell. Altogether, our results offer a theoretical basis for the biological evidence that regions of the tumor tissue near blood vessels are densely populated by proliferative phenotypic variants, while poorly oxygenated regions are sparsely populated by hypoxic phenotypic variants.

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.

Hybrid computational models of multicellular tumour growth considering glucose metabolism

Cancer cells metabolize glucose through metabolic pathways that differ from those used by healthy and differentiated cells. In particular, tumours have been shown to consume more glucose than their healthy counterparts and to use anaerobic metabolic pathways, even under aerobic conditions. Nevertheless, scientists have still not been able to explain why cancer cells evolved to present an altered metabolism and what evolutionary advantage this might provide them. Experimental and computational models have been increasingly used in recent years to understand some of these biological questions. Multicellular tumour spheroids are effective experimental models as they replicate the initial stages of avascular solid tumour growth. Furthermore, these experiments generate data which can be used to calibrate and validate computational studies that aim to simulate tumour growth. Hybrid models are of particular relevance in this field of research because they model cells as individual agents while also incorporating continuum representations of the substances present in the surrounding microenvironment that may participate in intracellular metabolic networks as concentration or density distributions. Henceforth, in this review, we explore the potential of computational modelling to reveal the role of metabolic reprogramming in tumour growth.

Examining the efficacy of localised gemcitabine therapy for the treatment of pancreatic cancer using a hybrid agent-based model

The prognosis for pancreatic ductal adenocarcinoma (PDAC) patients has not significantly improved in the past 3 decades, highlighting the need for more effective treatment approaches. Poor patient outcomes and lack of response to therapy can be attributed, in part, to a lack of uptake of perfusion of systemically administered chemotherapeutic drugs into the tumour. Wet-spun alginate fibres loaded with the chemotherapeutic agent gemcitabine have been developed as a potential tool for overcoming the barriers in delivery of systemically administrated drugs to the PDAC tumour microenvironment by delivering high concentrations of drug to the tumour directly over an extended period. While exciting, the practicality, safety, and effectiveness of these devices in a clinical setting requires further investigation. Furthermore, an in-depth assessment of the drug-release rate from these devices needs to be undertaken to determine whether an optimal release profile exists. Using a hybrid computational model (agent-based model and partial differential equation system), we developed a simulation of pancreatic tumour growth and response to treatment with gemcitabine loaded alginate fibres. The model was calibrated using in vitro and in vivo data and simulated using a finite volume method discretisation. We then used the model to compare different intratumoural implantation protocols and gemcitabine-release rates. In our model, the primary driver of pancreatic tumour growth was the rate of tumour cell division. We were able to demonstrate that intratumoural placement of gemcitabine loaded fibres was more effective than peritumoural placement. Additionally, we quantified the efficacy of different release profiles from the implanted fibres that have not yet been tested experimentally. Altogether, the model developed here is a tool that can be used to investigate other drug delivery devices to improve the arsenal of treatments available for PDAC and other difficult-to-treat cancers in the future.

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth.

A Review of Mathematical and Computational Methods in Cancer Dynamics

Cancers are complex adaptive diseases regulated by the nonlinear feedback systems between genetic instabilities, environmental signals, cellular protein flows, and gene regulatory networks. Understanding the cybernetics of cancer requires the integration of information dynamics across multidimensional spatiotemporal scales, including genetic, transcriptional, metabolic, proteomic, epigenetic, and multi-cellular networks. However, the time-series analysis of these complex networks remains vastly absent in cancer research. With longitudinal screening and time-series analysis of cellular dynamics, universally observed causal patterns pertaining to dynamical systems, may self-organize in the signaling or gene expression state-space of cancer triggering processes. A class of these patterns, strange attractors, may be mathematical biomarkers of cancer progression. The emergence of intracellular chaos and chaotic cell population dynamics remains a new paradigm in systems medicine. As such, chaotic and complex dynamics are discussed as mathematical hallmarks of cancer cell fate dynamics herein. Given the assumption that time-resolved single-cell datasets are made available, a survey of interdisciplinary tools and algorithms from complexity theory, are hereby reviewed to investigate critical phenomena and chaotic dynamics in cancer ecosystems. To conclude, the perspective cultivates an intuition for computational systems oncology in terms of nonlinear dynamics, information theory, inverse problems, and complexity. We highlight the limitations we see in the area of statistical machine learning but the opportunity at combining it with the symbolic computational power offered by the mathematical tools explored.

Tissue-Scale Biomechanical Testing of Brain Tissue for the Calibration of Nonlinear Material Models

Brain tissue is one of the most complex and softest tissues in the human body. Due to its ultrasoft and biphasic nature, it is difficult to control the deformation state during biomechanical testing and to quantify the highly nonlinear, time-dependent tissue response. In numerous experimental studies that have investigated the mechanical properties of brain tissue over the last decades, stiffness values have varied significantly. One reason for the observed discrepancies is the lack of standardized testing protocols and corresponding data analyses. The tissue properties have been tested on different length and time scales depending on the testing technique, and the corresponding data have been analyzed based on simplifying assumptions. In this review, we highlight the advantage of using nonlinear continuum mechanics based modeling and finite element simulations to carefully design experimental setups and protocols as well as to comprehensively analyze the corresponding experimental data. We review testing techniques and protocols that have been used to calibrate material model parameters and discuss artifacts that might falsify the measured properties. The aim of this work is to provide standardized procedures to reliably quantify the mechanical properties of brain tissue and to more accurately calibrate appropriate constitutive models for computational simulations of brain development, injury and disease. Computational models can not only be used to predictively understand brain tissue behavior, but can also serve as valuable tools to assist diagnosis and treatment of diseases or to plan neurosurgical procedures. © 2022 The Authors. Current Protocols published by Wiley Periodicals LLC.

Potential Role of Natural Products to Combat Radiotherapy and Their Future Perspectives

Cancer is the second leading cause of death in the world. Chemotherapy and radiotherapy (RT) are the common cancer treatments. In addition to these limitations, the development of adverse effects from chemotherapy and RT reduces the quality of life for cancer patients. Cellular radiosensitivity, or the ability to resist and overcome cell damage caused by ionizing radiation (IR), is directly related to cancer cells' response to RT. Therefore, radiobiological research is emphasizing chemical compounds 'radiosensitization of cancer cells so that they are more reactive in the IR spectrum. Recent years researchers have seen an increase in interest in natural products that have antitumor effects with minimal side effects. Natural products, on the other hand, are easy to recover and therefore less expensive. There have been several scientific studies done based on these compounds that have tested their ability in vitro and in vivo to induce tumor radiosensitization. The role of natural products in RT, as well as their usefulness and potential applications, is the goal of this current review.

Musiré: multimodal simulation and reconstruction framework for the radiological imaging sciences

A software-based workflow is proposed for managing the execution of simulation and image reconstruction for SPECT, PET, CBCT, MRI, BLI and FMI packages in single and multimodal biomedical imaging applications. The workflow is composed of a Bash script, the purpose of which is to provide an interface to the user, and to organize data flow between dedicated programs for simulation and reconstruction. The currently incorporated simulation programs comprise GATE for Monte Carlo simulation of SPECT, PET and CBCT, SpinScenario for simulating MRI, and Lipros for Monte Carlo simulation of BLI and FMI. Currently incorporated image reconstruction programs include CASToR for SPECT and PET as well as RTK for CBCT. MetaImage (mhd) standard is used for voxelized phantom and image data format. Meshlab project (mlp) containers incorporating polygon meshes and point clouds defined by the Stanford triangle format (ply) are employed to represent anatomical structures for optical simulation, and to represent tumour cell inserts. A number of auxiliary programs have been developed for data transformation and adaptive parameter assignment. The software workflow uses fully automatic distribution to, and consolidation from, any number of Linux workstations and CPU cores. Example data are presented for clinical SPECT, PET and MRI systems using the Mida head phantom and for preclinical X-ray, PET and BLI systems employing the Digimouse phantom. The presented method unifies and simplifies multimodal simulation setup and image reconstruction management and might be of value for synergistic image research. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 2'.

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

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.

BIO-LGCA: A cellular automaton modelling class for analysing collective cell migration

Collective dynamics in multicellular systems such as biological organs and tissues plays a key role in biological development, regeneration, and pathological conditions. Collective tissue dynamics—understood as population behaviour arising from the interplay of the constituting discrete cells—can be studied with on- and off-lattice agent-based models. However, classical on-lattice agent-based models, also known as cellular automata, fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. To overcome drawbacks of classical on-lattice models, we introduce an on-lattice, agent-based modelling class for collective cell migration, which we call biological lattice-gas cellular automaton (BIO-LGCA). The BIO-LGCA is characterised by synchronous time updates, and the explicit consideration of individual cell velocities. While rules in classical cellular automata are typically chosen ad hoc, rules for cell-cell and cell-environment interactions in the BIO-LGCA can also be derived from experimental cell migration data or biophysical laws for individual cell migration. We introduce elementary BIO-LGCA models of fundamental cell interactions, which may be combined in a modular fashion to model complex multicellular phenomena. We exemplify the mathematical mean-field analysis of specific BIO-LGCA models, which allows to explain collective behaviour. The first example predicts the formation of clusters in adhesively interacting cells. The second example is based on a novel BIO-LGCA combining adhesive interactions and alignment. For this model, our analysis clarifies the nature of the recently discovered invasion plasticity of breast cancer cells in heterogeneous environments.

Pattern formation during embryonic development and pathological tissue dynamics, such as cancer invasion, emerge from individual intercellular interactions. In order to study the impact of single cell dynamics and cell-cell interactions on tissue behaviour, one needs to develop space-time-dependent on- or off-lattice agent-based models (ABMs), which consider the behaviour of individual cells. However, classical on-lattice agent-based models also known as cellular automata fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. Here, we present the rule- and lattice-based BIO-LGCA modelling class which allows for (i) rigorous derivation of rules from biophysical laws and/or experimental data, (ii) mathematical analysis of collective migration, and (iii) computationally efficient simulations.

QLCA and Entangled States as Single-Neuron Activity Generators

Each neuron in the central nervous system has many dendrites, which provide input information through impulses. Assuming that a neuron's decision to continue or stop firing is made by rules applied to the dendrites' inputs, we associate neuron activity with a quantum like-cellular automaton (QLCA) concepts. Following a previous study that related the CA description with entangled states, we provide a quantum-like description of neuron activity. After reviewing and presenting the entanglement concept expressed by QLCA terminology, we propose a model that relates quantum-like measurement to consciousness. Then, we present a toy model that reviews the QLCA theory, which is adapted to our terminology. The study also focuses on implementing QLCA formalism to describe a single neuron activity.

In Silico Mathematical Modelling for Glioblastoma: A Critical Review and a Patient-Specific Case

Glioblastoma extensively infiltrates the brain; despite surgery and aggressive therapies, the prognosis is poor. A multidisciplinary approach combining mathematical, clinical and radiological data has the potential to foster our understanding of glioblastoma evolution in every single patient, with the aim of tailoring therapeutic weapons. In particular, the ultimate goal of biomathematics for cancer is the identification of the most suitable theoretical models and simulation tools, both to describe the biological complexity of carcinogenesis and to predict tumor evolution. In this report, we describe the results of a critical review about different mathematical models in neuro-oncology with their clinical implications. A comprehensive literature search and review for English-language articles concerning mathematical modelling in glioblastoma has been conducted. The review explored the different proposed models, classifying them and indicating the significative advances of each one. Furthermore, we present a specific case of a glioblastoma patient in which our recently proposed innovative mechanical model has been applied. The results of the mathematical models have the potential to provide a relevant benefit for clinicians and, more importantly, they might drive progress towards improving tumor control and patient's prognosis. Further prospective comparative trials, however, are still necessary to prove the impact of mathematical neuro-oncology in clinical practice.

A New Mathematical Model for Controlling Tumor Growth Based on Microenvironment Acidity and Oxygen Concentration

Hypoxia and the pH level of the tumor microenvironment have a great impact on the treatment of tumors. Here, the tumor growth is controlled by regulating the oxygen concentration and the acidity of the tumor microenvironment by introducing a two-dimensional multiscale cellular automata model of avascular tumor growth. The spatiotemporal evolution of tumor growth and metabolic variations is modeled based on biological assumptions, physical structure, states of cells, and transition rules. Each cell is allocated to one of the following states: proliferating cancer, nonproliferating cancer, necrotic, and normal cells. According to the response of the microenvironmental conditions, each cell consumes/produces metabolic factors and updates its state based on some stochastic rules. The input parameters are compatible with cancer biology using experimental data. The effect of neighborhoods during mitosis and simulating spatial heterogeneity is studied by considering multicellular layer structure of tumor. A simple Darwinist mutation is considered by introducing a critical parameter (Nmm) that affects division probability of the proliferative tumor cells based on the microenvironmental conditions and cancer hallmarks. The results show that Nmm regulation has a significant influence on the dynamics of tumor growth, the growth fraction, necrotic fraction, and the concentration levels of the metabolic factors. The model not only is able to simulate the in vivo tumor growth quantitatively and qualitatively but also can simulate the concentration of metabolic factors, oxygen, and acidity graphically. The results show the spatial heterogeneity effects on the proliferation of cancer cells and the rest of the system. By increasing Nmm, tumor shrinkage and significant increasing in the oxygen concentration and the pH value of the tumor microenvironment are observed. The results demonstrate the model's ability, providing an essential tool for simulating different tumor evolution scenarios of a patient and reliable prediction of spatiotemporal progression of tumors for utilizing in personalized therapy.

An Integrative and Modular Framework to Recapitulate Emergent Behavior in Cell Migration

Cell migration has been a subject of study in a broad variety of biological systems, from morphogenetic events during development to cancer progression. In this work, we describe single-cell movement in a modular framework from which we simulate the collective behavior of glioblastoma cells, the most prevalent and malignant primary brain tumor. We used the U87 cell line, which can be grown as a monolayer or spatially closely packed and organized in 3D structures called spheroids. Our integrative model considers the most relevant mechanisms involved in cell migration: chemotaxis of attractant factor, mechanical interactions and random movement. The effect of each mechanism is integrated into the overall probability of the cells to move in a particular direction, in an automaton-like approach. Our simulations fit and reproduced the emergent behavior of the spheroids in a set of migration assays where single-cell trajectories were tracked. We also predicted the effect of migration inhibition on the colonies from simple experimental characterization of single treated cell tracks. The development of tools that allow complementing molecular knowledge in migratory cell behavior is relevant for understanding essential cellular processes, both physiological (such as organ formation, tissue regeneration among others) and pathological perspectives. Overall, this is a versatile tool that has been proven to predict individual and collective behavior in U87 cells, but that can be applied to a broad variety of scenarios.

Dynamic mechanical characterization and viscoelastic modeling of bovine brain tissue

Brain tissue is vulnerable and sensitive, predisposed to potential damage under various conditions of mechanical loading. Although its material properties have been investigated extensively, the frequency-dependent viscoelastic characterization is currently limited. Computational models can provide a non-invasive method by which to analyze brain injuries and predict the mechanical response of the tissue. The brain injuries are expected to be induced by dynamic loading, mostly in compression and measurement of dynamic viscoelastic properties are essential to improve the accuracy and variety of finite element simulations on brain tissue. Thus, the aim of this study was to investigate the compressive frequency-dependent properties of brain tissue and present a mathematical model in the frequency domain to capture the tissue behavior based on experimental results. Bovine brain specimens, obtained from four locations of corona radiata, corpus callosum, basal ganglia and cortex, were tested under compression using dynamic mechanical analysis over a range of frequencies between 0.5 and 35 Hz to characterize the regional and directional response of the tissue. The compressive dynamic properties of bovine brain tissue were heterogenous for regions but not sensitive to orientation showing frequency dependent statistical results, with viscoelastic properties increasing with frequency. The mean storage and loss modulus were found to be 12.41 kPa and 5.54 kPa, respectively. The material parameters were obtained using the linear viscoelastic model in the frequency domain and the numeric simulation can capture the compressive mechanical behavior of bovine brain tissue across a range of frequencies. The frequency-dependent viscoelastic characterization of brain tissue will improve the fidelity of the computational models of the head and provide essential information to the prediction and analysis of brain injuries in clinical treatments.

A System Biology-Based Approach for Designing Combination Therapy in Cancer Precision Medicine

In this paper, we have used an agent-based stochastic tumor growth model and presented a mathematical and theoretical perspective to cancer therapy. This perspective can be used to theoretical study of precision medicine and combination therapy in individuals. We have conducted a series of in silico combination therapy experiments. Based on cancer drugs and new findings of cancer biology, we hypothesize relationships between model parameters which in some cases represent individual genome characteristics and cancer drugs, i.e., in our approach, therapy players are delegated by biologically reasonable parameters. In silico experiments showed that combined therapies are more effective when players affect tumor via different mechanisms and have different physical dimensions. This research presents for the first time an algorithm as a theoretical viewpoint for the prediction of effectiveness and classification of therapy sets.

Unification of aggregate growth models by emergence from cellular and intracellular mechanisms

Multicellular aggregate growth is regulated by nutrient availability and removal of metabolites, but the specifics of growth dynamics are dependent on cell type and environment. Classical models of growth are based on differential equations. While in some cases these classical models match experimental observations, they can only predict growth of a limited number of cell types and so can only be selectively applied. Currently, no classical model provides a general mathematical representation of growth for any cell type and environment. This discrepancy limits their range of applications, which a general modelling framework can enhance. In this work, a hybrid cellular Potts model is used to explain the discrepancy between classical models as emergent behaviours from the same mathematical system. Intracellular processes are described using probability distributions of local chemical conditions for proliferation and death and simulated. By fitting simulation results to a generalization of the classical models, their emergence is demonstrated. Parameter variations elucidate how aggregate growth may behave like one classical growth model or another. Three classical growth model fits were tested, and emergence of the Gompertz equation was demonstrated. Effects of shape changes are demonstrated, which are significant for final aggregate size and growth rate, and occur stochastically.

Mathematical modeling of PDGF-driven glioma reveals the dynamics of immune cells infiltrating into tumors

Background: Tumor-infiltrated immune cells compose a significant component of many cancers. They have been observed to have contradictory impacts on tumors. Although the primary reasons for these observations remain elusive, it is important to understand how immune cells infiltrating into tumors is regulated. Recently our group conducted a series of experimental studies, which showed that muIDH1 gliomas have a significant global reduction of immune cells and suggested that the longer survival time of mice with CIMP gliomas may be due to the IDH mutation and its effect on reducing of the tumor-infiltrated immune cells. However, to comprehend how IDH1 mutants regulate infiltration of immune cells into gliomas and how they affect the aggressiveness of gliomas, it is necessary to integrate our experimental data into a dynamical system to acquire a much deeper understanding of subtle regulation of immune cell infiltration. Methods: The method is integration of mathematical modeling and experiments. According to mass conservation laws and assumption that immune cells migrate into the tumor site along a chemotactic gradient field, a mathematical model is formulated. Parameters are estimated from our experiments. Numerical methods are developed to solve the problem. Numerical predictions are compared with experimental results. Results: Our analysis shows that the net rate of increase of immune cells infiltrated into the tumor is approximately proportional to the 4/5 power of the chemoattractant production rate, and it is an increasing function of time while the percentage of immune cells infiltrated into the tumor is a decreasing function of time. Our model predicts that wtIDH1 mice will survive longer if the immune cells are blocked by reducing chemotactic coefficient. For more aggressive gliomas, our model shows that there is little difference in their survivals between wtIDH1 and muIDH1 tumors, and the percentage of immune cells infiltrated into the tumor is much lower. These predictions are verified by our experimental results. In addition, wtIDH1 and muIDH1 can be quantitatively distinguished by their chemoattractant production rates, and the chemotactic coefficient determines possibilities of immune cells migration along chemoattractant gradient fields. Conclusions: The chemoattractant gradient field produced by tumor cells may facilitate immune cells migration to the tumor cite. The chemoattractant production rate may be utilized to classify wtIDH1 and muIDH1 tumors. The dynamics of immune cells infiltrating into tumors is largely determined by tumor cell chemoattractant production rate and chemotactic coefficient.

A minimal modeling framework of radiation and immune system synergy to assist radiotherapy planning

Recent evidence indicates the ability of radiotherapy to induce local and systemic tumor-specific immune responses as a result of immunogenic cell death. However, fractionation regimes routinely used in clinical practice typically ignore the synergy between radiation and the immune system, and instead attempt to completely eradicate tumors by the direct lethal effect of radiation on cancer cells. This paradigm is expected to change in the near future due to the potential benefits of considering radiation-induced antitumor immunity during treatment planning. Towards this goal, we propose a minimal modeling framework based on key aspects of the tumor-immune system interplay to simulate the effects of radiation on tumors and the immunological consequences of radiotherapy. The impacts of tumor-associated vasculature and intratumoral oxygen-mediated heterogeneity on treatment outcomes are ininvestigated. The model provides estimates of the minimum radiation doses required for tumor eradication given a certain number of treatment fractions. Moreover, estimates of treatment duration for disease control given predetermined fractional radiation doses can be also obtained. Although theoretical in nature, this study motivates the development and establishment of immune-based decision-support tools in radiotherapy planning.

Uncertainty quantification on a spatial Markov-chain model for the progression of skin cancer

A spatial Markov-chain model is formulated for the progression of skin cancer. The model is based on the division of the computational domain into nodal points, that can be in a binary state: either in 'cancer state' or in 'non-cancer state'. The model assigns probabilities for the non-reversible transition from 'non-cancer' state to the 'cancer state' that depend on the states of the neighbouring nodes. The likelihood of transition further depends on the life burden intensity of the UV-rays that the skin is exposed to. The probabilistic nature of the process and the uncertainty in the input data is assessed by the use of Monte Carlo simulations. A good fit between experiments on mice and our model has been obtained.

Mathematical Models of Cancer Cell Plasticity

Quantitative modelling is increasingly important in cancer research, helping to integrate myriad diverse experimental data into coherent pictures of the disease and able to discriminate between competing hypotheses or suggest specific experimental lines of enquiry and new approaches to therapy. Here, we review a diverse set of mathematical models of cancer cell plasticity (a process in which, through genetic and epigenetic changes, cancer cells survive in hostile environments and migrate to more favourable environments, respectively), tumour growth, and invasion. Quantitative models can help to elucidate the complex biological mechanisms of cancer cell plasticity. In this review, we discuss models of plasticity, tumour progression, and metastasis under three broadly conceived mathematical modelling techniques: discrete, continuum, and hybrid, each with advantages and disadvantages. An emerging theme from the predictions of many of these models is that cell escape from the tumour microenvironment (TME) is encouraged by a combination of physiological stress locally (e.g., hypoxia), external stresses (e.g., the presence of immune cells), and interactions with the extracellular matrix. We also discuss the value of mathematical modelling for understanding cancer more generally.

Enhancing oncolytic virotherapy: Observations from a Voronoi Cell-Based model

Oncolytic virotherapy is a promising cancer treatment using genetically modified viruses. Unfortunately, virus particles rapidly decay inside the body, significantly hindering their efficacy. In this article, treatment perturbations that could overcome obstacles to oncolytic virotherapy are investigated through the development of a Voronoi Cell-Based model (VCBM). The VCBM derived captures the interaction between an oncolytic virus and cancer cells in a 2-dimensional setting by using an agent-based model, where cell edges are designated by a Voronoi tessellation. Here, we investigate the sensitivity of treatment efficacy to the configuration of the treatment injections for different tumour shapes: circular, rectangular and irregular. The model predicts that multiple off-centre injections improve treatment efficacy irrespective of tumour shape. Additionally, we investigate delaying the infection of cancer cells by modifying viral particles with a substance such as alginate (a hydrogel polymer used in a range of cancer treatments). Simulations of the VCBM show that delaying the infection of cancer cells, and thus allowing more time for virus dissemination, can improve the efficacy of oncolytic virotherapy. The simulated treatment noticeably decreases the tumour size with no increase in toxicity. Improving oncolytic virotherapy in this way allows for a more effective treatment without changing its fundamental essence.

Agent-based model of multicellular tumor spheroid evolution including cell metabolism

Computational models aiming at the spatio-temporal description of cancer evolution are a suitable framework for testing biological hypotheses from experimental data, and generating new ones. Building on our recent work (J. Theor. Biol. 389, 146 (2016)) we develop a 3D agent-based model, capable of tracking hundreds of thousands of interacting cells, over time scales ranging from seconds to years. Cell dynamics is driven by a Monte Carlo solver, incorporating partial differential equations to describe chemical pathways and the activation/repression of "genes", leading to the up- or down-regulation of specific cell markers. Each cell-agent of different kind (stem, cancer, stromal etc.) runs through its cycle, undergoes division, can exit to a dormant, senescent, necrotic state, or apoptosis, according to the inputs from its systemic network. The basic network at this stage describes glucose/oxygen/ATP cycling, and can be readily extended to cancer-cell specific markers. Eventual accumulation of chemical/radiation damage to each cell's DNA is described by a Markov chain of internal states, and by a damage-repair network, whose evolution is linked to the cell systemic network. Aimed at a direct comparison with experiments of tumorsphere growth from stem cells, the present model will allow to quantitatively study the role of transcription factors involved in the reprogramming and variable radio-resistance of simulated cancer-stem cells, evolving in a realistic computer simulation of a growing multicellular tumorsphere.

Potential of Zerumbone as an Anti-Cancer Agent

Cancer is still a major risk factor to public health globally, causing approximately 9.8 million deaths worldwide in 2018. Despite advances in conventional treatment modalities for cancer treatment, there are still few effective therapies available due to the lack of selectivity, adverse side effects, non-specific toxicities, and tumour recurrence. Therefore, there is an immediate need for essential alternative therapeutics, which can prove to be beneficial and safe against cancer. Various phytochemicals from natural sources have been found to exhibit beneficial medicinal properties against various human diseases. Zerumbone is one such compound isolated from Zingiber zerumbet Smith that possesses diverse pharmacological properties including those of antioxidant, antibacterial, antipyretic, anti-inflammatory, immunomodulatory, as well as anti-neoplastic. Zerumbone has shown its anti-cancer effects by causing significant suppression of proliferation, survival, angiogenesis, invasion, and metastasis through the molecular modulation of different pathways such as NF-κB, Akt, and IL-6/JAK2/STAT3 (interleukin-6/janus kinase-2/signal transducer and activator of transcription 3) and their downstream target proteins. The current review briefly summarizes the modes of action and therapeutic potential of zerumbone against various cancers.

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.

Mathematical models of tumor cell proliferation: A review of the literature

INTRODUCTION

A defining hallmark of cancer is aberrant cell proliferation. Efforts to understand the generative properties of cancer cells span all biological scales: from genetic deviations and alterations of metabolic pathways to physical stresses due to overcrowding, as well as the effects of therapeutics and the immune system. While these factors have long been studied in the laboratory, mathematical and computational techniques are being increasingly applied to help understand and forecast tumor growth and treatment response. Advantages of mathematical modeling of proliferation include the ability to simulate and predict the spatiotemporal development of tumors across multiple experimental scales. Central to proliferation modeling is the incorporation of available biological data and validation with experimental data. Areas covered: We present an overview of past and current mathematical strategies directed at understanding tumor cell proliferation. We identify areas for mathematical development as motivated by available experimental and clinical evidence, with a particular emphasis on emerging, non-invasive imaging technologies. Expert commentary: The data required to legitimize mathematical models are often difficult or (currently) impossible to obtain. We suggest areas for further investigation to establish mathematical models that more effectively utilize available data to make informed predictions on tumor cell proliferation.

Modeling three-dimensional invasive solid tumor growth in heterogeneous microenvironment under chemotherapy

A systematic understanding of the evolution and growth dynamics of invasive solid tumors in response to different chemotherapy strategies is crucial for the development of individually optimized oncotherapy. Here, we develop a hybrid three-dimensional (3D) computational model that integrates pharmacokinetic model, continuum diffusion-reaction model and discrete cell automaton model to investigate 3D invasive solid tumor growth in heterogeneous microenvironment under chemotherapy. Specifically, we consider the effects of heterogeneous environment on drug diffusion, tumor growth, invasion and the drug-tumor interaction on individual cell level. We employ the hybrid model to investigate the evolution and growth dynamics of avascular invasive solid tumors under different chemotherapy strategies. Our simulations indicate that constant dosing is generally more effective in suppressing primary tumor growth than periodic dosing, due to the resulting continuous high drug concentration. In highly heterogeneous microenvironment, the malignancy of the tumor is significantly enhanced, leading to inefficiency of chemotherapies. The effects of geometrically-confined microenvironment and non-uniform drug dosing are also investigated. Our computational model, when supplemented with sufficient clinical data, could eventually lead to the development of efficient in silico tools for prognosis and treatment strategy optimization.

Scalable population-level modelling of biological cells incorporating mechanics and kinetics in continuous time

The processes taking place inside the living cell are now understood to the point where predictive computational models can be used to gain detailed understanding of important biological phenomena. A key challenge is to extrapolate this detailed knowledge of the individual cell to be able to explain at the population level how cells interact and respond with each other and their environment. In particular, the goal is to understand how organisms develop, maintain and repair functional tissues and organs. In this paper, we propose a novel computational framework for modelling populations of interacting cells. Our framework incorporates mechanistic, constitutive descriptions of biomechanical properties of the cell population, and uses a coarse-graining approach to derive individual rate laws that enable propagation of the population through time. Thanks to its multiscale nature, the resulting simulation algorithm is extremely scalable and highly efficient. As highlighted in our computational examples, the framework is also very flexible and may straightforwardly be coupled with continuous-time descriptions of biochemical signalling within, and between, individual cells.

Physiological random processes in precision cancer therapy

Many different physiological processes affect the growth of malignant lesions and their response to therapy. Each of these processes is spatially and genetically heterogeneous; dynamically evolving in time; controlled by many other physiological processes, and intrinsically random and unpredictable. The objective of this paper is to show that all of these properties of cancer physiology can be treated in a unified, mathematically rigorous way via the theory of random processes. We treat each physiological process as a random function of position and time within a tumor, defining the joint statistics of such functions via the infinite-dimensional characteristic functional. The theory is illustrated by analyzing several models of drug delivery and response of a tumor to therapy. To apply the methodology to precision cancer therapy, we use maximum-likelihood estimation with Emission Computed Tomography (ECT) data to estimate unknown patient-specific physiological parameters, ultimately demonstrating how to predict the probability of tumor control for an individual patient undergoing a proposed therapeutic regimen.

Modeling small cell lung cancer (SCLC) biology through deterministic and stochastic mathematical models

Mathematical cancer models are immensely powerful tools that are based in part on the fractal nature of biological structures, such as the geometry of the lung. Cancers of the lung provide an opportune model to develop and apply algorithms that capture changes and disease phenotypes. We reviewed mathematical models that have been developed for biological sciences and applied them in the context of small cell lung cancer (SCLC) growth, mutational heterogeneity, and mechanisms of metastasis. The ultimate goal is to develop the stochastic and deterministic nature of this disease, to link this comprehensive set of tools back to its fractalness and to provide a platform for accurate biomarker development. These techniques may be particularly useful in the context of drug development research, such as combination with existing omics approaches. The integration of these tools will be important to further understand the biology of SCLC and ultimately develop novel therapeutics.

Creation of Three-Dimensional Liver Tissue Models from Experimental Images for Systems Medicine

In this chapter, we illustrate how three-dimensional liver tissue models can be created from experimental image modalities by utilizing a well-established processing chain of experiments, microscopic imaging, image processing, image analysis and model construction. We describe how key features of liver tissue architecture are quantified and translated into model parameterizations, and show how a systematic iteration of experiments and model simulations often leads to a better understanding of biological phenomena in systems biology and systems medicine.

A model of the effects of cancer cell motility and cellular adhesion properties on tumour-immune dynamics

We present a three-dimensional model simulating the dynamics of an anti-cancer T-cell response against a small, avascular, early-stage tumour. Interactions at the tumour site are accounted for using an agent-based model (ABM), while immune cell dynamics in the lymph node are modelled as a system of delay differential equations (DDEs). We combine these separate approaches into a two-compartment hybrid ABM-DDE system to capture the T-cell response against the tumour. In the ABM at the tumour site, movement of tumour cells is modelled using effective physical forces with a specific focus on cell-to-cell adhesion properties and varying levels of tumour cell motility, thus taking into account the ability of cancer cells to spread and form clusters. We consider the effectiveness of the immune response over a range of parameters pertaining to tumour cell motility, cell-to-cell adhesion strength and growth rate. We also investigate the dependence of outcomes on the distribution of tumour cells. Low tumour cell motility is generally a good indicator for successful tumour eradication before relapse, while high motility leads, almost invariably, to relapse and tumour escape. In general, the effect of cell-to-cell adhesion on prognosis is dependent on the level of tumour cell motility, with an often unpredictable cross influence between adhesion and motility, which can lead to counterintuitive effects. In terms of overall tumour shape and structure, the spatial distribution of cancer cells in clusters of various sizes has shown to be strongly related to the likelihood of extinction.

Clinical implications of in silico mathematical modeling for glioblastoma: a critical review

Glioblastoma remains a clinical challenge in spite of years of extensive research. Novel approaches are needed in order to integrate the existing knowledge. This is the potential role of mathematical oncology. This paper reviews mathematical models on glioblastoma from the clinical doctor's point of view, with focus on 3D modeling approaches of radiation response of in vivo glioblastomas based on contemporary imaging techniques. As these models aim to provide a clinically useful tool in the era of personalized medicine, the integration of the latest advances in molecular and imaging science and in clinical practice by the in silico models is crucial for their clinical relevance. Our aim is to indicate areas of GBM research that have not yet been addressed by in silico models and to point out evidence that has come up from in silico experiments, which may be worth considering in the clinic. This review examines how close these models have come in predicting the outcome of treatment protocols and in shaping the future of radiotherapy treatments.

A Model for Adult Organ Resizing Demonstrates Stem Cell Scaling through a Tunable Commitment Rate

Many adult organs grow or shrink to accommodate different physiological demands. Often, as total cell number changes, stem cell number changes proportionally in a phenomenon called "stem cell scaling". The cellular behaviors that give rise to scaling are unknown. Here we study two complementary theoretical models of the adult Drosophila midgut, a stem cell-based organ with known resizing dynamics. First, we derive a differential equations model of midgut resizing and show that the in vivo kinetics of growth can be recapitulated if the rate of fate commitment depends on the tissue's stem cell proportion. Second, we develop a 2D simulation of the midgut and find that proportion-dependent commitment rate and stem cell scaling can arise phenomenologically from the stem cells' exploration of physical tissue space during its lifetime. Together, these models provide a biophysical understanding of how stem cell scaling is maintained during organ growth and shrinkage.

Mechanical role of a growing solid tumor on cortical folding

Cortical folding, or convolution of the brain, is a vital process in mammals that causes the brain to have a wrinkled appearance. The existence of different types of prenatal solid tumors may alter this complex phenomenon and cause severe brain disorders. Here we interpret the effects of a growing solid tumor on the cortical folding in the fetal brain by virtue of theoretical analyses and computational modeling. The developing fetal brain is modeled as a simple, double-layered, and soft structure with an outer cortex and an inner core, in combination with a circular tumor model imbedded in the structure to investigate the developmental mechanism of cortical convolution. Analytical approaches offer introductory insight into the deformation field and stress distribution of a developing brain. After the onset of instability, analytical approaches fail to capture complex secondary evolution patterns, therefore a series of non-linear finite element simulations are carried out to study the crease formation and the influence from a growing solid tumor inside the structure. Parametric studies show the dependency of the cortical folding pattern on the size, location, and growth speed of a solid tumor in fetal brain. It is noteworthy to mention that there is a critical distance from the cortex/core interface where the growing tumor shows its pronounced effect on the cortical convolution, and that a growing tumor decreases the gyrification index of cortical convolution while its stiffness does not have a profound effect on the gyrification process.

A Numerical Handling of the Boundary Conditions Imposed by the Skull on an Inhomogeneous Diffusion-Reaction Model of Glioblastoma Invasion Into the Brain: Clinical Validation Aspects

A novel explicit triscale reaction-diffusion numerical model of glioblastoma multiforme tumor growth is presented. The model incorporates the handling of Neumann boundary conditions imposed by the cranium and takes into account both the inhomogeneous nature of human brain and the complexity of the skull geometry. The finite-difference time-domain method is adopted. To demonstrate the workflow of a possible clinical validation procedure, a clinical case/scenario is addressed. A good agreement of the in silico calculated value of the doubling time (ie, the time for tumor volume to double) with the value of the same quantity based on tomographic imaging data has been observed. A theoretical exploration suggests that a rough but still quite informative value of the doubling time may be calculated based on a homogeneous brain model. The model could serve as the main component of a continuous mathematics-based glioblastoma oncosimulator aiming at supporting the clinician in the optimal patient-individualized design of treatment using the patient's multiscale data and experimenting in silico (ie, on the computer).

Mathematical modeling and optimal control problems in brain tumor targeted drug delivery strategies

In this paper, we present a mathematical model that describes tumor-normal cells interaction dynamics focusing on role of drugs in treatment of brain tumors. The goal is to predict distribution and necessary quantity of drugs delivered in drug-therapy by using optimal control framework. The model describes interactions of tumor and normal cells using a system of reactions–diffusion equations involving the drug concentration, tumor cells and normal tissues. The control estimates simultaneously blood perfusion rate, reabsorption rate of drug and drug dosage administered, which affect the effects of brain tumor chemotherapy. First, we develop mathematical framework which models the competition between tumor and normal cells under chemotherapy constraints. Then, existence, uniqueness and regularity of solution of state equations are proved as well as stability results. Afterwards, optimal control problems are formulated in order to minimize the drug delivery and tumor cell burden in different situations. We show existence and uniqueness of optimal solution, and we derive necessary conditions for optimality. Finally, to solve numerically optimal control and optimization problems, we propose and investigate an adjoint multiple-relaxation-time lattice Boltzmann method for a general nonlinear coupled anisotropic convection–diffusion system (which includes the developed model for brain tumor targeted drug delivery system).

Development of an in silico stochastic 4D model of tumor growth with angiogenesis

PURPOSE

A stochastic computer model of tumour growth with spatial and temporal components that includes tumour angiogenesis was developed. In the current work it was used to simulate head and neck tumour growth. The model also provides the foundation for a 4D cellular radiotherapy simulation tool.

METHODS

The model, developed in Matlab, contains cell positions randomised in 3D space without overlap. Blood vessels are represented by strings of blood vessel units which branch outwards to achieve the desired tumour relative vascular volume. Hypoxic cells have an increased cell cycle time and become quiescent at oxygen tensions less than 1 mmHg. Necrotic cells are resorbed. A hierarchy of stem cells, transit cells and differentiated cells is considered along with differentiated cell loss. Model parameters include the relative vascular volume (2-10%), blood oxygenation (20-100 mmHg), distance from vessels to the onset of necrosis (80-300 μm) and probability for stem cells to undergo symmetric division (2%). Simulations were performed to observe the effects of hypoxia on tumour growth rate for head and neck cancers. Simulations were run on a supercomputer with eligible parts running in parallel on 12 cores.

RESULTS

Using biologically plausible model parameters for head and neck cancers, the tumour volume doubling time varied from 45 ± 5 days (n = 3) for well oxygenated tumours to 87 ± 5 days (n = 3) for severely hypoxic tumours.

CONCLUSIONS

The main achievements of the current model were randomised cell positions and the connected vasculature structure between the cells. These developments will also be beneficial when irradiating the simulated tumours using Monte Carlo track structure methods.

Extracting cellular automaton rules from physical Langevin equation models for single and collective cell migration

Cellular automata (CA) are discrete time, space, and state models which are extensively used for modeling biological phenomena. CA are "on-lattice" models with low computational demands. In particular, lattice-gas cellular automata (LGCA) have been introduced as models of single and collective cell migration. The interaction rule dictates the behavior of a cellular automaton model and is critical to the model's biological relevance. The LGCA model's interaction rule has been typically chosen phenomenologically. In this paper, we introduce a method to obtain lattice-gas cellular automaton interaction rules from physically-motivated "off-lattice" Langevin equation models for migrating cells. In particular, we consider Langevin equations related to single cell movement (movement of cells independent of each other) and collective cell migration (movement influenced by cell-cell interactions). As examples of collective cell migration, two different alignment mechanisms are studied: polar and nematic alignment. Both kinds of alignment have been observed in biological systems such as swarms of amoebae and myxobacteria. Polar alignment causes cells to align their velocities parallel to each other, whereas nematic alignment drives cells to align either parallel or antiparallel to each other. Under appropriate assumptions, we have derived the LGCA transition probability rule from the steady-state distribution of the off-lattice Fokker-Planck equation. Comparing alignment order parameters between the original Langevin model and the derived LGCA for both mechanisms, we found different areas of agreement in the parameter space. Finally, we discuss potential reasons for model disagreement and propose extensions to the CA rule derivation methodology.

Mechanical characterization of human brain tissue

Mechanics are increasingly recognized to play an important role in modulating brain form and function. Computational simulations are a powerful tool to predict the mechanical behavior of the human brain in health and disease. The success of these simulations depends critically on the underlying constitutive model and on the reliable identification of its material parameters. Thus, there is an urgent need to thoroughly characterize the mechanical behavior of brain tissue and to identify mathematical models that capture the tissue response under arbitrary loading conditions. However, most constitutive models have only been calibrated for a single loading mode. Here, we perform a sequence of multiple loading modes on the same human brain specimen - simple shear in two orthogonal directions, compression, and tension - and characterize the loading-mode specific regional and directional behavior. We complement these three individual tests by combined multiaxial compression/tension-shear tests and discuss effects of conditioning and hysteresis. To explore to which extent the macrostructural response is a result of the underlying microstructural architecture, we supplement our biomechanical tests with diffusion tensor imaging and histology. We show that the heterogeneous microstructure leads to a regional but not directional dependence of the mechanical properties. Our experiments confirm that human brain tissue is nonlinear and viscoelastic, with a pronounced compression-tension asymmetry. Using our measurements, we compare the performance of five common constitutive models, neo-Hookean, Mooney-Rivlin, Demiray, Gent, and Ogden, and show that only the isotropic modified one-term Ogden model is capable of representing the hyperelastic behavior under combined shear, compression, and tension loadings: with a shear modulus of 0.4-1.4kPa and a negative nonlinearity parameter it captures the compression-tension asymmetry and the increase in shear stress under superimposed compression but not tension. Our results demonstrate that material parameters identified for a single loading mode fail to predict the response under arbitrary loading conditions. Our systematic characterization of human brain tissue will lead to more accurate computational simulations, which will allow us to determine criteria for injury, to develop smart protection systems, and to predict brain development and disease progression.

STATEMENT OF SIGNIFICANCE

There is a pressing need to characterize the mechanical behavior of human brain tissue under multiple loading conditions, and to identify constitutive models that are able to capture the tissue response under these conditions. We perform a sequence of experimental tests on the same brain specimen to characterize the regional and directional behavior, and we supplement our tests with DTI and histology to explore to which extent the macrostructural response is a result of the underlying microstructure. Results demonstrate that human brain tissue is nonlinear and viscoelastic, with a pronounced compression-tension asymmetry, and we show that the multiaxial data can best be captured by a modified version of the one-term Ogden model.

A multiscale agent-based framework integrated with a constraint-based metabolic network model of cancer for simulating avascular tumor growth

The integration of an agent-based framework with a constraint-based metabolic network model of cancer for simulating avascular tumor growth.

Spatial avascular growth of tumor in a homogeneous environment

Describing tumor growth is a key issue in oncology for correctly understanding the underlying mechanisms leading to deleterious cancers. In order to take into account the micro-environment in tumor growth, we used a model describing - at the tissue level - the interactions between host (non malignant), effector immune and tumor cells to simulate the evolution of cancer. The spatial growth is described by a Laplacian operator for the diffusion of tumor cells. We investigated how the evolution of the tumor diameter is related to the dynamics (periodic or chaotic oscillations, stable singular points) underlying the interactions between the different populations of cells in proliferation sites. The sensitivity of this evolution to the key parameter responsible for the immuno-evasion, namely the growth rate of effector immune cells and their inhibition rate by tumor cells, is also investigated.

The tumor as an organ: comprehensive spatial and temporal modeling of the tumor and its microenvironment

BACKGROUND

Research related to cancer is vast, and continues in earnest in many directions. Due to the complexity of cancer, a better understanding of tumor growth dynamics can be gleaned from a dynamic computational model. We present a comprehensive, fully executable, spatial and temporal 3D computational model of the development of a cancerous tumor together with its environment.

RESULTS

The model was created using Statecharts, which were then connected to an interactive animation front-end that we developed especially for this work, making it possible to visualize on the fly the on-going events of the system's execution, as well as the effect of various input parameters. We were thus able to gain a better understanding of, e.g., how different amounts or thresholds of oxygen and VEGF (vascular endothelial growth factor) affect the progression of the tumor. We found that the tumor has a critical turning point, where it either dies or recovers. If minimum conditions are met at that time, it eventually develops into a full, active, growing tumor, regardless of the actual amount; otherwise it dies.

CONCLUSIONS

This brings us to the conclusion that the tumor is in fact a very robust system: changing initial values of VEGF and oxygen can increase the time it takes to become fully developed, but will not necessarily completely eliminate it.