Oscillations in a Spatial Oncolytic Virus Model.

Virotherapy treatment is a new and promising target therapy that selectively attacks cancer cells without harming normal cells. Mathematical models of oncolytic viruses have shown predator-prey like oscillatory patterns as result of an underlying Hopf bifurcation. In a spatial context, these oscillations can lead to different spatio-temporal phenomena such as hollow-ring patterns, target patterns, and dispersed patterns. In this paper we continue the systematic analysis of these spatial oscillations and discuss their relevance in the clinical context. We consider a bifurcation analysis of a spatially explicit reaction-diffusion model to find the above mentioned spatio-temporal virus infection patterns. The desired pattern for tumor eradication is the hollow ring pattern and we find exact conditions for its occurrence. Moreover, we derive the minimal speed of travelling invasion waves for the cancer and for the oncolytic virus. Our numerical simulations in 2-D reveal complex spatial interactions of the virus infection and a new phenomenon of a periodic peak splitting. An effect that we cannot explain with our current methods.

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.

A computational model for the cancer field effect.

Introduction: The Cancer Field Effect describes an area of pre-cancerous cells that results from continued exposure to carcinogens. Cells in the cancer field can easily develop into cancer. Removal of the main tumor mass might leave the cancer field behind, increasing risk of recurrence. Methods: The model we propose for the cancer field effect is a hybrid cellular automaton (CA), which includes a multi-layer perceptron (MLP) to compute the effects of the carcinogens on the gene expression of the genes related to cancer development. We use carcinogen interactions that are typically associated with smoking and alcohol consumption and their effect on cancer fields of the tongue. Results: Using simulations we support the understanding that tobacco smoking is a potent carcinogen, which can be reinforced by alcohol consumption. The effect of alcohol alone is significantly less than the effect of tobacco. We further observe that pairing tumor excision with field removal delays recurrence compared to tumor excision alone. We track cell lineages and find that, in most cases, a polyclonal field develops, where the number of distinct cell lineages decreases over time as some lineages become dominant over others. Finally, we find tumor masses rarely form via monoclonal origin.

Detecting minimum energy states and multi-stability in nonlocal advection-diffusion models for interacting species.

Deriving emergent patterns from models of biological processes is a core concern of mathematical biology. In the context of partial differential equations, these emergent patterns sometimes appear as local minimisers of a corresponding energy functional. Here we give methods for determining the qualitative structure of local minimum energy states of a broad class of multi-species nonlocal advection-diffusion models, recently proposed for modelling the spatial structure of ecosystems. We show that when each pair of species respond to one another in a symmetric fashion (i.e. via mutual avoidance or mutual attraction, with equal strength), the system admits an energy functional that decreases in time and is bounded below. This suggests that the system will eventually reach a local minimum energy steady state, rather than fluctuating in perpetuity. We leverage this energy functional to develop tools, including a novel application of computational algebraic geometry, for making conjectures about the number and qualitative structure of local minimum energy solutions. These conjectures give a guide as to where to look for numerical steady state solutions, which we verify through numerical analysis. Our technique shows that even with two species, multi-stability with up to four classes of local minimum energy states can emerge. The associated dynamics include spatial sorting via aggregation and repulsion both within and between species. The emerging spatial patterns include a mixture of territory-like segregation as well as narrow spike-type solutions. Overall, our study reveals a general picture of rich multi-stability in systems of moving and interacting species.

The Tumor Invasion Paradox in Cancer Stem Cell-Driven Solid Tumors.

Cancer stem cells (CSCs) are key in understanding tumor growth and tumor progression. A counterintuitive effect of CSCs is the so-called tumor growth paradox: the effect where a tumor with a higher death rate may grow larger than a tumor with a lower death rate. Here we extend the modeling of the tumor growth paradox by including spatial structure and considering cancer invasion. Using agent-based modeling and a corresponding partial differential equation model, we demonstrate and prove mathematically a tumor invasion paradox: a larger cell death rate can lead to a faster invasion speed. We test this result on a generic hypothetical cancer with typical growth rates and typical treatment sensitivities. We find that the tumor invasion paradox may play a role for continuous and intermittent treatments, while it does not seem to be essential in fractionated treatments. It should be noted that no attempt was made to fit the model to a specific cancer, thus, our results are generic and theoretical.

Personalized Virus Load Curves for Acute Viral Infections.

We introduce an explicit function that describes virus-load curves on a patient-specific level. This function is based on simple and intuitive model parameters. It allows virus load analysis of acute viral infections without solving a full virus load dynamic model. We validate our model on data from mice influenza A, human rhinovirus data, human influenza A data, and monkey and human SARS-CoV-2 data. We find wide distributions for the model parameters, reflecting large variability in the disease outcomes between individuals. Further, we compare the virus load function to an established target model of virus dynamics, and we provide a new way to estimate the exponential growth rates of the corresponding infection phases. The virus load function, the target model, and the exponential approximations show excellent fits for the data considered. Our virus-load function offers a new way to analyze patient-specific virus load data, and it can be used as input for higher level models for the physiological effects of a virus infection, for models of tissue damage, and to estimate patient risks.

Homogenization of a reaction diffusion equation can explain influenza A virus load data.

We study the influence of spatial heterogeneity on the antiviral activity of mouse embryonic fibroblasts (MEF) infected with influenza A. MEF of type Ube1L-/- are composed of two distinct sub-populations, the strong type that sustains a strong viral infection and the weak type, sustaining a weak viral load. We present new data on the virus load infection of Ube1L-/-, which have been micro-printed in a checker board pattern of different sizes of the inner squares. Surprisingly, the total viral load at one day after inoculation significantly depends on the sizes of the inner squares. We explain this observation by using a reaction diffusion model and we show that mathematical homogenization can explain the observed inhomogeneities. If the individual patches are large, then the growth rate and the carrying capacity will be the arithmetic means of the patches. For finer and finer patches the average growth rate is still the arithmetic mean, however, the carrying capacity uses the harmonic mean. While fitting the PDE to the experimental data, we also predict that a discrepancy in virus load would be unobservable after only half a day. Furthermore, we predict the viral load in different inner squares that had not been measured in our experiment and the travelling distance the virions can reach after one day.

Iterative community-driven development of a SARS-CoV-2 tissue simulator.

The 2019 novel coronavirus, SARS-CoV-2, is a pathogen of critical significance to international public health. Knowledge of the interplay between molecular-scale virus-receptor interactions, single-cell viral replication, intracellular-scale viral transport, and emergent tissue-scale viral propagation is limited. Moreover, little is known about immune system-virus-tissue interactions and how these can result in low-level (asymptomatic) infections in some cases and acute respiratory distress syndrome (ARDS) in others, particularly with respect to presentation in different age groups or pre-existing inflammatory risk factors. Given the nonlinear interactions within and among each of these processes, multiscale simulation models can shed light on the emergent dynamics that lead to divergent outcomes, identify actionable "choke points" for pharmacologic interventions, screen potential therapies, and identify potential biomarkers that differentiate patient outcomes. Given the complexity of the problem and the acute need for an actionable model to guide therapy discovery and optimization, we introduce and iteratively refine a prototype of a multiscale model of SARS-CoV-2 dynamics in lung tissue. The first prototype model was built and shared internationally as open source code and an online interactive model in under 12 hours, and community domain expertise is driving regular refinements. In a sustained community effort, this consortium is integrating data and expertise across virology, immunology, mathematical biology, quantitative systems physiology, cloud and high performance computing, and other domains to accelerate our response to this critical threat to international health. More broadly, this effort is creating a reusable, modular framework for studying viral replication and immune response in tissues, which can also potentially be adapted to related problems in immunology and immunotherapy.

Implications of immune-mediated metastatic growth on metastatic dormancy, blow-up, early detection, and treatment.

Metastatic seeding of distant organs can occur in the very early stages of primary tumor development. Once seeded, these micrometastases may enter a dormant phase that can last decades. Curiously, the surgical removal of the primary tumor can stimulate the accelerated growth of distant metastases, a phenomenon known as metastatic blow-up. Recent clinical evidence has shown that the immune response can have strong tumor promoting effects. In this work, we investigate if the pro-tumor effects of the immune response can have a significant contribution to metastatic dormancy and metastatic blow-up. We develop an ordinary differential equation model of the immune-mediated theory of metastasis. We include both anti- and pro-tumor immune effects, in addition to the experimentally observed phenomenon of tumor-induced immune cell phenotypic plasticity. Using geometric singular perturbation analysis, we derive a rather simple model that captures the main processes and, at the same time, can be fully analyzed. Literature-derived parameter estimates are obtained, and model robustness is demonstrated through a time dependent sensitivity analysis. We determine conditions under which the parameterized model can successfully explain both metastatic dormancy and blow-up. The results confirm the significant active role of the immune system in the metastatic process. Numerical simulations suggest a novel measure to predict the occurrence of future metastatic blow-up in addition to new potential avenues for treatment of clinically undetectable micrometastases.

Multiscale phenomena and patterns in biological systems: special issue in honour of Hans Othmer.

This special issue on "Multiscale phenomena and patterns in biological systems" is an homage to the seminal contributions of Hans Othmer. He has remained at the forefront of multiscale modelling and pattern formation in biology for over half a century, developing models for molecular signalling networks, the mechanics of cellular movements, the interactions between multiple cells and their contributions to tissue patterning and dynamics. The contributions in this special issue follow Hans' legacy in using advanced mathematics to understand complex biological processes.

A stochastic model for cancer metastasis: branching stochastic process with settlement.

We introduce a new stochastic model for metastatic growth, which takes the form of a branching stochastic process with settlement. The moving particles are interpreted as clusters of cancer cells, while stationary particles correspond to micro-tumours and metastases. The analysis of expected particle location, their locational variance, the furthest particle distribution and the extinction probability leads to a common type of differential equation, namely, a non-local integro-differential equation with distributed delay. We prove global existence and uniqueness results for this type of equation. The solutions' asymptotic behaviour for long time is characterized by an explicit index, a metastatic reproduction number $R_0$: metastases spread for $R_{0}>1$ and become extinct for $R_{0}<1$. Using metastatic data from mouse experiments, we show the suitability of our framework to model metastatic cancer.

A mathematical model for the immune-mediated theory of metastasis.

Accumulating experimental and clinical evidence suggest that the immune response to cancer is not exclusively anti-tumor. Indeed, the pro-tumor roles of the immune system  -  as suppliers of growth and pro-angiogenic factors or defenses against cytotoxic immune attacks, for example  -  have been long appreciated, but relatively few theoretical works have considered their effects. Inspired by the recently proposed "immune-mediated" theory of metastasis, we develop a mathematical model for tumor-immune interactions at two anatomically distant sites, which includes both anti- and pro-tumor immune effects, and the experimentally observed tumor-induced phenotypic plasticity of immune cells (tumor "education" of the immune cells). Upon confrontation of our model to experimental data, we use it to evaluate the implications of the immune-mediated theory of metastasis. We find that tumor education of immune cells may explain the relatively poor performance of immunotherapies, and that many metastatic phenomena, including metastatic blow-up, dormancy, and metastasis to sites of injury, can be explained by the immune-mediated theory of metastasis. Our results suggest that further work is warranted to fully elucidate the pro-tumor effects of the immune system in metastatic cancer.

Karl-Peter Hadeler: His legacy in mathematical biology.

Karl-Peter Hadeler is a first-generation pioneer in mathematical biology. His work inspired the contributions to this special issue. In this preface we give a brief biographical sketch of K.P. Hadelers scientific life and highlight his impact to the field.

Effects of G2-checkpoint dynamics on low-dose hyper-radiosensitivity.

In experimental studies, it has been found that certain cell lines are more sensitive to low-dose radiation than would be expected from the classical Linear-Quadratic model (LQ model). In fact, it is frequently observed that cells incur more damage at low dose (say 0.3 Gy) than at higher dose (say 1 Gy). This effect has been termed hyper-radiosensitivity (HRS). The effect depends on the type of cells and on their phase in the cell cycle when radiation is applied. Experiments have shown that the G2-checkpoint plays an important role in the HRS effects. Here we design and analyze a differential equation model for the cell cycle that includes G2-checkpoint dynamics and radiation treatment. We fit the model to surviving fraction data for different cell lines including glioma cells, prostate cancer cells, as well as to cell populations that are enriched in certain phases of the cell cycle. The HRS effect is measured in the literature through [Formula: see text], the ratio of slope [Formula: see text] of the surviving fraction curve at zero dose to slope [Formula: see text] of the corresponding LQ model. We derive an explicit formula for this ratio and we show that it corresponds very closely to experimental observations. Finally, we identify the dependence of this ratio on the surviving fraction at 2 Gy. It was speculated in the literature that such dependence exists. Our theoretical analysis will help to more systematically identify the HRS in cell lines, and opens doors to analyze its use in cancer treatment.

A Patient-Specific Anisotropic Diffusion Model for Brain Tumour Spread.

Gliomas are primary brain tumours arising from the glial cells of the nervous system. The diffuse nature of spread, coupled with proximity to critical brain structures, makes treatment a challenge. Pathological analysis confirms that the extent of glioma spread exceeds the extent of the grossly visible mass, seen on conventional magnetic resonance imaging (MRI) scans. Gliomas show faster spread along white matter tracts than in grey matter, leading to irregular patterns of spread. We propose a mathematical model based on Diffusion Tensor Imaging, a new MRI imaging technique that offers a methodology to delineate the major white matter tracts in the brain. We apply the anisotropic diffusion model of Painter and Hillen (J Thoer Biol 323:25-39, 2013) to data from 10 patients with gliomas. Moreover, we compare the anisotropic model to the state-of-the-art Proliferation-Infiltration (PI) model of Swanson et al. (Cell Prolif 33:317-329, 2000). We find that the anisotropic model offers a slight improvement over the standard PI model. For tumours with low anisotropy, the predictions of the two models are virtually identical, but for patients whose tumours show higher anisotropy, the results differ. We also suggest using the data from the contralateral hemisphere to further improve the model fit. Finally, we discuss the potential use of this model in clinical treatment planning.

A space-jump derivation for non-local models of cell-cell adhesion and non-local chemotaxis.

Cellular adhesion provides one of the fundamental forms of biological interaction between cells and their surroundings, yet the continuum modelling of cellular adhesion has remained mathematically challenging. In 2006, Armstrong et al. proposed a mathematical model in the form of an integro-partial differential equation. Although successful in applications, a derivation from an underlying stochastic random walk has remained elusive. In this work we develop a framework by which non-local models can be derived from a space-jump process. We show how the notions of motility and a cell polarization vector can be naturally included. With this derivation we are able to include microscopic biological properties into the model. We show that particular choices yield the original Armstrong model, while others lead to more general models, including a doubly non-local adhesion model and non-local chemotaxis models. Finally, we use random walk simulations to confirm that the corresponding continuum model represents the mean field behaviour of the stochastic random walk.

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.

Existence and uniqueness for a coupled PDE model for motor-induced microtubule organization.

Microtubules (MTs) are protein filaments that provide structure to the cytoskeleton of cells and a platform for the movement of intracellular substances. The spatial organization of MTs is crucial for a cell's form and function. MTs interact with a class of proteins called motor proteins that can transport and position individual filaments, thus contributing to overall organization. In this paper, we study the mathematical properties of a coupled partial differential equation (PDE) model, introduced by White et al. in 2015, that describes the motor-induced organization of MTs. The model consists of a nonlinear coupling of a hyperbolic PDE for bound motor proteins, a parabolic PDE for unbound motor proteins, and a transport equation for MT dynamics. We locally smooth the motor drift velocity in the equation for bound motor proteins. The mollification is not only critical for the analysis of the model, but also adds biological realism. We then use a Banach Fixed Point argument to show local existence and uniqueness of mild solutions. We highlight the applicability of the model by showing numerical simulations that are consistent with in vitro experiments.

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.

A stochastic model for the normal tissue complication probability (NTCP) and applicationss.

The normal tissue complication probability (NTCP) is a measure for the estimated side effects of a given radiation treatment schedule. Here we use a stochastic logistic birth-death process to define an organ-specific and patient-specific NTCP. We emphasize an asymptotic simplification which relates the NTCP to the solution of a logistic differential equation. This framework is based on simple modelling assumptions and it prepares a framework for the use of the NTCP model in clinical practice. As example, we consider side effects of prostate cancer brachytherapy such as increase in urinal frequency, urinal retention and acute rectal dysfunction.