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.

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.

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.

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.

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 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.

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.

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 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.

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.

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.

A reaction-diffusion model for radiation-induced bystander effects.

We develop and analyze a reaction-diffusion model to investigate the dynamics of the lifespan of a bystander signal emitted when cells are exposed to radiation. Experimental studies by Mothersill and Seymour 1997, using malignant epithelial cell lines, found that an emitted bystander signal can still cause bystander effects in cells even 60 h after its emission. Several other experiments have also shown that the signal can persist for months and even years. Also, bystander effects have been hypothesized as one of the factors responsible for the phenomenon of low-dose hyper-radiosensitivity and increased radioresistance (HRS/IRR). Here, we confirm this hypothesis with a mathematical model, which we fit to Joiner's data on HRS/IRR in a T98G glioma cell line. Furthermore, we use phase plane analysis to understand the full dynamics of the signal's lifespan. We find that both single and multiple radiation exposure can lead to bystander signals that either persist temporarily or permanently. We also found that, in an heterogeneous environment, the size of the domain exposed to radiation and the number of radiation exposures can determine whether a signal will persist temporarily or permanently. Finally, we use sensitivity analysis to identify those cell parameters that affect the signal's lifespan and the signal-induced cell death the most.

Mathematical Modeling of the Role of Survivin on Dedifferentiation and Radioresistance in Cancer.

We use a mathematical model to investigate cancer resistance to radiation, based on dedifferentiation of non-stem cancer cells into cancer stem cells. Experimental studies by Iwasa 2008, using human non-small cell lung cancer (NSCLC) cell lines in mice, have implicated the inhibitor of apoptosis protein survivin in cancer resistance to radiation. A marked increase in radio-sensitivity was observed, after inhibiting survivin expression with a specific survivin inhibitor YM155 (sepantronium bromide). It was suggested that these observations are due to survivin-dependent dedifferentiation of non-stem cancer cells into cancer stem cells. Here, we confirm this hypothesis with a mathematical model, which we fit to Iwasa's data on NSCLC in mice. We investigate the timing of combination therapies of YM155 administration and radiation. We find an interesting dichotomy. Sometimes it is best to hit a cancer with a large radiation dose right at the beginning of the YM155 treatment, while in other cases, it appears advantageous to wait a few days until most cancer cells are sensitized and then radiate. The optimal strategy depends on the nature of the cancer and the dose of radiation administered.

Feedback Regulation in a Cancer Stem Cell Model can Cause an Allee Effect.

The exact mechanisms of spontaneous tumor remission or complete response to treatment are phenomena in oncology that are not completely understood. We use a concept from ecology, the Allee effect, to help explain tumor extinction in a model of tumor growth that incorporates feedback regulation of stem cell dynamics, which occurs in many tumor types where certain signaling molecules, such as Wnts, are upregulated. Due to feedback and the Allee effect, a tumor may become extinct spontaneously or after therapy even when the entire tumor has not been eradicated by the end of therapy. We quantify the Allee effect using an 'Allee index' that approximates the area of the basin of attraction for tumor extinction. We show that effectiveness of combination therapy in cancer treatment may occur due to the increased probability that the system will be in the Allee region after combination treatment versus monotherapy. We identify therapies that can attenuate stem cell self-renewal, alter the Allee region and increase its size. We also show that decreased response of tumor cells to growth inhibitors can reduce the size of the Allee region and increase stem cell densities, which may help to explain why this phenomenon is a hallmark of cancer.

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

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

The tumor growth paradox and immune system-mediated selection for cancer stem cells.

Cancer stem cells (CSCs) drive tumor progression, metastases, treatment resistance, and recurrence. Understanding CSC kinetics and interaction with their nonstem counterparts (called tumor cells, TCs) is still sparse, and theoretical models may help elucidate their role in cancer progression. Here, we develop a mathematical model of a heterogeneous population of CSCs and TCs to investigate the proposed "tumor growth paradox"--accelerated tumor growth with increased cell death as, for example, can result from the immune response or from cytotoxic treatments. We show that if TCs compete with CSCs for space and resources they can prevent CSC division and drive tumors into dormancy. Conversely, if this competition is reduced by death of TCs, the result is a liberation of CSCs and their renewed proliferation, which ultimately results in larger tumor growth. Here, we present an analytical proof for this tumor growth paradox. We show how numerical results from the model also further our understanding of how the fraction of cancer stem cells in a solid tumor evolves. Using the immune system as an example, we show that induction of cell death can lead to selection of cancer stem cells from a minor subpopulation to become the dominant and asymptotically the entire cell type in tumors.

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.

Assessing the prevalence of mental health disorders and mental health needs among preschool children in care in England.

Although school-aged children living in foster care have been identified as a high-risk group for mental health and developmental disorders, there is a paucity of data relating to preschool children in care (CIC). This study aimed to identify the prevalence of mental health and developmental disorders along with corresponding need for interventions in preschool CIC. All CIC aged 0 to 5 years in an inner city local authority underwent comprehensive, multifaceted assessments consisting of the Ages and Stages Questionnaire (J. Squires, D. Bricker, & E. Twombly, 2003), interviews with caregivers based on the Preschool Age Psychiatric Assessment (H.L. Egger & A. Angold, 2006), Mullen Scales of Early Learning (E.M. Mullen, 1995), and systematic clinical observation. Of 58 eligible preschoolers, 43 completed the assessment. At least one mental health disorder was found in 26 (60.5%) participants, and at least one developmental disorder was found in 11 (25.6%). When mental health and/or developmental disorders were considered together, 30 (69.8%) preschoolers fulfilled criteria for at least one diagnosis, and 18 (41.9%) had two or more comorbid conditions. Whereas 36 (83.7%) of the preschoolers needed an intervention, only 3 of these had received adequate input. In conclusion, preschool CIC constitute a high-risk group for mental health and developmental disorders. Without age-appropriate assessments, their needs go undetected, and opportunities for early intervention are being missed.