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Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

Expanded parameter analysis for model parameter c. Greater T cell sensitivity to Tregs reduces the size of the Goldilocks Window.

Drug resistance is the single most important driver of cancer treatment failure for modern targeted therapies, and the dialog between tumor and stroma has been shown to modulate the response to molecularly targeted therapies through proliferative and survival signaling. In this work, we investigate interactions between a growing tumor and its surrounding stroma and their role in facilitating the emergence of drug resistance. We used mathematical modeling as a theoretical framework to bridge between experimental models and scales, with the aim of separating intrinsic and extrinsic components of resistance in BRAF-mutated melanoma; the model describes tumor–stroma dynamics both with and without treatment. Integration of experimental data into our model revealed significant variation in either the intensity of stromal promotion or intrinsic tissue carrying capacity across animal replicates. Cancer Res; 77(19); 5409–18. ©2017 AACR.

Supplementary Material

Detailed legends for supplementary figures S1 - S7. This includes comparisons to baseline model values as well as the evaluation of the impacts of parameter variation.

The immune system is a robust and often untapped accomplice of many standard cancer therapies. A majority of tumors exist in a state of immune tolerance where the patient's immune system has become insensitive to the cancer cells. Because of its lymphodepleting effects, chemotherapy has the potential to break this tolerance. To investigate this, we created a mathematical modeling framework of tumor-immune dynamics. Our results suggest that optimal chemotherapy scheduling must balance two opposing objectives: maximizing tumor reduction while preserving patient immune function. Successful treatment requires therapy to operate in a “Goldilocks Window” where patient immune health is not overly compromised. By keeping therapy “just right,” we show that the synergistic effects of immune activation and chemotherapy can maximize tumor reduction and control.Significance:To maximize the synergy between chemotherapy and antitumor immune response, lymphodepleting therapy must be balanced in a “Goldilocks Window” of optimal dosing.

This file contains: results of the sensitivity analysis for parameter estimation shown in the main document (Figures S1-S4), results of curve fitting for mouse growth data under dual inhibition (Figure S5), examples for growth dynamics under dual inhibition, calibrated on representative mice from 2 cases identified in the Results section of the main document (Figures S6 and S7).

Adaptive therapy seeks to exploit intratumoral competition to avoid, or at least delay, the emergence of therapy resistance in cancer. Motivated by promising results in prostate cancer, there is growing interest in extending this approach to other neoplasms. As such, it is urgent to understand the characteristics of a cancer that determine whether or not it will respond well to adaptive therapy. A plausible candidate for such a selection criterion is the fitness cost of resistance. In this article, we study a general, but simple, mathematical model to investigate whether the presence of a cost is necessary for adaptive therapy to extend the time to progression beyond that of a standard-of-care continuous therapy. Tumor cells were divided into sensitive and resistant populations and we model their competition using a system of two ordinary differential equations based on the Lotka–Volterra model. For tumors close to their environmental carrying capacity, a cost was not required. However, for tumors growing far from carrying capacity, a cost may be required to see meaningful gains. Notably, it is important to consider cell turnover in the tumor, and we discuss its role in modulating the impact of a resistance cost. To conclude, we present evidence for the predicted cost–turnover interplay in data from 67 patients with prostate cancer undergoing intermittent androgen deprivation therapy. Our work helps to clarify under which circumstances adaptive therapy may be beneficial and suggests that turnover may play an unexpectedly important role in the decision-making process.Significance:Tumor cell turnover modulates the speed of selection against drug resistance by amplifying the effects of competition and resistance costs; as such, turnover is an important factor in resistance management via adaptive therapy.See related commentary by Strobl et al., p. 811

This file includes a detailed description of the methods used to obtain numerical solution of the equations, and method for the estimation of model parameters.

Free Article from Systems Biologists Seek Fuller Integration of Systems Biology Approaches in New Cancer Research Programs

Systems biology takes an interdisciplinary approach to the systematic study of complex interactions in biological systems. This approach seeks to decipher the emergent behaviors of complex systems rather than focusing only on their constituent properties. As an increasing number of examples illustrate the value of systems biology approaches to understand the initiation, progression, and treatment of cancer, systems biologists from across Europe and the United States hope for changes in the way their field is currently perceived among cancer researchers. In a recent EU-US workshop, supported by the European Commission, the German Federal Ministry for Education and Research, and the National Cancer Institute of the NIH, the participants discussed the strengths, weaknesses, hurdles, and opportunities in cancer systems biology. Cancer Res; 70(1); 12–3

Background: Collective and discrete neural crest cell (NCC) migratory streams are crucial to vertebrate head patterning. However, the factors that confine NCC trajectories and promote collective cell migration remain unclear. Results Computational simulations predicted that confinement is required only along the initial one-third of the cranial NCC migratory pathway. This guided our study of Colec12 (Collectin-12, a transmembrane scavenger receptor C-type lectin) and Trail (tumor necrosis factor-related apoptosis-inducing ligand, CD253) which we show expressed in chick cranial NCC-free zones. NCC trajectories are confined by Colec12 or Trail protein stripes in vitro and show significant and distinct changes in cell morphology and dynamic migratory characteristics when cocultured with either protein. Gain- or loss-of-function of either factor or in combination enhanced NCC confinement or diverted cell trajectories as observed in vivo with three-dimensional confocal microscopy, respectively, resulting in disrupted collective migration. Conclusions These data provide evidence for Colec12 and Trail as novel NCC microenvironmental factors playing a role to confine cranial NCC trajectories and promote collective cell migration.

In March 2020 mathematics became a key part of the scientific advice to the UK government on the pandemic response to COVID-19. Mathematical and statistical modelling provided critical information on the spread of the virus and the potential impact of different interventions. The unprecedented scale of the challenge led the epidemiological modelling community in the UK to be pushed to its limits. At the same time, mathematical modellers across the country were keen to use their knowledge and skills to support the COVID-19 modelling effort. However, this sudden great interest in epidemiological modelling needed to be coordinated to provide much-needed support, and to limit the burden on epidemiological modellers already very stretched for time. In this paper we describe three initiatives set up in the UK in spring 2020 to coordinate the mathematical sciences research community in supporting mathematical modelling of COVID-19. Each initiative had different primary aims and worked to maximise synergies between the various projects. We reflect on the lessons learnt, highlighting the key roles of pre-existing research collaborations and focal centres of coordination in contributing to the success of these initiatives. We conclude with recommendations about important ways in which the scientific research community could be better prepared for future pandemics. This manuscript was submitted as part of a theme issue on “Modelling COVID-19 and Preparedness for Future Pandemics”.

PARP inhibitors (PARPis) are revolutionizing the treatment of ovarian cancer. Yet for many patients these improvements come at the cost of physical and financial toxicity and responses are typically temporary due to emerging drug resistance. A growing body of pre-clinical and clinical work suggests that when cure is unlikely, it is possible to delay progression and reduce drug use through patient-specific drug scheduling. So-called 'adaptive therapy' dynamically adjusts treatment to preserve drug-sensitive cells which interfere with resistant cells through competition. In a prior study, we developed a mathematical model to describe the treatment response of ovarian cancer cells to the PARPi olaparib in vitro, and we proposed a candidate adaptive PARPi algorithm. Here, we extend our model to capture the dynamics in patients and use it to study the feasibility and potential benefit of adaptive PARPi administration in practice. Our prior model posited that treatment induces cell cycle arrest that moves cells from the proliferating subset of the population to an arrested compartment. The model predicted that while there is scope for treatment reductions, these need to be carefully timed and prolonged treatment breaks should be avoided. Based on these results we proposed an adaptive treatment algorithm in which the olaparib dose is switched between high and low doses, depending on the tumor’s growth rate. To test the translational potential of this strategy, we retrospectively collected data from 53 ovarian cancer patients who received olaparib at the H Lee Moffitt Cancer Center between 2014 and 2021. Using serum CA-125 as a proxy for tumor burden, we first examined whether our mathematical model could capture the patients’ longitudinal dynamics. While the response of some patients was consistent with what we had observed in vitro, in others there was evidence of the emergence of a distinct drug-resistant population, and we extended our mathematical model to account for this. After calibrating and validating the model with the patient data, we tested whether these patients would have benefited from adaptive PARPi treatment. Our simulations suggest that our proposed algorithm is feasible and provides a means for reducing treatment in a patient-specific manner. In addition, in a subset of patients we predict that adaptive therapy could delay progression. Overall, this work corroborates the potential for adaptive PARPi therapy and helps to identify outstanding challenges on the way to clinical translation. Citation Format: Maximilian A. Strobl, Alexandra L. Martin, Christopher Gallagher, Mehdi Damaghi, Mark Robertson-Tessi, Robert Gatenby, Robert M. Wenham, Philip K. Maini, Alexander R. Anderson. Adaptive treatment scheduling of PARP inhibitors in ovarian cancer: Using mathematical modeling to assess clinical feasibility and estimate potential benefits. [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 5694.

We present a mechanical model of tissue homeostasis that is specialised to the intestinal crypt. Growth and deformation of the crypt, idealised as a line of cells on a substrate, are modelled using morphoelastic rod theory. Alternating between Lagrangian and Eulerian mechanical descriptions enables us precisely to characterise the dynamic nature of tissue homeostasis, whereby the proliferative structure and morphology are static in the Eulerian frame, but there is active migration of Lagrangian material points out of the crypt. Assuming mechanochemical growth, we identify the necessary conditions for homeostasis, reducing the full, time-dependent system to a static boundary value problem characterising a spatially heterogeneous "treadmilling" state. We extract essential features of crypt homeostasis, such as the morphology, the proliferative structure, the migration velocity, and the sloughing rate. We also derive closed-form solutions for growth and sloughing dynamics in homeostasis, and show that mechanochemical growth is sufficient to generate the observed proliferative structure of the crypt. Key to this is the concept of threshold-dependent mechanical feedback, that regulates an established Wnt signal for biochemical growth. Numerical solutions demonstrate the importance of crypt morphology on homeostatic growth, migration, and sloughing, and highlight the value of this framework as a foundation for studying the role of mechanics in homeostasis., Comment: 21 pages, 7 figures

The intestinal epithelium is crucial to maintaining a healthy gut. Central to this are the crypts of Lieberkuhn, which coordinate the rapid self-renewal of the epithelium to protect the small intestine and colon during digestion. Further, mutations in crypt cells may initiate colorectal cancer. It is, therefore, important to understand how crypts function during homeostasis and disease. Mathematical and computational modeling has contributed to increasing our understanding of crypt dynamics. However, many open questions remain to be addressed, particularly regarding the role of mechanics in intestinal crypt dynamics. In this article, we review the state-of-the-art in crypt modeling and explain why further progress requires the integration of new theory from continuum mechanics with cell-based computational models and experimental data.

Vertebrate head morphogenesis involves carefully-orchestrated tissue growth and cell movements of the mesoderm and neural crest to form the distinct craniofacial pattern. To better understand structural birth defects, it is important that we characterize the dynamics of these processes and learn how they rely on each other. Here we examine this question during chick head morphogenesis using time-lapse imaging, computational modeling, and experiments. We find that head mesodermal cells in culture move in random directions as individuals and move faster in the presence of neural crest cells. In vivo, mesodermal cells migrate in a directed manner and maintain neighbor relationships; neural crest cells travel through the mesoderm at a faster speed. The mesoderm grows with a non-uniform spatio-temporal profile determined by BrdU labeling during the period of faster and more-directed neural crest collective migration through this domain. We use computer simulations to probe the robustness of neural crest stream formation by varying the spatio-temporal growth profile of the mesoderm. We follow this with experimental manipulations that either stop mesoderm growth or prevent neural crest migration and observe changes in the non-manipulated cell population, implying a dynamic feedback between tissue growth and neural crest cell signaling to confer robustness to the system. Overall, we present a novel descriptive analysis of mesoderm and neural crest cell dynamics that reveals the coordination and co-dependence of these two cell populations during head morphogenesis.

The processes underpinning solid tumour growth involve the interactions between various healthy and tumour tissue components and the vasculature, and can be affected in different ways by cancer treatment. In particular, the growth-limiting mechanisms at play may influence tumour responses to treatment. In this paper, we propose a simple ordinary differential equation model of solid tumour growth to investigate how tumour-specific mechanisms of growth arrest may affect tumour response to different combination cancer therapies. We consider the interactions of tumour cells with the physical space in which they proliferate and a nutrient supplied by the tumour vasculature, with the aim of representing two distinct growth arrest mechanisms. More specifically, we wish to consider growth arrest due to (1) nutrient deficiency, which corresponds to balancing cell proliferation and death rates, and (2) competition for space, which corresponds to cessation of proliferation without cell death. We perform numerical simulations of the model and a steady-state analysis to determine the possible tumour growth scenarios described by the model. We find that there are three distinct growth regimes: the nutrient- and spatially limited regimes and a bi-stable regime, in which both growth arrest mechanisms are simultaneously active. Thus, the proposed model has the features required to investigate and distinguish tumour responses to different cancer treatments.