Integrative mathematical oncology

A mathematical model for predicting the spatiotemporal response of breast cancer cells treated with doxorubicin

Tumor heterogeneity contributes significantly to chemoresistance, a leading cause of treatment failure. To better personalize therapies, it is essential to develop tools capable of identifying and predicting intra- and inter-tumor heterogeneities. Biology-inspired mathematical models are capable of attacking this problem, but tumor heterogeneity is often overlooked in in-vivo modeling studies, while phenotypic considerations capturing spatial dynamics are not typically included in in-vitro modeling studies. We present a data assimilation-prediction pipeline with a two-phenotype model that includes a spatiotemporal component to characterize and predict the evolution of in-vitro breast cancer cells and their heterogeneous response to chemotherapy. Our model assumes that the cells can be divided into two subpopulations: surviving cells unaffected by the treatment, and irreversibly damaged cells undergoing treatment-induced death. MCF7 breast cancer cells were previously cultivated in wells for up to 1000 hours, treated with various concentrations of doxorubicin and imaged with time-resolved microscopy to record spatiotemporally-resolved cell count data. Images were used to generate cell density maps. Treatment response predictions were initialized by a training set and updated by weekly measurements. Our mathematical model successfully calibrated the spatiotemporal cell growth dynamics, achieving median [range] concordance correlation coefficients of > .99 [.88, >.99] and .73 [.58, .85] across the whole well and individual pixels, respectively. Our proposed data assimilation-prediction approach achieved values of .97 [.44, >.99] and .69 [.35, .79] for the whole well and individual pixels, respectively. Thus, our model can capture and predict the spatiotemporal dynamics of MCF7 cells treated with doxorubicin in an in-vitro setting.

Engineering cancer's end: An interdisciplinary approach to confront the complexities of cancer

Cancer engineering is an interdisciplinary approach that promises to confront the complexities of cancer and accelerate transformative discoveries by integrating innovative fields across engineering and the physical sciences with a focus on cancer. We offer a conceptual framework for the hallmarks of cancer engineering, integrating 12 fields: system dynamics; imaging, radiation, and spectroscopy; robotics and controls; solid mechanics; fluid mechanics; chemistry and nanomaterials; mathematics and simulation; cellular and protein engineering; kinetics and thermodynamics; materials science; manufacturing and biofabrication; and microsystems.

Quantifying Geobacter sulfurreducens growth: A mathematical model based on acetate concentration as an oxidizing substrate

We developed a mathematical model to simulate dynamics associated with the proliferation of Geobacter and ultimately optimize cellular operation by analyzing the interaction of its components. The model comprises two segments: an initial part comprising a logistic form and a subsequent segment that incorporates acetate oxidation as a saturation term for the microbial nutrient medium. Given that four parameters can be obtained by minimizing the square root of the mean square error between experimental Geobacter growth and the mathematical model, the model underscores the importance of incorporating nonlinear terms. The determined parameter values closely align with experimental data, providing insights into the mechanisms that govern Geobacter proliferation. Furthermore, the model has been transformed into a scaleless equation with only two parameters to simplify the exploration of qualitative properties. This allowed us to conduct stability analysis of the fixed point and construct a co-dimension two bifurcation diagram.

Modeling tumors as species-rich ecological communities

Many advanced cancers resist therapeutic intervention. This process is fundamentally related to intra-tumor heterogeneity: multiple cell populations, each with different mutational and phenotypic signatures, coexist within a tumor and its metastatic nodes. Like species in an ecosystem, many cancer cell populations are intertwined in a complex network of ecological interactions. Most mathematical models of tumor ecology, however, cannot account for such phenotypic diversity nor are able to predict its consequences. Here we propose that the Generalized Lotka-Volterra model (GLV), a standard tool to describe complex, species-rich ecological communities, provides a suitable framework to describe the ecology of heterogeneous tumors. We develop a GLV model of tumor growth and discuss how its emerging properties, such as outgrowth and multistability, provide a new understanding of the disease. Additionally, we discuss potential extensions of the model and their application to three active areas of cancer research, namely phenotypic plasticity, the cancer-immune interplay and the resistance of metastatic tumors to treatment. Our work outlines a set of questions and a tentative road map for further research in cancer ecology.

Selected aspects of avascular tumor growth reproduced by a hybrid model of cell dynamics and chemical kinetics

We here propose a hybrid computational framework to reproduce and analyze aspects of the avascular progression of a generic solid tumor. Our method first employs an individual-based approach to represent the population of tumor cells, which are distinguished in viable and necrotic agents. The active part of the disease is in turn differentiated according to a set of metabolic states. We then describe the spatio-temporal evolution of the concentration of oxygen and of tumor-secreted proteolytic enzymes using partial differential equations (PDEs). A differential equation finally governs the local degradation of the extracellular matrix (ECM) by the malignant mass. Numerical realizations of the model are run to reproduce tumor growth and invasion in a number scenarios that differ for cell properties (adhesiveness, duplication potential, proteolytic activity) and/or environmental conditions (level of tissue oxygenation and matrix density pattern). In particular, our simulations suggest that tumor aggressiveness, in terms of invasive depth and extension of necrotic tissue, can be reduced by (i) stable cell-cell contact interactions, (ii) poor tendency of malignant agents to chemotactically move upon oxygen gradients, and (iii) presence of an overdense matrix, if coupled by a disrupted proteolytic activity of the disease.

Second-Order Effects of Chemotherapy Pharmacodynamics and Pharmacokinetics on Tumor Regression and Cachexia

Drug dose response curves are ubiquitous in cancer biology, but these curves are often used to measure differential response in first-order effects: the effectiveness of increasing the cumulative dose delivered. In contrast, second-order effects (the variance of drug dose) are often ignored. Knowledge of second-order effects may improve the design of chemotherapy scheduling protocols, leading to improvements in tumor response without changing the total dose delivered. By considering treatment schedules with identical cumulative dose delivered, we characterize differential treatment outcomes resulting from high variance schedules (e.g. high dose, low dose) and low variance schedules (constant dose). We extend a previous framework used to quantify second-order effects, known as antifragility theory, to investigate the role of drug pharmacokinetics. Using a simple one-compartment model, we find that high variance schedules are effective for a wide range of cumulative dose values. Next, using a mouse-parameterized two-compartment model of 5-fluorouracil, we show that schedule viability depends on initial tumor volume. Finally, we illustrate the trade-off between tumor response and lean mass preservation. Mathematical modeling indicates that high variance dose schedules provide a potential path forward in mitigating the risk of chemotherapy-associated cachexia by preserving lean mass without sacrificing tumor response.

Leveraging multi-omics data to empower quantitative systems pharmacology in immuno-oncology

Understanding the intricate interactions of cancer cells with the tumor microenvironment (TME) is a pre-requisite for the optimization of immunotherapy. Mechanistic models such as quantitative systems pharmacology (QSP) provide insights into the TME dynamics and predict the efficacy of immunotherapy in virtual patient populations/digital twins but require vast amounts of multimodal data for parameterization. Large-scale datasets characterizing the TME are available due to recent advances in bioinformatics for multi-omics data. Here, we discuss the perspectives of leveraging omics-derived bioinformatics estimates to inform QSP models and circumvent the challenges of model calibration and validation in immuno-oncology.

Fractal calculus in tumor growth simulations: The proof is in the pudding

Mathematical modeling in oncology has a long history. Recently, mathematical models and their predictions have made inroads into prospective clinical trials with encouraging results. The goal of many such modeling efforts is to make predictions, either to clinician's choice therapy or into "optimal" therapy - often for individual patients. The mathematical oncology community rightfully puts great hope into predictive modeling and mechanistic digital twins - but with this great opportunity comes great responsibility. Mathematical models need to be rigorously calibrated and validated, and their predictive performance ascertained, before conclusions about predictions into the unknown can be drawn. The recent article "Modeling tumor growth using fractal calculus: Insights into tumor dynamics" (Golmankhaneh et al., 2023), applied fractal calculus to tumor growth data. In this short commentary, I raise concerns about the study design and interpretation. In its current form, this study is poised to put cancer patients at risk if interpreted as concluded by the authors.

Use of Individual-Based Mathematical Modelling to Understand More About Antibiotic Resistance Within-Host

To model complex systems, individual-based models (IBMs), sometimes called "agent-based models" (ABMs), describe a simplification of the system through an adequate representation of the elements. IBMs simulate the actions and interaction of discrete individuals/agents within a system in order to discover the pattern of behavior that comes from these interactions. Examples of individuals/agents in biological systems are individual immune cells and bacteria that act independently with their own unique attributes defined by behavioral rules. In IBMs, each of these agents resides in a spatial environment and interactions are guided by predefined rules. These rules are often simple and can be easily implemented. It is expected that following the interaction guided by these rules we will have a better understanding of agent-agent interaction as well as agent-environment interaction. Stochasticity described by probability distributions must be accounted for. Events that seldom occur such as the accumulation of rare mutations can be easily modeled.Thus, IBMs are able to track the behavior of each individual/agent within the model while also obtaining information on the results of their collective behaviors. The influence of impact of one agent with another can be captured, thus allowing a full representation of both direct and indirect causation on the aggregate results. This means that important new insights can be gained and hypotheses tested.

Mathematical modeling predicts pathways to successful implementation of combination TRAIL-producing oncolytic virus and PAC-1 to treat granulosa cell tumors of the ovary

The development of new cancer therapies requires multiple rounds of validation from in vitro and in vivo experiments before they can be considered for clinical trials. Mathematical models assist in this preclinical phase by combining experimental data with human parameters to provide guidance about potential therapeutic regimens to bring forward into trials. However, granulosa cell tumors of the ovary lack a relevant mouse model, complexifying preclinical drug development for this rare tumor. To bridge this gap, we established a mathematical model as a framework to explore the potential of using a tumor necrosis factor-related apoptosis-inducing ligand (TRAIL)-producing oncolytic vaccinia virus in combination with the chemotherapeutic agent first procaspase activating compound (PAC-1). We have previously shown that TRAIL and PAC-1 act synergistically on granulosa tumor cells. In line with our previous results, our current model predicts that, although it is unable to stop the tumor from growing in its current form, combination oral PAC-1 with oncolytic virus (OV) provides the best result compared to monotherapies. Encouragingly, our results suggest that increases to the OV infection rate can lead to the success of this combination therapy within a year. The model developed here can continue to be improved as more data become available, allowing for regimen-tailoring via virtual clinical trials, ultimately shepherding effective regimens into trials.

Stimuli-sensitive nano-drug delivery with programmable size changes to enhance accumulation of therapeutic agents in tumors

Nano-based drug delivery systems hold significant promise for cancer therapies. Presently, the poor accumulation of drug-carrying nanoparticles in tumors has limited their success. In this study, based on a combination of the paradigms of intravascular and extravascular drug release, an efficient nanosized drug delivery system with programmable size changes is introduced. Drug-loaded smaller nanoparticles (secondary nanoparticles), which are loaded inside larger nanoparticles (primary nanoparticles), are released within the microvascular network due to temperature field resulting from focused ultrasound. This leads to the scale of the drug delivery system decreasing by 7.5 to 150 times. Subsequently, smaller nanoparticles enter the tissue at high transvascular rates and achieve higher accumulation, leading to higher penetration depths. In response to the acidic pH of tumor microenvironment (according to the distribution of oxygen), they begin to release the drug doxorubicin at very slow rates (i.e., sustained release). To predict the performance and distribution of therapeutic agents, a semi-realistic microvascular network is first generated based on a sprouting angiogenesis model and the transport of therapeutic agents is then investigated based on a developed multi-compartment model. The results show that reducing the size of the primary and secondary nanoparticles can lead to higher cell death rate. In addition, tumor growth can be inhibited for a longer time by enhancing the bioavailability of the drug in the extracellular space. The proposed drug delivery system can be very promising in clinical applications. Furthermore, the proposed mathematical model is applicable to broader applications to predict the performance of drug delivery systems.

FitMultiCell: simulating and parameterizing computational models of multi-scale and multi-cellular processes

MOTIVATION

Biological tissues are dynamic and highly organized. Multi-scale models are helpful tools to analyse and understand the processes determining tissue dynamics. These models usually depend on parameters that need to be inferred from experimental data to achieve a quantitative understanding, to predict the response to perturbations, and to evaluate competing hypotheses. However, even advanced inference approaches such as approximate Bayesian computation (ABC) are difficult to apply due to the computational complexity of the simulation of multi-scale models. Thus, there is a need for a scalable pipeline for modeling, simulating, and parameterizing multi-scale models of multi-cellular processes.

RESULTS

Here, we present FitMultiCell, a computationally efficient and user-friendly open-source pipeline that can handle the full workflow of modeling, simulating, and parameterizing for multi-scale models of multi-cellular processes. The pipeline is modular and integrates the modeling and simulation tool Morpheus and the statistical inference tool pyABC. The easy integration of high-performance infrastructure allows to scale to computationally expensive problems. The introduction of a novel standard for the formulation of parameter inference problems for multi-scale models additionally ensures reproducibility and reusability. By applying the pipeline to multiple biological problems, we demonstrate its broad applicability, which will benefit in particular image-based systems biology.

AVAILABILITY AND IMPLEMENTATION

FitMultiCell is available open-source at https://gitlab.com/fitmulticell/fit.

Mathematical modeling of radiotherapy: impact of model selection on estimating minimum radiation dose for tumor control

Introduction

Radiation therapy (RT) is one of the most common anticancer therapies. Yet, current radiation oncology practice does not adapt RT dose for individual patients, despite wide interpatient variability in radiosensitivity and accompanying treatment response. We have previously shown that mechanistic mathematical modeling of tumor volume dynamics can simulate volumetric response to RT for individual patients and estimation personalized RT dose for optimal tumor volume reduction. However, understanding the implications of the choice of the underlying RT response model is critical when calculating personalized RT dose.

Methods

In this study, we evaluate the mathematical implications and biological effects of 2 models of RT response on dose personalization: (1) cytotoxicity to cancer cells that lead to direct tumor volume reduction (DVR) and (2) radiation responses to the tumor microenvironment that lead to tumor carrying capacity reduction (CCR) and subsequent tumor shrinkage. Tumor growth was simulated as logistic growth with pre-treatment dynamics being described in the proliferation saturation index (PSI). The effect of RT was simulated according to each respective model for a standard schedule of fractionated RT with 2 Gy weekday fractions. Parameter sweeps were evaluated for the intrinsic tumor growth rate and the radiosensitivity parameter for both models to observe the qualitative impact of each model parameter. We then calculated the minimum RT dose required for locoregional tumor control (LRC) across all combinations of the full range of radiosensitvity and proliferation saturation values.

Results

Both models estimate that patients with higher radiosensitivity will require a lower RT dose to achieve LRC. However, the two models make opposite estimates on the impact of PSI on the minimum RT dose for LRC: the DVR model estimates that tumors with higher PSI values will require a higher RT dose to achieve LRC, while the CCR model estimates that higher PSI values will require a lower RT dose to achieve LRC.

Discussion

Ultimately, these results show the importance of understanding which model best describes tumor growth and treatment response in a particular setting, before using any such model to make estimates for personalized treatment recommendations.

Treatment of evolving cancers will require dynamic decision support

Cancer research has traditionally focused on developing new agents, but an underexplored question is that of the dose and frequency of existing drugs. Based on the modus operandi established in the early days of chemotherapies, most drugs are administered according to predetermined schedules that seek to deliver the maximum tolerated dose and are only adjusted for toxicity. However, we believe that the complex, evolving nature of cancer requires a more dynamic and personalized approach. Chronicling the milestones of the field, we show that the impact of schedule choice crucially depends on processes driving treatment response and failure. As such, cancer heterogeneity and evolution dictate that a one-size-fits-all solution is unlikely-instead, each patient should be mapped to the strategy that best matches their current disease characteristics and treatment objectives (i.e. their 'tumorscape'). To achieve this level of personalization, we need mathematical modeling. In this perspective, we propose a five-step 'Adaptive Dosing Adjusted for Personalized Tumorscapes (ADAPT)' paradigm to integrate data and understanding across scales and derive dynamic and personalized schedules. We conclude with promising examples of model-guided schedule personalization and a call to action to address key outstanding challenges surrounding data collection, model development, and integration.

Reuniting philosophy and science to advance cancer research

Cancers rely on multiple, heterogeneous processes at different scales, pertaining to many biomedical fields. Therefore, understanding cancer is necessarily an interdisciplinary task that requires placing specialised experimental and clinical research into a broader conceptual, theoretical, and methodological framework. Without such a framework, oncology will collect piecemeal results, with scant dialogue between the different scientific communities studying cancer. We argue that one important way forward in service of a more successful dialogue is through greater integration of applied sciences (experimental and clinical) with conceptual and theoretical approaches, informed by philosophical methods. By way of illustration, we explore six central themes: (i) the role of mutations in cancer; (ii) the clonal evolution of cancer cells; (iii) the relationship between cancer and multicellularity; (iv) the tumour microenvironment; (v) the immune system; and (vi) stem cells. In each case, we examine open questions in the scientific literature through a philosophical methodology and show the benefit of such a synergy for the scientific and medical understanding of cancer.

Mathematical Modeling for the Assessment of Public Policies in the Cancer Health-Care System Implemented for the Colombian Case

The incidence of cancer has been constantly growing worldwide, placing pressure on health systems and increasing the costs associated with the treatment of cancer. In particular, low- and middle-income countries are expected to face serious challenges related to caring for the majority of the world's new cancer cases in the next 10 years. In this study, we propose a mathematical model that allows for the simulation of different strategies focused on public policies by combining spending and epidemiological indicators. In this way, strategies aimed at efficient spending management with better epidemiological indicators can be determined. For validation and calibration of the model, we use data from Colombia-which, according to the World Bank, is an upper-middle-income country. The results of the simulations using the proposed model, calibrated and validated for Colombia, indicate that the most effective strategy for reducing mortality and financial burden consists of a combination of early detection and greater efficiency of treatment in the early stages of cancer. This approach is found to present a 38% reduction in mortality rate and a 20% reduction in costs (% GDP) when compared to the baseline scenario. Hence, Colombia should prioritize comprehensive care models that focus on patient-centered care, prevention, and early detection.

Modelling the Tumour Microenvironment, but What Exactly Do We Mean by "Model"?

The Oxford English Dictionary includes 17 definitions for the word "model" as a noun and another 11 as a verb. Therefore, context is necessary to understand the meaning of the word model. For instance, "model railways" refer to replicas of railways and trains at a smaller scale and a "model student" refers to an exemplary individual. In some cases, a specific context, like cancer research, may not be sufficient to provide one specific meaning for model. Even if the context is narrowed, specifically, to research related to the tumour microenvironment, "model" can be understood in a wide variety of ways, from an animal model to a mathematical expression. This paper presents a review of different "models" of the tumour microenvironment, as grouped by different definitions of the word into four categories: model organisms, in vitro models, mathematical models and computational models. Then, the frequencies of different meanings of the word "model" related to the tumour microenvironment are measured from numbers of entries in the MEDLINE database of the United States National Library of Medicine at the National Institutes of Health. The frequencies of the main components of the microenvironment and the organ-related cancers modelled are also assessed quantitatively with specific keywords. Whilst animal models, particularly xenografts and mouse models, are the most commonly used "models", the number of these entries has been slowly decreasing. Mathematical models, as well as prognostic and risk models, follow in frequency, and these have been growing in use.

Comparing mechanism-based and machine learning models for predicting the effects of glucose accessibility on tumor cell proliferation

Glucose plays a central role in tumor metabolism and development and is a target for novel therapeutics. To characterize the response of cancer cells to blockade of glucose uptake, we collected time-resolved microscopy data to track the growth of MDA-MB-231 breast cancer cells. We then developed a mechanism-based, mathematical model to predict how a glucose transporter (GLUT1) inhibitor (Cytochalasin B) influences the growth of the MDA-MB-231 cells by limiting access to glucose. The model includes a parameter describing dose dependent inhibition to quantify both the total glucose level in the system and the glucose level accessible to the tumor cells. Four common machine learning models were also used to predict tumor cell growth. Both the mechanism-based and machine learning models were trained and validated, and the prediction error was evaluated by the coefficient of determination (R²). The random forest model provided the highest accuracy predicting cell dynamics (R² = 0.92), followed by the decision tree (R² = 0.89), k-nearest-neighbor regression (R² = 0.84), mechanism-based (R² = 0.77), and linear regression model (R² = 0.69). Thus, the mechanism-based model has a predictive capability comparable to machine learning models with the added benefit of elucidating biological mechanisms.

Fractional calculus in mathematical oncology

Even though, nowadays, cancer is one of the leading causes of death, too little is known about the behavior of this disease due to its unpredictability from one patient to another. Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, improvements are still necessary on these models. The primary objective of the present research is to prove the efficiency of fractional order calculus in mathematical oncology, more specifically in tumor growth modeling. For this, a generalization of the four most used differential equation models in tumor volume measurements fitting is realized, using the corresponding fractional order equivalent. Are established the fractional order Exponential, Logistic, Gompertz, General Bertalanffy-Pütter and Classical Bertalanffy-Pütter models for a treated and untreated dataset. The obtained results are compared by Mean Squared Error (MSE) with the integer order correspondent of each model. The results prove the superiority of the fractional order models. The MSE of fractional order models are reduced at least at half in comparison with the MSE of the integer order equivalent. It is demonstrated in this way that fractional order deterministic models can offer a good starting point in finding a proper mathematical model for tumor evolution prediction. Fractional calculus is a suitable method in this case due to its memory property, aspect that particularly characterizes biological processes.

Agent-based methods facilitate integrative science in cancer

In this opinion, we highlight agent-based modeling as a key tool for exploration of cell-cell and cell-environment interactions that drive cancer progression, therapeutic resistance, and metastasis. These biological phenomena are particularly suited to be captured at the cell-scale resolution possible only within agent-based or individual-based mathematical models. These modeling approaches complement experimental work (in vitro and in vivo systems) through parameterization and data extrapolation but also feed forward to drive new experiments that test model-generated predictions.

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.

Practical Understanding of Cancer Model Identifiability in Clinical Applications

Mathematical models are a core component in the foundation of cancer theory and have been developed as clinical tools in precision medicine. Modeling studies for clinical applications often assume an individual's characteristics can be represented as parameters in a model and are used to explain, predict, and optimize treatment outcomes. However, this approach relies on the identifiability of the underlying mathematical models. In this study, we build on the framework of an observing-system simulation experiment to study the identifiability of several models of cancer growth, focusing on the prognostic parameters of each model. Our results demonstrate that the frequency of data collection, the types of data, such as cancer proxy, and the accuracy of measurements all play crucial roles in determining the identifiability of the model. We also found that highly accurate data can allow for reasonably accurate estimates of some parameters, which may be the key to achieving model identifiability in practice. As more complex models required more data for identification, our results support the idea of using models with a clear mechanism that tracks disease progression in clinical settings. For such a model, the subset of model parameters associated with disease progression naturally minimizes the required data for model identifiability.

A hybrid spatiotemporal model of PCa dynamics and insights into optimal therapeutic strategies

Using a hybrid cellular automaton with stochastic elements, we investigate the effectiveness of multiple drug therapies on prostate cancer (PCa) growth. The ability of Androgen Deprivation Therapy to reduce PCa growth represents a milestone in prostate cancer treatment, nonetheless most patients eventually become refractory and develop castration-resistant prostate cancer. In recent years, a "second generation" drug called enzalutamide has been used to treat advanced PCa, or patients already exposed to chemotherapy that stopped responding to it. However, tumour resistance to enzalutamide is not well understood, and in this context, preclinical models and in silico experiments (numerical simulations) are key to understanding the mechanisms of resistance and to assessing therapeutic settings that may delay or prevent the onset of resistance. In our mathematical system, we incorporate cell phenotype switching to model the development of increased drug resistance, and consider the effect of the micro-environment dynamics on necrosis and apoptosis of the tumour cells. The therapeutic strategies that we explore include using a single drug (enzalutamide), and drug combinations (enzalutamide and everolimus or cabazitaxel) with different treatment schedules. Our results highlight the effectiveness of alternating therapies, especially alternating enzalutamide and cabazitaxel over a year, and a comparison is made with data taken from TRAMP mice to verify our findings.

The impact of tumor associated macrophages on tumor biology under the lens of mathematical modelling: A review

In this article, we review the role of mathematical modelling to elucidate the impact of tumor-associated macrophages (TAMs) in tumor progression and therapy design. We first outline the biology of TAMs, and its current application in tumor therapies, and their experimental methods that provide insights into tumor cell-macrophage interactions. We then focus on the mechanistic mathematical models describing the role of macrophages as drug carriers, the impact of macrophage polarized activation on tumor growth, and the role of tumor microenvironment (TME) parameters on the tumor-macrophage interactions. This review aims to identify the synergies between biological and mathematical approaches that allow us to translate knowledge on fundamental TAMs biology in addressing current clinical challenges.

Can the Kuznetsov Model Replicate and Predict Cancer Growth in Humans?

Several mathematical models to predict tumor growth over time have been developed in the last decades. A central aspect of such models is the interaction of tumor cells with immune effector cells. The Kuznetsov model (Kuznetsov et al. in Bull Math Biol 56(2):295–321, 1994) is the most prominent of these models and has been used as a basis for many other related models and theoretical studies. However, none of these models have been validated with large-scale real-world data of human patients treated with cancer immunotherapy. In addition, parameter estimation of these models remains a major bottleneck on the way to model-based and data-driven medical treatment. In this study, we quantitatively fit Kuznetsov’s model to a large dataset of 1472 patients, of which 210 patients have more than six data points, by estimating the model parameters of each patient individually. We also conduct a global practical identifiability analysis for the estimated parameters. We thus demonstrate that several combinations of parameter values could lead to accurate data fitting. This opens the potential for global parameter estimation of the model, in which the values of all or some parameters are fixed for all patients. Furthermore, by omitting the last two or three data points, we show that the model can be extrapolated and predict future tumor dynamics. This paves the way for a more clinically relevant application of mathematical tumor modeling, in which the treatment strategy could be adjusted in advance according to the model’s future predictions.

Integrating a dynamic central metabolism model of cancer cells with a hybrid 3D multiscale model for vascular hepatocellular carcinoma growth

We develop here a novel modelling approach with the aim of closing the conceptual gap between tumour-level metabolic processes and the metabolic processes occurring in individual cancer cells. In particular, the metabolism in hepatocellular carcinoma derived cell lines (HEPG2 cells) has been well characterized but implementations of multiscale models integrating this known metabolism have not been previously reported. We therefore extend a previously published multiscale model of vascular tumour growth, and integrate it with an experimentally verified network of central metabolism in HEPG2 cells. This resultant combined model links spatially heterogeneous vascular tumour growth with known metabolic networks within tumour cells and accounts for blood flow, angiogenesis, vascular remodelling and nutrient/growth factor transport within a growing tumour, as well as the movement of, and interactions between normal and cancer cells. Model simulations report for the first time, predictions of spatially resolved time courses of core metabolites in HEPG2 cells. These simulations can be performed at a sufficient scale to incorporate clinically relevant features of different tumour systems using reasonable computational resources. Our results predict larger than expected temporal and spatial heterogeneity in the intracellular concentrations of glucose, oxygen, lactate pyruvate, f16bp and Acetyl-CoA. The integrated multiscale model developed here provides an ideal quantitative framework in which to study the relationship between dosage, timing, and scheduling of anti-neoplastic agents and the physiological effects of tumour metabolism at the cellular level. Such models, therefore, have the potential to inform treatment decisions when drug response is dependent on the metabolic state of individual cancer cells.

SNAIL driven by a feed forward loop motif promotes TGFβinduced epithelial to mesenchymal transition

Epithelial to Mesenchymal Transition (EMT) plays an important role in tissue regeneration, embryonic development, and cancer metastasis. Several signaling pathways are known to regulate EMT, among which the modulation of TGFβ(Transforming Growth Factor-β) induced EMT is crucial in several cancer types. Several mathematical models were built to explore the role of core regulatory circuit of ZEB/miR-200, SNAIL/miR-34 double negative feedback loops in modulating TGFβinduced EMT. Different emergent behavior including tristability, irreversible switching, existence of hybrid EMT states were inferred though these models. Some studies have explored the role of TGFβreceptor activation, SMADs nucleocytoplasmic shuttling and complex formation. Recent experiments have revealed that MDM2 along with SMAD complex regulates SNAIL expression driven EMT. Encouraged by this, in the present study we developed a mathematical model for p53/MDM2 dependent TGFβinduced EMT regulation. Inclusion of p53 brings in an additional mechanistic perspective in exploring the EM transition. The network formulated comprises a C1FFL moderating SNAIL expression involving MDM2 and SMAD complex, which functions as a noise filter and persistent detector. The C1FFL was also observed to operate as a coincidence detector driving the SNAIL dependent downstream signaling into phenotypic switching decision. Systems modelling and analysis of the devised network, displayed interesting dynamic behavior, systems response to various inputs stimulus, providing a better understanding of p53/MDM2 dependent TGF-βinduced Epithelial to Mesenchymal Transition.

A global method for fast simulations of molecular dynamics in multiscale agent-based models of biological tissues

Agent-based models (ABMs) are a natural platform for capturing the multiple time and spatial scales in biological processes. However, these models are computationally expensive, especially when including molecular-level effects. The traditional approach to simulating this type of multiscale ABM is to solve a system of ordinary differential equations for the molecular events per cell. This significantly adds to the computational cost of simulations as the number of agents grows, which contributes to many ABMs being limited to around10 5 cells. We propose an approach that requires the same computational time independent of the number of agents. This speeds up the entire simulation by orders of magnitude, allowing for more thorough explorations of ABMs with even larger numbers of agents. We use two systems to show that the new method strongly agrees with the traditionally used approach. This computational strategy can be applied to a wide range of biological investigations.

Mathematical modeling of radiotherapy and its impact on tumor interactions with the immune system

Radiotherapy is a primary therapeutic modality widely utilized with curative intent. Traditionally tumor response was hypothesized to be due to high levels of cell death induced by irreparable DNA damage. However, the immunomodulatory aspect of radiation is now widely accepted. As such, interest into the combination of radiotherapy and immunotherapy is increasing, the synergy of which has the potential to improve tumor regression beyond that observed after either treatment alone. However, questions regarding the timing (sequential vs concurrent) and dose fractionation (hyper-, standard-, or hypo-fractionation) that result in improved anti-tumor immune responses, and thus potentially enhanced tumor inhibition, remain. Here we discuss the biological response to radiotherapy and its immunomodulatory properties before giving an overview of pre-clinical data and clinical trials concerned with answering these questions. Finally, we review published mathematical models of the impact of radiotherapy on tumor-immune interactions. Ranging from considering the impact of properties of the tumor microenvironment on the induction of anti-tumor responses, to the impact of choice of radiation site in the setting of metastatic disease, these models all have an underlying feature in common: the push towards personalized therapy.

A multicompartment mathematical model based on host immunity for dissecting COVID-19 heterogeneity

The determinants underlying the heterogeneity of coronavirus disease 2019 (COVID-19) remain to be elucidated. To systemically analyze the immunopathogenesis of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection, we built a multicompartment mathematical model based on immunological principles and typical COVID-19-related characteristics. This model integrated the trafficking of immune cells and cytokines among the secondary lymphoid organs, peripheral blood and lungs. Our results suggested that early-stage lymphopenia was related to lymphocyte chemotaxis, while prolonged lymphopenia in critically ill patients was associated with myeloid-derived suppressor cells. Furthermore, our model predicted that insufficient SARS-CoV-2-specific naïve T/B cell pools and ineffective activation of antigen-presenting cells (APCs) would cause delayed immunity activation, resulting in elevated viral load, low immunoglobulin level, etc. Overall, we provided a comprehensive view of the dynamics of host immunity after SARS-CoV-2 infection that enabled us to understand COVID-19 heterogeneity from systemic perspective.

Integrated computational and in vivo models reveal Key Insights into macrophage behavior during bone healing

Myeloid-derived monocyte and macrophages are key cells in the bone that contribute to remodeling and injury repair. However, their temporal polarization status and control of bone-resorbing osteoclasts and bone-forming osteoblasts responses is largely unknown. In this study, we focused on two aspects of monocyte/macrophage dynamics and polarization states over time: 1) the injury-triggered pro- and anti-inflammatory monocytes/macrophages temporal profiles, 2) the contributions of pro- versus anti-inflammatory monocytes/macrophages in coordinating healing response. Bone healing is a complex multicellular dynamic process. While traditional in vitro and in vivo experimentation may capture the behavior of select populations with high resolution, they cannot simultaneously track the behavior of multiple populations. To address this, we have used an integrated coupled ordinary differential equations (ODEs)-based framework describing multiple cellular species to in vivo bone injury data in order to identify and test various hypotheses regarding bone cell populations dynamics. Our approach allowed us to infer several biological insights including, but not limited to,: 1) anti-inflammatory macrophages are key for early osteoclast inhibition and pro-inflammatory macrophage suppression, 2) pro-inflammatory macrophages are involved in osteoclast bone resorptive activity, whereas osteoblasts promote osteoclast differentiation, 3) Pro-inflammatory monocytes/macrophages rise during two expansion waves, which can be explained by the anti-inflammatory macrophages-mediated inhibition phase between the two waves. In addition, we further tested the robustness of the mathematical model by comparing simulation results to an independent experimental dataset. Taken together, this novel comprehensive mathematical framework allowed us to identify biological mechanisms that best recapitulate bone injury data and that explain the coupled cellular population dynamics involved in the process. Furthermore, our hypothesis testing methodology could be used in other contexts to decipher mechanisms in complex multicellular processes.

Myeloid-derived monocytes/macrophages are key cells for bone remodeling and injury repair. However, their temporal polarization status and control of bone-resorbing osteoclasts and bone-forming osteoblasts responses is largely unknown. In this study, we focused on two aspects of monocyte/macrophage population dynamics: 1) the injury-triggered pro- and anti-inflammatory monocytes/macrophages temporal profiles, 2) the contributions of pro- versus anti-inflammatory monocytes/macrophages in coordinating healing response. In order to test various hypotheses regarding bone cell populations dynamics, we have integrated a coupled ordinary differential equations-based framework describing multiple cellular species to in vivo bone injury data. Our approach allowed us to infer several biological insights including: 1) anti-inflammatory macrophages are key for early osteoclast inhibition and pro-inflammatory macrophage suppression, 2) pro-inflammatory macrophages are involved in osteoclast bone resorptive activity, whereas osteoblasts promote osteoclast differentiation, 3) Pro-inflammatory monocytes/macrophages rise during two expansion waves, which can be explained by the anti-inflammatory macrophages-mediated inhibition phase between the two waves. Taken together, this mathematical framework allowed us to identify biological mechanisms that recapitulate bone injury data and that explain the coupled cellular population dynamics involved in the process.

Numerical Investigation on the Anti-Angiogenic Therapy-Induced Normalization in Solid Tumors

This study numerically analyzes the fluid flow and solute transport in a solid tumor to comprehensively examine the consequence of normalization induced by anti-angiogenic therapy on drug delivery. The current study leads to a more accurate model in comparison to previous research, as it incorporates a non-homogeneous real-human solid tumor including necrotic, semi-necrotic, and well-vascularized regions. Additionally, the model considers the effects of concurrently chemotherapeutic agents (three macromolecules of IgG, F(ab′)2, and F(ab′)) and different normalization intensities in various tumor sizes. Examining the long-term influence of normalization on the quality of drug uptake by necrotic area is another contribution of the present study. Results show that normalization decreases the interstitial fluid pressure (IFP) and spreads the pressure gradient and non-zero interstitial fluid velocity (IFV) into inner areas. Subsequently, wash-out of the drug from the tumor periphery is decreased. It is also demonstrated that normalization can improve the distribution of solute concentration in the interstitium. The efficiency of normalization is introduced as a function of the time course of perfusion, which depends on the tumor size, drug type, as well as normalization intensity, and consequently on the dominant mechanism of drug delivery. It is suggested to accompany anti-angiogenic therapy by F(ab′) in large tumor size (Req=2.79  cm) to improve reservoir behavior benefit from normalization. However, IgG is proposed as the better option in the small tumor (Req=0.46  cm), in which normalization finds the opportunity of enhancing uniformity of IgG average exposure by 22%. This study could provide a perspective for preclinical and clinical trials on how to take advantage of normalization, as an adjuvant treatment, in improving drug delivery into a non-homogeneous solid tumor.

Classical mathematical models for prediction of response to chemotherapy and immunotherapy

Classical mathematical models of tumor growth have shaped our understanding of cancer and have broad practical implications for treatment scheduling and dosage. However, even the simplest textbook models have been barely validated in real world-data of human patients. In this study, we fitted a range of differential equation models to tumor volume measurements of patients undergoing chemotherapy or cancer immunotherapy for solid tumors. We used a large dataset of 1472 patients with three or more measurements per target lesion, of which 652 patients had six or more data points. We show that the early treatment response shows only moderate correlation with the final treatment response, demonstrating the need for nuanced models. We then perform a head-to-head comparison of six classical models which are widely used in the field: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. Several models provide a good fit to tumor volume measurements, with the Gompertz model providing the best balance between goodness of fit and number of parameters. Similarly, when fitting to early treatment data, the general Bertalanffy and Gompertz models yield the lowest mean absolute error to forecasted data, indicating that these models could potentially be effective at predicting treatment outcome. In summary, we provide a quantitative benchmark for classical textbook models and state-of-the art models of human tumor growth. We publicly release an anonymized version of our original data, providing the first benchmark set of human tumor growth data for evaluation of mathematical models.

Mathematical oncology uses quantitative models for prediction of tumor growth and treatment response. The theoretical foundation of mathematical oncology is provided by six classical mathematical models: the Exponential, Logistic, Classic Bertalanffy, General Bertalanffy, Classic Gompertz and General Gompertz model. These models have been introduced decades ago, have been used in thousands of scientific articles and are part of textbooks and curricula in mathematical oncology. However, these models have not been systematically tested in clinical data from actual patients. In this study, we have collected quantitative tumor volume measurements from thousands of patients in five large clinical trials of cancer immunotherapy. We use this dataset to systematically investigate how accurately mathematical models can describe tumor growth, showing that there are pronounced differences between models. In addition, we show that two of these models can predict tumor response to immunotherapy and chemotherapy at later time points when trained on early tumor growth dynamics. Thus, our article closes a conceptual gap in the literature and at the same time provides a simple tool to predict response to chemotherapy and immunotherapy on the level of individual patients.

Optimal regulation of tumour-associated neutrophils in cancer progression

In a tumour microenvironment, tumour-associated neutrophils could display two opposing differential phenotypes: anti-tumour (N1) and pro-tumour (N2) effector cells. Converting N2 to N1 neutrophils provides innovative therapies for cancer treatment. In this study, a mathematical model for N1-N2 dynamics describing the cancer survival and immune inhibition in response to TGF-β and IFN-β is considered. The effects of exogenous intervention of TGF-β inhibitor and IFN-β are examined in order to enhance N1 recruitment to combat tumour progression. Our approach employs optimal control theory to determine drug infusion protocols that could minimize tumour volume with least administration cost possible. Four optimal control scenarios corresponding to different therapeutic strategies are explored, namely, TGF-β inhibitor control only, IFN-β control only, concomitant TGF-β inhibitor and IFN-β controls, and alternating TGF-β inhibitor and IFN-β controls. For each scheme, different initial conditions are varied to depict different pathophysiological condition of a cancer patient, leading to adaptive treatment schedule. TGF-β inhibitor and IFN-β drug dosages, total drug amount, infusion times and relative cost of drug administrations are obtained under various circumstances. The control strategies achieved could guide in designing individualized therapeutic protocols.

Predictive Radiation Oncology - A New NCI-DOE Scientific Space and Community

With a widely attended virtual kickoff event on January 29, 2021, the National Cancer Institute (NCI) and the Department of Energy (DOE) launched a series of 4 interactive, interdisciplinary workshops-and a final concluding "World Café" on March 29, 2021-focused on advancing computational approaches for predictive oncology in the clinical and research domains of radiation oncology. These events reflect 3,870 human hours of virtual engagement with representation from 8 DOE national laboratories and the Frederick National Laboratory for Cancer Research (FNL), 4 research institutes, 5 cancer centers, 17 medical schools and teaching hospitals, 5 companies, 5 federal agencies, 3 research centers, and 27 universities. Here we summarize the workshops by first describing the background for the workshops. Participants identified twelve key questions-and collaborative parallel ideas-as the focus of work going forward to advance the field. These were then used to define short-term and longer-term "Blue Sky" goals. In addition, the group determined key success factors for predictive oncology in the context of radiation oncology, if not the future of all of medicine. These are: cross-discipline collaboration, targeted talent development, development of mechanistic mathematical and computational models and tools, and access to high-quality multiscale data that bridges mechanisms to phenotype. The workshop participants reported feeling energized and highly motivated to pursue next steps together to address the unmet needs in radiation oncology specifically and in cancer research generally and that NCI and DOE project goals align at the convergence of radiation therapy and advanced computing.

A mathematical model for phenotypic heterogeneity in breast cancer with implications for therapeutic strategies

Inevitably, almost all cancer patients develop resistance to targeted therapy. Intratumour heterogeneity is a major cause of drug resistance. Mathematical models that explain experiments quantitatively are useful in understanding the origin of intratumour heterogeneity, which then could be used to explore scenarios for efficacious therapy. Here, we develop a mathematical model to investigate intratumour heterogeneity in breast cancer by exploiting the observation that HER2+ and HER2- cells could divide symmetrically or asymmetrically. Our predictions for the evolution of cell fractions are in quantitative agreement with single-cell experiments. Remarkably, the colony size of HER2+ cells emerging from a single HER2- cell (or vice versa), which occurs in about four cell doublings, also agrees with experimental results, without tweaking any parameter in the model. The theory explains experimental data on the responses of breast tumours under different treatment protocols. We then used the model to predict that, not only the order of two drugs, but also the treatment period for each drug and the tumour cell plasticity could be manipulated to improve the treatment efficacy. Mathematical models, when integrated with data on patients, make possible exploration of a broad range of parameters readily, which might provide insights in devising effective therapies.

A phenotype-structured model to reproduce the avascular growth of a tumor and its interaction with the surrounding environment

We here propose a one-dimensional spatially explicit phenotype-structured model to analyze selected aspects of avascular tumor progression. In particular, our approach distinguishes viable and necrotic cell fractions. The metabolically active part of the disease is, in turn, differentiated according to a continuous trait, that identifies cell variants with different degrees of motility and proliferation potential. A parabolic partial differential equation (PDE) then governs the spatio-temporal evolution of the phenotypic distribution of active cells within the host tissue. In this respect, active tumor agents are allowed to duplicate, move upon haptotactic and pressure stimuli, and eventually undergo necrosis. The mutual influence between the emerging malignancy and its environment (in terms of molecular landscape) is implemented by coupling the evolution law of the viable tumor mass with a parabolic PDE for oxygen kinetics and a differential equation that accounts for local consumption of extracellular matrix (ECM) elements. The resulting numerical realizations reproduce tumor growth and invasion in a number scenarios that differ for cell properties (i.e., individual migratory behavior, duplication and mutation potential) and environmental conditions (i.e., level of tissue oxygenation and homogeneity in the initial matrix profile). In particular, our simulations show that, in all cases, more mobile cell variants occupy the front edge of the tumor, whereas more proliferative clones are selected at the more internal regions. A necrotic core constantly occupies the bulk of the mass due to nutrient deprivation. This work may eventually suggest some biomedical strategies to partially reduce tumor aggressiveness, i.e., to enhance necrosis of malignant tissue and to promote the presence of more proliferative cell phenotypes over more invasive ones.

Forecasting Individual Patient Response to Radiation Therapy in Head and Neck Cancer With a Dynamic Carrying Capacity Model

Purpose:

To model and predict individual patient responses to radiation therapy.

Methods and Materials:

We modeled tumor dynamics as logistic growth and the effect of radiation as a reduction in the tumor carrying capacity, motivated by the effect of radiation on the tumor microenvironment. The model was assessed on weekly tumor volume data collected for 2 independent cohorts of patients with head and neck cancer from the H. Lee Moffitt Cancer Center (MCC) and the MD Anderson Cancer Center (MDACC) who received 66 to 70 Gy in standard daily fractions or with accelerated fractionation. To predict response to radiation therapy for individual patients, we developed a new forecasting framework that combined the learned tumor growth rate and carrying capacity reduction fraction (δ) distribution with weekly measurements of tumor volume reduction for a given test patient to estimate δ, which was used to predict patient-specific outcomes.

Results:

The model fit data from MCC with high accuracy with patient-specific δ and a fixed tumor growth rate across all patients. The model fit data from an independent cohort from MDACC with comparable accuracy using the tumor growth rate learned from the MCC cohort, showing transferability of the growth rate. The forecasting framework predicted patient-specific outcomes with 76% sensitivity and 83% specificity for locoregional control and 68% sensitivity and 85% specificity for disease-free survival with the inclusion of 4 on-treatment tumor volume measurements.

Conclusions:

These results demonstrate that our simple mathematical model can describe a variety of tumor volume dynamics. Furthermore, combining historically observed patient responses with a few patient-specific tumor volume measurements allowed for the accurate prediction of patient outcomes, which may inform treatment adaptation and personalization.

Tumor-immune ecosystem dynamics define an individual Radiation Immune Score to predict pan-cancer radiocurability

Radiotherapy efficacy is the result of radiation-mediated cytotoxicity coupled with stimulation of antitumor immune responses. We develop an in silico 3-dimensional agent-based model of diverse tumor-immune ecosystems (TIES) represented as anti- or pro-tumor immune phenotypes. We validate the model in 10,469 patients across 31 tumor types by demonstrating that clinically detected tumors have pro-tumor TIES. We then quantify the likelihood radiation induces antitumor TIES shifts toward immune-mediated tumor elimination by developing the individual Radiation Immune Score (iRIS). We show iRIS distribution across 31 tumor types is consistent with the clinical effectiveness of radiotherapy, and in combination with a molecular radiosensitivity index (RSI) combines to predict pan-cancer radiocurability. We show that iRIS correlates with local control and survival in a separate cohort of 59 lung cancer patients treated with radiation. In combination, iRIS and RSI predict radiation-induced TIES shifts in individual patients and identify candidates for radiation de-escalation and treatment escalation. This is the first clinically and biologically validated computational model to simulate and predict pan-cancer response and outcomes via the perturbation of the TIES by radiotherapy.

Darwinian Approaches for Cancer Treatment: Benefits of Mathematical Modeling

One of the major problems of traditional anti-cancer treatments is that they lead to the emergence of treatment-resistant cells, which results in treatment failure. To avoid or delay this phenomenon, it is relevant to take into account the eco-evolutionary dynamics of tumors. Designing evolution-based treatment strategies may help overcoming the problem of drug resistance. In particular, a promising candidate is adaptive therapy, a containment strategy which adjusts treatment cycles to the evolution of the tumors in order to keep the population of treatment-resistant cells under control. Mathematical modeling is a crucial tool to understand the dynamics of cancer in response to treatments, and to make predictions about the outcomes of these treatments. In this review, we highlight the benefits of in silico modeling to design adaptive therapy strategies, and to assess whether they could effectively improve treatment outcomes. Specifically, we review how two main types of models (i.e., mathematical models based on Lotka-Volterra equations and agent-based models) have been used to model tumor dynamics in response to adaptive therapy. We give examples of the advances they permitted in the field of adaptive therapy and discuss about how these models can be integrated in experimental approaches and clinical trial design.

Mechanical Pressure Driving Proteoglycan Expression in Mammographic Density: a Self-perpetuating Cycle?

Regions of high mammographic density (MD) in the breast are characterised by a proteoglycan (PG)-rich fibrous stroma, where PGs mediate aligned collagen fibrils to control tissue stiffness and hence the response to mechanical forces. Literature is accumulating to support the notion that mechanical stiffness may drive PG synthesis in the breast contributing to MD. We review emerging patterns in MD and other biological settings, of a positive feedback cycle of force promoting PG synthesis, such as in articular cartilage, due to increased pressure on weight bearing joints. Furthermore, we present evidence to suggest a pro-tumorigenic effect of increased mechanical force on epithelial cells in contexts where PG-mediated, aligned collagen fibrous tissue abounds, with implications for breast cancer development attributable to high MD. Finally, we summarise means through which this positive feedback mechanism of PG synthesis may be intercepted to reduce mechanical force within tissues and thus reduce disease burden.

A mathematical framework for modelling 3D cell motility: applications to glioblastoma cell migration

The collection of 3D cell tracking data from live images of micro-tissues is a recent innovation made possible due to advances in imaging techniques. As such there is increased interest in studying cell motility in 3D in vitro model systems but a lack of rigorous methodology for analysing the resulting data sets. One such instance of the use of these in vitro models is in the study of cancerous tumours. Growing multicellular tumour spheroids in vitro allows for modelling of the tumour microenvironment and the study of tumour cell behaviours, such as migration, which improves understanding of these cells and in turn could potentially improve cancer treatments. In this paper, we present a workflow for the rigorous analysis of 3D cell tracking data, based on the persistent random walk model, but adaptable to other biologically informed mathematical models. We use statistical measures to assess the fit of the model to the motility data and to estimate model parameters and provide confidence intervals for those parameters, to allow for parametrization of the model taking correlation in the data into account. We use in silico simulations to validate the workflow in 3D before testing our method on cell tracking data taken from in vitro experiments on glioblastoma tumour cells, a brain cancer with a very poor prognosis. The presented approach is intended to be accessible to both modellers and experimentalists alike in that it provides tools for uncovering features of the data set that may suggest amendments to future experiments or modelling attempts.

A Mathematical Study of the Influence of Hypoxia and Acidity on the Evolutionary Dynamics of Cancer

Hypoxia and acidity act as environmental stressors promoting selection for cancer cells with a more aggressive phenotype. As a result, a deeper theoretical understanding of the spatio-temporal processes that drive the adaptation of tumour cells to hypoxic and acidic microenvironments may open up new avenues of research in oncology and cancer treatment. We present a mathematical model to study the influence of hypoxia and acidity on the evolutionary dynamics of cancer cells in vascularised tumours. The model is formulated as a system of partial integro-differential equations that describe the phenotypic evolution of cancer cells in response to dynamic variations in the spatial distribution of three abiotic factors that are key players in tumour metabolism: oxygen, glucose and lactate. The results of numerical simulations of a calibrated version of the model based on real data recapitulate the eco-evolutionary spatial dynamics of tumour cells and their adaptation to hypoxic and acidic microenvironments. Moreover, such results demonstrate how nonlinear interactions between tumour cells and abiotic factors can lead to the formation of environmental gradients which select for cells with phenotypic characteristics that vary with distance from intra-tumour blood vessels, thus promoting the emergence of intra-tumour phenotypic heterogeneity. Finally, our theoretical findings reconcile the conclusions of earlier studies by showing that the order in which resistance to hypoxia and resistance to acidity arise in tumours depend on the ways in which oxygen and lactate act as environmental stressors in the evolutionary dynamics of cancer cells.

Superlinear growth reveals the Allee effect in tumors

Integrating experimental data into ecological models plays a central role in understanding biological mechanisms that drive tumor progression where such knowledge can be used to develop new therapeutic strategies. While the current studies emphasize the role of competition among tumor cells, they fail to explain recently observed superlinear growth dynamics across human tumors. Here we study tumor growth dynamics by developing a model that incorporates evolutionary dynamics inside tumors with tumor-microenvironment interactions. Our results reveal that tumor cells' ability to manipulate the environment and induce angiogenesis drives superlinear growth-a process compatible with the Allee effect. In light of this understanding, our model suggests that, for high-risk tumors that have a higher growth rate, suppressing angiogenesis can be the appropriate therapeutic intervention.

Identifying key questions in the ecology and evolution of cancer

The application of evolutionary and ecological principles to cancer prevention and treatment, as well as recognizing cancer as a selection force in nature, has gained impetus over the last 50 years. Following the initial theoretical approaches that combined knowledge from interdisciplinary fields, it became clear that using the eco-evolutionary framework is of key importance to understand cancer. We are now at a pivotal point where accumulating evidence starts to steer the future directions of the discipline and allows us to underpin the key challenges that remain to be addressed. Here, we aim to assess current advancements in the field and to suggest future directions for research. First, we summarize cancer research areas that, so far, have assimilated ecological and evolutionary principles into their approaches and illustrate their key importance. Then, we assembled 33 experts and identified 84 key questions, organized around nine major themes, to pave the foundations for research to come. We highlight the urgent need for broadening the portfolio of research directions to stimulate novel approaches at the interface of oncology and ecological and evolutionary sciences. We conclude that progressive and efficient cross-disciplinary collaborations that draw on the expertise of the fields of ecology, evolution and cancer are essential in order to efficiently address current and future questions about cancer.

Computational modeling reveals a key role for polarized myeloid cells in controlling osteoclast activity during bone injury repair

Bone-forming osteoblasts and -resorbing osteoclasts control bone injury repair, and myeloid-derived cells such as monocytes and macrophages are known to influence their behavior. However, precisely how these multiple cell types coordinate and regulate each other over time within the bone marrow to restore bone is difficult to dissect using biological approaches. Conversely, mathematical modeling lends itself well to this challenge. Therefore, we generated an ordinary differential equation (ODE) model powered by experimental data (osteoblast, osteoclast, bone volume, pro- and anti-inflammatory myeloid cells) obtained from intra-tibially injured mice. Initial ODE results using only osteoblast/osteoclast populations demonstrated that bone homeostasis could not be recovered after injury, but this issue was resolved upon integration of pro- and anti-inflammatory myeloid population dynamics. Surprisingly, the ODE revealed temporal disconnects between the peak of total bone mineralization/resorption, and osteoblast/osteoclast numbers. Specifically, the model indicated that osteoclast activity must vary greatly (> 17-fold) to return the bone volume to baseline after injury and suggest that osteoblast/osteoclast number alone is insufficient to predict bone the trajectory of bone repair. Importantly, the values of osteoclast activity fall within those published previously. These data underscore the value of mathematical modeling approaches to understand and reveal new insights into complex biological processes.

Fractional Mathematical Oncology: On the potential of non-integer order calculus applied to interdisciplinary models

Mathematical Oncology investigates cancer-related phenomena through mathematical models as comprehensive as possible. Accordingly, an interdisciplinary approach involving concepts from biology to materials science can provide a deeper understanding of biological systems pertaining the disease. In this context, fractional calculus (also referred to as non-integer order) is a branch in mathematical analysis whose tools can describe complex phenomena comprising different time and space scales. Fractional-order models may allow a better description and understanding of oncological particularities, potentially contributing to decision-making in areas of interest such as tumor evolution, early diagnosis techniques and personalized treatment therapies. By following a phenomenological (i.e. mechanistic) approach, the present study surveys and explores different aspects of Fractional Mathematical Oncology, reviewing and discussing recent developments in view of their prospective applications.