The 2019 mathematical oncology roadmap

Agent-based modeling in cancer biomedicine: applications and tools for calibration and validation

Computational models are not just appealing because they can simulate and predict the development of biological phenomena across multiple spatial and temporal scales, but also because they can integrate information from well-established in vitro and in vivo models and test new hypotheses in cancer biomedicine. Agent-based models and simulations are especially interesting candidates among computational modeling procedures in cancer research due to the capability to, for instance, recapitulate the dynamics of neoplasia and tumor - host interactions. Yet, the absence of methods to validate the consistency of the results across scales can hinder adoption by turning fine-tuned models into black boxes. This review compiles relevant literature that explores strategies to leverage high-fidelity simulations of multi-scale, or multi-level, cancer models with a focus on verification approached as simulation calibration. We consolidate our review with an outline of modern approaches for agent-based models' validation and provide an ambitious outlook toward rigorous and reliable calibration.

Harnessing Flex Point Symmetry to Estimate Logistic Tumor Population Growth

The observed time evolution of a population is well approximated by a logistic growth function in many research fields, including oncology, ecology, chemistry, demography, economy, linguistics, and artificial neural networks. Initial growth is exponential, then decelerates as the population approaches its limit size, i.e., the carrying capacity. In mathematical oncology, the tumor carrying capacity has been postulated to be dynamically evolving as the tumor overcomes several evolutionary bottlenecks and, thus, to be patient specific. As the relative tumor-over-carrying capacity ratio may be predictive and prognostic for tumor growth and treatment response dynamics, it is paramount to estimate it from limited clinical data. We show that exploiting the logistic function's rotation symmetry can help estimate the population's growth rate and carry capacity from fewer data points than conventional regression approaches. We test this novel approach against published pan-cancer animal and human breast cancer data, achieving a 30% to 40% reduction in the time at which subsequent data collection is necessary to estimate the logistic growth rate and carrying capacity correctly. These results could improve tumor dynamics forecasting and augment the clinical decision-making process.

Data Science Opportunities To Improve Radiotherapy Planning and Clinical Decision Making

Radiotherapy aims to achieve a high tumor control probability while minimizing damage to normal tissues. Personalizing radiotherapy treatments for individual patients, therefore, depends on integrating physical treatment planning with predictive models of tumor control and normal tissue complications. Predictive models could be improved using a wide range of rich data sources, including tumor and normal tissue genomics, radiomics, and dosiomics. Deep learning will drive improvements in classifying normal tissue tolerance, predicting intra-treatment tumor changes, tracking accumulated dose distributions, and quantifying the tumor response to radiotherapy based on imaging. Mechanistic patient-specific computer simulations ('digital twins') could also be used to guide adaptive radiotherapy. Overall, we are entering an era where improved modeling methods will allow the use of newly available data sources to better guide radiotherapy treatments.

Merging Metabolic Modeling and Imaging for Screening Therapeutic Targets in Colorectal Cancer

Cancer-associated fibroblasts (CAFs) play a key role in metabolic reprogramming and are well-established contributors to drug resistance in colorectal cancer (CRC). To exploit this metabolic crosstalk, we integrated a systems biology approach that identified key metabolic targets in a data-driven method and validated them experimentally. This process involved a novel machine learning-based method to computationally screen, in a high-throughput manner, the effects of enzyme perturbations predicted by a computational model of CRC metabolism. This approach reveals the network-wide effects of metabolic perturbations. Our results highlighted hexokinase (HK) as the crucial target, which subsequently became our focus for experimental validation using patient-derived tumor organoids (PDTOs). Through metabolic imaging and viability assays, we found that PDTOs cultured in CAF-conditioned media exhibited increased sensitivity to HK inhibition, confirming the model predictions. Our approach emphasizes the critical role of integrating computational and experimental techniques in exploring and exploiting CRC-CAF crosstalk.

Pediatric Leukemia Roadmaps Are a Guide for Positive Metastatic Bone Sarcoma Trials

ALL cures require many MRD therapies. This strategy should drive experiments and trials in metastatic bone sarcomas.

Integrating frontiers: a holistic, quantum and evolutionary approach to conquering cancer through systems biology and multidisciplinary synergy

Cancer therapy is facing increasingly significant challenges, marked by a wide range of techniques and research efforts centered around somatic mutations, precision oncology, and the vast amount of big data. Despite this abundance of information, the quest to cure cancer often seems more elusive, with the "war on cancer" yet to deliver a definitive victory. A particularly pressing issue is the development of tumor treatment resistance, highlighting the urgent need for innovative approaches. Evolutionary, Quantum Biology and System Biology offer a promising framework for advancing experimental cancer research. By integrating theoretical studies, translational methods, and flexible multidisciplinary clinical research, there's potential to enhance current treatment strategies and improve outcomes for cancer patients. Establishing stronger links between evolutionary, quantum, entropy and chaos principles and oncology could lead to more effective treatments that leverage an understanding of the tumor's evolutionary dynamics, paving the way for novel methods to control and mitigate cancer. Achieving these objectives necessitates a commitment to multidisciplinary and interprofessional collaboration at the heart of both research and clinical endeavors in oncology. This entails dismantling silos between disciplines, encouraging open communication and data sharing, and integrating diverse viewpoints and expertise from the outset of research projects. Being receptive to new scientific discoveries and responsive to how patients react to treatments is also crucial. Such strategies are key to keeping the field of oncology at the forefront of effective cancer management, ensuring patients receive the most personalized and effective care. Ultimately, this approach aims to push the boundaries of cancer understanding, treating it as a manageable chronic condition, aiming to extend life expectancy and enhance patient quality of life.

Nicotinamide metabolism face-off between macrophages and fibroblasts manipulates the microenvironment in gastric cancer

Immune checkpoint blockade has led to breakthroughs in the treatment of advanced gastric cancer. However, the prominent heterogeneity in gastric cancer, notably the heterogeneity of the tumor microenvironment, highlights the idea that the antitumor response is a reflection of multifactorial interactions. Through transcriptomic analysis and dynamic plasma sample analysis, we identified a metabolic "face-off" mechanism within the tumor microenvironment, as shown by the dual prognostic significance of nicotinamide metabolism. Specifically, macrophages and fibroblasts expressing the rate-limiting enzymes nicotinamide phosphoribosyltransferase and nicotinamide N-methyltransferase, respectively, regulate the nicotinamide/1-methylnicotinamide ratio and CD8+ T cell function. Mechanistically, nicotinamide N-methyltransferase is transcriptionally activated by the NOTCH pathway transcription factor RBP-J and is further inhibited by macrophage-derived extracellular vesicles containing nicotinamide phosphoribosyltransferase via the SIRT1/NICD axis. Manipulating nicotinamide metabolism through autologous injection of extracellular vesicles restored CD8+ T cell cytotoxicity and the anti-PD-1 response in gastric cancer.

Patient-Specific, Mechanistic Models of Tumor Growth Incorporating Artificial Intelligence and Big Data

Despite the remarkable advances in cancer diagnosis, treatment, and management over the past decade, malignant tumors remain a major public health problem. Further progress in combating cancer may be enabled by personalizing the delivery of therapies according to the predicted response for each individual patient. The design of personalized therapies requires the integration of patient-specific information with an appropriate mathematical model of tumor response. A fundamental barrier to realizing this paradigm is the current lack of a rigorous yet practical mathematical theory of tumor initiation, development, invasion, and response to therapy. We begin this review with an overview of different approaches to modeling tumor growth and treatment, including mechanistic as well as data-driven models based on big data and artificial intelligence. We then present illustrative examples of mathematical models manifesting their utility and discuss the limitations of stand-alone mechanistic and data-driven models. We then discuss the potential of mechanistic models for not only predicting but also optimizing response to therapy on a patient-specific basis. We describe current efforts and future possibilities to integrate mechanistic and data-driven models. We conclude by proposing five fundamental challenges that must be addressed to fully realize personalized care for cancer patients driven by computational models.

Mathematical model of MMC chemotherapy for non-invasive bladder cancer treatment

Mitomycin-C (MMC) chemotherapy is a well-established anti-cancer treatment for non-muscle-invasive bladder cancer (NMIBC). However, despite comprehensive biological research, the complete mechanism of action and an ideal regimen of MMC have not been elucidated. In this study, we present a theoretical investigation of NMIBC growth and its treatment by continuous administration of MMC chemotherapy. Using temporal ordinary differential equations (ODEs) to describe cell populations and drug molecules, we formulated the first mathematical model of tumor-immune interactions in the treatment of MMC for NMIBC, based on biological sources. Several hypothetical scenarios for NMIBC under the assumption that tumor size correlates with cell count are presented, depicting the evolution of tumors classified as small, medium, and large. These scenarios align qualitatively with clinical observations of lower recurrence rates for tumor size ≤ 30[mm] with MMC treatment, demonstrating that cure appears up to a theoretical x[mm] tumor size threshold, given specific parameters within a feasible biological range. The unique use of mole units allows to introduce a new method for theoretical pre-treatment assessments by determining MMC drug doses required for a cure. In this way, our approach provides initial steps toward personalized MMC chemotherapy for NMIBC patients, offering the possibility of new insights and potentially holding the key to unlocking some of its mysteries.

A war on many fronts: cross disciplinary approaches for novel cancer treatment strategies

Cancer is a disease characterized by uncontrolled cellular growth where cancer cells take advantage of surrounding cellular populations to obtain resources and promote invasion. Carcinomas are the most common type of cancer accounting for almost 90% of cancer cases. One of the major subtypes of carcinomas are adenocarcinomas, which originate from glandular cells that line certain internal organs. Cancers such as breast, prostate, lung, pancreas, colon, esophageal, kidney are often adenocarcinomas. Current treatment strategies include surgery, chemotherapy, radiation, targeted therapy, and more recently immunotherapy. However, patients with adenocarcinomas often develop resistance or recur after the first line of treatment. Understanding how networks of tumor cells interact with each other and the tumor microenvironment is crucial to avoid recurrence, resistance, and high-dose therapy toxicities. In this review, we explore how mathematical modeling tools from different disciplines can aid in the development of effective and personalized cancer treatment strategies. Here, we describe how concepts from the disciplines of ecology and evolution, economics, and control engineering have been applied to mathematically model cancer dynamics and enhance treatment strategies.

Theranostic digital twins: Concept, framework and roadmap towards personalized radiopharmaceutical therapies

Radiopharmaceutical therapy (RPT) is a rapidly developing field of nuclear medicine, with several RPTs already well established in the treatment of several different types of cancers. However, the current approaches to RPTs often follow a somewhat inflexible "one size fits all" paradigm, where patients are administered the same amount of radioactivity per cycle regardless of their individual characteristics and features. This approach fails to consider inter-patient variations in radiopharmacokinetics, radiation biology, and immunological factors, which can significantly impact treatment outcomes. To address this limitation, we propose the development of theranostic digital twins (TDTs) to personalize RPTs based on actual patient data. Our proposed roadmap outlines the steps needed to create and refine TDTs that can optimize radiation dose to tumors while minimizing toxicity to organs at risk. The TDT models incorporate physiologically-based radiopharmacokinetic (PBRPK) models, which are additionally linked to a radiobiological optimizer and an immunological modulator, taking into account factors that influence RPT response. By using TDT models, we envisage the ability to perform virtual clinical trials, selecting therapies towards improved treatment outcomes while minimizing risks associated with secondary effects. This framework could empower practitioners to ultimately develop tailored RPT solutions for subgroups and individual patients, thus improving the precision, accuracy, and efficacy of treatments while minimizing risks to patients. By incorporating TDT models into RPTs, we can pave the way for a new era of precision medicine in cancer treatment.

Discrete and continuous mathematical models of sharp-fronted collective cell migration and invasion

Mathematical models describing the spatial spreading and invasion of populations of biological cells are often developed in a continuum modelling framework using reaction-diffusion equations. While continuum models based on linear diffusion are routinely employed and known to capture key experimental observations, linear diffusion fails to predict well-defined sharp fronts that are often observed experimentally. This observation has motivated the use of nonlinear degenerate diffusion; however, these nonlinear models and the associated parameters lack a clear biological motivation and interpretation. Here, we take a different approach by developing a stochastic discrete lattice-based model incorporating biologically inspired mechanisms and then deriving the reaction-diffusion continuum limit. Inspired by experimental observations, agents in the simulation deposit extracellular material, which we call a substrate, locally onto the lattice, and the motility of agents is taken to be proportional to the substrate density. Discrete simulations that mimic a two-dimensional circular barrier assay illustrate how the discrete model supports both smooth and sharp-fronted density profiles depending on the rate of substrate deposition. Coarse-graining the discrete model leads to a novel partial differential equation (PDE) model whose solution accurately approximates averaged data from the discrete model. The new discrete model and PDE approximation provide a simple, biologically motivated framework for modelling the spreading, growth and invasion of cell populations with well-defined sharp fronts. Open-source Julia code to replicate all results in this work is available on GitHub.

A novel multiscale framework for delineating cancer evolution from subclonal compositions

OBJECTIVE

Owing to the heterogeneity in the evolution of cancer, distinguishing between diverse growth patterns and predicting long-term outcomes based on short-term measurements poses a great challenge.

METHODS

A novel multiscale framework is proposed to unravel the connections between the population dynamics of cancer growth (i.e., aggressive, bounded, and indolent) and the cellular-subclonal dynamics of cancer evolution. This framework employs the non-negative lasso (NN-LASSO) algorithm to forge a link between an ordinary differential equation (ODE)-based population model and a cellular evolution model.

RESULTS

The findings of our current work not only affirm the impact of subclonal composition on growth dynamics but also identify two significant subclones within heterogeneous growth patterns. Moreover, the subclonal compositions at the initial time are able to accurately discriminate diverse growth patterns through a machine learning algorithm.

CONCLUSION

The proposed multiscale framework successfully delineates the intricate landscape of cancer evolution, bridging the gap between long-term growth dynamics and short-term measurements, both in simulated and real-world data. This methodology provides a novel avenue for thorough exploration into the realm of cancer evolution.

A review of mechanistic learning in mathematical oncology

Mechanistic learning refers to the synergistic combination of mechanistic mathematical modeling and data-driven machine or deep learning. This emerging field finds increasing applications in (mathematical) oncology. This review aims to capture the current state of the field and provides a perspective on how mechanistic learning may progress in the oncology domain. We highlight the synergistic potential of mechanistic learning and point out similarities and differences between purely data-driven and mechanistic approaches concerning model complexity, data requirements, outputs generated, and interpretability of the algorithms and their results. Four categories of mechanistic learning (sequential, parallel, extrinsic, intrinsic) of mechanistic learning are presented with specific examples. We discuss a range of techniques including physics-informed neural networks, surrogate model learning, and digital twins. Example applications address complex problems predominantly from the domain of oncology research such as longitudinal tumor response predictions or time-to-event modeling. As the field of mechanistic learning advances, we aim for this review and proposed categorization framework to foster additional collaboration between the data- and knowledge-driven modeling fields. Further collaboration will help address difficult issues in oncology such as limited data availability, requirements of model transparency, and complex input data which are embraced in a mechanistic learning framework.

Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling

Longitudinal tumour volume data from head-and-neck cancer patients show that tumours of comparable pre-treatment size and stage may respond very differently to the same radiotherapy fractionation protocol. Mathematical models are often proposed to predict treatment outcome in this context, and have the potential to guide clinical decision-making and inform personalised fractionation protocols. Hindering effective use of models in this context is the sparsity of clinical measurements juxtaposed with the model complexity required to produce the full range of possible patient responses. In this work, we present a compartment model of tumour volume and tumour composition, which, despite relative simplicity, is capable of producing a wide range of patient responses. We then develop novel statistical methodology and leverage a cohort of existing clinical data to produce a predictive model of both tumour volume progression and the associated level of uncertainty that evolves throughout a patient's course of treatment. To capture inter-patient variability, all model parameters are patient specific, with a bootstrap particle filter-like Bayesian approach developed to model a set of training data as prior knowledge. We validate our approach against a subset of unseen data, and demonstrate both the predictive ability of our trained model and its limitations.

Connecting Agent-Based Models with High-Dimensional Parameter Spaces to Multidimensional Data Using SMoRe ParS: A Surrogate Modeling Approach

Across a broad range of disciplines, agent-based models (ABMs) are increasingly utilized for replicating, predicting, and understanding complex systems and their emergent behavior. In the biological and biomedical sciences, researchers employ ABMs to elucidate complex cellular and molecular interactions across multiple scales under varying conditions. Data generated at these multiple scales, however, presents a computational challenge for robust analysis with ABMs. Indeed, calibrating ABMs remains an open topic of research due to their own high-dimensional parameter spaces. In response to these challenges, we extend and validate our novel methodology, Surrogate Modeling for Reconstructing Parameter Surfaces (SMoRe ParS), arriving at a computationally efficient framework for connecting high dimensional ABM parameter spaces with multidimensional data. Specifically, we modify SMoRe ParS to initially confine high dimensional ABM parameter spaces using unidimensional data, namely, single time-course information of in vitro cancer cell growth assays. Subsequently, we broaden the scope of our approach to encompass more complex ABMs and constrain parameter spaces using multidimensional data. We explore this extension with in vitro cancer cell inhibition assays involving the chemotherapeutic agent oxaliplatin. For each scenario, we validate and evaluate the effectiveness of our approach by comparing how well ABM simulations match the experimental data when using SMoRe ParS-inferred parameters versus parameters inferred by a commonly used direct method. In so doing, we show that our approach of using an explicitly formulated surrogate model as an interlocutor between the ABM and the experimental data effectively calibrates the ABM parameter space to multidimensional data. Our method thus provides a robust and scalable strategy for leveraging multidimensional data to inform multiscale ABMs and explore the uncertainty in their parameters.

Pairwise and higher-order epistatic effects among somatic cancer mutations across oncogenesis

Cancer occurs as a consequence of multiple somatic mutations that lead to uncontrolled cell growth. Mutual exclusivity and co-occurrence of mutations imply-but do not prove-that mutations exert synergistic or antagonistic epistatic effects on oncogenesis. Knowledge of these interactions, and the consequent trajectories of mutation and selection that lead to cancer has been a longstanding goal within the cancer research community. Recent research has revealed mutation rates and scaled selection coefficients for specific recurrent variants across many cancer types. However, there are no current methods to quantify the strength of selection incorporating pairwise and higher-order epistatic effects on selection within the trajectory of likely cancer genotoypes. Therefore, we have developed a continuous-time Markov chain model that enables the estimation of mutation origination and fixation (flux), dependent on somatic cancer genotype. Coupling this continuous-time Markov chain model with a deconvolution approach provides estimates of underlying mutation rates and selection across the trajectory of oncogenesis. We demonstrate computation of fluxes and selection coefficients in a somatic evolutionary model for the four most frequently variant driver genes (TP53, LRP1B, KRAS and STK11) from 565 cases of lung adenocarcinoma. Our analysis reveals multiple antagonistic epistatic effects that reduce the possible routes of oncogenesis, and inform cancer research regarding viable trajectories of somatic evolution whose progression could be forestalled by precision medicine. Synergistic epistatic effects are also identified, most notably in the somatic genotype TP53 LRP1B for mutations in the KRAS gene, and in somatic genotypes containing KRAS or TP53 mutations for mutations in the STK11 gene. Large positive fluxes of KRAS variants were driven by large selection coefficients, whereas the flux toward LRP1B mutations was substantially aided by a large mutation rate for this gene. The approach enables inference of the most likely routes of site-specific variant evolution and estimation of the strength of selection operating on each step along the route, a key component of what we need to know to develop and implement personalized cancer therapies.

Digitize your Biology! Modeling multicellular systems through interpretable cell behavior

Cells are fundamental units of life, constantly interacting and evolving as dynamical systems. While recent spatial multi-omics can quantitate individual cells' characteristics and regulatory programs, forecasting their evolution ultimately requires mathematical modeling. We develop a conceptual framework-a cell behavior hypothesis grammar-that uses natural language statements (cell rules) to create mathematical models. This allows us to systematically integrate biological knowledge and multi-omics data to make them computable. We can then perform virtual "thought experiments" that challenge and extend our understanding of multicellular systems, and ultimately generate new testable hypotheses. In this paper, we motivate and describe the grammar, provide a reference implementation, and demonstrate its potential through a series of examples in tumor biology and immunotherapy. Altogether, this approach provides a bridge between biological, clinical, and systems biology researchers for mathematical modeling of biological systems at scale, allowing the community to extrapolate from single-cell characterization to emergent multicellular behavior.

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.

An adaptive information-theoretic experimental design procedure for high-to-low fidelity calibration of prostate cancer models

The use of mathematical models to make predictions about tumor growth and response to treatment has become increasingly prevalent in the clinical setting. The level of complexity within these models ranges broadly, and the calibration of more complex models requires detailed clinical data. This raises questions about the type and quantity of data that should be collected and when, in order to maximize the information gain about the model behavior while still minimizing the total amount of data used and the time until a model can be calibrated accurately. To address these questions, we propose a Bayesian information-theoretic procedure, using an adaptive score function to determine the optimal data collection times and measurement types. The novel score function introduced in this work eliminates the need for a penalization parameter used in a previous study, while yielding model predictions that are superior to those obtained using two potential pre-determined data collection protocols for two different prostate cancer model scenarios: one in which we fit a simple ODE system to synthetic data generated from a cellular automaton model using radiotherapy as the imposed treatment, and a second scenario in which a more complex ODE system is fit to clinical patient data for patients undergoing intermittent androgen suppression therapy. We also conduct a robust analysis of the calibration results, using both error and uncertainty metrics in combination to determine when additional data acquisition may be terminated.

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.

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.

Advances in computational and translational approaches for malignant glioma

Gliomas are the most common primary brain tumors in adults and carry a dismal prognosis for patients. Current standard-of-care for gliomas is comprised of maximal safe surgical resection following by a combination of chemotherapy and radiation therapy depending on the grade and type of tumor. Despite decades of research efforts directed towards identifying effective therapies, curative treatments have been largely elusive in the majority of cases. The development and refinement of novel methodologies over recent years that integrate computational techniques with translational paradigms have begun to shed light on features of glioma, previously difficult to study. These methodologies have enabled a number of point-of-care approaches that can provide real-time, patient-specific and tumor-specific diagnostics that may guide the selection and development of therapies including decision-making surrounding surgical resection. Novel methodologies have also demonstrated utility in characterizing glioma-brain network dynamics and in turn early investigations into glioma plasticity and influence on surgical planning at a systems level. Similarly, application of such techniques in the laboratory setting have enhanced the ability to accurately model glioma disease processes and interrogate mechanisms of resistance to therapy. In this review, we highlight representative trends in the integration of computational methodologies including artificial intelligence and modeling with translational approaches in the study and treatment of malignant gliomas both at the point-of-care and outside the operative theater in silico as well as in the laboratory setting.

Data driven model discovery and interpretation for CAR T-cell killing using sparse identification and latent variables

In the development of cell-based cancer therapies, quantitative mathematical models of cellular interactions are instrumental in understanding treatment efficacy. Efforts to validate and interpret mathematical models of cancer cell growth and death hinge first on proposing a precise mathematical model, then analyzing experimental data in the context of the chosen model. In this work, we present the first application of the sparse identification of non-linear dynamics (SINDy) algorithm to a real biological system in order discover cell-cell interaction dynamics in in vitro experimental data, using chimeric antigen receptor (CAR) T-cells and patient-derived glioblastoma cells. By combining the techniques of latent variable analysis and SINDy, we infer key aspects of the interaction dynamics of CAR T-cell populations and cancer. Importantly, we show how the model terms can be interpreted biologically in relation to different CAR T-cell functional responses, single or double CAR T-cell-cancer cell binding models, and density-dependent growth dynamics in either of the CAR T-cell or cancer cell populations. We show how this data-driven model-discovery based approach provides unique insight into CAR T-cell dynamics when compared to an established model-first approach. These results demonstrate the potential for SINDy to improve the implementation and efficacy of CAR T-cell therapy in the clinic through an improved understanding of CAR T-cell dynamics.

Novel computational biology modeling system can accurately forecast response to neoadjuvant therapy in early breast cancer

BACKGROUND

Generalizable population-based studies are unable to account for individual tumor heterogeneity that contributes to variability in a patient's response to physician-chosen therapy. Although molecular characterization of tumors has advanced precision medicine, in early-stage and locally advanced breast cancer patients, predicting a patient's response to neoadjuvant therapy (NAT) remains a gap in current clinical practice. Here, we perform a study in an independent cohort of early-stage and locally advanced breast cancer patients to forecast tumor response to NAT and assess the stability of a previously validated biophysical simulation platform.

METHODS

A single-blinded study was performed using a retrospective database from a single institution (9/2014-12/2020). Patients included: ≥ 18 years with breast cancer who completed NAT, with pre-treatment dynamic contrast enhanced magnetic resonance imaging. Demographics, chemotherapy, baseline (pre-treatment) MRI and pathologic data were input into the TumorScope Predict (TS) biophysical simulation platform to generate predictions. Primary outcomes included predictions of pathological complete response (pCR) versus residual disease (RD) and final volume for each tumor. For validation, post-NAT predicted pCR and tumor volumes were compared to actual pathological assessment and MRI-assessed volumes. Predicted pCR was pre-defined as residual tumor volume ≤ 0.01 cm³ (≥ 99.9% reduction).

RESULTS

The cohort consisted of eighty patients; 36 Caucasian and 40 African American. Most tumors were high-grade (54.4% grade 3) invasive ductal carcinomas (90.0%). Receptor subtypes included hormone receptor positive (HR+)/human epidermal growth factor receptor 2 positive (HER2+, 30%), HR+/HER2- (35%), HR-/HER2+ (12.5%) and triple negative breast cancer (TNBC, 22.5%). Simulated tumor volume was significantly correlated with post-treatment radiographic MRI calculated volumes (r = 0.53, p = 1.3 × 10^-7, mean absolute error of 6.57%). TS prediction of pCR compared favorably to pathological assessment (pCR: TS n = 28; Path n = 27; RD: TS n = 52; Path n = 53), for an overall accuracy of 91.2% (95% CI: 82.8% - 96.4%; Clopper-Pearson interval). Five-year risk of recurrence demonstrated similar prognostic performance between TS predictions (Hazard ratio (HR): - 1.99; 95% CI [- 3.96, - 0.02]; p = 0.043) and clinically assessed pCR (HR: - 1.76; 95% CI [- 3.75, 0.23]; p = 0.054).

CONCLUSION

We demonstrated TS ability to simulate and model tumor in vivo conditions in silico and forecast volume response to NAT across breast tumor subtypes.

Spatial cumulant models enable spatially informed treatment strategies and analysis of local interactions in cancer systems

Theoretical and applied cancer studies that use individual-based models (IBMs) have been limited by the lack of a mathematical formulation that enables rigorous analysis of these models. However, spatial cumulant models (SCMs), which have arisen from theoretical ecology, describe population dynamics generated by a specific family of IBMs, namely spatio-temporal point processes (STPPs). SCMs are spatially resolved population models formulated by a system of differential equations that approximate the dynamics of two STPP-generated summary statistics: first-order spatial cumulants (densities), and second-order spatial cumulants (spatial covariances). We exemplify how SCMs can be used in mathematical oncology by modelling theoretical cancer cell populations comprising interacting growth factor-producing and non-producing cells. To formulate model equations, we use computational tools that enable the generation of STPPs, SCMs and mean-field population models (MFPMs) from user-defined model descriptions (Cornell et al. Nat Commun 10:4716, 2019). To calculate and compare STPP, SCM and MFPM-generated summary statistics, we develop an application-agnostic computational pipeline. Our results demonstrate that SCMs can capture STPP-generated population density dynamics, even when MFPMs fail to do so. From both MFPM and SCM equations, we derive treatment-induced death rates required to achieve non-growing cell populations. When testing these treatment strategies in STPP-generated cell populations, our results demonstrate that SCM-informed strategies outperform MFPM-informed strategies in terms of inhibiting population growths. We thus demonstrate that SCMs provide a new framework in which to study cell-cell interactions, and can be used to describe and perturb STPP-generated cell population dynamics. We, therefore, argue that SCMs can be used to increase IBMs' applicability in cancer research.

Quantification of spatial and phenotypic heterogeneity in an agent-based model of tumour-macrophage interactions

We introduce a new spatial statistic, the weighted pair correlation function (wPCF). The wPCF extends the existing pair correlation function (PCF) and cross-PCF to describe spatial relationships between points marked with combinations of discrete and continuous labels. We validate its use through application to a new agent-based model (ABM) which simulates interactions between macrophages and tumour cells. These interactions are influenced by the spatial positions of the cells and by macrophage phenotype, a continuous variable that ranges from anti-tumour to pro-tumour. By varying model parameters that regulate macrophage phenotype, we show that the ABM exhibits behaviours which resemble the 'three Es of cancer immunoediting': Equilibrium, Escape, and Elimination. We use the wPCF to analyse synthetic images generated by the ABM. We show that the wPCF generates a 'human readable' statistical summary of where macrophages with different phenotypes are located relative to both blood vessels and tumour cells. We also define a distinct 'PCF signature' that characterises each of the three Es of immunoediting, by combining wPCF measurements with the cross-PCF describing interactions between vessels and tumour cells. By applying dimension reduction techniques to this signature, we identify its key features and train a support vector machine classifier to distinguish between simulation outputs based on their PCF signature. This proof-of-concept study shows how multiple spatial statistics can be combined to analyse the complex spatial features that the ABM generates, and to partition them into interpretable groups. The intricate spatial features produced by the ABM are similar to those generated by state-of-the-art multiplex imaging techniques which distinguish the spatial distribution and intensity of multiple biomarkers in biological tissue regions. Applying methods such as the wPCF to multiplex imaging data would exploit the continuous variation in biomarker intensities and generate more detailed characterisation of the spatial and phenotypic heterogeneity in tissue samples.

Modeling cancer's ecological and evolutionary dynamics

In this didactic paper, we present a theoretical modeling framework, called the G-function, that integrates both the ecology and evolution of cancer to understand oncogenesis. The G-function has been used in evolutionary ecology, but has not been widely applied to problems in cancer. Here, we build the G-function framework from fundamental Darwinian principles and discuss how cancer can be seen through the lens of ecology, evolution, and game theory. We begin with a simple model of cancer growth and add on components of cancer cell competition and drug resistance. To aid in exploration of eco-evolutionary modeling with this approach, we also present a user-friendly software tool. By the end of this paper, we hope that readers will be able to construct basic G function models and grasp the usefulness of the framework to understand the games cancer plays in a biologically mechanistic fashion.

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.

In-silico study of asymmetric remodeling of tumors in response to external biochemical stimuli

Among different hallmarks of cancer, understanding biomechanics of tumor growth and remodeling benefits the most from the theoretical framework of continuum mechanics. Tumor remodeling initiates when cancer cells seek new homeostasis in response to the microenvironmental stimuli. Cells within a growing tumor are capable to remodel their inter- and intra-connections and become more mobile to achieve a new homeostasis. This mobility enables the tumor to undergo large deformation. In this work, we studied the remodeling of homogeneous tumors, at their early stage of growth, in the context of continuum mechanics. We developed an evolution law for the remodeling-associated deformation which correlates the remodeling to a characteristic tensor of external stimuli. The asymmetric remodeling and the induced mechanical stresses were analyzed for different types of biochemical distributions. To experimentally investigate the model, we studied the remodeling of human glioblastoma (hGB) tumoroids in response to the gradient of nutrients. Using a tumoroid-on-a-chip platform, the degree of remodeling was estimated for the ellipsoidal tumoroids over time. It was observed that higher gradient of nutrients induces higher degree of ellipticity suggesting that the gradient of nutrient is a characteristic property of nutrient distribution that derives the remodeling. We also showed that remodeling gives rise to heterogeneity in cell distribution forming circumferentially aligned cells within the tumors. Compared to the existing studies on tumor growth, our work provides a biomechanical module that relates the remodeling to biochemical stimuli, and allows for large deformation. It also includes experimental component, a necessary but challenging step, that connects the theory and reality to evaluate the practicability of the model.

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

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

Towards integration of time-resolved confocal microscopy of a 3D in vitro microfluidic platform with a hybrid multiscale model of tumor angiogenesis

The goal of this study is to calibrate a multiscale model of tumor angiogenesis with time-resolved data to allow for systematic testing of mathematical predictions of vascular sprouting. The multi-scale model consists of an agent-based description of tumor and endothelial cell dynamics coupled to a continuum model of vascular endothelial growth factor concentration. First, we calibrate ordinary differential equation models to time-resolved protein concentration data to estimate the rates of secretion and consumption of vascular endothelial growth factor by endothelial and tumor cells, respectively. These parameters are then input into the multiscale tumor angiogenesis model, and the remaining model parameters are then calibrated to time resolved confocal microscopy images obtained within a 3D vascularized microfluidic platform. The microfluidic platform mimics a functional blood vessel with a surrounding collagen matrix seeded with inflammatory breast cancer cells, which induce tumor angiogenesis. Once the multi-scale model is fully parameterized, we forecast the spatiotemporal distribution of vascular sprouts at future time points and directly compare the predictions to experimentally measured data. We assess the ability of our model to globally recapitulate angiogenic vasculature density, resulting in an average relative calibration error of 17.7% ± 6.3% and an average prediction error of 20.2% ± 4% and 21.7% ± 3.6% using one and four calibrated parameters, respectively. We then assess the model's ability to predict local vessel morphology (individualized vessel structure as opposed to global vascular density), initialized with the first time point and calibrated with two intermediate time points. In this study, we have rigorously calibrated a mechanism-based, multiscale, mathematical model of angiogenic sprouting to multimodal experimental data to make specific, testable predictions.

Theranostic digital twins for personalized radiopharmaceutical therapies: Reimagining theranostics via computational nuclear oncology

This work emphasizes that patient data, including images, are not operable (clinically), but that digital twins are. Based on the former, the latter can be created. Subsequently, virtual clinical operations can be performed towards selection of optimal therapies. Digital twins are beginning to emerge in the field of medicine. We suggest that theranostic digital twins (TDTs) are amongst the most natural and feasible flavors of digitals twins. We elaborate on the importance of TDTs in a future where 'one-size-fits-all' therapeutic schemes, as prevalent nowadays, are transcended in radiopharmaceutical therapies (RPTs). Personalized RPTs will be deployed, including optimized intervention parameters. Examples include optimization of injected radioactivities, sites of injection, injection intervals and profiles, and combination therapies. Multi-modal multi-scale images, combined with other data and aided by artificial intelligence (AI) techniques, will be utilized towards routine digital twinning of our patients, and will enable improved deliveries of RPTs and overall healthcare.

Optimizing combination therapy in a murine model of HER2+ breast cancer

Human epidermal growth factor receptor 2 positive (HER2+) breast cancer is frequently treated with drugs that target the HER2 receptor, such as trastuzumab, in combination with chemotherapy, such as doxorubicin. However, an open problem in treatment design is to determine the therapeutic regimen that optimally combines these two treatments to yield optimal tumor control. Working with data quantifying temporal changes in tumor volume due to different trastuzumab and doxorubicin treatment protocols in a murine model of human HER2+ breast cancer, we propose a complete framework for model development, calibration, selection, and treatment optimization to find the optimal treatment protocol. Through different assumptions for the drug-tumor interactions, we propose ten different models to characterize the dynamic relationship between tumor volume and drug availability, as well as the drug-drug interaction. Using a Bayesian framework, each of these models are calibrated to the dataset and the model with the highest Bayesian information criterion weight is selected to represent the biological system. The selected model captures the inhibition of trastuzumab due to pre-treatment with doxorubicin, as well as the increase in doxorubicin efficacy due to pre-treatment with trastuzumab. We then apply optimal control theory (OCT) to this model to identify two optimal treatment protocols. In the first optimized protocol, we fix the maximum dosage for doxorubicin and trastuzumab to be the same as the maximum dose delivered experimentally, while trying to minimize tumor burden. Within this constraint, optimal control theory indicates the optimal regimen is to first deliver two doses of trastuzumab on days 35 and 36, followed by two doses of doxorubicin on days 37 and 38. This protocol predicts an additional 45% reduction in tumor burden compared to that achieved with the experimentally delivered regimen. In the second optimized protocol we fix the tumor control to be the same as that obtained experimentally, and attempt to reduce the doxorubicin dose. Within this constraint, the optimal regimen is the same as the first optimized protocol but uses only 43% of the doxorubicin dose used experimentally. This protocol predicts tumor control equivalent to that achieved experimentally. These results strongly suggest the utility of mathematical modeling and optimal control theory for identifying therapeutic regimens maximizing efficacy and minimizing toxicity.

A novel interpretable machine learning algorithm to identify optimal parameter space for cancer growth

Recent years have seen an increase in the application of machine learning to the analysis of physical and biological systems, including cancer progression. A fundamental downside to these tools is that their complexity and nonlinearity makes it almost impossible to establish a deterministic, a priori relationship between their input and output, and thus their predictions are not wholly accountable. We begin with a series of proofs establishing that this holds even for the simplest possible model of a neural network; the effects of specific loss functions are explored more fully in Appendices. We return to first principles and consider how to construct a physics-inspired model of tumor growth without resorting to stochastic gradient descent or artificial nonlinearities. We derive an algorithm which explores the space of possible parameters in a model of tumor growth and identifies candidate equations much faster than a simulated annealing approach. We test this algorithm on synthetic tumor-growth trajectories and show that it can efficiently and reliably narrow down the area of parameter space where the correct values are located. This approach has the potential to greatly improve the speed and reliability with which patient-specific models of cancer growth can be identified in a clinical setting.

Mathematical characterization of population dynamics in breast cancer cells treated with doxorubicin

The development of chemoresistance remains a significant cause of treatment failure in breast cancer. We posit that a mathematical understanding of chemoresistance could assist in developing successful treatment strategies. Towards that end, we have developed a model that describes the cytotoxic effects of the standard chemotherapeutic drug doxorubicin on the MCF-7 breast cancer cell line. We assume that treatment with doxorubicin induces a compartmentalization of the breast cancer cell population into surviving cells, which continue proliferating after treatment, and irreversibly damaged cells, which gradually transition from proliferating to treatment-induced death. The model is fit to experimental data including variations in drug concentration, inter-treatment interval, and number of doses. Our model recapitulates tumor cell dynamics in all these scenarios (as quantified by the concordance correlation coefficient, CCC > 0.95). In particular, superior tumor control is observed with higher doxorubicin concentrations, shorter inter-treatment intervals, and a higher number of doses (p < 0.05). Longer inter-treatment intervals require adapting the model parameterization after each doxorubicin dose, suggesting the promotion of chemoresistance. Additionally, we propose promising empirical formulas to describe the variation of model parameters as functions of doxorubicin concentration (CCC > 0.78). Thus, we conclude that our mathematical model could deepen our understanding of the cytotoxic effects of doxorubicin and could be used to explore practical drug regimens achieving optimal tumor control.

Quantification of spatial subclonal interactions enhancing the invasive phenotype of pediatric glioma

Diffuse midline gliomas (DMGs) are highly aggressive, incurable childhood brain tumors. They present a clinical challenge due to many factors, including heterogeneity and diffuse infiltration, complicating disease management. Recent studies have described the existence of subclonal populations that may co-operate to drive pro-tumorigenic processes such as cellular invasion. However, a precise quantification of subclonal interactions is lacking, a problem that extends to other cancers. In this study, we combine spatial computational modeling of cellular interactions during invasion with co-evolution experiments of clonally disassembled patient-derived DMG cells. We design a Bayesian inference framework to quantify spatial subclonal interactions between molecular and phenotypically distinct lineages with different patterns of invasion. We show how this approach could discriminate genuine interactions, where one clone enhanced the invasive phenotype of another, from those apparently only due to the complex dynamics of spatially restricted growth. This study provides a framework for the quantification of subclonal interactions in DMG.

The role of evolutionary game theory in spatial and non-spatial models of the survival of cooperation in cancer: a review

Evolutionary game theory (EGT) is a branch of mathematics which considers populations of individuals interacting with each other to receive pay-offs. An individual’s pay-off is dependent on the strategy of its opponent(s) as well as on its own, and the higher its pay-off, the higher its reproductive fitness. Its offspring generally inherit its interaction strategy, subject to random mutation. Over time, the composition of the population shifts as different strategies spread or are driven extinct. In the last 25 years there has been a flood of interest in applying EGT to cancer modelling, with the aim of explaining how cancerous mutations spread through healthy tissue and how intercellular cooperation persists in tumour-cell populations. This review traces this body of work from theoretical analyses of well-mixed infinite populations through to more realistic spatial models of the development of cooperation between epithelial cells. We also consider work in which EGT has been used to make experimental predictions about the evolution of cancer, and discuss work that remains to be done before EGT can make large-scale contributions to clinical treatment and patient outcomes.

Opportunities for improving brain cancer treatment outcomes through imaging-based mathematical modeling of the delivery of radiotherapy and immunotherapy

Immunotherapy has become a fourth pillar in the treatment of brain tumors and, when combined with radiation therapy, may improve patient outcomes and reduce the neurotoxicity. As with other combination therapies, the identification of a treatment schedule that maximizes the synergistic effect of radiation- and immune-therapy is a fundamental challenge. Mechanism-based mathematical modeling is one promising approach to systematically investigate therapeutic combinations to maximize positive outcomes within a rigorous framework. However, successful clinical translation of model-generated combinations of treatment requires patient-specific data to allow the models to be meaningfully initialized and parameterized. Quantitative imaging techniques have emerged as a promising source of high quality, spatially and temporally resolved data for the development and validation of mathematical models. In this review, we will present approaches to personalize mechanism-based modeling frameworks with patient data, and then discuss how these techniques could be leveraged to improve brain cancer outcomes through patient-specific modeling and optimization of treatment strategies.

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.

Data-Driven Simulation of Fisher-Kolmogorov Tumor Growth Models Using Dynamic Mode Decomposition

The computer simulation of organ-scale biomechanistic models of cancer personalized via routinely collected clinical and imaging data enables to obtain patient-specific predictions of tumor growth and treatment response over the anatomy of the patient's affected organ. However, the simulation of the underlying spatiotemporal models can entail a prohibitive computational cost, which constitutes a barrier to the successful development of clinically-actionable computational technologies for personalized tumor forecasting. Here we propose to utilize Dynamic-Mode Decomposition (DMD), an unsupervised machine learning method, to construct a low-dimensional representation of cancer models and accelerate their simulation. We show that DMD may be applied to Fisher-Kolmogorov models, which constitute an established formulation to represent untreated solid tumor growth that can further accommodate other relevant cancer phenomena. Our results show that a DMD implementation of this model over a clinically-relevant parameter space can yield impressive predictions, with short to medium-term errors remaining under 1% and long-term errors remaining under 20%, despite very short training periods. In particular, we have found that, for moderate to high tumor cell diffusivity and low to moderate tumor cell proliferation rate, DMD reconstructions provide accurate, bounded-error reconstructions for all tested training periods. We posit that this data-driven approach has the potential to greatly reduce the computational overhead of personalized simulations of cancer models, thereby facilitating tumor forecasting, parameter identification, uncertainty quantification, and treatment optimization.

Comparison of cell state models derived from single-cell RNA sequencing data: graph versus multi-dimensional space

Single-cell sequencing technologies have revolutionized molecular and cellular biology and stimulated the development of computational tools to analyze the data generated from these technology platforms. However, despite the recent explosion of computational analysis tools, relatively few mathematical models have been developed to utilize these data. Here we compare and contrast two cell state geometries for building mathematical models of cell state-transitions with single-cell RNA-sequencing data with hematopoeisis as a model system; (i) by using partial differential equations on a graph representing intermediate cell states between known cell types, and (ii) by using the equations on a multi-dimensional continuous cell state-space. As an application of our approach, we demonstrate how the calibrated models may be used to mathematically perturb normal hematopoeisis to simulate, predict, and study the emergence of novel cell states during the pathogenesis of acute myeloid leukemia. We particularly focus on comparing the strength and weakness of the graph model and multi-dimensional model.

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.

Integrating mechanism-based modeling with biomedical imaging to build practical digital twins for clinical oncology

Digital twins employ mathematical and computational models to virtually represent a physical object (e.g., planes and human organs), predict the behavior of the object, and enable decision-making to optimize the future behavior of the object. While digital twins have been widely used in engineering for decades, their applications to oncology are only just emerging. Due to advances in experimental techniques quantitatively characterizing cancer, as well as advances in the mathematical and computational sciences, the notion of building and applying digital twins to understand tumor dynamics and personalize the care of cancer patients has been increasingly appreciated. In this review, we present the opportunities and challenges of applying digital twins in clinical oncology, with a particular focus on integrating medical imaging with mechanism-based, tissue-scale mathematical modeling. Specifically, we first introduce the general digital twin framework and then illustrate existing applications of image-guided digital twins in healthcare. Next, we detail both the imaging and modeling techniques that provide practical opportunities to build patient-specific digital twins for oncology. We then describe the current challenges and limitations in developing image-guided, mechanism-based digital twins for oncology along with potential solutions. We conclude by outlining five fundamental questions that can serve as a roadmap when designing and building a practical digital twin for oncology and attempt to provide answers for a specific application to brain cancer. We hope that this contribution provides motivation for the imaging science, oncology, and computational communities to develop practical digital twin technologies to improve the care of patients battling cancer.

OPTIMAL CONTROL OF GENETIC DIVERSITY IN THE MORAN MODEL WITH POPULATION GROWTH

In the Moran model of drift and selection of a mutant allele with population growth, instead of examining the consequences of pre-specified selection and population growth, the coexistence of the wild allele and the mutant allele becomes the maximization of the expected sojourn time in a given set. The process is controlled by the additional mortality of the mutant and by population growth. This makes it possible to retroactively assign fitness values as functions of the constraints, thus guiding a conservation policy or the achievement of a wishful proportion of mutants. This also gives the optimal conditions that have allowed an observed coexistence.

Modelling glioma progression, mass effect and intracranial pressure in patient anatomy

Increased intracranial pressure is the source of most critical symptoms in patients with glioma, and often the main cause of death. Clinical interventions could benefit from non-invasive estimates of the pressure distribution in the patient's parenchyma provided by computational models. However, existing glioma models do not simulate the pressure distribution and they rely on a large number of model parameters, which complicates their calibration from available patient data. Here we present a novel model for glioma growth, pressure distribution and corresponding brain deformation. The distinct feature of our approach is that the pressure is directly derived from tumour dynamics and patient-specific anatomy, providing non-invasive insights into the patient's state. The model predictions allow estimation of critical conditions such as intracranial hypertension, brain midline shift or neurological and cognitive impairments. A diffuse-domain formalism is employed to allow for efficient numerical implementation of the model in the patient-specific brain anatomy. The model is tested on synthetic and clinical cases. To facilitate clinical deployment, a high-performance computing implementation of the model has been publicly released.

Combining Evolution and Cancer Therapy: A Review of the Mathematical Approach

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Conventional cancer therapy kills tumors by applying the maximum tolerable dose of therapy. However, it leads to the development of tumoral heterogeneity and resistance, hence leading to therapy failure and progression. It is necessary to design therapies keeping in mind the evolutionary dynamics of tumors to minimize resistance and delay progression. Mathematical models are of great importance in oncology as they assist in the recreation of the tumor microenvironment, predict the outcomes of treatment strategies and elucidate fundamentals of tumor growth and resistance development. The body of literature covering models which incorporate evolutionary dynamics is vast. This paper provides an overview of existing models of “evolutionary therapy”, including ordinary differential equations, fitness, and probability functions.