Classical mathematical models for description and prediction of experimental tumor growth

Identifiability and model selection frameworks for models of high-grade glioma response to chemoradiation

We have developed a family of biology-based mathematical models of high-grade glioma (HGG), capturing the key features of tumour growth and response to chemoradiation. We now seek to quantify the accuracy of parameter estimation and determine, when given a virtual patient cohort, which model was used to generate the tumours. In this way, we systematically test both the parameter and model identifiability. Virtual patients are generated from unique growth parameters whose growth dynamics are determined by the model family. We then assessed the ability to recover model parameters and select the model used to generate the tumour. We then evaluated the accuracy of predictions using the selected model at four weeks post-chemoradiation. We observed median parameter errors from 0.04% to 72.96%. Our model selection framework selected the model that was used to generate the data in 82% of the cases. Finally, we predicted the growth of the virtual tumours using the selected model resulting in low error at the voxel-level (concordance correlation coefficient (CCC) ranged from 0.66 to 0.99) and global level (percentage error in total tumour cellularity ranged from -12.35% to 0.07%). These results demonstrate the reliability of our framework to identify the most appropriate model under noisy conditions expected in the clinical setting.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 2)'.

Exploring trade-offs in drug administration for cancer treatment: A multi-criteria optimisation approach

This study addresses the combination of immunotherapy and chemotherapy in cancer treatment, recognising its promising effectiveness but highlighting the challenges of complex interactions between these therapeutic modalities. The central objective is to determine guidelines for the optimal administration of drugs, using an optimal control model that considers interactions in tumour dynamics, including cancer cells, the immune system, and therapeutic agents. The optimal control model is transformed into a multi-objective optimisation problem with treatment constraints. This is achieved by introducing adjustable trade-offs, allowing personalised adaptations in drug administration to achieve an optimal balance between established objectives. Various optimisation problems are addressed, considering two and three simultaneous objectives, such as optimising the number of cancer cells and the density of effector cells at the final treatment time. The diverse combinations presented reflect the model's flexibility in the face of multi-objective optimisation, providing a range of approaches to meet specific medical needs. The analysis of Pareto optimal fronts in in silico investigation offers an additional resource for decision-makers, enabling a more effective determination of the optimal administration of cytotoxic and immunotherapeutic agents. By leveraging an optimal control model, we have demonstrated the effectiveness of considering interactions in tumour dynamics, including the integration of immunotherapy and chemotherapy. Our findings underscore the importance of tailored treatment plans to achieve optimal outcomes, showcasing the versatility of our approach in addressing individual patient needs. The insights gained from our analysis offer valuable guidance for future research and clinical practice, paving the way for more effective and personalised cancer therapies.

Validating the predictions of mathematical models describing tumor growth and treatment response

Despite advances in methods to interrogate tumor biology, the observational and population-based approach of classical cancer research and clinical oncology does not enable anticipation of tumor outcomes to hasten the discovery of cancer mechanisms and personalize disease management. To address these limitations, individualized cancer forecasts have been shown to predict tumor growth and therapeutic response, inform treatment optimization, and guide experimental efforts. These predictions are obtained via computer simulations of mathematical models that are constrained with data from a patient's cancer and experiments. This book chapter addresses the validation of these mathematical models to forecast tumor growth and treatment response. We start with an overview of mathematical modeling frameworks, model selection techniques, and fundamental metrics. We then describe the usual strategies employed to validate cancer forecasts in preclinical and clinical scenarios. Finally, we discuss existing barriers in validating these predictions along with potential strategies to address them.

Proper likelihood functions for parameter estimation in S-shaped models of unperturbed tumor growth

The analysis of unperturbed tumor growth kinetics, particularly the estimation of parameters for S-shaped equations used to describe growth, requires an appropriate likelihood function that accounts for the increasing error in solid tumor measurements as tumor size grows over time. This study aims to propose suitable likelihood functions for parameter estimation in S-shaped models of unperturbed tumor growth. Five different likelihood functions are evaluated and compared using three Bayesian criteria (the Bayesian Information Criterion, Deviance Information Criterion, and Bayes Factor) along with hypothesis tests on residuals. These functions are applied to fit data from unperturbed Ehrlich, fibrosarcoma Sa-37, and F3II tumors using the Gompertz equation, though they are generalizable to other S-shaped growth models for solid tumors or analogous systems (e.g., microorganisms, viruses). Results indicate that error models with tumor volume-dependent dispersion outperform standard constant-variance models in capturing the variability of tumor measurements, particularly the Thres model, which provides interpretable parameters for tumor growth. Additionally, constant-variance models, such as those assuming a normal error distribution, remain valuable as complementary benchmarks in analysis. It is concluded that models incorporating volume-dependent dispersion are preferred for accurate and clinically meaningful tumor growth modeling, whereas constant-dispersion models serve as useful complements for consistency and historical comparability.

Assessment of Cell Viability in Drug Therapy: IC50 and Other New Time-Independent Indices for Evaluating Chemotherapy Efficacy

Background/Objectives: Cell viability assays play a crucial role in cancer research and the development of effective treatments. Evaluating the efficacy of conventional treatments across different tumor profiles is essential for understanding patient resistance to chemotherapy and relapse. The IC50 index has been commonly used as a guide in these assays. The idea behind the IC50 index is to compare cell proliferation under treatment with respect to a control population exposed to the same treatment. The index requires normalization to a control and is time dependent. These aspects are disadvantages, as small variations yield different results. In this article, we propose a new method to analyze cell viability assays. Methods: This method involves calculating the effective growth rate for both control (untreated) cells and cells exposed to a range of drug doses for short times, during which exponential proliferation can be assumed. The concentration dependence of the effective growth rate gives a real estimate of the treatment on cell proliferation. A curve fit of the effective growth rate related to concentration yields the concentration corresponding to a given effective growth rate. Results: We use this estimation to calculate the IC50 index and introduce two new parameters (ICr0 and ICrmed) to compare treatment efficacy under different culture conditions or cell lines. Conclusions: In summary, this study presents a new method to analyze cell viability assays and introduces two more precise parameters, improving the comparison and evaluation of efficacy under different conditions.

Effects of Non-linear Interaction Between Oxygen and Lactate on Solid Tumor Growth Under Cyclic Hypoxia

Oxygen is a crucial element for cellular respiration. Based on oxygen concentration, tumor regions can be categorized as normoxic, hypoxic, and necrotic. Hypoxic tumor cells switch their metabolism from aerobic glycolysis to anaerobic glycolysis. As a result, lactate is produced in hypoxic regions and is used as an alternative metabolic fuel by normoxic tumor cells. The consumption of lactate and oxygen by tumor cells does not follow a linear pattern. Scientific studies suggest that oxygen consumption and lactate production are non-linear phenomena. In this study, we propose a two-dimensional mathematical model to investigate lactate dynamics in avascular tumors with various initial shapes, such as circular, elliptical, and petal, and to explore its growth patterns in the context of non-linear interactions between oxygen and lactate. In certain human tumors, particularly in kidney, skin, and liver, multiple tumors may emerge within a tissue domain simultaneously. We also examine how the growth patterns of multiple tumors evolve within a shared domain. Cyclic hypoxia, commonly observed in solid tumors, results from oxygen fluctuations over time at the tumor site. Additionally, we analyze lactate dynamics and tumor growth patterns in environments with cyclic hypoxia. In order to simulate the proposed model, we use finite element based COMSOL Multiphysics 6.0 interface. The simulated results show excellent agreement with experimental data. Our findings reveal that the initial tumor shape significantly influences the lactate distribution and the tumor's internal structure. Furthermore, the simulations indicate that multiple tumors eventually merge into a single tumor. We also observe that cyclic hypoxia with short periodicity increases tumor volume.

Evolutionary rescue model informs strategies for driving cancer cell populations to extinction

Cancers exhibit a remarkable ability to develop resistance to a range of treatments, often resulting in relapse following first-line therapies and significantly worse outcomes for subsequent treatments. While our understanding of the mechanisms and dynamics of the emergence of resistance during cancer therapy continues to advance, questions remain about how to minimize the probability that resistance will evolve, thereby improving long-term patient outcomes. Here, we present an evolutionary simulation model of a clonal population of cells that can acquire resistance mutations to one or more treatments. We leverage this model to examine the efficacy of a two-strike "extinction therapy" protocol, in which two treatments are applied sequentially to first contract the population to a vulnerable state and then push it to extinction, and compare it to a combination therapy protocol. We investigate how factors such as the timing of the switch between the two strikes, the rate of emergence of resistant mutations, the doses of the applied drugs, the presence of cross-resistance, and whether resistance is a binary or a quantitative trait affect the outcome. Our results show that the timing of switching to the second strike has a marked effect on the likelihood of driving the cancer to extinction, and that extinction therapy outperforms combination therapy when cross-resistance is present. We conduct an in silico trial that reveals when and why a second strike will succeed or fail. Finally, we demonstrate that our conclusions hold whether we model resistance as a binary trait or as a quantitative, multi-locus trait.

Negligible Long-Term Impact of Nonlinear Growth Dynamics on Heterogeneity in Models of Cancer Cell Populations

Linear compartmental models are often employed to capture the change in cell type composition of cancer cell populations. Yet, these populations usually grow in a nonlinear fashion. This begs the question of how linear compartmental models can successfully describe the dynamics of cell types. Here, we propose a general modeling framework with a nonlinear part capturing growth dynamics and a linear part capturing cell type transitions. We prove that dynamics in this general model are asymptotically equivalent to those governed only by its linear part under a wide range of assumptions for nonlinear growth.

An in-depth examination of the fuzzy fractional cancer tumor model and its numerical solution by implicit finite difference method

The cancer tumor model serves a s a crucial instrument for understanding the behavior of different cancer tumors. Researchers have employed fractional differential equations to describe these models. In the context of time fractional cancer tumor models, there's a need to introduce fuzzy quantities instead of crisp quantities to accommodate the inherent uncertainty and imprecision in this model, giving rise to a formulation known as fuzzy time fractional cancer tumor models. In this study, we have developed an implicit finite difference method to solve a fuzzy time-fractional cancer tumor model. Instead of utilizing classical time derivatives in fuzzy cancer models, we have examined the effect of employing fuzzy time-fractional derivatives. To assess the stability of our proposed model, we applied the von Neumann method, considering the cancer cell killing rate as time-dependent and utilizing Caputo's derivative for the time-fractional derivative. Additionally, we conducted various numerical experiments to assess the viability of this new approach and explore relevant aspects. Furthermore, our study identified specific needs in researching the cancer tumor model with fuzzy fractional derivative, aiming to enhance our inclusive understanding of tumor behavior by considering diverse fuzzy cases for the model's initial conditions. It was found that the presented approach provides the ability to encompass all scenarios for the fuzzy time fractional cancer tumor model and handle all potential cases specifically focusing on scenarios where the net cell-killing rate is time-dependent.

Spatiotemporal spread of oncolytic virus in a heterogeneous cell population

Oncolytic (cancer-killing) virus treatment is a promising new therapy for cancer, with many viruses currently being tested for their ability to eradicate tumors. One of the major stumbling blocks to the development of this treatment modality has been preventing spread of the virus to non-cancerous cells. Our recent ability to manipulate RNA and DNA now allows for the possibility of creating designer viruses specifically targeted to cancer cells, thereby significantly reducing unwanted side effects in patients. In this study, we use a partial differential equation model to determine the characteristics of a virus needed to contain spread of an oncolytic virus within a spherical tumor and prevent it from spreading to non-cancerous cells outside the tumor. We find that oncolytic viruses that have different infection rates or different cell death rates in cancer and non-cancerous cells can be made to stay within the tumor. We find that there is a minimum difference in infection rates or cell death rates that will contain the virus and that this threshold value depends on the growth rate of the cancer. Identification of these types of thresholds can help researchers develop safer strains of oncolytic viruses allowing further development of this promising treatment.

AMBER: A Modular Model for Tumor Growth, Vasculature and Radiation Response

Computational models of tumor growth are valuable for simulating the dynamics of cancer progression and treatment responses. In particular, agent-based models (ABMs) tracking individual agents and their interactions are useful for their flexibility and ability to model complex behaviors. However, ABMs have often been confined to small domains or, when scaled up, have neglected crucial aspects like vasculature. Additionally, the integration into tumor ABMs of precise radiation dose calculations using gold-standard Monte Carlo (MC) methods, crucial in contemporary radiotherapy, has been lacking. Here, we introduce AMBER, an Agent-based fraMework for radioBiological Effects in Radiotherapy that computationally models tumor growth and radiation responses. AMBER is based on a voxelized geometry, enabling realistic simulations at relevant pre-clinical scales by tracking temporally discrete states stepwise. Its hybrid approach, combining traditional ABM techniques with continuous spatiotemporal fields of key microenvironmental factors such as oxygen and vascular endothelial growth factor, facilitates the generation of realistic tortuous vascular trees. Moreover, AMBER is integrated with TOPAS, an MC-based particle transport algorithm that simulates heterogeneous radiation doses. The impact of radiation on tumor dynamics considers the microenvironmental factors that alter radiosensitivity, such as oxygen availability, providing a full coupling between the biological and physical aspects. Our results show that simulations with AMBER yield accurate tumor evolution and radiation treatment outcomes, consistent with established volumetric growth laws and radiobiological understanding. Thus, AMBER emerges as a promising tool for replicating essential features of tumor growth and radiation response, offering a modular design for future expansions to incorporate specific biological traits.

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.

A mathematical framework for the statistical interpretation of biological growth models

Biological entities are inherently dynamic. As such, various ecological disciplines use mathematical models to describe temporal evolution. Typically, growth curves are modelled as sigmoids, with the evolution modelled by ordinary differential equations. Among the various sigmoid models, the logistic, Gompertz and Richards equations are well-established and widely used for the purpose of fitting growth data in the fields of biology and ecology. The present paper puts forth a mathematical framework for the statistical analysis of population growth models. The analysis is based on a mathematical model of the population-environment relationship, the theoretical foundations of which are discussed in detail. By applying this theory, stochastic evolutionary equations are obtained, for which the logistic, Gompertz, Richards and Birch equations represent a limiting case. To substantiate the models of population growth dynamics, the results of numerical simulations are presented. It is demonstrated that a variety of population growth models can be addressed in a comparable manner. It is suggested that the discussed mathematical framework for statistical interpretation of the joint population-environment evolution represents a promising avenue for further research.

Spatial interactions modulate tumor growth and immune infiltration

Direct observation of tumor-immune interactions is unlikely in tumors with currently available technology, but computational simulations based on clinical data can provide insight to test hypotheses. It is hypothesized that patterns of collagen evolve as a mechanism of immune escape, but the exact nature of immune-collagen interactions is poorly understood. Spatial data quantifying collagen fiber alignment in squamous cell carcinomas indicates that late-stage disease is associated with highly aligned fibers. Our computational modeling framework discriminates between two hypotheses: immune cell migration that moves (1) parallel or (2) perpendicular to collagen fiber orientation. The modeling recapitulates immune-extracellular matrix interactions where collagen patterns provide immune protection, leading to an emergent inverse relationship between disease stage and immune coverage. Here, computational modeling provides important mechanistic insights by defining a kernel cell-cell interaction function that considers a spectrum of local (cell-scale) to global (tumor-scale) spatial interactions. Short-range interaction kernels provide a mechanism for tumor cell survival under conditions with strong Allee effects, while asymmetric tumor-immune interaction kernels lead to poor immune response. Thus, the length scale of tumor-immune interaction kernels drives tumor growth and infiltration.

Shell-by-Shell functionalized nanoparticles as radiosensitizers and radioprotectors in radiation therapy of cancer cells and tumor spheroids

Shell-by-Shell (SbS)-functionalized NPs can be tailor-made by combining a metal oxide NP core of choice with any desired phosphonic acids and amphiphiles as 1st or 2nd ligand shell building blocks. The complementary composition of such highly hierarchical structures makes them interesting candidates for various biomedical applications, as certain active ingredients can be incorporated into the structure. Here, we used TiO2 and CoFe2O4 NPs as drug delivery tools and coated them with a hexadecylphosphonic acid and with hexadecyl ammonium phenolates (caffeate, p-coumarate, ferulate), that possess anticancer as well as antioxidant properties. These architectures were then incubated in 2D and 3D cell cultures of non-tumorigenic and tumorigenic breast cells and irradiated to study their anticancer effect. It was found that both, the functionalized TiO2 and CoFe2O4 NPs acted as strong protective agents in non-tumorigenic spheroids. In contrast, the functionalized CoFe2O4 NPs induce a higher damage in irradiated tumor spheroids compared to the functionalized TiO2 NPs. CoFe3O4 NPs act additionally as radiosensitizing agents to the tumor spheroids. The radio-enhancement of the CoFe2O4 NPs is due to the generation of highly toxic hydroxyl radicals during X-ray irradiation. The irradiation exposed the CoFe2O4 surface, releasing the anticancer drugs into the cytoplasm and making the surface Co2+ ions accessible. These surface ions catalyze the Fenton reaction. This combination of radiosensitizer and anticancer drug delivery proved to be a very effective nanotherapeutic in 2D and 3D cell cultures of breast cancer cells.

Practical parameter identifiability and handling of censored data with Bayesian inference in mathematical tumour models

Mechanistic mathematical models (MMs) are a powerful tool to help us understand and predict the dynamics of tumour growth under various conditions. In this work, we use 5 MMs with an increasing number of parameters to explore how certain (often overlooked) decisions in estimating parameters from data of experimental tumour growth affect the outcome of the analysis. In particular, we propose a framework for including tumour volume measurements that fall outside the upper and lower limits of detection, which are normally discarded. We demonstrate how excluding censored data results in an overestimation of the initial tumour volume and the MM-predicted tumour volumes prior to the first measurements, and an underestimation of the carrying capacity and the MM-predicted tumour volumes beyond the latest measurable time points. We show in which way the choice of prior for the MM parameters can impact the posterior distributions, and illustrate that reporting the most likely parameters and their 95% credible interval can lead to confusing or misleading interpretations. We hope this work will encourage others to carefully consider choices made in parameter estimation and to adopt the approaches we put forward herein.

Predicting efficacy assessment of combined treatment of radiotherapy and nivolumab for NSCLC patients through virtual clinical trials using QSP modeling

Non-Small Cell Lung Cancer (NSCLC) remains one of the main causes of cancer death worldwide. In the urge of finding an effective approach to treat cancer, enormous therapeutic targets and treatment combinations are explored in clinical studies, which are not only costly, suffer from a shortage of participants, but also unable to explore all prospective therapeutic solutions. Within the evolving therapeutic landscape, the combined use of radiotherapy (RT) and checkpoint inhibitors (ICIs) emerged as a promising avenue. Exploiting the power of quantitative system pharmacology (QSP), we undertook a study to anticipate the therapeutic outcomes of these interventions, aiming to address the limitations of clinical trials. After enhancing a pre-existing QSP platform and accurately replicating clinical data outcomes, we conducted an in-depth study, examining different treatment protocols with nivolumab and RT, both as monotherapy and in combination, by assessing their efficacy through clinical endpoints, namely time to progression (TTP) and duration of response (DOR). As result, the synergy of combined protocols showcased enhanced TTP and extended DOR, suggesting dual advantages of extended response and slowed disease progression with certain combined regimens. Through the lens of QSP modeling, our findings highlight the potential to fine-tune combination therapies for NSCLC, thereby providing pivotal insights for tailoring patient-centric therapeutic interventions.

Mathematical Modeling of Tumor Growth in Preclinical Mouse Models with Applications in Biomarker Discovery and Drug Mechanism Studies

Oncology drug efficacy is evaluated in mouse models by continuously monitoring tumor volumes, which can be mathematically described by growth kinetic models. Although past studies have investigated various growth models, their reliance on small datasets raises concerns about whether their findings are truly representative of tumor growth in diverse mouse models under different vehicle or drug treatments. In this study, we systematically evaluated six parametric models (exponential, exponential quadratic, monomolecular, logistic, Gompertz, and von Bertalanffy) and the semiparametric generalized additive model (GAM) on fitting tumor volume data from more than 30,000 mice in 930 experiments conducted in patient-derived xenografts, cell line-derived xenografts, and syngeneic models. We found that the exponential quadratic model is the best parametric model and can adequately model 87% studies, higher than other models including von Bertalanffy (82%) and Gompertz (80%) models; the latter is often considered the standard growth model. At the mouse group level, 7.5% of growth data could not be fit by any parametric model and were fitted by GAM. We show that endpoint gain integrated in time, a GAM-derived efficacy metric, is equivalent to exponential growth rate, a metric we previously proposed and conveniently calculated by simple algebra. Using five studies on paclitaxel, anti-PD1 antibody, cetuximab, irinotecan, and sorafenib, we showed that exponential and exponential quadratic models achieve similar performance in uncovering drug mechanism and biomarkers. We also compared exponential growth rate-based association analysis and exponential modeling approach in biomarker discovery and found that they complement each other. Modeling methods herein are implemented in an open-source R package freely available at https://github.com/hjzhou988/TuGroMix.

SIGNIFICANCE

We present a general strategy for mathematically modeling tumor growth in mouse models using data from 30,000 mice and show that modeling and nonmodeling approaches are complementary in biomarker discovery and drug mechanism studies.

New solutions of time-fractional cancer tumor models using modified He-Laplace algorithm

Cancer develops through cells when mutations build up in different genes that control cell proliferation. To treat these abnormal cells and minimize their growth, various cancer tumor samples have been modeled and analyzed in literature. The current study is focused on the investigation of more generalized cancer tumor model in fractional environment, where net killing rate is taken into account in different domains. Three types of killing rates are considered in the current study including time and position dependent killing rates, and concentration of cells based killing rate. A hybrid mechanism is proposed in which different homotopies are used with perturbation technique and Laplace transform. This leads to a convenient algorithm to tackle all types of fractional derivatives efficiently. The convergence and error bounds of the proposed scheme are computed theoretically by proving related theorems. In the next phase, convergence and validity is analyzed numerically by calculating residual errors round the fractional domain. It is observed that computed errors are very less in the entire fractional domain. Moreover, comparative analysis of Caputo, Caputo-Fabrizio (CF), and Atangana-Baleanu (AB) fractional derivatives is also performed graphically to discern the effect of different fractional approaches on the solution profile. Analysis asserts the reliability of proposed methodology in the matter of intricate fractional tumor models, and hence can be used to other complex physical phenomena.

Modeling free tumor growth: Discrete, continuum, and hybrid approaches to interpreting cancer development

Tumor growth dynamics serve as a critical aspect of understanding cancer progression and treatment response to mitigate one of the most pressing challenges in healthcare. The in silico approach to understanding tumor behavior computationally provides an efficient, cost-effective alternative to wet-lab examinations and are adaptable to different environmental conditions, time scales, and unique patient parameters. As a result, this paper explored modeling of free tumor growth in cancer, surveying contemporary literature on continuum, discrete, and hybrid approaches. Factors like predictive power and high-resolution simulation competed against drawbacks like simulation load and parameter feasibility in these models. Understanding tumor behavior in different scenarios and contexts became the first step in advancing cancer research and revolutionizing clinical outcomes.

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.

The combined treatment with ketogenic diet and metformin slows tumor growth in two mouse models of triple negative breast cancer

Background

Many tumors contain hypoxic microenvironments caused by inefficient tumor vascularization. Hypoxic tumors have been shown to resist conventional cancer therapies. Hypoxic cancer cells rely on glucose to meet their energetic and anabolic needs to fuel uncontrolled proliferation and metastasis. This glucose dependency is linked to a metabolic shift in response to hypoxic conditions.

Methods

To leverage the glucose dependency of hypoxic tumor cells, we assessed the effects of a mild reduction in systemic glucose by controlling both dietary carbohydrates with a ketogenic diet and endogenous glucose production by using metformin on two mouse models of triple-negative breast cancer (TNBC).

Results

Here, we showed that animals with TNBC treated with the combination regimen of ketogenic diet and metformin (a) had their tumor burden lowered by two-thirds, (b) displayed 38% slower tumor growth, and (c) showed 36% longer latency, compared to the animals treated with a ketogenic diet or metformin alone. As a result, lowering systemic glucose by this combined dietary and pharmacologic approach improved overall survival in our mouse TNBC models by 31 days, approximately equivalent to 3 years of life extension in human terms.

Conclusion

This preclinical study demonstrates that reducing systemic glucose by combining a ketogenic diet and metformin significantly inhibits tumor proliferation and increases overall survival. Our findings suggest a possible treatment for a broad range of hypoxic and glycolytic tumor types that can augment existing treatment options to improve patient outcomes.

Spatial interactions modulate tumor growth and immune infiltration

Lenia, a cellular automata framework used in artificial life, provides a natural setting to implement mathematical models of cancer incorporating features such as morphogenesis, homeostasis, motility, reproduction, growth, stimuli response, evolvability, and adaptation. Historically, agent-based models of cancer progression have been constructed with rules that govern birth, death and migration, with attempts to map local rules to emergent global growth dynamics. In contrast, Lenia provides a flexible framework for considering a spectrum of local (cell-scale) to global (tumor-scale) dynamics by defining an interaction kernel governing density-dependent growth dynamics. Lenia can recapitulate a range of cancer model classifications including local or global, deterministic or stochastic, non-spatial or spatial, single or multi-population, and off or on-lattice. Lenia is subsequently used to develop data-informed models of 1) single-population growth dynamics, 2) multi-population cell-cell competition models, and 3) cell migration or chemotaxis. Mathematical modeling provides important mechanistic insights. First, short-range interaction kernels provide a mechanism for tumor cell survival under conditions with strong Allee effects. Next, we find that asymmetric interaction tumor-immune kernels lead to poor immune response. Finally, modeling recapitulates immune-ECM interactions where patterns of collagen formation provide immune protection, indicated by an emergent inverse relationship between disease stage and immune coverage.

Selective sweep probabilities in spatially expanding populations

Evolution during range expansions shapes biological systems from microbial communities and tumours up to invasive species. A fundamental question is whether, when a beneficial mutation arises during a range expansion, it will evade clonal interference and sweep through the population to fixation. However, most theoretical investigations of range expansions have been confined to regimes in which selective sweeps are effectively impossible, while studies of selective sweeps have either assumed constant population size or have ignored spatial structure. Here we use mathematical modelling and analysis to investigate selective sweep probabilities in the alternative yet biologically relevant scenario in which mutants can outcompete and displace a slowly spreading wildtype. Assuming constant radial expansion speed, we derive probability distributions for the arrival time and location of the first surviving mutant and hence find surprisingly simple approximate and exact expressions for selective sweep probabilities in one, two and three dimensions, which are independent of mutation rate. Namely, the selective sweep probability is approximately 1 - c w t / c m d , where c w t and c m are the wildtype and mutant radial expansion speeds, and d the spatial dimension. Using agent-based simulations, we show that our analytical results accurately predict selective sweep frequencies in the two-dimensional spatial Moran process. We further compare our results with those obtained for alternative growth laws. Parameterizing our model for human tumours, we find that selective sweeps are predicted to be rare except during very early solid tumour growth, thus providing a general, pan-cancer explanation for findings from recent sequencing studies.

Long-term in vivo three-photon imaging reveals region-specific differences in healthy and regenerative oligodendrogenesis

The generation of new myelin-forming oligodendrocytes in the adult central nervous system is critical for cognitive function and regeneration following injury. Oligodendrogenesis varies between gray and white matter regions, suggesting that local cues drive regional differences in myelination and the capacity for regeneration. However, the layer- and region-specific regulation of oligodendrocyte populations is unclear due to the inability to monitor deep brain structures in vivo. Here we harnessed the superior imaging depth of three-photon microscopy to permit long-term, longitudinal in vivo three-photon imaging of the entire cortical column and subcortical white matter in adult mice. We find that cortical oligodendrocyte populations expand at a higher rate in the adult brain than those of the white matter. Following demyelination, oligodendrocyte replacement is enhanced in the white matter, while the deep cortical layers show deficits in regenerative oligodendrogenesis and the restoration of transcriptional heterogeneity. Together, our findings demonstrate that regional microenvironments regulate oligodendrocyte population dynamics and heterogeneity in the healthy and diseased brain.

The Gompertz model and its applications in microbial growth and bioproduction kinetics: Past, present and future

The Gompertz model, initially proposed for human mortality rates, has found various applications in growth analysis across the biotechnological field. This paper presents a comprehensive review of the Gompertz model's applications in the biotechnological field, examining its past, present, and future. The past of the Gompertz model was examined by tracing its origins to 1825, and then it underwent various modifications throughout the 20th century to increase its applicability in biotechnological fields. The Zwietering-modified version has proven to be a versatile tool for calculating the lag-time and maximum growth rate/quantity in microbial growth. In addition, the present applications of the Gompertz model to microbial growth kinetics and bioproduction (e.g., hydrogen, methane, caproate, butanol, and hexanol production) kinetics have been comprehensively summarized and discussed. We highlighted the importance of standardized citations and guidance on model selection. The Zwietering-modified Gompertz model and the Lay-modified Gompertz model are recommended for describing microbial growth kinetics and bioproduction kinetics, recognized for their widespread use and provision of valuable kinetics information. Finally, in response to the current Gompertz models' focus on internal mortality, the modified Makeham-Gompertz models that consider both internal/external mortality were introduced and validated for microbial growth and bioproduction kinetics with good fitting performance. This paper provides a perspective of the Gompertz model and offers valuable insights that facilitate the diverse applications of this model in microbial growth and bioproduction kinetics.

Balancing selection and the functional effects of shared polymorphism in cryptic Daphnia species

The patterns of genetic variation within and between related taxa represent the genetic history of a species. Shared polymorphisms, loci with identical alleles across species, are of unique interest as they may represent cases of ancient selection maintaining functional variation post-speciation. In this study, we investigate the abundance of shared polymorphism in the Daphnia pulex species complex. We test whether shared mutations are consistent with the action of balancing selection or alternative hypotheses such as hybridization, incomplete lineage sorting, or convergent evolution. We analyzed over 2,000 genomes from North American and European D. pulex and several outgroup species to examine the prevalence and distribution of shared alleles between the focal species pair, North American and European D. pulex. We show that while North American and European D. pulex diverged over ten million years ago, they retained tens of thousands of shared alleles. We found that the number of shared polymorphisms between North American and European D. pulex cannot be explained by hybridization or incomplete lineage sorting alone. Instead, we show that most shared polymorphisms could be the product of convergent evolution, that a limited number appear to be old trans-specific polymorphisms, and that balancing selection is affecting young and ancient mutations alike. Finally, we provide evidence that a blue wavelength opsin gene with trans-specific polymorphisms has functional effects on behavior and fitness in the wild. Ultimately, our findings provide insights into the genetic basis of adaptation and the maintenance of genetic diversity between species.

Explaining in-vitro to in-vivo efficacy correlations in oncology pre-clinical development via a semi-mechanistic mathematical model

In-vitro to in-vivo correlations (IVIVC), relating in-vitro parameters like IC50 to in-vivo drug exposure in plasma and tumour growth, are widely used in oncology for experimental design and dose decisions. However, they lack a deeper understanding of the underlying mechanisms. Our paper therefore focuses on linking empirical IVIVC relations for small-molecule kinase inhibitors with a semi-mechanistic tumour-growth model. We develop an approach incorporating parameters like the compound's peak-trough ratio (PTR), Hill coefficient of in-vitro dose-response curves, and xenograft-specific properties. This leads to formulas for determining efficacious doses for tumor stasis under linear pharmacokinetics equivalent to traditional empirical IVIVC relations, but enabling more systematic analysis. Our findings reveal that in-vivo xenograft-specific parameters, specifically the growth rate (g) and decay rate (d), along with the average exposure, are generally more significant determinants of tumor stasis and effective dose than the compound's peak-trough ratio. However, as the Hill coefficient increases, the dependency of tumor stasis on the PTR becomes more pronounced, indicating that the compound is more influenced by its maximum or trough values rather than the average exposure. Furthermore, we discuss the translation of our method to predict population dose ranges in clinical studies and propose a resistance mechanism that solely relies on specific in-vivo xenograft parameters instead of IC50 exposure coverage. In summary, our study aims to provide a more mechanistic understanding of IVIVC relations, emphasizing the importance of xenograft-specific parameters and PTR on tumor stasis.

Modeling of Mouse Experiments Suggests that Optimal Anti-Hormonal Treatment for Breast Cancer is Diet-Dependent

Estrogen receptor positive breast cancer is frequently treated with anti-hormonal treatment such as aromatase inhibitors (AI). Interestingly, a high body mass index has been shown to have a negative impact on AI efficacy, most likely due to disturbances in steroid metabolism and adipokine production. Here, we propose a mathematical model based on a system of ordinary differential equations to investigate the effect of high-fat diet on tumor growth. We inform the model with data from mouse experiments, where the animals are fed with high-fat or control (normal) diet. By incorporating AI treatment with drug resistance into the model and by solving optimal control problems we found differential responses for control and high-fat diet. To the best of our knowledge, this is the first attempt to model optimal anti-hormonal treatment for breast cancer in the presence of drug resistance. Our results underline the importance of considering high-fat diet and obesity as factors influencing clinical outcomes during anti-hormonal therapies in breast cancer patients.

Spatial interactions modulate tumor growth and immune infiltration

Direct observation of immune cell trafficking patterns and tumor-immune interactions is unlikely in human tumors with currently available technology, but computational simulations based on clinical data can provide insight to test hypotheses. It is hypothesized that patterns of collagen formation evolve as a mechanism of immune escape, but the exact nature of the interaction between immune cells and collagen is poorly understood. Spatial data quantifying the degree of collagen fiber alignment in squamous cell carcinomas indicates that late stage disease is associated with highly aligned fibers. Here, we introduce a computational modeling framework (called Lenia) to discriminate between two hypotheses: immune cell migration that moves 1) parallel or 2) perpendicular to collagen fiber orientation. The modeling recapitulates immune-ECM interactions where collagen patterns provide immune protection, leading to an emergent inverse relationship between disease stage and immune coverage. We also illustrate the capabilities of Lenia to model the evolution of tumor progression and immune predation. Lenia provides a flexible framework for considering a spectrum of local (cell-scale) to global (tumor-scale) dynamics by defining a kernel cell-cell interaction function that governs tumor growth dynamics under immune predation with immune cell migration. Mathematical modeling provides important mechanistic insights into cell interactions. Short-range interaction kernels provide a mechanism for tumor cell survival under conditions with strong Allee effects, while asymmetric tumor-immune interaction kernels lead to poor immune response. Thus, the length scale of tumor-immune interactions drives tumor growth and infiltration.

Tumor containment: a more general mathematical analysis

Clinical and pre-clinical data suggest that treating some tumors at a mild, patient-specific dose might delay resistance to treatment and increase survival time. A recent mathematical model with sensitive and resistant tumor cells identified conditions under which a treatment aiming at tumor containment rather than eradication is indeed optimal. This model however neglected mutations from sensitive to resistant cells, and assumed that the growth-rate of sensitive cells is non-increasing in the size of the resistant population. The latter is not true in standard models of chemotherapy. This article shows how to dispense with this assumption and allow for mutations from sensitive to resistant cells. This is achieved by a novel mathematical analysis comparing tumor sizes across treatments not as a function of time, but as a function of the resistant population size.

Calibration of agent based models for monophasic and biphasic tumour growth using approximate Bayesian computation

Agent-based models (ABMs) are readily used to capture the stochasticity in tumour evolution; however, these models are often challenging to validate with experimental measurements due to model complexity. The Voronoi cell-based model (VCBM) is an off-lattice agent-based model that captures individual cell shapes using a Voronoi tessellation and mimics the evolution of cancer cell proliferation and movement. Evidence suggests tumours can exhibit biphasic growth in vivo. To account for this phenomena, we extend the VCBM to capture the existence of two distinct growth phases. Prior work primarily focused on point estimation for the parameters without consideration of estimating uncertainty. In this paper, approximate Bayesian computation is employed to calibrate the model to in vivo measurements of breast, ovarian and pancreatic cancer. Our approach involves estimating the distribution of parameters that govern cancer cell proliferation and recovering outputs that match the experimental data. Our results show that the VCBM, and its biphasic extension, provides insight into tumour growth and quantifies uncertainty in the switching time between the two phases of the biphasic growth model. We find this approach enables precise estimates for the time taken for a daughter cell to become a mature cell. This allows us to propose future refinements to the model to improve accuracy, whilst also making conclusions about the differences in cancer cell characteristics.

On the possibility of engineering social evolution in microfluidic environments

Many species of microbes cooperate by producing public goods from which they collectively benefit. However, these populations are under the risk of being taken over by cheating mutants that do not contribute to the pool of public goods. Here we present theoretical findings that address how the social evolution of microbes can be manipulated by external perturbations to inhibit or promote the fixation of cheaters. To control social evolution, we determine the effects of fluid-dynamical properties such as flow rate or domain geometry. We also study the social evolutionary consequences of introducing beneficial or harmful chemicals at steady state and in a time-dependent fashion. We show that by modulating the flow rate and by applying pulsed chemical signals, we can modulate the spatial structure and dynamics of the population in a way that can select for more or less cooperative microbial populations.

Computational markers for personalized prediction of outcomes in non-small cell lung cancer patients with brain metastases

Intracranial progression after curative treatment of early-stage non-small cell lung cancer (NSCLC) occurs from 10 to 50% and is difficult to manage, given the heterogeneity of clinical presentations and the variability of treatments available. The objective of this study was to develop a mechanistic model of intracranial progression to predict survival following a first brain metastasis (BM) event occurring at a time [Formula: see text]. Data included early-stage NSCLC patients treated with a curative intent who had a BM as the first and single relapse site (N = 31). We propose a mechanistic mathematical model able to derive computational markers from primary tumor and BM data at [Formula: see text] and estimate the amount and sizes of (visible and invisible) BMs, as well as their future behavior. These two key computational markers are [Formula: see text], the proliferation rate of a single tumor cell; and [Formula: see text], the per day, per cell, probability to metastasize. The predictive value of these individual computational biomarkers was evaluated. The model was able to correctly describe the number and size of metastases at [Formula: see text] for 20 patients. Parameters [Formula: see text] and [Formula: see text] were significantly associated with overall survival (OS) (HR 1.65 (1.07-2.53) p = 0.0029 and HR 1.95 (1.31-2.91) p = 0.0109, respectively). Adding the computational markers to the clinical ones significantly improved the predictive value of OS (c-index increased from 0.585 (95% CI 0.569-0.602) to 0.713 (95% CI 0.700-0.726), p < 0.0001). We demonstrated that our model was applicable to brain oligoprogressive patients in NSCLC and that the resulting computational markers had predictive potential. This may help lung cancer physicians to guide and personalize the management of NSCLC patients with intracranial oligoprogression.

A computational tumor growth model experience based on molecular dynamics point of view using deep cellular automata

Cancer, as identified by the World Health Organization, stands as the second leading cause of death globally. Its intricate nature makes it challenging to study solely based on biological knowledge, often leading to expensive research endeavors. While tremendous strides have been made in understanding cancer, gaps remain, especially in predicting tumor behavior across various stages. The integration of artificial intelligence in oncology research has accelerated our insights into tumor behavior, right from its genesis to metastasis. Nevertheless, there's a pressing need for a holistic understanding of the interactions between cancer cells, their microenvironment, and their subsequent interplay with the broader body environment. In this landscape, deep learning emerges as a potent tool with its multifaceted applications in diverse scientific challenges. Motivated by this, our study presents a novel approach to modeling cancer tumor growth from a molecular dynamics' perspective, harnessing the capabilities of deep-learning cellular automata. This not only facilitates a microscopic examination of tumor behavior and growth but also delves deeper into its overarching behavioral patterns. Our work primarily focused on evaluating the developed tumor growth model through the proposed network, followed by a rigorous compatibility check with traditional mathematical tumor growth models using R and Matlab software. The outcomes notably aligned with the Gompertz growth model, accentuating the robustness of our approach. Our validated model stands out by offering adaptability to diverse tumor growth datasets, positioning itself as a valuable tool for predictions and further research.

Simulating tumor volume dynamics in response to radiotherapy: Implications of model selection

From the beginning of the usage of radiotherapy (RT) for cancer treatment, mathematical modeling has been integral to understanding radiobiology and for designing treatment approaches and schedules. There has been extensive modeling of response to RT with the inclusion of various degrees of biological complexity. In this study, we compare three models of tumor volume dynamics: (1) exponential growth with RT directly reducing tumor volume, (2) logistic growth with direct tumor volume reduction, and (3) logistic growth with RT reducing the tumor carrying capacity with the objective of understanding the implications of model selection and informing the process of model calibration and parameterization. For all three models, we: examined the rates of change in tumor volume during and RT treatment course; performed parameter sensitivity and identifiability analyses; and investigated the impact of the parameter sensitivity on the tumor volume trajectories. In examining the tumor volume dynamics trends, we coined a new metric - the point of maximum reduction of tumor volume (MRV) - to quantify the magnitude and timing of the expected largest impact of RT during a treatment course. We found distinct timing differences in MRV, dependent on model selection. The parameter identifiability and sensitivity analyses revealed the interdependence of the different model parameters and that it is only possible to independently identify tumor growth and radiation response parameters if the underlying tumor growth rate is sufficiently large. Ultimately, the results of these analyses help us to better understand the implications of model selection while simultaneously generating falsifiable hypotheses about MRV timing that can be tested on longitudinal measurements of tumor volume from pre-clinical or clinical data with high acquisition frequency. Although, our study only compares three particular models, the results demonstrate that caution is necessary in selecting models of response to RT, given the artifacts imposed by each model.

Modeling tumor growth using fractal calculus: Insights into tumor dynamics

Important concepts like fractal calculus and fractal analysis, the sum of squared residuals, and Aikaike's information criterion must be thoroughly understood in order to correctly fit cancer-related data using the proposed models. The fractal growth models employed in this work are classified in three main categories: Sigmoidal growth models (Logistic, Gompertz, and Richards models), Power Law growth model, and Exponential growth models (Exponential and Exponential-Lineal models)". We fitted the data, computed the sum of squared residuals, and determined Aikaike's information criteria using Matlab and the web tool WebPlotDigitizer. In addition, the research investigates "double-size cancer" in the fractal temporal dimension with respect to various mathematical models.

A calibration and uncertainty quantification analysis of classical, fractional and multiscale logistic models of tumour growth

BACKGROUND AND OBJECTIVE

The validation of mathematical models of tumour growth is frequently hampered by the lack of sufficient experimental data, resulting in qualitative rather than quantitative studies. Recent approaches to this problem have attempted to extract information about tumour growth by integrating multiscale experimental measurements, such as longitudinal cell counts and gene expression data. In the present study, we investigated the performance of several mathematical models of tumour growth, including classical logistic, fractional and novel multiscale models, in terms of quantifying in-vitro tumour growth in the presence and absence of therapy. We further examined the effect of genes associated with changes in chemosensitivity in cell death rates.

METHODS

The multiscale expansion of logistic growth models was performed by coupling gene expression profiles to the cell death rates. State-of-the-art Bayesian inference, likelihood maximisation and uncertainty quantification techniques allowed a thorough evaluation of model performance.

RESULTS

The results suggest that the classical single-cell population model (SCPM) was the best fit for the untreated and low-dose treatment conditions, while the multiscale model with a cell death rate symmetric with the expression profile of OCT4 (Sym-SCPM) yielded the best fit for the high-dose treatment data. Further identifiability analysis showed that the multiscale model was both structurally and practically identifiable under the condition of known OCT4 expression profiles.

CONCLUSIONS

Overall, the present study demonstrates that model performance can be improved by incorporating multiscale measurements of tumour growth when high-dose treatment is involved.

The Combined Treatment with Ketogenic Diet and Metformin Slows Tumor Growth in Two Mouse Models of Triple Negative Breast Cancer

BACKGROUND

Many tumors contain hypoxic microenvironments caused by inefficient tumor vascularization. Hypoxic tumors have been shown to resist conventional cancer therapies. Hypoxic cancer cells rely on glucose to meet their energetic and anabolic needs to fuel uncontrolled proliferation and metastasis. This glucose dependency is linked to a metabolic shift in response to hypoxic conditions.

METHODS

To leverage the glucose dependency of hypoxic tumor cells, we assessed the effects of a controlled reduction in systemic glucose by combining dietary carbohydrate restriction, using a ketogenic diet, with gluconeogenesis inhibition, using metformin, on two mouse models of triple-negative breast cancer (TNBC).

RESULTS

We confirmed that MET - 1 breast cancer cells require abnormally high glucose concentrations to survive in a hypoxic environment in vitro. Then, we showed that, compared to a ketogenic diet or metformin alone, animals treated with the combination regimen showed significantly lower tumor burden, higher tumor latency and slower tumor growth. As a result, lowering systemic glucose by this combined dietary and pharmacologic approach improved overall survival in our mouse model by 31 days, which is approximately equivalent to 3 human years.

CONCLUSION

This is the first preclinical study to demonstrate that reducing systemic glucose by combining a ketogenic diet and metformin significantly inhibits tumor proliferation and increases overall survival. Our findings suggest a possible treatment for a broad range of hypoxic and glycolytic tumor types, one that can also augment existing treatment options to improve patient outcomes.

Long-term in vivo three-photon imaging reveals region-specific differences in healthy and regenerative oligodendrogenesis

The generation of new myelin-forming oligodendrocytes in the adult CNS is critical for cognitive function and regeneration following injury. Oligodendrogenesis varies between gray and white matter regions suggesting that local cues drive regional differences in myelination and the capacity for regeneration. Yet, the determination of regional variability in oligodendrocyte cell behavior is limited by the inability to monitor the dynamics of oligodendrocytes and their transcriptional subpopulations in white matter of the living brain. Here, we harnessed the superior imaging depth of three-photon microscopy to permit long-term, longitudinal in vivo three-photon imaging of an entire cortical column and underlying subcortical white matter without cellular damage or reactivity. Using this approach, we found that the white matter generated substantially more new oligodendrocytes per volume compared to the gray matter, yet the rate of population growth was proportionally higher in the gray matter. Following demyelination, the white matter had an enhanced population growth that resulted in higher oligodendrocyte replacement compared to the gray matter. Finally, deep cortical layers had pronounced deficits in regenerative oligodendrogenesis and restoration of the MOL5/6-positive oligodendrocyte subpopulation following demyelinating injury. Together, our findings demonstrate that regional microenvironments regulate oligodendrocyte population dynamics and heterogeneity in the healthy and diseased brain.

Predictive digital twin for optimizing patient-specific radiotherapy regimens under uncertainty in high-grade gliomas

We develop a methodology to create data-driven predictive digital twins for optimal risk-aware clinical decision-making. We illustrate the methodology as an enabler for an anticipatory personalized treatment that accounts for uncertainties in the underlying tumor biology in high-grade gliomas, where heterogeneity in the response to standard-of-care (SOC) radiotherapy contributes to sub-optimal patient outcomes. The digital twin is initialized through prior distributions derived from population-level clinical data in the literature for a mechanistic model's parameters. Then the digital twin is personalized using Bayesian model calibration for assimilating patient-specific magnetic resonance imaging data. The calibrated digital twin is used to propose optimal radiotherapy treatment regimens by solving a multi-objective risk-based optimization under uncertainty problem. The solution leads to a suite of patient-specific optimal radiotherapy treatment regimens exhibiting varying levels of trade-off between the two competing clinical objectives: (i) maximizing tumor control (characterized by minimizing the risk of tumor volume growth) and (ii) minimizing the toxicity from radiotherapy. The proposed digital twin framework is illustrated by generating an in silico cohort of 100 patients with high-grade glioma growth and response properties typically observed in the literature. For the same total radiation dose as the SOC, the personalized treatment regimens lead to median increase in tumor time to progression of around six days. Alternatively, for the same level of tumor control as the SOC, the digital twin provides optimal treatment options that lead to a median reduction in radiation dose by 16.7% (10 Gy) compared to SOC total dose of 60 Gy. The range of optimal solutions also provide options with increased doses for patients with aggressive cancer, where SOC does not lead to sufficient tumor control.

A phase-field model for non-small cell lung cancer under the effects of immunotherapy

Formulating mathematical models that estimate tumor growth under therapy is vital for improving patient-specific treatment plans. In this context, we present our recent work on simulating non-small-scale cell lung cancer (NSCLC) in a simple, deterministic setting for two different patients receiving an immunotherapeutic treatment. At its core, our model consists of a Cahn-Hilliard-based phase-field model describing the evolution of proliferative and necrotic tumor cells. These are coupled to a simplified nutrient model that drives the growth of the proliferative cells and their decay into necrotic cells. The applied immunotherapy decreases the proliferative cell concentration. Here, we model the immunotherapeutic agent concentration in the entire lung over time by an ordinary differential equation (ODE). Finally, reaction terms provide a coupling between all these equations. By assuming spherical, symmetric tumor growth and constant nutrient inflow, we simplify this full 3D cancer simulation model to a reduced 1D model. We can then resort to patient data gathered from computed tomography (CT) scans over several years to calibrate our model. Our model covers the case in which the immunotherapy is successful and limits the tumor size, as well as the case predicting a sudden relapse, leading to exponential tumor growth. Finally, we move from the reduced model back to the full 3D cancer simulation in the lung tissue. Thereby, we demonstrate the predictive benefits that a more detailed patient-specific simulation including spatial information as a possible generalization within our framework could yield in the future.

Modeling of oncolytic viruses in a heterogeneous cell population to predict spread into non-cancerous cells

New cancer treatment modalities that limit patient discomfort need to be developed. One possible new therapy is the use of oncolytic (cancer-killing) viruses. It is only recently that our ability to manipulate viral genomes has allowed us to consider deliberately infecting cancer patients with viruses. One key consideration is to ensure that the virus exclusively targets cancer cells and does not harm nearby non-cancerous cells. Here, we use a mathematical model of viral infection to determine the characteristics a virus would need to have in order to eradicate a tumor, but leave non-cancerous cells untouched. We conclude that the virus must differ in its ability to infect the two different cell types, with the infection rate of non-cancerous cells needing to be less than one hundredth of the infection rate of cancer cells. Differences in viral production rate or infectious cell death rate alone are not sufficient to protect non-cancerous cells.

Mechanistic characterization of oscillatory patterns in unperturbed tumor growth dynamics: The interplay between cancer cells and components of tumor microenvironment

Mathematical modeling of unperturbed and perturbed tumor growth dynamics (TGD) in preclinical experiments provides an opportunity to establish translational frameworks. The most commonly used unperturbed tumor growth models (i.e. linear, exponential, Gompertz and Simeoni) describe a monotonic increase and although they capture the mean trend of the data reasonably well, systematic model misspecifications can be identified. This represents an opportunity to investigate possible underlying mechanisms controlling tumor growth dynamics through a mathematical framework. The overall goal of this work is to develop a data-driven semi-mechanistic model describing non-monotonic tumor growth in untreated mice. For this purpose, longitudinal tumor volume profiles from different tumor types and cell lines were pooled together and analyzed using the population approach. After characterizing the oscillatory patterns (oscillator half-periods between 8-11 days) and confirming that they were systematically observed across the different preclinical experiments available (p<10-9), a tumor growth model was built including the interplay between resources (i.e. oxygen or nutrients), angiogenesis and cancer cells. The new structure, in addition to improving the model diagnostic compared to the previously used tumor growth models (i.e. AIC reduction of 71.48 and absence of autocorrelation in the residuals (p>0.05)), allows the evaluation of the different oncologic treatments in a mechanistic way. Drug effects can potentially, be included in relevant processes taking place during tumor growth. In brief, the new model, in addition to describing non-monotonic tumor growth and the interaction between biological factors of the tumor microenvironment, can be used to explore different drug scenarios in monotherapy or combination during preclinical drug development.

A two-stages global sensitivity analysis by using the δ sensitivity index in presence of correlated inputs: application on a tumor growth inhibition model based on the dynamic energy budget theory

Global sensitivity analysis (GSA) evaluates the impact of variability and/or uncertainty of the model parameters on given model outputs. GSA is useful for assessing the quality of Pharmacometric model inference. Indeed, model parameters can be affected by high (estimation) uncertainty due to the sparsity of data. Independence between model parameters is a common assumption of GSA methods. However, ignoring (known) correlations between parameters may alter model predictions and, then, GSA results. To address this issue, a novel two-stages GSA technique based on the δ index, which is well-defined also in presence of correlated parameters, is here proposed. In the first step, statistical dependencies are neglected to identify parameters exerting causal effects. Correlations are introduced in the second step to consider the real distribution of the model output and investigate also the 'indirect' effects due to the correlation structure. The proposed two-stages GSA strategy was applied, as case study, to a preclinical tumor-in-host-growth inhibition model based on the Dynamic Energy Budget theory. The aim is to evaluate the impact of the model parameter estimate uncertainty (including correlations) on key model-derived metrics: the drug threshold concentration for tumor eradication, the tumor volume doubling time and a new index evaluating the drug efficacy-toxicity trade-off. This approach allowed to rank parameters according to their impact on the output, discerning whether a parameter mainly exerts a causal or 'indirect' effect. Thus, it was possible to identify uncertainties that should be necessarily reduced to obtain robust predictions for the outputs of interest.

Logistic tumor-population growth and ghost-points symmetry

The observed time evolution of a population is well approximated by a logistic function in many research fields, including oncology, ecology, chemistry, demography, economy, linguistics, and artificial neural networks. Initial growth is exponential at a constant rate and capped at a limit size, i.e., the carrying capacity. In mathematical oncology, the carrying capacity has been postulated to be co-evolving and thus 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 a classic oncology database of logistic tumor growth, achieving a 30% to 40% reduction in the time necessary to correctly estimate the logistic growth rate and carrying capacity. Our results will improve tumor dynamics forecasting and augment the clinical decision-making process.

ConvLSTM coordinated longitudinal transformer under spatio-temporal features for tumor growth prediction

Accurate quantification of tumor growth patterns can indicate the development process of the disease. According to the important features of tumor growth rate and expansion, physicians can intervene and diagnose patients more efficiently to improve the cure rate. However, the existing longitudinal growth model can not well analyze the dependence between tumor growth pixels in the long space-time, and fail to effectively fit the nonlinear growth law of tumors. So, we propose the ConvLSTM coordinated longitudinal Transformer (LCTformer) under spatiotemporal features for tumor growth prediction. We design the Adaptive Edge Enhancement Module (AEEM) to learn static spatial features of different size tumors under time series and make the depth model more focused on tumor edge regions. In addition, we propose the Growth Prediction Module (GPM) to characterize the future growth trend of tumors. It consists of a Longitudinal Transformer and ConvLSTM. Based on the adaptive abstract features of current tumors, Longitudinal Transformer explores the dynamic growth patterns between spatiotemporal CT sequences and learns the future morphological features of tumors under the dual views of residual information and sequence motion relationship in parallel. ConvLSTM can better learn the location information of target tumors, and it complements Longitudinal Transformer to jointly predict future imaging of tumors to reduce the loss of growth information. Finally, Channel Enhancement Fusion Module (CEFM) performs the dense fusion of the generated tumor feature images in the channel and spatial dimensions and realizes accurate quantification of the whole tumor growth process. Our model has been strictly trained and tested on the NLST dataset. The average prediction accuracy can reach 88.52% (Dice score), 89.64% (Recall), and 11.06 (RMSE), which can improve the work efficiency of doctors.

Acriflavine-Functionalized Silica@Manganese Ferrite Nanostructures for Synergistic Radiation and Hypoxia Therapies

Mesoporous and nonmesoporous SiO2@MnFe2O4 nanostructures were loaded with the hypoxia-inducible factor-1 inhibitor acriflavine for combined radiation and hypoxia therapies. The X-ray irradiation of the drug-loaded nanostructures not only triggered the release of the acriflavine inside the cells but also initiated an energy transfer from the nanostructures to surface-adsorbed oxygen to generate singlet oxygen. While the drug-loaded mesoporous nanostructures showed an initial drug release before the irradiation, the drug was primarily released upon X-ray radiation in the case of the nonmesoporous nanostructures. However, the drug loading capacity was less efficient for the nonmesoporous nanostructures. Both drug-loaded nanostructures proved to be very efficient in irradiated MCF-7 multicellular tumor spheroids. The damage of these nanostructures toward the nontumorigenic MCF-10A multicellular spheroids was very limited because of the small number of nanostructures that entered the MCF-10A spheroids, while similar concentrations of acriflavine without nanostructures were toxic for the MCF-10A spheroids.

Evaluation of four tumour growth models to describe the natural history of meningiomas

BACKGROUND

The incidence of newly diagnosed meningiomas, particularly those diagnosed incidentally, is continually increasing. The indication for treatment is empirical because, despite numerous studies, the natural history of these tumours remains difficult to describe and predict.

METHODS

This retrospective single-centre study included 294 consecutive patients with 333 meningiomas who underwent three or more brain imaging scans. Linear, exponential, power, and Gompertz models were constructed to derive volume-time curves, by using a mixed-effect approach. The most accurate model was used to analyse tumour growth and predictors of rapid growth.

FINDINGS

The Gompertz model provided the best results. Hierarchical clustering at the time of diagnosis and at the end of follow-up revealed at least three distinct groups, which can be described as pseudoexponential, linear, and slowing growth with respect to their parameters. Younger patients and smaller tumours were more frequent in the pseudo-exponential clusters. We found that the more "aggressive" the cluster, the higher the proportion of patients with grade II meningiomas and who have had a cranial radiotherapy. Over a mean observation period of 56.5 months, 21% of the tumours moved to a cluster with a lower growth rate, consistent with the Gompertz's law.

INTERPRETATION

Meningiomas exhibit multiple growth phases, as described by the Gompertz model. The management of meningiomas should be discussed according to the growth phase, comorbidities, tumour location, size, and growth rate. Further research is needed to evaluate the associations between radiomics features and the growth phases of meningiomas.

FUNDING

No funding.

Growth exponents reflect evolutionary processes and treatment response in brain metastases

Tumor growth is the result of the interplay of complex biological processes in huge numbers of individual cells living in changing environments. Effective simple mathematical laws have been shown to describe tumor growth in vitro, or simple animal models with bounded-growth dynamics accurately. However, results for the growth of human cancers in patients are scarce. Our study mined a large dataset of 1133 brain metastases (BMs) with longitudinal imaging follow-up to find growth laws for untreated BMs and recurrent treated BMs. Untreated BMs showed high growth exponents, most likely related to the underlying evolutionary dynamics, with experimental tumors in mice resembling accurately the disease. Recurrent BMs growth exponents were smaller, most probably due to a reduction in tumor heterogeneity after treatment, which may limit the tumor evolutionary capabilities. In silico simulations using a stochastic discrete mesoscopic model with basic evolutionary dynamics led to results in line with the observed data.