Mathematical modelling reveals cellular dynamics within tumour spheroids

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.

3D cell aggregates amplify diffusion signals

Biophysical models can predict the behavior of cell cultures including 3D cell aggregates (3DCAs), thereby reducing the need for costly and time-consuming experiments. Specifically, mass transfer models enable studying the transport of nutrients, oxygen, signaling molecules, and drugs in 3DCA. These models require the defining of boundary conditions (BC) between the 3DCA and surrounding medium. However, accurately modeling the BC that relates the inner and outer boundary concentrations at the border between the 3DCA and the medium remains a challenge that this paper addresses using both theoretical and experimental methods. The provided biophysical analysis indicates that the concentration of molecules inside boundary is higher than that at the outer boundary, revealing an amplification factor that is confirmed by a particle-based simulator (PBS). Due to the amplification factor, the PBS confirms that when a 3DCA with a low concentration of target molecules is introduced to a culture medium with a higher concentration, the molecule concentration in the medium rapidly decreases. The theoretical model and PBS simulations were used to design a pilot experiment with liver spheroids as the 3DCA and glucose as the target molecule. Experimental results agree with the proposed theory and derived properties.

Determining Rates of Molecular Secretion from Supernatant Concentration Measurements in a 3D-Bioprinted Human Liver Tissue Model

The secretion rate of albumin is a key indicator of function in liver tissue models used for hepatotoxicity and pharmacokinetic testing. However, it is not generally clear how to determine molecular secretion rates from measurements of the molecular concentration in supernatant media. Here, we develop computational and analytical models of molecular transport in an experimental system that enable determination of albumin secretion rates based on measurements of albumin concentration in supernatant media. The experimental system is a 3D-bioprinted human liver tissue construct embedded in a 3D culture environment made from packed microgel particles swollen in liquid growth media. The mathematical models reveal that the range of albumin synthesis rates necessary to match experimentally measured albumin concentrations corresponds to reaction-limited conditions, where a steady state of albumin spatial distribution is rapidly reached between media exchanges. Our results show that temporally resolved synthesis rates can be inferred from serial concentration measurements of collected supernatant media. This link is critical to confidently assessing in vitro tissue performance in applications where critical quality attributes must be quantified, like in drug development and screening.

The influence of spheroid maturity on fusion dynamics and micro-tissue assembly in 3D tumor models

Three-dimensional (3D) cell culture has been used in many fields of biology because of its unique advantages. As a representative of the 3D systems, 3D spheroids are used as building blocks for tissue construction. Larger tumor aggregates can be assembled by manipulating or stacking the tumor spheroids. The motivation of this study is to investigate the behavior of the cells distributed at different locations of the spheroids in the fusion process and the mechanism behind it. To this aim, spheroids with varying grades of maturity or age were generated for fusion to assemble micro-tumor tissues. The dynamics of the fusion process, the motility of the cells distributed in different heterogeneous architecture sites, and their reactive oxygen species profiles were studied. We found that the larger the spheroid necrotic core, the slower the fusion rate of the spheroid. The cells that move were mainly distributed on the spheroid's surface during fusion. In addition to dense microfilament distribution and low microtubule content, the reactive oxygen content was high in the fusion site, while the non-fusion site was the opposite. Last, multi-spheroids with different maturities were fused to complex micro-tissues to mimic solid tumors and evaluate Doxorubicin's anti-tumor efficacy.

Metabolic symbiosis between oxygenated and hypoxic tumour cells: An agent-based modelling study

Deregulated metabolism is one of the hallmarks of cancer. It is well-known that tumour cells tend to metabolize glucose via glycolysis even when oxygen is available and mitochondrial respiration is functional. However, the lower energy efficiency of aerobic glycolysis with respect to mitochondrial respiration makes this behaviour, namely the Warburg effect, counter-intuitive, although it has now been recognized as source of anabolic precursors. On the other hand, there is evidence that oxygenated tumour cells could be fuelled by exogenous lactate produced from glycolysis. We employed a multi-scale approach that integrates multi-agent modelling, diffusion-reaction, stoichiometric equations, and Boolean networks to study metabolic cooperation between hypoxic and oxygenated cells exposed to varying oxygen, nutrient, and inhibitor concentrations. The results show that the cooperation reduces the depletion of environmental glucose, resulting in an overall advantage of using aerobic glycolysis. In addition, the oxygen level was found to be decreased by symbiosis, promoting a further shift towards anaerobic glycolysis. However, the oxygenated and hypoxic populations may gradually reach quasi-equilibrium. A sensitivity analysis using Latin hypercube sampling and partial rank correlation shows that the symbiotic dynamics depends on properties of the specific cell such as the minimum glucose level needed for glycolysis. Our results suggest that strategies that block glucose transporters may be more effective to reduce tumour growth than those blocking lactate intake transporters.

Modeling the extracellular matrix in cell migration and morphogenesis: a guide for the curious biologist

The extracellular matrix (ECM) is a highly complex structure through which biochemical and mechanical signals are transmitted. In processes of cell migration, the ECM also acts as a scaffold, providing structural support to cells as well as points of potential attachment. Although the ECM is a well-studied structure, its role in many biological processes remains difficult to investigate comprehensively due to its complexity and structural variation within an organism. In tandem with experiments, mathematical models are helpful in refining and testing hypotheses, generating predictions, and exploring conditions outside the scope of experiments. Such models can be combined and calibrated with in vivo and in vitro data to identify critical cell-ECM interactions that drive developmental and homeostatic processes, or the progression of diseases. In this review, we focus on mathematical and computational models of the ECM in processes such as cell migration including cancer metastasis, and in tissue structure and morphogenesis. By highlighting the predictive power of these models, we aim to help bridge the gap between experimental and computational approaches to studying the ECM and to provide guidance on selecting an appropriate model framework to complement corresponding experimental studies.

Formation and Growth of Co-Culture Tumour Spheroids: New Compartment-Based Mathematical Models and Experiments

Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and test anti-cancer therapies. Therefore, methods to quantitatively characterise and interpret co-culture spheroid growth are of great interest. However, co-culture spheroid growth is complex. Multiple biological processes occur on overlapping timescales and different cell types within the spheroid may have different characteristics, such as differing proliferation rates or responses to nutrient availability. At present there is no standard, widely-accepted mathematical model of such complex spatio-temporal growth processes. Typical approaches to analyse these experiments focus on the late-time temporal evolution of spheroid size and overlook early-time spheroid formation, spheroid structure and geometry. Here, using a range of ordinary differential equation-based mathematical models and parameter estimation, we interpret new co-culture experimental data. We provide new biological insights about spheroid formation, growth, and structure. As part of this analysis we connect Greenspan's seminal mathematical model to co-culture data for the first time. Furthermore, we generalise a class of compartment-based spheroid mathematical models that have previously been restricted to one population so they can be applied to multiple populations. As special cases of the general model, we explore multiple natural two population extensions to Greenspan's seminal model and reveal biological mechanisms that can describe the internal dynamics of growing co-culture spheroids and those that cannot. This mathematical and statistical modelling-based framework is well-suited to analyse spheroids grown with multiple different cell types and the new class of mathematical models provide opportunities for further mathematical and biological insights.

Luciferase Expressing Preclinical Model Systems Representing the Different Molecular Subtypes of Colorectal Cancer

Colorectal cancer (CRC) is a heterogeneous disease. More insight into the biological diversity of CRC is needed to improve therapeutic outcomes. Established CRC cell lines are frequently used and were shown to be representative models of the main subtypes of CRC at the genomic and transcriptomic level. In the present work, we established stable, luciferase expressing derivatives from 10 well-established CRC cell lines, generated spheroids and subcutaneous xenograft tumors in nude mice, and performed comparative characterization of these model systems. Transcriptomic analyses revealed the close relation of cell lines with their derived spheroids and xenograft tumors. The preclinical model systems clustered with patient tumor samples when compared to normal tissue thereby confirming that cell-line-based tumor models retain specific characteristics of primary tumors. Xenografts showed different differentiation patterns and bioluminescence imaging revealed metastatic spread to the lungs. In addition, the models were classified according to the CMS classification system, with further sub-classification according to the recently identified two intrinsic epithelial tumor cell states of CRC, iCMS2 and iCMS3. The combined data showed that regarding primary tumor characteristics, 3D-spheroid cultures resemble xenografts more closely than 2D-cultured cells do. Furthermore, we set up a bioluminescence-based spheroid cytotoxicity assay in order to be able to perform dose-response relationship studies in analogy to typical monolayer assays. Applying the established assay, we studied the efficacy of oxaliplatin. Seven of the ten used cell lines showed a significant reduction in the response to oxaliplatin in the 3D-spheroid model compared to the 2D-monolayer model. Therapy studies in selected xenograft models confirmed the response or lack of response to oxaliplatin treatment. Analyses of differentially expressed genes in these models identified CAV1 as a possible marker of oxaliplatin resistance. In conclusion, we established a combined 2D/3D, in vitro/in vivo model system representing the heterogeneity of CRC, which can be used in preclinical research applications.

Gell: A GPU-powered 3D hybrid simulator for large-scale multicellular system

As a powerful but computationally intensive method, hybrid computational models study the dynamics of multicellular systems by evolving discrete cells in reacting and diffusing extracellular microenvironments. As the scale and complexity of studied biological systems continuously increase, the exploding computational cost starts to limit large-scale cell-based simulations. To facilitate the large-scale hybrid computational simulation and make it feasible on easily accessible computational devices, we develop Gell (GPU Cell), a fast and memory-efficient open-source GPU-based hybrid computational modeling platform for large-scale system modeling. We fully parallelize the simulations on GPU for high computational efficiency and propose a novel voxel sorting method to further accelerate the modeling of massive cell-cell mechanical interaction with negligible additional memory footprint. As a result, Gell efficiently handles simulations involving tens of millions of cells on a personal computer. We compare the performance of Gell with a state-of-the-art paralleled CPU-based simulator on a hanging droplet spheroid growth task and further demonstrate Gell with a ductal carcinoma in situ (DCIS) simulation. Gell affords ~150X acceleration over the paralleled CPU method with one-tenth of the memory requirement.

Calibrating agent-based models to tumor images using representation learning

Agent-based models (ABMs) have enabled great advances in the study of tumor development and therapeutic response, allowing researchers to explore the spatiotemporal evolution of the tumor and its microenvironment. However, these models face serious drawbacks in the realm of parameterization-ABM parameters are typically set individually based on various data and literature sources, rather than through a rigorous parameter estimation approach. While ABMs can be fit to simple time-course data (such as tumor volume), that type of data loses the spatial information that is a defining feature of ABMs. Tumor images provide spatial information; however, such images only represent individual timepoints, limiting their utility in calibrating the tumor dynamics predicted by ABMs. Furthermore, it is exceedingly difficult to compare tumor images to ABM simulations beyond a qualitative visual comparison. Without a quantitative method of comparing the similarity of tumor images to ABM simulations, a rigorous parameter fitting is not possible. Here, we present a novel approach that applies neural networks to represent both tumor images and ABM simulations as low dimensional points, with the distance between points acting as a quantitative measure of difference between the two. This enables a quantitative comparison of tumor images and ABM simulations, where the distance between simulated and experimental images can be minimized using standard parameter-fitting algorithms. Here, we describe this method and present two examples to demonstrate the application of the approach to estimate parameters for two distinct ABMs. Overall, we provide a novel method to robustly estimate ABM parameters.

Robust formation of optimal single spheroids towards cost-effective in vitro three-dimensional tumor models

While useful for fundamental in vitro studies, monolayer cell cultures are not physiologically relevant. Spheroids, a complex three-dimensional (3D) structure, more closely resemble in vivo tumor growth. Spheroids allow the results obtained relating to proliferation, cell death, differentiation, metabolism, and various antitumor therapies to be more predictive of in vivo outcomes. In the protocol herein, a rapid and high-throughput method is discussed for the generation of single spheroids using various cancer cell lines, including brain cancer cells (U87 MG, SEBTA-027, SF188), prostate cancer cells (DU-145, TRAMP-C1), and breast cancer cells (BT-549, Py230) in 96-round bottom-well plates. The proposed method is associated with significantly low costs per plate without requiring refining or transferring. Homogeneous compact spheroid morphology was evidenced as early as 1 day after following this protocol. Proliferating cells were traced in the rim, while dead cells were found to be located inside the core region of the spheroid using confocal microscopy and the Incucyte^® live imaging system. H&E staining of spheroid sections was utilized to investigate the tightness of the cell packaging. Through western blotting analyses, it was revealed that a stem cell-like phenotype was adopted by these spheroids. This method was also used to obtain the EC50 of the anticancer dipeptide carnosine on U87 MG 3D culture. This affordable, easy-to-follow five-step protocol allows for the robust generation of various uniform spheroids with 3D morphology characteristics.

A mechanobiological model for tumor spheroid evolution with application to glioblastoma: A continuum multiphysics approach

BACKGROUND

Spheroids are in vitro quasi-spherical structures of cell aggregates, eventually cultured within a hydrogel matrix, that are used, among other applications, as a technological platform to investigate tumor formation and evolution. Several interesting features can be replicated using this methodology, such as cell communication mechanisms, the effect of gradients of nutrients, or the creation of realistic 3D biological structures. The main objective of this work is to link the spheroid evolution with the mechanical activity of cells, coupled with nutrient consumption and the subsequent cell dynamics.

METHOD

We propose a continuum mechanobiological model which accounts for the most relevant phenomena that take place in tumor spheroid evolution under in vitro suspension, namely, nutrient diffusion in the spheroid, kinetics of cellular growth and death, and mechanical interactions among the cells. The model is qualitatively validated, after calibration of the model parameters, versus in vitro experiments of spheroids of different glioblastoma cell lines.

RESULTS

Our model is able to explain in a novel way quite different setups, such as spheroid growth (up to six times the initial configuration for U-87 MG cell line) or shrinking (almost half of the initial configuration for U-251 MG cell line); as the result of the mechanical interplay of cells driven by cellular evolution.

CONCLUSIONS

Glioblastoma tumor spheroid evolution is driven by mechanical interactions of the cell aggregate and the dynamical evolution of the cell population. All this information can be used to further investigate mechanistic effects in the evolution of tumors and their role in cancer disease.

An in silico study on the influence of extracellular matrix mechanics on vasculogenesis

BACKGROUND AND OBJECTIVES

Blood vessels form a network of capillaries throughout the body that perform essential functions for life. Vasculogenesis, i.e. the formation of new blood vessels, is regulated by many factors, biochemical ones being among the most important. However, others such as the biomechanical influence on shape, organization and structure of vessel networks require further investigation. In this paper, we develop a 3D agent-based mechanobiological model of vasculogenesis with the aim of analyzing how the mechanics of the extracellular matrix (ECM) affects vasculogenesis.

METHODS

For this purpose, we consider a growing domain composed of different cells: tip cells, which are the driving cells located at the end of the vessels and stalk cells, which are found in the interior of the vascular network. ECM is considered as particles (agents) that surround the growth of the vascular network. Depending on the cell type, different sets of forces are considered, such as chemotactic, mechanical, random and viscoelastic forces among others.

RESULTS

The growth of the network is iteratively analyzed and updated at each time step based on a mechanically-driven proliferation rule. The influence of different biomechanical factors, such as ECM stiffness or viscoelasticity are explored through in silico simulations. A number of indicators are defined along the algorithm, like number of cells, branches, tortuosity and anisotropy, in order to compare topological differences of the vascular network during vasculogenesis under different ECM conditions. The obtained results are qualitatively compared with other related works in the literature.

CONCLUSIONS

The present study sheds some light and partially explain, from an in silico perspective, the role of ECM mechanics on vasculogenesis. The main conclusions of this work are: (i) increased stiffness increases proliferation, (ii) the network tends to migrate towards stiffer areas, and (iii) increased viscoelasticity decreases proliferation.

Replacement, Reduction, and Refinement of Animal Experiments in Anticancer Drug Development: The Contribution of 3D In Vitro Cancer Models in the Drug Efficacy Assessment

In the last decades three-dimensional (3D) in vitro cancer models have been proposed as a bridge between bidimensional (2D) cell cultures and in vivo animal models, the gold standards in the preclinical assessment of anticancer drug efficacy. 3D in vitro cancer models can be generated through a multitude of techniques, from both immortalized cancer cell lines and primary patient-derived tumor tissue. Among them, spheroids and organoids represent the most versatile and promising models, as they faithfully recapitulate the complexity and heterogeneity of human cancers. Although their recent applications include drug screening programs and personalized medicine, 3D in vitro cancer models have not yet been established as preclinical tools for studying anticancer drug efficacy and supporting preclinical-to-clinical translation, which remains mainly based on animal experimentation. In this review, we describe the state-of-the-art of 3D in vitro cancer models for the efficacy evaluation of anticancer agents, focusing on their potential contribution to replace, reduce and refine animal experimentations, highlighting their strength and weakness, and discussing possible perspectives to overcome current challenges.

Interplay of adherens junctions and matrix proteolysis determines the invasive pattern and growth of squamous cell carcinoma

Cancers, such as squamous cell carcinoma, frequently invade as multicellular units. However, these invading units can be organised in a variety of ways, ranging from thin discontinuous strands to thick 'pushing' collectives. Here we employ an integrated experimental and computational approach to identify the factors that determine the mode of collective cancer cell invasion. We find that matrix proteolysis is linked to the formation of wide strands but has little effect on the maximum extent of invasion. Cell-cell junctions also favour wide strands, but our analysis also reveals a requirement for cell-cell junctions for efficient invasion in response to uniform directional cues. Unexpectedly, the ability to generate wide invasive strands is coupled to the ability to grow effectively when surrounded by extracellular matrix in three-dimensional assays. Combinatorial perturbation of both matrix proteolysis and cell-cell adhesion demonstrates that the most aggressive cancer behaviour, both in terms of invasion and growth, is achieved at high levels of cell-cell adhesion and high levels of proteolysis. Contrary to expectation, cells with canonical mesenchymal traits - no cell-cell junctions and high proteolysis - exhibit reduced growth and lymph node metastasis. Thus, we conclude that the ability of squamous cell carcinoma cells to invade effectively is also linked to their ability to generate space for proliferation in confined contexts. These data provide an explanation for the apparent advantage of retaining cell-cell junctions in squamous cell carcinomas.

Development of a coupled modeling for tumor growth, angiogenesis, oxygen delivery, and phenotypic heterogeneity

Analysis of the evolution and growth dynamics of tumors is crucial for understanding cancer and the development of individually optimized therapies. During tumor growth, a hypoxic microenvironment around cancer cells caused by excessive non-vascular tumor growth induces tumor angiogenesis that plays a key role in the ensuing tumor growth and its progression into higher stages. Various mathematical simulation models have been introduced to simulate these biologically and physically complex hallmarks of cancer. Here, we developed a hybrid two-dimensional computational model that integrates spatiotemporally different components of the tumor system to investigate both angiogenesis and tumor growth/proliferation. This spatiotemporal evolution is based on partial diffusion equations, the cellular automation method, transition and probabilistic rules, and biological assumptions. The new vascular network provided by angiogenesis affects tumor microenvironmental conditions and drives individual cells to adapt themselves to spatiotemporal conditions. Furthermore, some stochastic rules are involved besides microenvironmental conditions. Overall, the conditions promote some commonly observed cellular states, i.e., proliferative, migrative, quiescent, and cell death, depending on the condition of each cell. Altogether, our results offer a theoretical basis for the biological evidence that regions of the tumor tissue near blood vessels are densely populated by proliferative phenotypic variants, while poorly oxygenated regions are sparsely populated by hypoxic phenotypic variants.

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.

Data-driven spatio-temporal modelling of glioblastoma

Mathematical oncology provides unique and invaluable insights into tumour growth on both the microscopic and macroscopic levels. This review presents state-of-the-art modelling techniques and focuses on their role in understanding glioblastoma, a malignant form of brain cancer. For each approach, we summarize the scope, drawbacks and assets. We highlight the potential clinical applications of each modelling technique and discuss the connections between the mathematical models and the molecular and imaging data used to inform them. By doing so, we aim to prime cancer researchers with current and emerging computational tools for understanding tumour progression. By providing an in-depth picture of the different modelling techniques, we also aim to assist researchers who seek to build and develop their own models and the associated inference frameworks. Our article thus strikes a unique balance. On the one hand, we provide a comprehensive overview of the available modelling techniques and their applications, including key mathematical expressions. On the other hand, the content is accessible to mathematicians and biomedical scientists alike to accommodate the interdisciplinary nature of cancer research.

Hybrid computational models of multicellular tumour growth considering glucose metabolism

Cancer cells metabolize glucose through metabolic pathways that differ from those used by healthy and differentiated cells. In particular, tumours have been shown to consume more glucose than their healthy counterparts and to use anaerobic metabolic pathways, even under aerobic conditions. Nevertheless, scientists have still not been able to explain why cancer cells evolved to present an altered metabolism and what evolutionary advantage this might provide them. Experimental and computational models have been increasingly used in recent years to understand some of these biological questions. Multicellular tumour spheroids are effective experimental models as they replicate the initial stages of avascular solid tumour growth. Furthermore, these experiments generate data which can be used to calibrate and validate computational studies that aim to simulate tumour growth. Hybrid models are of particular relevance in this field of research because they model cells as individual agents while also incorporating continuum representations of the substances present in the surrounding microenvironment that may participate in intracellular metabolic networks as concentration or density distributions. Henceforth, in this review, we explore the potential of computational modelling to reveal the role of metabolic reprogramming in tumour growth.

Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability

Tumours are subject to external environmental variability. However, in vitro tumour spheroid experiments, used to understand cancer progression and develop cancer therapies, have been routinely performed for the past fifty years in constant external environments. Furthermore, spheroids are typically grown in ambient atmospheric oxygen (normoxia), whereas most in vivo tumours exist in hypoxic environments. Therefore, there are clear discrepancies between in vitro and in vivo conditions. We explore these discrepancies by combining tools from experimental biology, mathematical modelling, and statistical uncertainty quantification. Focusing on oxygen variability to develop our framework, we reveal key biological mechanisms governing tumour spheroid growth. Growing spheroids in time-dependent conditions, we identify and quantify novel biological adaptation mechanisms, including unexpected necrotic core removal, and transient reversal of the tumour spheroid growth phases.

Spatio-temporal modelling of phenotypic heterogeneity in tumour tissues and its impact on radiotherapy treatment

We present a mathematical model that describes how tumour heterogeneity evolves in a tissue slice that is oxygenated by a single blood vessel. Phenotype is identified with the stemness level of a cell and determines its proliferative capacity, apoptosis propensity and response to treatment. Our study is based on numerical bifurcation analysis and dynamical simulations of a system of coupled, non-local (in phenotypic "space") partial differential equations that link the phenotypic evolution of the tumour cells to local tissue oxygen levels. In our formulation, we consider a 1D geometry where oxygen is supplied by a blood vessel located on the domain boundary and consumed by the tumour cells as it diffuses through the tissue. For biologically relevant parameter values, the system exhibits multiple steady states; in particular, depending on the initial conditions, the tumour is either eliminated ("tumour-extinction") or it persists ("tumour-invasion"). We conclude by using the model to investigate tumour responses to radiotherapy, and focus on identifying radiotherapy strategies which can eliminate the tumour. Numerical simulations reveal how phenotypic heterogeneity evolves during treatment and highlight the critical role of tissue oxygen levels on the efficacy of radiation protocols that are commonly used in the clinic.

Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates

An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth experiments. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, classify non-identifiable parameter spaces into identifiable parameter combinations that relate to features in the data characterised by parameters in a surrogate model, and overall provide additional biological insight from complex non-identifiable models.

Two-phase model of compressive stress induced on a surrounding hyperelastic medium by an expanding tumour

In vitro experiments in which tumour cells are seeded in a gelatinous medium, or hydrogel, show how mechanical interactions between tumour cells and the tissue in which they are embedded, together with local levels of an externally-supplied, diffusible nutrient (e.g., oxygen), affect the tumour's growth dynamics. In this article, we present a mathematical model that describes these in vitro experiments. We use the model to understand how tumour growth generates mechanical deformations in the hydrogel and how these deformations in turn influence the tumour's growth. The hydrogel is viewed as a nonlinear hyperelastic material and the tumour is modelled as a two-phase mixture, comprising a viscous tumour cell phase and an isotropic, inviscid interstitial fluid phase. Using a combination of numerical and analytical techniques, we show how the tumour's growth dynamics change as the mechanical properties of the hydrogel vary. When the hydrogel is soft, nutrient availability dominates the dynamics: the tumour evolves to a large equilibrium configuration where the proliferation rate of nutrient-rich cells on the tumour boundary balances the death rate of nutrient-starved cells in the central, necrotic core. As the hydrogel stiffness increases, mechanical resistance to growth increases and the tumour's equilibrium size decreases. Indeed, for small tumours embedded in stiff hydrogels, the inhibitory force experienced by the tumour cells may be so large that the tumour is eliminated. Analysis of the model identifies parameter regimes in which the presence of the hydrogel drives tumour elimination.

Thermodynamically-motivated chemo-mechanical models and multicellular simulation to provide new insight into active cell and tumour remodelling

Computational models can shape our understanding of cell and tissue remodelling, from cell spreading, to active force generation, adhesion, and growth. In this mini-review, we discuss recent progress in modelling of chemo-mechanical cell behaviour and the evolution of multicellular systems. In particular, we highlight recent advances in (i) free-energy based single cell models that can provide new fundamental insight into cell spreading, cancer cell invasion, stem cell differentiation, and remodelling in disease, and (ii) mechanical agent-based models to simulate large numbers of discrete interacting cells in proliferative tumours. We describe how new biological understanding has emerged from such theoretical models, and the trade-offs and constraints associated with current approaches. Ultimately, we aim to make a case for why theory should be integrated with an experimental workflow to optimise new in-vitro studies, to predict feedback between cells and their microenvironment, and to deepen understanding of active cell behaviour.

Network Analysis Identifies Phase Transitions for Tumor With Interacting Cells

Metastasis is the process by which cancer cells acquire the capability to leave the primary tumor and travel to distant sites. Recent experiments have suggested that the epithelial–mesenchymal transition can regulate invasion and metastasis. Another possible scenario is the collective motion of cells. Recent studies have also proposed a jamming–unjamming transition for epithelial cells based on physical forces. Here, we assume that there exists a short-range chemical attraction between cancer cells and employ the Brownian dynamics to simulate tumor growth. Applying the network analysis, we suggest three possible phases for a given tumor and study the transition between these phases by adjusting the attraction strength.

A stochastic mathematical model of 4D tumour spheroids with real-time fluorescent cell cycle labelling

In vitro tumour spheroids have been used to study avascular tumour growth and drug design for over 50 years. Tumour spheroids exhibit heterogeneity within the growing population that is thought to be related to spatial and temporal differences in nutrient availability. The recent development of real-time fluorescent cell cycle imaging allows us to identify the position and cell cycle status of individual cells within the growing spheroid, giving rise to the notion of a four-dimensional (4D) tumour spheroid. We develop the first stochastic individual-based model (IBM) of a 4D tumour spheroid and show that IBM simulation data compares well with experimental data using a primary human melanoma cell line. The IBM provides quantitative information about nutrient availability within the spheroid, which is important because it is difficult to measure these data experimentally.

In vitro assays for investigating the FLASH effect

FLASH radiotherapy is a novel technique that has been shown in numerous preclinical in vivo studies to have the potential to be the next important improvement in cancer treatment. However, the biological mechanisms responsible for the selective FLASH sparing effect of normal tissues are not yet known. An optimal translation of FLASH radiotherapy into the clinic would require a good understanding of the specific beam parameters that induces a FLASH effect, environmental conditions affecting the response, and the radiobiological mechanisms involved. Even though the FLASH effect has generally been considered as an in vivo effect, studies finding these answers would be difficult and ethically challenging to carry out solely in animals. Hence, suitable in vitro studies aimed towards finding these answers are needed. In this review, we describe and summarise several in vitro assays that have been used or could be used to finally elucidate the mechanisms behind the FLASH effect.

Designing and interpreting 4D tumour spheroid experiments

Tumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal sampling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.

Bayesian calibration of a stochastic, multiscale agent-based model for predicting in vitro tumor growth

Hybrid multiscale agent-based models (ABMs) are unique in their ability to simulate individual cell interactions and microenvironmental dynamics. Unfortunately, the high computational cost of modeling individual cells, the inherent stochasticity of cell dynamics, and numerous model parameters are fundamental limitations of applying such models to predict tumor dynamics. To overcome these challenges, we have developed a coarse-grained two-scale ABM (cgABM) with a reduced parameter space that allows for an accurate and efficient calibration using a set of time-resolved microscopy measurements of cancer cells grown with different initial conditions. The multiscale model consists of a reaction-diffusion type model capturing the spatio-temporal evolution of glucose and growth factors in the tumor microenvironment (at tissue scale), coupled with a lattice-free ABM to simulate individual cell dynamics (at cellular scale). The experimental data consists of BT474 human breast carcinoma cells initialized with different glucose concentrations and tumor cell confluences. The confluence of live and dead cells was measured every three hours over four days. Given this model, we perform a time-dependent global sensitivity analysis to identify the relative importance of the model parameters. The subsequent cgABM is calibrated within a Bayesian framework to the experimental data to estimate model parameters, which are then used to predict the temporal evolution of the living and dead cell populations. To this end, a moment-based Bayesian inference is proposed to account for the stochasticity of the cgABM while quantifying uncertainties due to limited temporal observational data. The cgABM reduces the computational time of ABM simulations by 93% to 97% while staying within a 3% difference in prediction compared to ABM. Additionally, the cgABM can reliably predict the temporal evolution of breast cancer cells observed by the microscopy data with an average error and standard deviation for live and dead cells being 7.61±2.01 and 5.78±1.13, respectively.

The calibration of agent-based models of tumor cell growth to experimental data remains a challenge in computational oncology. Besides the computational cost of modeling thousands of agents, the model’s intrinsic stochasticity demands numerous realizations of the simulations to accurately represent the statistical features of the model predictions. We developed a hybrid, multiscale, coarse-grain, agent-based model that captures the growth and decline of human breast carcinoma cells under different initial conditions. We determined the effects of coarse-graining the ABM on the multiscale model output and the number of repetitions necessary to capture the stochastic transitions present in the model. We identified the most influential parameters on the model prediction through a sensitivity analysis and selected which parameters can be fixed and which ones should be calibrated. Using Bayesian calibration, we show that the model can accurately represent the experimental data. The validation step indicates that our model can reliably predict the in vitro temporal data, depending on the choice of the training (calibration data) sets.

Quantitative analysis of tumour spheroid structure

Tumour spheroids are common in vitro experimental models of avascular tumour growth. Compared with traditional two-dimensional culture, tumour spheroids more closely mimic the avascular tumour microenvironment where spatial differences in nutrient availability strongly influence growth. We show that spheroids initiated using significantly different numbers of cells grow to similar limiting sizes, suggesting that avascular tumours have a limiting structure; in agreement with untested predictions of classical mathematical models of tumour spheroids. We develop a novel mathematical and statistical framework to study the structure of tumour spheroids seeded from cells transduced with fluorescent cell cycle indicators, enabling us to discriminate between arrested and cycling cells and identify an arrested region. Our analysis shows that transient spheroid structure is independent of initial spheroid size, and the limiting structure can be independent of seeding density. Standard experimental protocols compare spheroid size as a function of time; however, our analysis suggests that comparing spheroid structure as a function of overall size produces results that are relatively insensitive to variability in spheroid size. Our experimental observations are made using two melanoma cell lines, but our modelling framework applies across a wide range of spheroid culture conditions and cell lines.

A rigid body framework for multicellular modeling

Off-lattice models are a well-established approach in multicellular modeling, where cells are represented as points that are free to move in space. The representation of cells as point objects is useful in a wide range of settings, particularly when large populations are involved; however, a purely point-based representation is not naturally equipped to deal with objects that have length, such as cell boundaries or external membranes. Here we introduce an off-lattice modeling framework that exploits rigid body mechanics to represent objects using a collection of conjoined one-dimensional edges in a viscosity-dominated system. This framework can be used to represent cells as free moving polygons, to allow epithelial layers to smoothly interact with themselves, to model rod-shaped cells such as bacteria and to robustly represent membranes. We demonstrate that this approach offers solutions to the problems that limit the scope of current off-lattice multicellular models.

Integrating single cell sequencing with a spatial quantitative systems pharmacology model spQSP for personalized prediction of triple-negative breast cancer immunotherapy response

Response to cancer immunotherapies depends on the complex and dynamic interactions between T cell recognition and killing of cancer cells that are counteracted through immunosuppressive pathways in the tumor microenvironment. Therefore, while measurements such as tumor mutational burden provide biomarkers to select patients for immunotherapy, they neither universally predict patient response nor implicate the mechanisms that underlie immunotherapy resistance. Recent advances in single-cell RNA sequencing technology measure cellular heterogeneity within cells of an individual tumor but have yet to realize the promise of predictive oncology. In addition to data, mechanistic multiscale computational models are developed to predict treatment response. Incorporating single-cell data from tumors to parameterize these computational models provides deeper insights into prediction of clinical outcome in individual patients. Here, we integrate whole-exome sequencing and scRNA-seq data from Triple-Negative Breast Cancer patients to model neoantigen burden in tumor cells as input to a spatial Quantitative System Pharmacology model. The model comprises a four-compartmental Quantitative System Pharmacology sub-model to represent a whole patient and a spatial agent-based sub-model to represent tumor volumes at the cellular scale. We use the high-throughput single-cell data to model the role of antigen burden and heterogeneity relative to the tumor microenvironment composition on predicted immunotherapy response. We demonstrate how this integrated modeling and single-cell analysis framework can be used to relate neoantigen heterogeneity to immunotherapy treatment outcomes. Our results demonstrate feasibility of merging single-cell data to initialize cell states in multiscale computational models such as the spQSP for personalized prediction of clinical outcomes to immunotherapy.

Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors

Highly resolved spatial data of complex systems encode rich and nonlinear information. Quantification of heterogeneous and noisy data-often with outliers, artifacts, and mislabeled points-such as those from tissues, remains a challenge. The mathematical field that extracts information from the shape of data, topological data analysis (TDA), has expanded its capability for analyzing real-world datasets in recent years by extending theory, statistics, and computation. An extension to the standard theory to handle heterogeneous data is multiparameter persistent homology (MPH). Here we provide an application of MPH landscapes, a statistical tool with theoretical underpinnings. MPH landscapes, computed for (noisy) data from agent-based model simulations of immune cells infiltrating into a spheroid, are shown to surpass existing spatial statistics and one-parameter persistent homology. We then apply MPH landscapes to study immune cell location in digital histology images from head and neck cancer. We quantify intratumoral immune cells and find that infiltrating regulatory T cells have more prominent voids in their spatial patterns than macrophages. Finally, we consider how TDA can integrate and interrogate data of different types and scales, e.g., immune cell locations and regions with differing levels of oxygenation. This work highlights the power of MPH landscapes for quantifying, characterizing, and comparing features within the tumor microenvironment in synthetic and real datasets.

A Computational Study on the Role of Parameters for Identification of Thyroid Nodules by Infrared Images (and Comparison with Real Data)

According to experts and medical literature, healthy thyroids and thyroids containing benign nodules tend to be less inflamed and less active than those with malignant nodules. It seems to be a consensus that malignant nodules have more blood veins and more blood circulation. This may be related to the maintenance of the nodule's heat at a higher level compared with neighboring tissues. If the internal heat modifies the skin radiation, then it could be detected by infrared sensors. The goal of this work is the investigation of the factors that allow this detection, and the possible relation with any pattern referent to nodule malignancy. We aim to consider a wide range of factors, so a great number of numerical simulations of the heat transfer in the region under analysis, based on the Finite Element method, are performed to study the influence of each nodule and patient characteristics on the infrared sensor acquisition. To do so, the protocol for infrared thyroid examination used in our university's hospital is simulated in the numerical study. This protocol presents two phases. In the first one, the body under observation is in steady state. In the second one, it is submitted to thermal stress (transient state). Both are simulated in order to verify if it is possible (by infrared sensors) to identify different behavior referent to malignant nodules. Moreover, when the simulation indicates possible important aspects, patients with and without similar characteristics are examined to confirm such influences. The results show that the tissues between skin and thyroid, as well as the nodule size, have an influence on superficial temperatures. Other thermal parameters of thyroid nodules show little influence on surface infrared emissions, for instance, those related to the vascularization of the nodule. All details of the physical parameters used in the simulations, characteristics of the real nodules and thermal examinations are publicly available, allowing these simulations to be compared with other types of heat transfer solutions and infrared examination protocols. Among the main contributions of this work, we highlight the simulation of the possible range of parameters, and definition of the simulation approach for mapping the used infrared protocol, promoting the investigation of a possible relation between the heat transfer process and the data obtained by infrared acquisitions.

An integrated framework for quantifying immune-tumour interactions in a 3D co-culture model

Investigational in vitro models that reflect the complexity of the interaction between the immune system and tumours are limited and difficult to establish. Herein, we present a platform to study the tumour-immune interaction using a co-culture between cancer spheroids and activated immune cells. An algorithm was developed for analysis of confocal images of the co-culture to evaluate the following quantitatively; immune cell infiltration, spheroid roundness and spheroid growth. As a proof of concept, the effect of the glucocorticoid stress hormone, cortisol was tested on 66CL4 co-culture model. Results were comparable to 66CL4 syngeneic in vivo mouse model undergoing psychological stress. Furthermore, administration of glucocorticoid receptor antagonists demonstrated the use of this model to determine the effect of treatments on the immune-tumour interplay. In conclusion, we provide a method of quantifying the interaction between the immune system and cancer, which can become a screening tool in immunotherapy design.

Challenges of applying multicellular tumor spheroids in preclinical phase

The three-dimensional (3D) multicellular tumor spheroids (MCTs) model is becoming an essential tool in cancer research as it expresses an intermediate complexity between 2D monolayer models and in vivo solid tumors. MCTs closely resemble in vivo solid tumors in many aspects, such as the heterogeneous architecture, internal gradients of signaling factors, nutrients, and oxygenation. MCTs have growth kinetics similar to those of in vivo tumors, and the cells in spheroid mimic the physical interaction of the tumors, such as cell-to-cell and cell-to-extracellular matrix interactions. These similarities provide great potential for studying the biological properties of tumors and a promising platform for drug screening and therapeutic efficacy evaluation. However, MCTs are not well adopted as preclinical tools for studying tumor behavior and therapeutic efficacy up to now. In this review, we addressed the challenges with MCTs application and discussed various efforts to overcome the challenges.

Digital Twins for Tissue Culture Techniques—Concepts, Expectations, and State of the Art

Techniques to provide in vitro tissue culture have undergone significant changes during the last decades, and current applications involve interactions of cells and organoids, three-dimensional cell co-cultures, and organ/body-on-chip tools. Efficient computer-aided and mathematical model-based methods are required for efficient and knowledge-driven characterization, optimization, and routine manufacturing of tissue culture systems. As an alternative to purely experimental-driven research, the usage of comprehensive mathematical models as a virtual in silico representation of the tissue culture, namely a digital twin, can be advantageous. Digital twins include the mechanistic of the biological system in the form of diverse mathematical models, which describe the interaction between tissue culture techniques and cell growth, metabolism, and the quality of the tissue. In this review, current concepts, expectations, and the state of the art of digital twins for tissue culture concepts will be highlighted. In general, DT’s can be applied along the full process chain and along the product life cycle. Due to the complexity, the focus of this review will be especially on the design, characterization, and operation of the tissue culture techniques.

Impact of tumor-parenchyma biomechanics on liver metastatic progression: a multi-model approach

Colorectal cancer and other cancers often metastasize to the liver in later stages of the disease, contributing significantly to patient death. While the biomechanical properties of the liver parenchyma (normal liver tissue) are known to affect tumor cell behavior in primary and metastatic tumors, the role of these properties in driving or inhibiting metastatic inception remains poorly understood, as are the longer-term multicellular dynamics. This study adopts a multi-model approach to study the dynamics of tumor-parenchyma biomechanical interactions during metastatic seeding and growth. We employ a detailed poroviscoelastic model of a liver lobule to study how micrometastases disrupt flow and pressure on short time scales. Results from short-time simulations in detailed single hepatic lobules motivate constitutive relations and biological hypotheses for a minimal agent-based model of metastatic growth in centimeter-scale tissue over months-long time scales. After a parameter space investigation, we find that the balance of basic tumor-parenchyma biomechanical interactions on shorter time scales (adhesion, repulsion, and elastic tissue deformation over minutes) and longer time scales (plastic tissue relaxation over hours) can explain a broad range of behaviors of micrometastases, without the need for complex molecular-scale signaling. These interactions may arrest the growth of micrometastases in a dormant state and prevent newly arriving cancer cells from establishing successful metastatic foci. Moreover, the simulations indicate ways in which dormant tumors could "reawaken" after changes in parenchymal tissue mechanical properties, as may arise during aging or following acute liver illness or injury. We conclude that the proposed modeling approach yields insight into the role of tumor-parenchyma biomechanics in promoting liver metastatic growth, and advances the longer term goal of identifying conditions to clinically arrest and reverse the course of late-stage cancer.