Calibration of Multi-Parameter Models of Avascular Tumor Growth Using Time Resolved Microscopy Data

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

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

Calibration of stochastic, agent-based neuron growth models with approximate Bayesian computation

Understanding how genetically encoded rules drive and guide complex neuronal growth processes is essential to comprehending the brain's architecture, and agent-based models (ABMs) offer a powerful simulation approach to further develop this understanding. However, accurately calibrating these models remains a challenge. Here, we present a novel application of Approximate Bayesian Computation (ABC) to address this issue. ABMs are based on parametrized stochastic rules that describe the time evolution of small components-the so-called agents-discretizing the system, leading to stochastic simulations that require appropriate treatment. Mathematically, the calibration defines a stochastic inverse problem. We propose to address it in a Bayesian setting using ABC. We facilitate the repeated comparison between data and simulations by quantifying the morphological information of single neurons with so-called morphometrics and resort to statistical distances to measure discrepancies between populations thereof. We conduct experiments on synthetic as well as experimental data. We find that ABC utilizing Sequential Monte Carlo sampling and the Wasserstein distance finds accurate posterior parameter distributions for representative ABMs. We further demonstrate that these ABMs capture specific features of pyramidal cells of the hippocampus (CA1). Overall, this work establishes a robust framework for calibrating agent-based neuronal growth models and opens the door for future investigations using Bayesian techniques for model building, verification, and adequacy assessment.

Cancer cell sedimentation in 3D cultures reveals active migration regulated by self-generated gradients and adhesion sites

Cell sedimentation in 3D hydrogel cultures refers to the vertical migration of cells towards the bottom of the space. Understanding this poorly examined phenomenon may allow us to design better protocols to prevent it, as well as provide insights into the mechanobiology of cancer development. We conducted a multiscale experimental and mathematical examination of 3D cancer growth in triple negative breast cancer cells. Migration was examined in the presence and absence of Paclitaxel, in high and low adhesion environments and in the presence of fibroblasts. The observed behaviour was modeled by hypothesizing active migration due to self-generated chemotactic gradients. Our results did not reject this hypothesis, whereby migration was likely to be regulated by the MAPK and TGF-β pathways. The mathematical model enabled us to describe the experimental data in absence (normalized error<40%) and presence of Paclitaxel (normalized error<10%), suggesting inhibition of random motion and advection in the latter case. Inhibition of sedimentation in low adhesion and co-culture experiments further supported the conclusion that cells actively migrated downwards due to the presence of signals produced by cells already attached to the adhesive glass surface.

Tumour growth: An approach to calibrate parameters of a multiphase porous media model based on in vitro observations of Neuroblastoma spheroid growth in a hydrogel microenvironment

To unravel processes that lead to the growth of solid tumours, it is necessary to link knowledge of cancer biology with the physical properties of the tumour and its interaction with the surrounding microenvironment. Our understanding of the underlying mechanisms is however still imprecise. We therefore developed computational physics-based models, which incorporate the interaction of the tumour with its surroundings based on the theory of porous media. However, the experimental validation of such models represents a challenge to its clinical use as a prognostic tool. This study combines a physics-based model with in vitro experiments based on microfluidic devices used to mimic a three-dimensional tumour microenvironment. By conducting a global sensitivity analysis, we identify the most influential input parameters and infer their posterior distribution based on Bayesian calibration. The resulting probability density is in agreement with the scattering of the experimental data and thus validates the proposed workflow. This study demonstrates the huge challenges associated with determining precise parameters with usually only limited data for such complex processes and models, but also demonstrates in general how to indirectly characterise the mechanical properties of neuroblastoma spheroids that cannot feasibly be measured experimentally.

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

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

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

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

Environmental stress level to model tumor cell growth and survival

Survival of living tumor cells underlies many influences such as nutrient saturation, oxygen level, drug concentrations or mechanical forces. Data-supported mathematical modeling can be a powerful tool to get a better understanding of cell behavior in different settings. However, under consideration of numerous environmental factors mathematical modeling can get challenging. We present an approach to model the separate influences of each environmental quantity on the cells in a collective manner by introducing the "environmental stress level". It is an immeasurable auxiliary variable, which quantifies to what extent viable cells would get in a stressed state, if exposed to certain conditions. A high stress level can inhibit cell growth, promote cell death and influence cell movement. As a proof of concept, we compare two systems of ordinary differential equations, which model tumor cell dynamics under various nutrient saturations respectively with and without considering an environmental stress level. Particle-based Bayesian inversion methods are used to quantify uncertainties and calibrate unknown model parameters with time resolved measurements of in vitro populations of liver cancer cells. The calibration results of both models are compared and the quality of fit is quantified. While predictions of both models show good agreement with the data, there is indication that the model considering the stress level yields a better fitting. The proposed modeling approach offers a flexible and extendable framework for considering systems with additional environmental factors affecting the cell dynamics.

Modeling of Glioma Growth with Mass Effect by Longitudinal Magnetic Resonance Imaging

It is well-known that expanding glioblastomas typically induce significant deformations of the surrounding parenchyma (i.e., the so-called ?mass effect?). In this study, we evaluate the performance of three mathematical models of tumor growth: 1) a reaction-diffusion-advection model which accounts for mass effect (RDAM), 2) a reaction-diffusion model with mass effect that is consistent only in the case of small deformations (RDM), and 3) a reaction-diffusion model that does not include the mass effect (RD). The models were calibrated with magnetic resonance imaging (MRI) data obtained during tumor development in a murine model of glioma (n = 9). We obtained T₂-weighted and contrast-enhanced T₁-weighted MRI at 6 time points over 10 days to determine the spatiotemporal variation in the mass effect and tumor concentration, respectively. We calibrated the three models using data 1) at the first four, 2) only at the first and fourth, and 3) only at the third and fourth time points. Each of these calibrations were run forward in time to predict the volume fraction of tumor cells at the conclusion of the experiment. The diffusion coefficient for the RDAM model (median of 10.65 ? 10^-3 mm²d^-1) is significantly less than those for the RD and RDM models (17.46 ? 10^-3 mm²d^-1 and 19.38 ? 10^-3 mm²d^-1, respectively). The tumor concentrations for the RD, RDM, and RDAM models have medians of 40.2%, 32.1%, and 44.7%, respectively, for the calibration using data from the first four time points. The RDM model most accurately predicts tumor growth, while the RDAM model presents the least variation in its estimates of the diffusion coefficient and proliferation rate. This study demonstrates that the mathematical models capture both tumor development and mass effect observed in experiments.

Characterization of multicellular breast tumor spheroids using image data-driven biophysical mathematical modeling

Multicellular tumor spheroid (MCTS) systems provide an in vitro cell culture model system which mimics many of the complexities of an in vivo solid tumor and tumor microenvironment, and are often used to study cancer cell growth and drug efficacy. Here, we present a coupled experimental-computational framework to estimate phenotypic growth and biophysical tumor microenvironment properties. This novel framework utilizes standard microscopy imaging of MCTS systems to drive a biophysical mathematical model of MCTS growth and mechanical interactions. By extending our previous in vivo mechanically-coupled reaction–diffusion modeling framework we developed a microscopy image processing framework capable of mechanistic characterization of MCTS systems. Using MDA-MB-231 breast cancer MCTS, we estimated biophysical parameters of cellular diffusion, rate of cellular proliferation, and cellular tractions forces. We found significant differences in these model-based biophysical parameters throughout the treatment time course between untreated and treated MCTS systems, whereas traditional size-based morphometric parameters were inconclusive. The proposed experimental-computational framework estimates mechanistic MCTS growth and invasion parameters with significant potential to assist in better and more precise assessment of in vitro drug efficacy through the development of computational analysis methodologies for three-dimensional cell culture systems to improve the development and evaluation of antineoplastic drugs.

A Coupled Mass Transport and Deformation Theory of Multi-constituent Tumor Growth

We develop a general class of thermodynamically consistent, continuum models based on mixture theory with phase effects that describe the behavior of a mass of multiple interacting constituents. The constituents consist of solid species undergoing large elastic deformations and compressible viscous fluids. The fundamental building blocks framing the mixture theories consist of the mass balance law of diffusing species and microscopic (cellular scale) and macroscopic (tissue scale) force balances, as well as energy balance and the entropy production inequality derived from the first and second laws of thermodynamics. A general phase-field framework is developed by closing the system through postulating constitutive equations (i.e., specific forms of free energy and rate of dissipation potentials) to depict the growth of tumors in a microenvironment. A notable feature of this theory is that it contains a unified continuum mechanics framework for addressing the interactions of multiple species evolving in both space and time and involved in biological growth of soft tissues (e.g., tumor cells and nutrients). The formulation also accounts for the regulating roles of the mechanical deformation on the growth of tumors, through a physically and mathematically consistent coupled diffusion and deformation framework. A new algorithm for numerical approximation of the proposed model using mixed finite elements is presented. The results of numerical experiments indicate that the proposed theory captures critical features of avascular tumor growth in the various microenvironment of living tissue, in agreement with the experimental studies in the literature.