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.     

Sensitivity Analysis of an ENteric Immunity SImulator (ENISI)-Based Model of Immune Responses to Helicobacter pylori Infection

Agent-based models (ABM) are widely used to study immune systems, providing a procedural and interactive view of the underlying system. The interaction of components and the behavior of individual objects is described procedurally as a function of the internal states and the local interactions, which are often stochastic in nature. Such models typically have complex structures and consist of a large number of modeling parameters. Determining the key modeling parameters which govern the outcomes of the system is very challenging. Sensitivity analysis plays a vital role in quantifying the impact of modeling parameters in massively interacting systems, including large complex ABM. The high computational cost of executing simulations impedes running experiments with exhaustive parameter settings. Existing techniques of analyzing such a complex system typically focus on local sensitivity analysis, i.e. one parameter at a time, or a close “neighborhood” of particular parameter settings. However, such methods are not adequate to measure the uncertainty and sensitivity of parameters accurately because they overlook the global impacts of parameters on the system. In this article, we develop novel experimental design and analysis techniques to perform both global and local sensitivity analysis of large-scale ABMs. The proposed method can efficiently identify the most significant parameters and quantify their contributions to outcomes of the system. We demonstrate the proposed methodology for ENteric Immune SImulator (ENISI), a large-scale ABM environment, using a computational model of immune responses to Helicobacter pylori colonization of the gastric mucosa.     

Immune Response to Combination Therapy for Non-Hodgkin Lymphomas

A parametric model of tumor response to combination therapy in the presence of an immune system is described. Synergistic mechanisms which induce tumor regression are simulated with a coupled set of equations. The simulations are first compared to tumor history data obtained with a SCID mouse model to determine key parameters; predictions are then made for an immune-competent animal. The minimum immune cell birth rate relative to malignant B-cell birth rate necessary to induce tumor regression is determined, and optimization of drug combinations in the presence of an immune response is explored. The delayed effect of an immune response relative to drug scheduling is examined, and a mechanism for disease transformation in heterogeneous tumors is proposed.     

An “Age” Structured Model of Hematopoietic Stem Cell Organization with Application to Chronic Myeloid Leukemia

Previously, we have modeled hematopoietic stem cell organization by a stochastic, single cell-based approach. Applications to different experimental systems demonstrated that this model consistently explains a broad variety of in vivo and in vitro data. A major advantage of the agent-based model (ABM) is the representation of heterogeneity within the hematopoietic stem cell population. However, this advantage comes at the price of time-consuming simulations if the systems become large. One example in this respect is the modeling of disease and treatment dynamics in patients with chronic myeloid leukemia (CML), where the realistic number of individual cells to be considered exceeds 10⁶. To overcome this deficiency, without losing the representation of the inherent heterogeneity of the stem cell population, we here propose to approximate the ABM by a system of partial differential equations (PDEs). The major benefit of such an approach is its independence from the size of the system. Although this mean field approach includes a number of simplifying assumptions compared to the ABM, it retains the key structure of the model including the “age”-structure of stem cells. We show that the PDE model qualitatively and quantitatively reproduces the results of the agent-based approach.     

Integrating genealogical and dynamical modelling to infer escape and reversion rates in HIV epitopes

The rates of escape and reversion in response to selection pressure arising from the host immune system, notably the cytotoxic T-lymphocyte (CTL) response, are key factors determining the evolution of HIV. Existing methods for estimating these parameters from cross-sectional population data using ordinary differential equations (ODEs) ignore information about the genealogy of sampled HIV sequences, which has the potential to cause systematic bias and overestimate certainty. Here, we describe an integrated approach, validated through extensive simulations, which combines genealogical inference and epidemiological modelling, to estimate rates of CTL escape and reversion in HIV epitopes. We show that there is substantial uncertainty about rates of viral escape and reversion from cross-sectional data, which arises from the inherent stochasticity in the evolutionary process. By application to empirical data, we find that point estimates of rates from a previously published ODE model and the integrated approach presented here are often similar, but can also differ several-fold depending on the structure of the genealogy. The model-based approach we apply provides a framework for the statistical analysis and hypothesis testing of escape and reversion in population data and highlights the need for longitudinal and denser cross-sectional sampling to enable accurate estimate of these key parameters.     

Modeling and Estimation of Kinetic Parameters and Replicative Fitness of HIV-1 from Flow-Cytometry-Based Growth Competition Experiments

Growth competition assays have been developed to quantify the relative fitness of HIV-1 mutants. In this article, we develop mathematical models to describe viral/cellular dynamic interactions in the assay system from which the competitive fitness indices or parameters are defined. In our previous HIV-viral fitness experiments, the concentration of uninfected target cells was assumed to be constant (Wu et al. 2006). But this may not be true in some experiments. In addition, dual infection may frequently occur in viral fitness experiments and may not be ignorable. Here, we relax these two assumptions and extend our earlier viral fitness model (Wu et al. 2006). The resulting models then become nonlinear ODE systems for which closed-form solutions are not achievable. In the new model, the viral relative fitness is a function of time since it depends on the target cell concentration. First, we studied the structure identifiability of the nonlinear ODE models. The identifiability analysis showed that all parameters in the proposed models are identifiable from the flow-cytometry-based experimental data that we collected. We then employed a global optimization approach (the differential evolution algorithm) to directly estimate the kinetic parameters as well as the relative fitness index in the nonlinear ODE models using nonlinear least square regression based on the experimental data. Practical identifiability was investigated via Monte Carlo simulations.     

An integrative and practical evolutionary optimization for a complex, dynamic model of biological networks

Computer simulation is an important technique to capture the dynamics of biochemical networks. Numerical optimization is the key to estimate the values of kinetic parameters so that the dynamic model reproduces the behaviors of the existing experimental data. It is required to develop general strategies for the optimization of complex biochemical networks with a huge space of search parameters, under the condition that kinetic and quantitative data are hardly available. We propose an integrative and practical strategy for optimizing a complex dynamic model by using qualitative and incomplete experimental data. The key technologies are the divide and conquer method for reducing the search space, handling of multiple objective functions representing different types of biological behaviors, and design of rule-based objective functions that are suitable for qualitative and error-prone experimental data. This strategy is applied to optimizing a dynamic model of the yeast cell cycle to demonstrate the feasibility of it.     

A cellular automata model of tumor-immune system interactions

We present a hybrid cellular automata-partial differential equation model of moderate complexity to describe the interactions between a growing tumor next to a nutrient source and the immune system of the host organism. The model allows both temporal and two-dimensional spatial evolution of the system under investigation and is comprised of biological cell metabolism rules derived from both the experimental and mathematical modeling literature. We present numerical simulations that display behaviors which are qualitatively similar to those exhibited in tumor-immune system interaction experiments. These include spherical tumor growth, stable and unstable oscillatory tumor growth, satellitosis and tumor infiltration by immune cells. Finally, the relationship between these different growth regimes and key system parameters is discussed.     

Development and validation of an atlas-based finite element brain model

Traumatic brain injury is a leading cause of disability and injury-related death. To enhance our ability to prevent such injuries, brain response can be studied using validated finite element (FE) models. In the current study, a high-resolution, anatomically accurate FE model was developed from the International Consortium for Brain Mapping brain atlas. Due to wide variation in published brain material parameters, optimal brain properties were identified using a technique called Latin hypercube sampling, which optimized material properties against three experimental cadaver tests to achieve ideal biomechanics. Additionally, falx pretension and thickness were varied in a lateral impact variation. The atlas-based brain model (ABM) was subjected to the boundary conditions from three high-rate experimental cadaver tests with different material parameter combinations. Local displacements, determined experimentally using neutral density targets, were compared to displacements predicted by the ABM at the same locations. Error between the observed and predicted displacements was quantified using CORrelation and Analysis (CORA), an objective signal rating method that evaluates the correlation of two curves. An average CORA score was computed for each variation and maximized to identify the optimal combination of parameters. The strongest relationships between CORA and material parameters were observed for the shear parameters. Using properties obtained through the described multiobjective optimization, the ABM was validated in three impact configurations and shows good agreement with experimental data. The final model developed in this study consists of optimized brain material properties and was validated in three cadaver impacts against local brain displacement data.     

Estimating Incidence Rates from Population‐Based Case‐Control Studies in the Presence of Nonrespondents

In population‐based case‐control studies, it is of great public‐health importance to estimate the disease incidence rates associated with different levels of risk factors. This estimation is complicated by the fact that in such studies the selection probabilities for the cases and controls are unequal. A further complication arises when the subjects who are selected into the study do not participate (i.e. become nonrespondents) and nonrespondents differ systematically from respondents. In this paper, we show how to account for unequal selection probabilities as well as differential nonresponses in the incidence estimation. We use two logistic models, one relating the disease incidence rate to the risk factors, and one modelling the predictors that affect the nonresponse probability. After estimating the regression parameters in the nonresponse model, we estimate the regression parameters in the disease incidence model by a weighted estimating function that weights a respondent's contribution to the likelihood score function by the inverse of the product of his/her selection probability and his/her model‐predicted response probability. The resulting estimators of the regression parameters and the corresponding estimators of the incidence rates are shown to be consistent and asymptotically normal with easily estimated variances. Simulation results demonstrate that the asymptotic approximations are adequate for practical use and that failure to adjust for nonresponses could result in severe biases. An illustration with data from a cardiovascular study that motivated this work is presented.     

STAT1 is a key gene in a gene regulatory network related to immune phenotypes in bladder cancer: An integrative analysis of multi-omics data

The immunophenotype of bladder cancer plays a pivotal role in the prognosis of cancer, but the effect of different epigenetic factors on different immunophenotypes in bladder tumours remains unclear. This study used multi-omics data analysis to provide molecular basis support for different immune phenotypes. Unsupervised cluster analysis revealed distinct subclusters with higher (subcluster B2) or lower cytotoxic immune phenotypes (subcluster A1) related to PD-L1 and IFNG expression. Mutational landscape analyses showed that the mutation level of TP53 in subcluster B1 was highest than other subclusters, and subcluster B1 had a lower frequency of concurrent mutation than subcluster A2. A total of 2364 differentially expressed genes were identified between subclusters A2 and B1, and the main functions of the up-regulated genes in subcluster B1 were enriched in the activation of T cells and other related pathways. We found that STAT1 was a key gene in a gene regulatory network related to immune phenotypes in bladder cancer. Finally, we constructed a prognostic prediction model by LASSO Cox regression which could distinguish high-risk and low-risk cases significantly. In conclusion, the present study addressed a field synopsis between genetic and epigenetic events in immune phenotypes of bladder cancer.     

The Influence of the Global Gene Expression Shift on Downstream Analyses

The assumption that total abundance of RNAs in a cell is roughly the same in different cells is underlying most studies based on gene expression analyses. But experiments have shown that changes in the expression of some master regulators such as c-MYC can cause global shift in the expression of almost all genes in some cell types like cancers. Such shift will violate this assumption and can cause wrong or biased conclusions for standard data analysis practices, such as detection of differentially expressed (DE) genes and molecular classification of tumors based on gene expression. Most existing gene expression data were generated without considering this possibility, and are therefore at the risk of having produced unreliable results if such global shift effect exists in the data. To evaluate this risk, we conducted a systematic study on the possible influence of the global gene expression shift effect on differential expression analysis and on molecular classification analysis. We collected data with known global shift effect and also generated data to simulate different situations of the effect based on a wide collection of real gene expression data, and conducted comparative studies on representative existing methods. We observed that some DE analysis methods are more tolerant to the global shift while others are very sensitive to it. Classification accuracy is not sensitive to the shift and actually can benefit from it, but genes selected for the classification can be greatly affected.     

A hybrid multi-compartment model of granuloma formation and T cell priming in tuberculosis

Tuberculosis is a worldwide health problem with 2 billion people infected with Mycobacterium tuberculosis (Mtb, the bacteria causing TB). The hallmark of infection is the emergence of organized structures of immune cells forming primarily in the lung in response to infection. Granulomas physically contain and immunologically restrain bacteria that cannot be cleared. We have developed several models that spatially characterize the dynamics of the host-mycobacterial interaction, and identified mechanisms that control granuloma formation and development. In particular, we published several agent-based models (ABMs) of granuloma formation in TB that include many subtypes of T cell populations, macrophages as well as key cytokine and chemokine effector molecules. These ABM studies emphasize the important role of T-cell related mechanisms in infection progression, such as magnitude and timing of T cell recruitment, and macrophage activation. In these models, the priming and recruitment of T cells from the lung draining lymph node (LN) was captured phenomenologically. In addition to these ABM studies, we have also developed several multi-organ models using ODEs to examine trafficking of cells between, for example, the lung and LN. While we can predict temporal dynamic behaviors, those models are not coupled to the spatial aspects of granuloma. To this end, we have developed a multi-organ model that is hybrid: an ABM for the lung compartment and a non-linear system of ODE representing the lymph node compartment. This hybrid multi-organ approach to study TB granuloma formation in the lung and immune priming in the LN allows us to dissect protective mechanisms that cannot be achieved using the single compartment or multi-compartment ODE system. The main finding of this work is that trafficking of important cells known as antigen presenting cells from the lung to the lymph node is a key control mechanism for protective immunity: the entire spectrum of infection outcomes can be regulated by key immune cell migration rates. Our hybrid multi-organ implementation suggests that effector CD4+ T cells can rescue the system from a persistent infection and lead to clearance once a granuloma is fully formed. This could be effective as an immunotherapy strategy for latently infected individuals.     

A statistical model of the human core-temperature circadian rhythm

We formulate a statistical model of the human core-temperature circadian rhythm in which the circadian signal is modeled as a van der Pol oscillator, the thermoregulatory response is represented as a first-order autoregressive process, and the evoked effect of activity is modeled with a function specific for each circadian protocol. The new model directly links differential equation-based simulation models and harmonic regression analysis methods and permits statistical analysis of both static and dynamical properties of the circadian pacemaker from experimental data. We estimate the model parameters by using numerically efficient maximum likelihood algorithms and analyze human core-temperature data from forced desynchrony, free-run, and constant-routine protocols. By representing explicitly the dynamical effects of ambient light input to the human circadian pacemaker, the new model can estimate with high precision the correct intrinsic period of this oscillator ( approximately 24 h) from both free-run and forced desynchrony studies. Although the van der Pol model approximates well the dynamical features of the circadian pacemaker, the optimal dynamical model of the human biological clock may have a harmonic structure different from that of the van der Pol oscillator.     

Continuous modeling of metabolic networks with gene regulation in yeast and in vivo determination of rate parameters

A continuous model of a metabolic network including gene regulation to simulate metabolic fluxes during batch cultivation of yeast Saccharomyces cerevisiae was developed. The metabolic network includes reactions of glycolysis, gluconeogenesis, glycerol and ethanol synthesis and consumption, the tricarboxylic acid cycle, and protein synthesis. Carbon sources considered were glucose and then ethanol synthesized during growth on glucose. The metabolic network has 39 fluxes, which represent the action of 50 enzymes and 64 genes and it is coupled with a gene regulation network which defines enzyme synthesis (activities) and incorporates regulation by glucose (enzyme induction and repression), modeled using ordinary differential equations. The model includes enzyme kinetics, equations that follow both mass-action law and transport as well as inducible, repressible, and constitutive enzymes of metabolism. The model was able to simulate a fermentation of S. cerevisiae during the exponential growth phase on glucose and the exponential growth phase on ethanol using only one set of kinetic parameters. All fluxes in the continuous model followed the behavior shown by the metabolic flux analysis (MFA) obtained from experimental results. The differences obtained between the fluxes given by the model and the fluxes determined by the MFA do not exceed 25% in 75% of the cases during exponential growth on glucose, and 20% in 90% of the cases during exponential growth on ethanol. Furthermore, the adjustment of the fermentation profiles of biomass, glucose, and ethanol were 95%, 95%, and 79%, respectively. With these results the simulation was considered successful. A comparison between the simulation of the continuous model and the experimental data of the diauxic yeast fermentation for glucose, biomass, and ethanol, shows an extremely good match using the parameters found. The small discrepancies between the fluxes obtained through MFA and those predicted by the differential equations, as well as the good match between the profiles of glucose, biomass, and ethanol, and our simulation, show that this simple model, that does not rely on complex kinetic expressions, is able to capture the global behavior of the experimental data. Also, the determination of parameters using a straightforward minimization technique using data at only two points in time was sufficient to produce a relatively accurate model. Thus, even with a small amount of experimental data (rates and not concentrations) it was possible to estimate the parameters minimizing a simple objective function. The method proposed allows the obtention of reasonable parameters and concentrations in a system with a much larger number of unknowns than equations. Hence a contribution of this study is to present a convenient way to find in vivo rate parameters to model metabolic and genetic networks under different conditions.     

Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response

We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of Scale Relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of Scale Relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.     

Mathematical modeling of wastewater decolorization in a trickle-bed bioreactor

This work focuses on mathematical modeling of removal of organic dyes from textile industry waste waters by a white-rot fungus Irpex lacteus in a trickle-bed bioreactor. We developed a mathematical model of biomass and decolorization process dynamics. The model comprises mass balances of glucose and the dye in a fungal biofilm and a liquid film. The biofilm is modeled using a spatially two-dimensional domain. The liquid film is considered as homogeneous in the direction normal to the biofilm surface. The biomass growth, decay and the erosion of the biofilm are taken into account. Using experimental data, we identified values of key model parameters: the dye degradation rate constant, biofilm corrugation factor and liquid velocity. Considering the dye degradation rate constant 1×10?? kg m?3 s?1, we found optimal values of the corrugation factor 0.853 and 0.59 and values of the liquid velocity 5.23×10?3?m?s?1 and 6.2×10?3?m?s?1 at initial dye concentrations 0.09433 kg m?3 and 0.05284 kg m?3, respectively. A good agreement between the simulated and experimental data using estimated values of the model parameters was achieved. The model can be used to simulate the performance of laboratory scale trickle-bed bioreactor operated in a batch regime or to estimate values of principal parameters of the bioreactor system.