Hybrid Cellular Automaton Model Research Articles

A SCALING LAW RELATING THE RATE OF DESTRUCTION OF A SOLID TUMOR AND THE FRACTAL DIMENSION OF ITS BOUNDARY

In this paper, we investigate the scaling law relating the size of the boundary of a solid tumor and the rate at which it is lysed by a cell population of non-infiltrating cytotoxic lymphocytes. We do it in the context of enzyme kinetics through geometrical, analytical and numerical arguments. Following the Koch island fractal model, a scale-dependent function that describes the constant rate of the decay process and the fractal dimension is obtained. Then, in silico experiments are accomplished by means of a stochastic hybrid cellular automaton model. This model is used to grow several tumors with varying morphology and to test the power decay law when the cell-mediated immune response is effective, confirming its validity.

Landscapes of Biochemical Warfare: Spatial Self-Organization Woven from Allelopathic Interactions

Evidence shows that diversity and spatial distributions of biological communities are largely driven by the race of living organisms in their adaptation to chemicals synthesized by their neighbors. In this report, the emergence of mathematical models on pure spatial self-organization induced by biochemical suppression (allelopathy) and competition between species were investigated through numerical analysis. For both random and patched initial spatial distributions of species, we demonstrate that warfare survivors are self-organized on the landscape in Turing-like patterns driven by diffusive instabilities of allelochemicals. These patterns are simple; either all species coexist at low diffusion rates or are massively extinct, except for a few at high diffusivities, but they are complex and biodiversity-sustained at intermediate diffusion rates. "Defensive alliances" and ecotones seem to be basic mechanisms that sustain great biodiversity in our hybrid cellular automata model. Moreover, species coexistence and extinction exhibit multi-stationarity.

ДВА КОНКУРИРУЮЩИХ МЕХАНИЗМА ПРИ АГРЕГАЦИИ DICTYOSTELIUM DISCOIDEUM

In the present work, the hybrid cellular automaton model of the initial stages of aggregation of the cellular slime mold Dictyostelium discoideum, proposed by the authors earlier, is modified by taking into account the dependence of myxamoebae movement on the intracellular Ca2+ concentration. The cellular automaton is a three-dimensional grid the cells of which are either empty or contain one amoeba, the state of the automaton being determined by the distribution of amoebae over the grid. At each time step, the state of the automaton changes according to fixed rules which are determined by solutions to systems of equations that describe the distribution of aggregation factors in the medium and their concentrations inside cells. Movements of amoebae can be either directed along the concentration gradient of cyclic adenosine monophosphate, or random walk. The calculation results show that when the aggregate reaches a certain size its growth stops and the number of cells in it fluctuates around the average value. Such behavior suggests that aggregation is determined by two competing mechanisms: one promotes the growth of the aggregate while the other promotes detachment of cells or small clusters from it.

Abstract B008: Man vs machine: Crowdsourcing vs deep learning of adaptive therapy strategies

Abstract Cancer Crusade is a citizen science game in which players design treatment strategies to influence a virtual tumor growth model. The treatment regimens that the players construct are collected and sent to an online database for analysis, which may give insights into clinically effective adaptive strategies. In parallel, a machine learning-based approach was applied to the game to find optimal treatment policies, and the optimized therapy approaches generated are compared to those generated by the human players. For the core game engine in Cancer Crusade, we started with a previously published hybrid cellular automata model of tumor metabolism and growth. The original model was used to study how tumor cells evolve acid-mediated invasion and the impact of treatment on these metabolic processes. We expanded the model by adding more drugs and cell phenotypes, including drug-resistant cells. The game was released on mobile platforms and has been generating data from plays for several years. The treatment strategies were analyzed using dendrogram clustering to find key decision differences and profile their relative performance. We also performed network analysis to observe treatment transitions. These analyses indicated several effective adaptive strategies, which tended to oscillate between chemotherapy and a pro-angiogenic drug, or chemotherapy and a hypoxia-activated prodrug, or combinations of all three of these drugs. These results parallel a machine learning (Q-learning) approach to the problem, which yielded a preferred strategy based on chemotherapy combined with a pro-angiogenic drug. The existence of several human-discovered alternative strategies suggests that in general human players may offer greater variety of successful strategies via citizen science games than Q-learning. The human and machine learning approaches also excelled in different areas. Human players tended to use less drug overall, and perform best on less aggressive tumors, while the Q-learning approach used more drug overall and performed best on more aggressive tumors. Citation Format: Rafael R. Bravo, Brenden K. Peterson, Mark Robertson-Tessi, Daniel M. Faissol, Alexander R.A. Anderson. Man vs machine: Crowdsourcing vs deep learning of adaptive therapy strategies [abstract]. In: Proceedings of the AACR Special Conference on the Evolutionary Dynamics in Carcinogenesis and Response to Therapy; 2022 Mar 14-17. Philadelphia (PA): AACR; Cancer Res 2022;82(10 Suppl):Abstract nr B008.

A scalable solver for a stochastic, hybrid cellular automaton model of personalized breast cancer therapy.

Mathematical modeling and simulation is a promising approach to personalized cancer medicine. Yet, the complexity, heterogeneity and multi-scale nature of cancer pose significant computational challenges. Coupling discrete cell-based models with continuous models using hybrid cellular automata (CA) is a powerful approach for mimicking biological complexity and describing the dynamical exchange of information across different scales. However, when clinically relevant cancer portions are taken into account, such models become computationally very expensive. While efficient parallelization techniques for continuous models exist, their coupling with discrete models, particularly CA, necessitates more elaborate solutions. Building upon FEniCS, a popular and powerful scientific computing platform for solving partial differential equations, we developed parallel algorithms to link stochastic CA with differential equations (https://bitbucket.org/HTasken/cansim). The algorithms minimize the communication between processes that share CA neighborhood values while also allowing for reproducibility during stochastic updates. We demonstrated the potential of our solution on a complex hybrid cellular automaton model of breast cancer treated with combination chemotherapy. On a single-core processor, we obtained nearly linear scaling with an increasing problem size, whereas weak parallel scaling showed moderate growth in solving time relative to increase in problem size. Finally, we applied the algorithm to a problem that is 500 times larger than previous work, allowing us to run personalized therapy simulations based on heterogeneous cell density and tumor perfusion conditions estimated from magnetic resonance imaging data on an unprecedented scale.

Modelling urban expansion of a south-east Asian city, India: comparison between SLEUTH and a hybrid CA model

Accurate identification of the dominant driving factors in the expansion of a city is essential for cellular automata (CA)-based urban expansion modelling. The current study aimed at analyzing the drivers of change for Vijayawada city, a tier-II developing city in south-east Asia, India and then applies them to model the future scenario of the city in a CA-based environment. Furthermore, the study also aims at comparing the efficiency of the renowned CA-based urban growth model, SLEUTH (Slope, Land use, Exclusion, Urban extent, Transportation, Hill shade), and a self-designed CA-based hybrid model developed in combination with genetic algorithm (GA) for developing future scenario of the city. The models were built, calibrated, and validated for the city using land use maps derived from temporal Landsat satellite images. Analysis of the statistical significance of the driving factors shows that the terrain and population density are the two dominant factors influencing expansion of the city. Both the models output on predicting the urban growth indicate that the available open spaces within the existing city extent gets further converted to built-up indicating infill development, and more growth occurs along the fringes of the existing city. Also, emergence of new urban centers merged with the existing city areas for predictions till 2050 and extended into the dense vegetation on the north resulting in further reduction of green cover around the city. Validation of both the models showed overall accuracy higher than 85%. The inclusion of the demographic variable and utilization of the GA ensured that the hybrid model performs better than the SLEUTH model. Analysis on the built-up densification of the cityscape showed a shift in the pattern of built-up dynamics over time. The built urban expansion scenario for Vijayawada informs urban modelers and policymakers the possible evolution of the city.

A novel cellular automata approach: seed input/output of the alien species Leucaena leucocephala in the soil and the effects of climate changes

The potential impact of the soil seed bank of a given species and subsequent population persistence will depend on the robustness of the seeds against depletion. Invasiveness is frequently linked to this behavior and successful control of invasive species depends on reducing the size of their seed banks. We propose a hybrid model that simulates the seed bank dynamics in association with their physical dormancy (PD) based on experimental data on seed bank depletion of a Leucaena leucocephala population during the dispersal phase. The hybrid model uses cellular automata (CA) and Monte Carlo techniques. The sensitivity analysis revealed that the model delivers reproducible and reliable results. Then, we discuss the ability of the hybrid model to predict the effect of seasonal climate change. A 2.0 °C increase in temperature was enough to cause a considerable impact in the seed depletion curve, and after 1 year, ~ 75% of the seeds had already disappeared from the seed bank. The proposed hybrid CA model can be used to study soil-stored seed banks with physical dormancy, and could be applied to other Fabaceae species with similar seed traits. We believe that this computational model contributes to a better understanding of seed bank dynamics and will aid in the control of invasive species and for testing different scenarios to reduce or eliminate seed banks that have already been formed.

Size Matters: Metastatic Cluster Size and Stromal Recruitment in the Establishment of Successful Prostate Cancer to Bone Metastases.

Prostate cancer (PCa) impacts over 180,000 men every year in the USA alone, with 26,000 patients expected to succumb to the disease ( cancer.gov ). The primary cause of death is metastasis, with secondary lesions most commonly occurring in the skeleton. Prostate cancer to bone metastasis is an important, yet poorly understood, process that is difficult to explore with experimental techniques alone. To this end we have utilized a hybrid (discrete-continuum) cellular automaton model of normal bone matrix homeostasis that allowed us to investigate how metastatic PCa can disrupt the bone microenvironment. Our previously published results showed that PCa cells can recruit mesenchymal stem cells (MSCs) that give rise to bone-building osteoblasts. MSCs are also thought to be complicit in the establishment of successful bone metastases (Lu, in Mol Cancer Res 4(4):221-233, 2006). Here we have explored the aspects of early metastatic colonization and shown that the size of PCa clusters needs to be within a specific range to become successfully established: sufficiently large to maximize success, but not too large to risk failure through competition among cancer and stromal cells for scarce resources. Furthermore, we show that MSC recruitment can promote the establishment of a metastasis and compensate for relatively low numbers of PCa cells seeding the bone microenvironment. Combined, our results highlight the utility of biologically driven computational models that capture the complex and dynamic dialogue between cells during the initiation of active metastases.

Abstract A13: Predictive computational modeling to define effective treatment strategies for bone metastatic prostate cancer

Abstract The ability to rapidly assess the efficacy of therapeutic strategies for incurable bone metastatic prostate cancer is an important and urgent need. Pre-clinical in vivo models provide insights yet are limited in their capacity to interrogate temporal multi-cellular interactions occurring in the cancer-bone microenvironment. Computational and mathematical models that integrate and, are validated experimentally, can overcome this limitation and accelerate biological research. Here, we describe how a biologically driven discrete continuum hybrid cellular automaton (HCA) model approach can dissect the pleiotropic effects of inhibiting putative therapeutic targets such as, TGFβ. Using our HCA model, we tested five different therapeutic doses delivered in two therapeutic windows. In silico results predict that TGFβ inhibition, applied prior to tumor seeding acts by directly impacting prostate cancer cell viability but also by simultaneously restricting osteoclast formation and unexpectedly, promoting osteoblast differentiation. This effect was dependent on the prostate cancer cell expression of TGFβ receptors. In silico predictions were validated with two independent in vivo models of bone metastatic prostate cancer (PAIII and C4-2B). Using immunohistochemical information from human bone metastatic prostate cancer samples, we also demonstrate how the HCA can be used to predict the evolution of heterogeneous disease in response to applied therapies. Collectively, these data underscore the power of a combined HCA/biological approach in optimizing the efficacy of applied therapies and measuring their impact on bone metastatic prostate cancer. Citation Format: Leah Cook, Arturo Araujo, Julio Pow-Sang, Mikalai Budzevich, David Basanta, Conor C. Lynch. Predictive computational modeling to define effective treatment strategies for bone metastatic prostate cancer. [abstract]. In: Proceedings of the AACR Special Conference on Engineering and Physical Sciences in Oncology; 2016 Jun 25-28; Boston, MA. Philadelphia (PA): AACR; Cancer Res 2017;77(2 Suppl):Abstract nr A13.

Predictive computational modeling to define effective treatment strategies for bone metastatic prostate cancer.

The ability to rapidly assess the efficacy of therapeutic strategies for incurable bone metastatic prostate cancer is an urgent need. Pre-clinical in vivo models are limited in their ability to define the temporal effects of therapies on simultaneous multicellular interactions in the cancer-bone microenvironment. Integrating biological and computational modeling approaches can overcome this limitation. Here, we generated a biologically driven discrete hybrid cellular automaton (HCA) model of bone metastatic prostate cancer to identify the optimal therapeutic window for putative targeted therapies. As proof of principle, we focused on TGFβ because of its known pleiotropic cellular effects. HCA simulations predict an optimal effect for TGFβ inhibition in a pre-metastatic setting with quantitative outputs indicating a significant impact on prostate cancer cell viability, osteoclast formation and osteoblast differentiation. In silico predictions were validated in vivo with models of bone metastatic prostate cancer (PAIII and C4-2B). Analysis of human bone metastatic prostate cancer specimens reveals heterogeneous cancer cell use of TGFβ. Patient specific information was seeded into the HCA model to predict the effect of TGFβ inhibitor treatment on disease evolution. Collectively, we demonstrate how an integrated computational/biological approach can rapidly optimize the efficacy of potential targeted therapies on bone metastatic prostate cancer.

Simulation of Tumor Response to Immunotherapy Using a Hybrid Cellular Automata Model

A cellular automata model is presented to describe the interaction between a growing tumor next to the presence of nutrient elements, and the immune response as well as the immunotherapy intervention and its stimulator effect on immune cell activity. A comparison between tumor growth before and after immunotherapy effect is discussed, and many characteristics of simulated tumor growth patterns are extracted such as the fractal dimension, the fractality of tumor boundary and the number of cancerous cells on tumor periphery. The simulation results shed light on the factor that may play a central role in immunotherapy effectiveness and which is related to tumor morphology.

Crime Theory Evaluation Using Simulation Models of Residential Burglary

This paper provides new insights on the study of crime modeling through the development of a hybrid cellular automaton (CA) and Multi-agent System (MAS) simulation model that is able to combine components of multiple criminological theories to forecast the locations of residential burglary targets: journey to crime (JTC), social disorganization (SD) theory, and routine activity (RA) theory. In order to combine individual factors from each theory into a unified model, Analytic Hierarchy Process (AHP) was employed for hierarchical parameter selection. The model is then evaluated using data on offenders obtained from the Dallas Police Department to examine how different crime theories perform in the prediction of residential burglary. Compared to the SD- and RA-weighted models, the JTC-weighted model performed the best when comparisons were made to actual burglary locations. The findings demonstrate that the simulation models of crime provide test beds for research into the explanatory power of various crime theories.

Spatial Metrics of Tumour Vascular Organisation Predict Radiation Efficacy in a Computational Model.

Intratumoural heterogeneity is known to contribute to poor therapeutic response. Variations in oxygen tension in particular have been correlated with changes in radiation response in vitro and at the clinical scale with overall survival. Heterogeneity at the microscopic scale in tumour blood vessel architecture has been described, and is one source of the underlying variations in oxygen tension. We seek to determine whether histologic scale measures of the erratic distribution of blood vessels within a tumour can be used to predict differing radiation response. Using a two-dimensional hybrid cellular automaton model of tumour growth, we evaluate the effect of vessel distribution on cell survival outcomes of simulated radiation therapy. Using the standard equations for the oxygen enhancement ratio for cell survival probability under differing oxygen tensions, we calculate average radiation effect over a range of different vessel densities and organisations. We go on to quantify the vessel distribution heterogeneity and measure spatial organization using Ripley’s L function, a measure designed to detect deviations from complete spatial randomness. We find that under differing regimes of vessel density the correlation coefficient between the measure of spatial organization and radiation effect changes sign. This provides not only a useful way to understand the differences seen in radiation effect for tissues based on vessel architecture, but also an alternate explanation for the vessel normalization hypothesis.

Abstract 3751: Defining the temporal effects of TGFβ inhibition on the cellular heterogeneity of the bone metastatic prostate cancer microenvironment

Abstract Introduction. Bone metastatic prostate cancer is currently incurable. Traditional in vivo experimentation has provided insight into the circuitry driving the osteolytic and osteogenic nature of the disease but is limited in its ability to interrogate multiple simultaneous interactions occurring in the tumor-bone microenvironment. We hypothesize that this limitation can be overcome by combining the power of predictive computational modeling with biological approaches. To this end, we developed a clinically relevant hybrid cellular automaton model of bone metastatic prostate cancer to test the impact of putative targeted therapies. Herein, we examined the impact of inhibiting transforming growth factor beta (TGFβ) because of its well-described pleiotropic effects on the tumor-bone microenvironment. Results. In silico, simulations of 250 days (n = 27) were performed for five different levels of TGFβ inhibition (0-100%), applied either pre- or post- metastatic seeding. The computational outputs predicted that TGFβ inhibition will only work if administered as a pre-treatment. In this scenario, TGFβ inhibition (at 80% inhibition); 1) reduced tumor cell number by ∼25%, 2) prevented osteoclast precursor infiltration and maturation by 40% and surprisingly, 3) reduced tumor-induced osteogenesis by ∼14%; all data were significant with p<0.0001. Next we tested these predictions in vivo with an osteogenic and TGFβ responsive model of bone metastatic prostate cancer (PAIII). Pre-treatment of mice (n = 9/group) with a TGFβ inhibitor (1D11) prior to intratibial inoculation of luciferase expressing PAIII cells remarkably validated computational model predictions while post-treatment (n = 7/group) with 1D11 had little or no effect. Unlike our homogeneous in vivo model, analysis of human clinical specimens (n = 20) of bone metastatic prostate cancer revealed heterogeneous expression of TGFβ ligand and receptor in prostate cancer cells. Again, computational modeling is a powerful way to address the issue of heterogeneity and using TGFβ as an example, the computational model was seeded with equal ratios of TRP (ligand and receptor-producing), TR (receptor-expressing), and TN (neutral) cells. In silico control simulations (n = 30) reveal that TR cancer cells are the dominant population. However, application of a TGFβ inhibitor in silico results in the eventual emergence (∼100 days later) of the TN population. These data suggest that adaptive application of an inhibitor such as TGFβ could prevent the emergence of resistant populations over time. Conclusions. Integration of computational and biological approaches can be a powerful tactic in determining the temporal impact of putative therapies on heterogeneous tumor microenvironments. Further, the computational model can be of major benefit in optimizing treatments for the eradication of bone metastatic prostate cancer. Citation Format: Leah M. Cook, Arturo Araujo, David Basanta, Conor C. Lynch. Defining the temporal effects of TGFβ inhibition on the cellular heterogeneity of the bone metastatic prostate cancer microenvironment. [abstract]. In: Proceedings of the 106th Annual Meeting of the American Association for Cancer Research; 2015 Apr 18-22; Philadelphia, PA. Philadelphia (PA): AACR; Cancer Res 2015;75(15 Suppl):Abstract nr 3751. doi:10.1158/1538-7445.AM2015-3751

A High Fidelity Hybrid Cellular Automaton Model for Bone Adaptation with Cellular Rules for Bone Resorption

A High Fidelity Hybrid Cellular Automaton Model for Bone Adaptation with Cellular Rules for Bone Resorption

A Hybrid Cellular Automaton Model of Solid Tumor Growth and Bioreductive Drug Transport

Bioreductive drugs are a class of hypoxia selective drugs that are designed to eradicate the hypoxic fraction of solid tumors. Their activity depends upon a number of biological and pharmacological factors and we used a mathematical modeling approach to explore the dynamics of tumor growth, infusion, and penetration of the bioreductive drug Tirapazamine (TPZ). An in-silico model is implemented to calculate the tumor mass considering oxygen and glucose as key microenvironmental parameters. The next stage of the model integrated extra cellular matrix (ECM), cell-cell adhesion, and cell movement parameters as growth constraints. The tumor microenvironments strongly influenced tumor morphology and growth rates. Once the growth model was established, a hybrid model was developed to study drug dynamics inside the hypoxic regions of tumors. The model used 10, 50 and 100 \mu {\rm M} as TPZ initial concentrations and determined TPZ pharmacokinetic (PK) (transport) and pharmacodynamics (cytotoxicity) properties inside hypoxic regions of solid tumor. The model results showed that diminished drug transport is a reason for TPZ failure and recommend the optimization of the drug transport properties in the emerging TPZ generations. The modeling approach used in this study is novel and can be a step to explore the behavioral dynamics of TPZ.

Modeling dynamic urban growth using hybrid cellular automata and particle swarm optimization

Conventional raster-based cellular automata (CA) confront many difficulties because of cell size and neighborhood sensitivity. Alternatively, vector CA-based models are very complex and difficult to implement. We present a hybrid cellular automata (HCA) model as a combination of cellular structure and vector concept. The space is still defined by a set of cells, but rasterized spatial objects are also utilized in the structure of transition rules. Particle swarm optimization (PSO) is also used to calculate the urbanization probability of cells based on their distance from the development parameters. The proposed model is applied to Landsat satellite imagery of the city of Tehran, Iran with 28.5-m spatial resolution to simulate the urban growth from 1988 to 2010. Statistical comparison of the ground truth and the simulated image using a kappa coefficient shows an accuracy of 83.42% in comparison to the 81.13% accuracy for the conventional Geo-CA model. Moreover, decreasing the spatial resolution by a factor of one-fourth has reduced the accuracy of the HCA and Geo-CA models by 1.19% and 3.04%, respectively, which shows the lower scale sensitivity of the proposed model. The HCA model is developed to have the simplicity of cellular structure together with optimum features of vector models.

Avascular tumour growth dynamics and the constraints of protein binding for drug transportation

The potential for the use of in-silico models of disease in progression monitoring is becoming increasingly recognised, as well as its contribution to the development of complete curative processes. In this paper we report the development of a hybrid cellular automaton model to mimic the growth of avascular tumours, including the infusion of a bioreductive drug to study the effects of protein binding on drug transportation. The growth model is operated within an extracellular tumour microenvironment. An artificial Neural Network based scheme was implemented that modelled the behaviours of each cell (proliferation, quiescence, apoptosis and/or movement) based on the complex heterogeneous microenvironment; consisting of oxygen, glucose, hydrogen ions, inhibitory factors and growth factors. To validate the growth model results, we conducted experiments with multicellular tumour spheroids. These results showed good agreement with the predicted growth dynamics. The outcome of the avascular tumour growth model suggested that tumour microenvironments have a strong impact on cell behaviour. To address the problem of cellular proteins acting as resistive factors preventing efficient drug penetration, a bioreactive drug (tirapazamine) was added to the system. This allowed us to study the drug penetration through multicellular layers of tissue after its binding to cellular proteins. The results of the in vitro model suggested that the proteins reduce the toxicity of the drug, reducing its efficacy for the most severely hypoxic fractions furthest from a functional blood vessel. Finally this research provides a unique comparison of in vitro tumour growth with an intelligent in silico model to measure bioreductive drug availability inside tumour tissue through a set of experiments.

Computational Modeling of Tumor Response to Vascular-Targeting Therapies—Part I: Validation

Mathematical modeling techniques have been widely employed to understand how cancer grows, and, more recently, such approaches have been used to understand how cancer can be controlled. In this manuscript, a previously validated hybrid cellular automaton model of tumor growth in a vascularized environment is used to study the antitumor activity of several vascular-targeting compounds of known efficacy. In particular, this model is used to test the antitumor activity of a clinically used angiogenesis inhibitor (both in isolation, and with a cytotoxic chemotherapeutic) and a vascular disrupting agent currently undergoing clinical trial testing. I demonstrate that the mathematical model can make predictions in agreement with preclinical/clinical data and can also be used to gain more insight into these treatment protocols. The results presented herein suggest that vascular-targeting agents, as currently administered, cannot lead to cancer eradication, although a highly efficacious agent may lead to long-term cancer control.

Diffusion-limited tumour growth: Simulations and analysis

The morphology of solid tumours is known to be affected by the background oxygen concentration of the tissue in which the tumour grows, and both computational and experimental studies have suggested that branched tumour morphology in low oxygen concentration is caused by diffusion-limited growth. In this paper we present a simple hybrid cellular automaton model of solid tumour growth aimed at investigating this phenomenon. Simulation results show that for high consumption rates (or equivalently low oxygen concentrations) the tumours exhibit branched morphologies, but more importantly the simplicity of the model allows for an analytic approach to the problem. By applying a steady-state assumption we derive an approximate solution of the oxygen equation, which closely matches the simulation results. Further, we derive a dispersion relation which reveals that the average branch width in the tumour depends on the width of the active rim, and that a smaller active rim gives rise to thinner branches. Comparison between the prediction of the stability analysis and the results from the simulations shows good agreement between theory and simulation.