A phenomenological approach to the simulation of metabolism and proliferation dynamics of large tumour cell populations

A Multidisciplinary Hyper-Modeling Scheme in Personalized In Silico Oncology: Coupling Cell Kinetics with Metabolism, Signaling Networks, and Biomechanics as Plug-In Component Models of a Cancer Digital Twin

The massive amount of human biological, imaging, and clinical data produced by multiple and diverse sources necessitates integrative modeling approaches able to summarize all this information into answers to specific clinical questions. In this paper, we present a hypermodeling scheme able to combine models of diverse cancer aspects regardless of their underlying method or scale. Describing tissue-scale cancer cell proliferation, biomechanical tumor growth, nutrient transport, genomic-scale aberrant cancer cell metabolism, and cell-signaling pathways that regulate the cellular response to therapy, the hypermodel integrates mutation, miRNA expression, imaging, and clinical data. The constituting hypomodels, as well as their orchestration and links, are described. Two specific cancer types, Wilms tumor (nephroblastoma) and non-small cell lung cancer, are addressed as proof-of-concept study cases. Personalized simulations of the actual anatomy of a patient have been conducted. The hypermodel has also been applied to predict tumor control after radiotherapy and the relationship between tumor proliferative activity and response to neoadjuvant chemotherapy. Our innovative hypermodel holds promise as a digital twin-based clinical decision support system and as the core of future in silico trial platforms, although additional retrospective adaptation and validation are necessary.

The control of acidity in tumor cells: a biophysical model

Acidosis of the tumor microenvironment leads to cancer invasion, progression and resistance to therapies. We present a biophysical model that describes how tumor cells regulate intracellular and extracellular acidity while they grow in a microenvironment characterized by increasing acidity and hypoxia. The model takes into account the dynamic interplay between glucose and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_2$$\end{document}O2 consumption with lactate and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {CO}_2$$\end{document}CO2 production and connects these processes to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}^+$$\end{document}H+ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {HCO}_3^-$$\end{document}HCO3- fluxes inside and outside cells. We have validated the model with independent experimental data and used it to investigate how and to which extent tumor cells can survive in adverse micro-environments characterized by acidity and hypoxia. The simulations show a dominance of the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {H}^+$$\end{document}H+ exchanges in well-oxygenated regions, and of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {HCO}_3^-$$\end{document}HCO3- exchanges in the inner hypoxic regions where tumor cells are known to acquire malignant phenotypes. The model also includes the activity of the enzyme Carbonic Anhydrase 9 (CA9), a known marker of tumor aggressiveness, and the simulations demonstrate that CA9 acts as a nonlinear \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {pH}_i$$\end{document}pHi equalizer at any \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbox {O}_2$$\end{document}O2 level in cells that grow in acidic extracellular environments.

Oxygen in the Tumor Microenvironment: Mathematical and Numerical Modeling

There are many reasons to try to achieve a good grasp of the distribution of oxygen in the tumor microenvironment. The lack of oxygen - hypoxia - is a main actor in the evolution of tumors and in their growth and appears to be just as important in tumor invasion and metastasis. Mathematical models of the distribution of oxygen in tumors which are based on reaction-diffusion equations provide partial but qualitatively significant descriptions of the measured oxygen concentrations in the tumor microenvironment, especially when they incorporate important elements of the blood vessel network such as the blood vessel size and spatial distribution and the pulsation of local pressure due to blood circulation. Here, we review our mathematical and numerical approaches to the distribution of oxygen that yield insights both on the role of the distribution of blood vessel density and size and on the fluctuations of blood pressure.

Dynamical Detection of Boundaries and Cavities in Biophysical Cell-Based Simulations of Growing Tumor Tissues

Cell-based lattice-free simulations of the growth of tumor tissues require the definition of geometrical and topological relations among cells and the other basic elements of the simulation (most notably the local and the global environments). This is necessary for the correct description of the biochemistry of tumor tissues, and to implement the biomechanical interactions among cells. Weak cell-cell forces and the necrosis of tumor tissues due to poor vascularization can lead to the formation of cavities - i.e., regions without viable cells and filled with cellular debris and fluids. It is important to give an accurate geometrical/topological description of the resulting microenvironment that plays an important role in the pathology of cancer. In this paper, we concentrate on simulations of the growth of avascular solid tumors and we describe the STAR (Shape of Tumors from Algorithmic Reconstruction) algorithm that defines the shape of clusters of cells and searches for the boundary and cavities in a 3D environment. The algorithm is GPU-based and exploits the high degree of parallelism of GPUs. The final implementation achieves a 30-fold speedup with respect to a previous CPU-based version.

ATP depletion during mitotic arrest induces mitotic slippage and APC/CCdh1-dependent cyclin B1 degradation

ATP depletion inhibits cell cycle progression, especially during the G1 phase and the G2 to M transition. However, the effect of ATP depletion on mitotic progression remains unclear. We observed that the reduction of ATP after prometaphase by simultaneous treatment with 2-deoxyglucose and NaN₃ did not arrest mitotic progression. Interestingly, ATP depletion during nocodazole-induced prometaphase arrest resulted in mitotic slippage, as indicated by a reduction in mitotic cells, APC/C-dependent degradation of cyclin B1, increased cell attachment, and increased nuclear membrane reassembly. Additionally, cells successfully progressed through the cell cycle after mitotic slippage, as indicated by EdU incorporation and time-lapse imaging. Although degradation of cyclin B during normal mitotic progression is primarily regulated by APC/C^Cdc20, we observed an unexpected decrease in Cdc20 prior to degradation of cyclin B during mitotic slippage. This decrease in Cdc20 was followed by a change in the binding partner preference of APC/C from Cdc20 to Cdh1; consequently, APC/C^Cdh1, but not APC/C^Cdc20, facilitated cyclin B degradation following ATP depletion. Pulse-chase analysis revealed that ATP depletion significantly abrogated global translation, including the translation of Cdc20 and Cdh1. Additionally, the half-life of Cdh1 was much longer than that of Cdc20. These data suggest that ATP depletion during mitotic arrest induces mitotic slippage facilitated by APC/C^Cdh1-dependent cyclin B degradation, which follows a decrease in Cdc20 resulting from reduced global translation and the differences in the half-lives of the Cdc20 and Cdh1 proteins.

An investigation into the effects of cellular energy depletion reveals a potential mechanism by which tumors evade chemotherapy. Adenosine triphosphate (ATP) is the primary energetic currency for many biological processes, and ATP depletion generally stalls the cell cycle that regulates proliferation. However, researchers led by Jae-Ho Lee of South Korea’s Ajou University School of Medicine discovered that ATP-depleted cells can sometimes bypass roadblocks in the cell division process. Before dividing, cells synthesize duplicates of every chromosome, and Lee’s team treated cells with chemotherapy agents that stall cell division by preventing separation of these duplicates. Surprisingly, subsequent ATP depletion allowed these cells to bypass this arrested state and re-enter the cell cycle, albeit with twice as much DNA as normal. Since many cancerous cells experience ATP depletion, this ‘escape hatch’ could help tumors survive treatment.

Computer-aided biophysical modeling: a quantitative approach to complex biological systems

When dealing with the biophysics of tumors, analytical and numerical modeling tools have long been regarded as potentially useful but practically immature tools. Further developments could not just overturn this predicament, but lead to completely new perspectives in biology. Here, we give an account of our own computational tool and how we have put it to good use, and we discuss a paradigmatic example to outline a path to making cell biology more quantitative and predictive.

Interplay between distribution of live cells and growth dynamics of solid tumours

Experiments show that simple diffusion of nutrients and waste molecules is not sufficient to explain the typical multilayered structure of solid tumours, where an outer rim of proliferating cells surrounds a layer of quiescent but viable cells and a central necrotic region. These experiments challenge models of tumour growth based exclusively on diffusion. Here we propose a model of tumour growth that incorporates the volume dynamics and the distribution of cells within the viable cell rim. The model is suggested by in silico experiments and is validated using in vitro data. The results correlate with in vivo data as well, and the model can be used to support experimental and clinical oncology.

COMPUTATIONAL CHALLENGES OF TUMOR SPHEROID MODELING

The speed and the versatility of today's computers open up new opportunities to simulate complex biological systems. Here we review a computational approach recently proposed by us to model large tumor cell populations and spheroids, and we put forward general considerations that apply to any fine-grained numerical model of tumors. We discuss ways to bypass computational limitations and discuss our incremental approach, where each step is validated by experimental observations on a quantitative basis. We present a few results on the growth of tumor cells in closed and open environments and of tumor spheroids. This study suggests new ways to explore the initial growth phase of solid tumors and to optimize antitumor treatments.

Emergent properties of tumor microenvironment in a real-life model of multicell tumor spheroids

Multicellular tumor spheroids are an important in vitro model of the pre-vascular phase of solid tumors, for sizes well below the diagnostic limit: therefore a biophysical model of spheroids has the ability to shed light on the internal workings and organization of tumors at a critical phase of their development. To this end, we have developed a computer program that integrates the behavior of individual cells and their interactions with other cells and the surrounding environment. It is based on a quantitative description of metabolism, growth, proliferation and death of single tumor cells, and on equations that model biochemical and mechanical cell-cell and cell-environment interactions. The program reproduces existing experimental data on spheroids, and yields unique views of their microenvironment. Simulations show complex internal flows and motions of nutrients, metabolites and cells, that are otherwise unobservable with current experimental techniques, and give novel clues on tumor development and strong hints for future therapies.

Metabolic properties of chicken embryonic stem cells

Cellular energy metabolism correlates with cell fate, but the metabolic properties of chicken embryonic stem (chES) cells are poorly understood. Using a previously established chES cell model and electron microscopy (EM), we found that undifferentiated chES cells stored glycogen. Additionally, undifferentiated chES cells expressed lower levels of glucose transporter 1 (GLUT1) and phosphofructokinase (PFK) mRNAs but higher levels of hexokinase 1 (HK1) and glycogen synthase (GYS) mRNAs compared with control primary chicken embryonic fibroblast (CEF) cells, suggesting that chES cells direct glucose flux towards the glycogenic pathway. Moreover, we demonstrated that undifferentiated chES cells block gluconeogenic outflow and impede the accumulation of glucose-6-phosphate (G6P) from this pathway, as evidenced by the barely detectable levels of pyruvate carboxylase (PCX) and mitochondrial phosphoenolpyruvate carboxykinase (PCK2) mRNAs. Additionally, cell death occurred in undifferentiated chES cells as shown by Hoechst 33342 and propidium iodide (PI) double staining, but it could be rescued by exogenous G6P. However, we found that differentiated chES cells decreased the glycogen reserve through the use of PAS staining. Moreover, differentiated chES cells expressed higher levels of GLUT1, HK1 and PFK mRNAs, while the level of GYS mRNA remained similar in control CEF cells. These data indicate that undifferentiated chES cells continue to synthesize glycogen from glucose at the expense of G6P, while differentiated chES cells have a decreased glycogen reserve, which suggests that the amount of glycogen is indicative of the chES cell state.

Essential operating principles for tumor spheroid growth

BACKGROUND

Our objective was to discover in silico axioms that are plausible representations of the operating principles realized during characteristic growth of EMT6/Ro mouse mammary tumor spheroids in culture. To reach that objective we engineered and iteratively falsified an agent-based analogue of EMT6 spheroid growth. EMT6 spheroids display consistent and predictable growth characteristics, implying that individual cell behaviors are tightly controlled and regulated. An approach to understanding how individual cell behaviors contribute to system behaviors is to discover a set of principles that enable abstract agents to exhibit closely analogous behaviors using only information available in an agent's immediate environment. We listed key attributes of EMT6 spheroid growth, which became our behavioral targets. Included were the development of a necrotic core surrounded by quiescent and proliferating cells, and growth data at two distinct levels of nutrient.

RESULTS

We then created an analogue made up of quasi-autonomous software agents and an abstract environment in which they could operate. The system was designed so that upon execution it could mimic EMT6 cells forming spheroids in culture. Each agent used an identical set of axiomatic operating principles. In sequence, we used the list of targeted attributes to falsify and revise these axioms, until the analogue exhibited behaviors and attributes that were within prespecified ranges of those targeted, thereby achieving a level of validation.

CONCLUSION

The finalized analogue required nine axioms. We posit that the validated analogue's operating principles are reasonable representations of those utilized by EMT6/Ro cells during tumor spheroid development.

Ab initio phenomenological simulation of the growth of large tumor cell populations

In a previous paper we have introduced a phenomenological model of cell metabolism and of the cell cycle to simulate the behavior of large tumor cell populations (Chignola and Milotti 2005 Phys. Biol. 2 8). Here we describe a refined and extended version of the model that includes some of the complex interactions between cells and their surrounding environment. The present version takes into consideration several additional energy-consuming biochemical pathways such as protein and DNA synthesis, the tuning of extracellular pH and of the cell membrane potential. The control of the cell cycle, which was previously modeled by means of ad hoc thresholds, has been directly addressed here by considering checkpoints from proteins that act as targets for phosphorylation on multiple sites. As simulated cells grow, they can now modify the chemical composition of the surrounding environment which in turn acts as a feedback mechanism to tune cell metabolism and hence cell proliferation: in this way we obtain growth curves that match quite well those observed in vitro with human leukemia cell lines. The model is strongly constrained and returns results that can be directly compared with actual experiments, because it uses parameter values in narrow ranges estimated from experimental data, and in perspective we hope to utilize it to develop in silico studies of the growth of very large tumor cell populations (10(6) cells or more) and to support experimental research. In particular, the program is used here to make predictions on the behavior of cells grown in a glucose-poor medium: these predictions are confirmed by experimental observation.