Mathematical modelling of tumour acidity

Computational model of the cancer necrotic core formation in a tumor-on-a-chip device

The mechanisms underlying the formation of necrotic regions within avascular tumors are complex and poorly understood. In this paper, we investigate the formation of a necrotic core in a 3D tumor cell culture within a microfluidic device, considering oxygen, nutrients, and the microenvironment acidification by means of a computational-mathematical model. Our objective is to simulate cell processes, including proliferation and death inside a microfluidic device, according to the microenvironmental conditions. We employed approximation utilizing finite element models taking into account glucose, oxygen, and hydrogen ions diffusion, consumption and production, as well as cell proliferation, migration and death, addressing how tumor cells evolve under different conditions. The resulting mathematical model was examined under different scenarios, being capable of reproducing cell death and proliferation under different cell concentrations, and the formation of a necrotic core, in good agreement with experimental data reported in the literature. This approach not only advances our fundamental understanding of necrotic core formation but also provides a robust computational platform to study personalized therapeutic strategies, offering an important tool in cancer research and treatment design.

pH as a potential therapeutic target to improve temozolomide antitumor efficacy : A mechanistic modeling study

Despite intensive treatments including temozolomide (TMZ) administration, glioblastoma patient prognosis remains dismal and innovative therapeutic strategies are urgently needed. A systems pharmacology approach was undertaken to investigate TMZ pharmacokinetics-pharmacodynamics (PK-PD) incorporating the effect of local pH, tumor spatial configuration and micro-environment. A hybrid mathematical framework was designed coupling ordinary differential equations describing the intracellular reactions, with a spatial cellular automaton to individualize the cells. A differential drug impact on tumor and healthy cells at constant extracellular pH was computationally demonstrated as TMZ-induced DNA damage was larger in tumor cells as compared to normal cells due to less acidic intracellular pH in cancer cells. Optimality of TMZ efficacy defined as maximum difference between damage in tumor and healthy cells was reached for extracellular pH between 6.8 and 7.5. Next, TMZ PK-PD in a solid tumor was demonstrated to highly depend on its spatial configuration as spread cancer cells or fragmented tumors presented higher TMZ-induced damage as compared to compact tumor spheroid. Simulations highlighted that smaller tumors were less acidic than bigger ones allowing for faster TMZ activation and their closer distance to blood capillaries allowed for better drug penetration. For model parameters corresponding to U87 glioma cells, inter-cell variability in TMZ uptake play no role regarding the mean drug-induced damage in the whole cell population whereas this quantity was increased by inter-cell variability in TMZ efflux which was thus a disadvantage in terms of drug resistance. Overall, this study revealed pH as a new potential target to significantly improve TMZ antitumor efficacy.

Glioma growth modeling based on the effect of vital nutrients and metabolic products

Glioma brain tumors exhibit considerably aggressive behavior leading to high mortality rates. Mathematical modeling of tumor growth aims to explore the interactions between glioma cells and tissue microenvironment, which affect tumor evolution. Leveraging this concept, we present a three-dimensional model of glioma spatio-temporal evolution based on existing continuum approaches, yet incorporating novel factors of the phenomenon. The proposed model involves the interactions between different tumor cell phenotypes and their microenvironment, investigating how tumor growth is affected by complex biological exchanges. It focuses on the separate and combined effect of vital nutrients and cellular wastes on tumor expansion, leading to the formation of cell populations with different metabolic, proliferative, and diffusive profiles. Several simulations were performed on a virtual and a real glioma, using combinations of proliferation and diffusion rates for different evolution times. The model results were validated on a glioma model available in the literature and a real case of tumor progression. The experimental observations indicate that our model estimates quite satisfactorily the expansion of each region and the overall tumor growth. Based on the individual results, the proposed model may provide an important research tool for patient-specific simulation of different tumor evolution scenarios and reliable estimation of glioma evolution. Graphical Abstract Outline of the mathematical model functionality and application to glioma growth with indicative results.

Model of vascular desmoplastic multispecies tumor growth

We present a three-dimensional nonlinear tumor growth model composed of heterogeneous cell types in a multicomponent-multispecies system, including viable, dead, healthy host, and extra-cellular matrix (ECM) tissue species. The model includes the capability for abnormal ECM dynamics noted in tumor development, as exemplified by pancreatic ductal adenocarcinoma, including dense desmoplasia typically characterized by a significant increase of interstitial connective tissue. An elastic energy is implemented to provide elasticity to the connective tissue. Cancer-associated fibroblasts (myofibroblasts) are modeled as key contributors to this ECM remodeling. The tumor growth is driven by growth factors released by these stromal cells as well as by oxygen and glucose provided by blood vasculature which along with lymphatics are stimulated to proliferate in and around the tumor based on pro-angiogenic factors released by hypoxic tissue regions. Cellular metabolic processes are simulated, including respiration and glycolysis with lactate fermentation. The bicarbonate buffering system is included for cellular pH regulation. This model system may be of use to simulate the complex interactions between tumor and stromal cells as well as the associated ECM and vascular remodeling that typically characterize malignant cancers notorious for poor therapeutic response.

A Glycolysis-Based In Silico Model for the Solid Tumor Growth

Cancer-tumor growth is a complex process depending on several biological factors, such as the chemical microenvironment of the tumor, the cellular metabolic profile, and its proliferation rate. Several mathematical models have been developed for identifying the interactions between tumor cells and tissue microenvironment, since they play an important role in tumor formation and progression. Toward this direction we propose a new continuum model of avascular glioma-tumor growth, which incorporates a new factor, namely, the glycolytic potential of cancer cells, to express the interactions of three different tumor-cell populations (proliferative, hypoxic, and necrotic) with their tissue microenvironment. The glycolytic potential engages three vital nutrients, i.e., oxygen, glucose, and lactate, which provide cells with the necessary energy for their survival and proliferation. Extensive simulations are performed for different evolution times and various proliferation rates, in order to investigate how the tumor growth is affected. According to medical experts, the experimental observations indicate that the model predicts quite satisfactorily the overall tumor growth as well as the expansion of each region separately. Following extensive evaluation, the proposed model may provide an essential tool for patient-specific tumor simulation and reliable prediction of glioma spatiotemporal expansion.

Model for acid-mediated tumour invasion with chemotherapy intervention II: Spatially heterogeneous populations

This paper extends the model for acid-mediated tumour invasion with chemotherapy intervention examined in part I. The model presented in part I considers the interaction between tumour cells, normal cells, acid and drug in a well mixed (i.e. spatially homogeneous) setting, which is governed by a system of nonlinear differential equations. The model examined here removes the assumption that the populations are spatially homogeneous resulting in a system of nonlinear partial differential equations. Numerical simulations of this model are presented for different treatment methods displaying several possible behaviours. Asymptotic approximations are also derived for a special case of the treatment method and set of parameter values. This analysis then allows us to draw conclusions about the effectiveness of treating acid-mediated tumours with chemotherapy.

Microenvironment acidity as a major determinant of tumor chemoresistance: Proton pump inhibitors (PPIs) as a novel therapeutic approach

Despite the major progresses in biomedical research and the development of novel therapeutics and treatment strategies, cancer is still among the dominant causes of death worldwide. One of the crucial challenges in the clinical management of cancer is primary (intrinsic) and secondary (acquired) resistance to both conventional and targeted chemotherapeutics. Multiple mechanisms have been identifiedthat underlie intrinsic and acquired chemoresistance: these include impaired drug uptake, increased drug efflux, deletion of receptors, altered drug metabolism, quantitative and qualitative alterations in drug targets, increased DNA damage repair and various mechanisms of anti-apoptosis. The fast efflux of anticancer drugs mediated by multidrug efflux pumps and the partial or complete reversibility of chemoresistance combined with the absence of genetic mutations suggests a multifactorial process. However, a growing body of recent evidence suggests that chemoresistance is often triggered by the highly acidic microenvironment of tumors. The vast majority of drugs, including conventional chemotherapeutics and more recent biological agents, are weak bases that are quickly protonated and neutralized in acidic environments, such as the extracellular microenvironment and the acidic organelles of tumor cells. It is therefore essential to develop new strategies to overcome the entrapment and neutralization of weak base drugs. One such strategy is the use of proton pump inhibitors which can enhance tumor chemosensitivity by increasing the pH of the tumor microenvironment. Recent clinical trials in animals with spontaneous tumors have indicated that patient alkalization is capable of reversing acquired chemoresistance in a large percentage of tumors that are refractory to chemotherapy. Of particular interest was the benefit of alkalization for patients undergoing metronomic regimens which are becoming more widely used in veterinary medicine. Overall, these results provide substantial new evidence that altering the acidic tumor microenvironment is an effective, well tolerated and low cost strategy for the overcoming of anticancer drug resistance.

Proton pump inhibitors while belonging to the same family of generic drugs show different anti-tumor effect

CONTEXT

Tumor acidity represents a major cause of chemoresistance. Proton pump inhibitors (PPIs) can neutralize tumor acidity, sensitizing cancer cells to chemotherapy.

OBJECTIVE

To compare the anti-tumor efficacy of different PPIs in vitro and in vivo.

MATERIALS AND METHODS

In vitro experiments PPIs anti-tumor efficacy in terms of cell proliferation and cell death/apoptosis/necrosis evaluation were performed. In vivo PPIs efficacy experiments were carried out using melanoma xenograft model in SCID mice.

RESULTS

Lansoprazole showed higher anti-tumor effect when compared to the other PPIs. The lansoprazole effect lasted even upon drug removal from the cell culture medium and it was independent from the lipophilicity of the PPIs formulation.

DISCUSSION

These PPIs have shown different anti-tumoral efficacy, and the most effective at low dose was lansoprazole.

CONCLUSION

The possibility to contrast tumor acidity by off-label using PPIs opens a new field of oncology investigation.

PET-specific parameters and radiotracers in theoretical tumour modelling

The innovation of computational techniques serves as an important step toward optimized, patient-specific management of cancer. In particular, in silico simulation of tumour growth and treatment response may eventually yield accurate information on disease progression, enhance the quality of cancer treatment, and explain why certain therapies are effective where others are not. In silico modelling is demonstrated to considerably benefit from information obtainable with PET and PET/CT. In particular, models have successfully integrated tumour glucose metabolism, cell proliferation, and cell oxygenation from multiple tracers in order to simulate tumour behaviour. With the development of novel radiotracers to image additional tumour phenomena, such as pH and gene expression, the value of PET and PET/CT data for use in tumour models will continue to grow. In this work, the use of PET and PET/CT information in in silico tumour models is reviewed. The various parameters that can be obtained using PET and PET/CT are detailed, as well as the radiotracers that may be used for this purpose, their utility, and limitations. The biophysical measures used to quantify PET and PET/CT data are also described. Finally, a list of in silico models that incorporate PET and/or PET/CT data is provided and reviewed.

Invasion and proliferation kinetics in enhancing gliomas predict IDH1 mutation status

BACKGROUND

Glioblastomas with a specific mutation in the isocitrate dehydrogenase 1 (IDH1) gene have a better prognosis than gliomas with wild-type IDH1.

METHODS

Here we compare the IDH1 mutational status in 172 contrast-enhancing glioma patients with the invasion profile generated by a patient-specific mathematical model we developed based on MR imaging.

RESULTS

We show that IDH1-mutated contrast-enhancing gliomas were relatively more invasive than wild-type IDH1 for all 172 contrast-enhancing gliomas as well as the subset of 158 histologically confirmed glioblastomas. The appearance of this relatively increased, model-predicted invasive profile appears to be determined more by a lower model-predicted net proliferation rate rather than an increased model-predicted dispersal rate of the glioma cells. Receiver operator curve analysis of the model-predicted MRI-based invasion profile revealed an area under the curve of 0.91, indicative of a predictive relationship. The robustness of this relationship was tested by cross-validation analysis of the invasion profile as a predictive metric for IDH1 status.

CONCLUSIONS

The strong correlation between IDH1 mutation status and the MRI-based invasion profile suggests that use of our tumor growth model may lead to noninvasive clinical detection of IDH1 mutation status and thus lead to better treatment planning, particularly prior to surgical resection, for contrast-enhancing gliomas.

Involvement of tumor macrophage HIFs in chemotherapy effectiveness: mathematical modeling of oxygen, pH, and glutathione

The four variables, hypoxia, acidity, high glutathione (GSH) concentration and fast reducing rate (redox) are distinct and varied characteristics of solid tumors compared to normal tissue. These parameters are among the most significant factors underlying the metabolism and physiology of solid tumors, regardless of their type or origin. Low oxygen tension contributes to both inhibition of cancer cell proliferation and therapeutic resistance of tumors; low extracellular pH, the reverse of normal cells, mainly enhances tumor invasion; and dysregulated GSH and redox potential within cancer cells favor their proliferation. In fact, cancer cells under these microenvironmental conditions appreciably alter tumor response to cytotoxic anti-cancer treatments. Recent experiments measured the in vivo longitudinal data of these four parameters with tumor development and the corresponding presence and absence of tumor macrophage HIF-1α or HIF-2α in a mouse model of breast cancer. In the current paper, we present a mathematical model-based system of (ordinary and partial) differential equations to monitor tumor growth and susceptibility to standard chemotherapy with oxygen level, pH, and intracellular GSH concentration. We first show that our model simulations agree with the corresponding experiments, and then we use our model to suggest treatments of tumors by altering these four parameters in tumor microenvironment. For example, the model qualitatively predicts that GSH depletion can raise the level of reactive oxygen species (ROS) above a toxic threshold and result in inhibition of tumor growth.

Metabolic imaging in multiple time scales

We report here a novel combination of time-resolved imaging methods for probing mitochondrial metabolism in multiple time scales at the level of single cells. By exploiting a mitochondrial membrane potential reporter fluorescence we demonstrate the single cell metabolic dynamics in time scales ranging from microseconds to seconds to minutes in response to glucose metabolism and mitochondrial perturbations in real time. Our results show that in comparison with normal human mammary epithelial cells, the breast cancer cells display significant alterations in metabolic responses at all measured time scales by single cell kinetics, fluorescence recovery after photobleaching and by scaling analysis of time-series data obtained from mitochondrial fluorescence fluctuations. Furthermore scaling analysis of time-series data in living cells with distinct mitochondrial dysfunction also revealed significant metabolic differences thereby suggesting the broader applicability (e.g. in mitochondrial myopathies and other metabolic disorders) of the proposed strategies beyond the scope of cancer metabolism. We discuss the scope of these findings in the context of developing portable, real-time metabolic measurement systems that can find applications in preclinical and clinical diagnostics.

Parameter non-identifiability of the Gyllenberg-Webb ODE model

An ODE model introduced by Gyllenberg and Webb (Growth Develop Aging 53:25-33, 1989) describes tumour growth in terms of the dynamics between proliferating and quiescent cell states. The passage from one state to another and vice versa is modelled by two functions r0 and ri depending on the total tumour size. As these functions do not represent any observable quantities, they have to be identified from the observations. In this paper we show that there is an infinite number of pairs (r0, ri) corresponding to the same solution of the ODE system and the functions (r0, ri) will be classified in terms of this equivalence. Surprisingly, the technique used for this classification permits a uniqueness proof of the solution of the ODE model in a non-Lipschitz case. The reasoning can be widened to a more general setting including an extension of the Gyllenberg-Webb model with a nonlinear birth rate. The relevance of this result is discussed in a preclinical application scenario.

Modeling the connection between primary and metastatic tumors

We put forward a model for cancer metastasis as a migration phenomenon between tumor cell populations coexisting and evolving in two different habitats. One of them is a primary tumor and the other one is a secondary or metastatic tumor. The evolution of the different cell phenotype populations in each habitat is described by means of a simple quasispecies model allowing for a cascade of mutations between the different phenotypes in each habitat. The cell migration event is supposed to be unidirectional and take place continuously in time. The possible clinical outcomes of the model depending on the parameter space are analyzed and the effect of the resection of the primary tumor is studied.

Multiscale cancer modeling

Simulating cancer behavior across multiple biological scales in space and time, i.e., multiscale cancer modeling, is increasingly being recognized as a powerful tool to refine hypotheses, focus experiments, and enable more accurate predictions. A growing number of examples illustrate the value of this approach in providing quantitative insights in the initiation, progression, and treatment of cancer. In this review, we introduce the most recent and important multiscale cancer modeling works that have successfully established a mechanistic link between different biological scales. Biophysical, biochemical, and biomechanical factors are considered in these models. We also discuss innovative, cutting-edge modeling methods that are moving predictive multiscale cancer modeling toward clinical application. Furthermore, because the development of multiscale cancer models requires a new level of collaboration among scientists from a variety of fields such as biology, medicine, physics, mathematics, engineering, and computer science, an innovative Web-based infrastructure is needed to support this growing community.

A mathematical model for the glucose-lactate metabolism of in vitro cancer cells

We propose a mathematical model of tumor cell nutrient uptake governed by the presence of two key biomolecular fuels: glucose and lactate. The model allows us to describe, in a remarkably simple way, different in vitro scenarios previously reported in experiments of tumor cell metabolism using distinct energy sources. The predictions of our model show good agreement with all the examined tumor cell lines (cervix, colon, and glioma) and provide a first step toward the development of more comprehensive frameworks accounting for in vivo cancer dynamics under complex spatial heterogeneities.

Discovering Molecular Targets in Cancer with Multiscale Modeling

Multiscale modeling is increasingly being recognized as a promising research area in computational cancer systems biology. Here, exemplified by two pioneering studies, we attempt to explain why and how such a multiscale approach paired with an innovative cross-scale analytical technique can be useful in identifying high-value molecular therapeutic targets. This novel, integrated approach has the potential to offer a more effective in silico framework for target discovery and represents an important technical step towards systems medicine.

Computational analysis of the influence of the microenvironment on carcinogenesis

The tumour microenvironment is known to play an important role in determining the sequence of acquired phenotypic traits that characterise cancer evolution. A more precise understanding of this role could have a major influence in the understanding of cancer growth and development, and potentially in the optimisation of innovative anti-cancer treatments delivery. However, to lead such an analysis in the basis of traditional biological experiments and observations is almost utopian given the complexity of the underlying biological processes and the typical time scales involved. In this context, computer models constitute a complementary exploratory tool. In this paper we introduce a two-dimensional cellular automaton that models key cancer cell capabilities. The model has been especially designed to mimic the behaviour of a cancer colony growing in vitro and to analyse the effect of different environmental conditions on the sequence of acquisition of phenotypic traits. Our results indicate that microenvironmental factors such as the local concentration of oxygen or nutrients and cell overcrowding may determine the expansion of the tumour colony. The results also show that tumour cells evolve and that their phenotypes adapt to the microenvironment so that environmental stress determines the dominance of particular phenotypical traits.

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

Mathematical modelling of the Warburg effect in tumour cords

The model proposed here links together two approaches to describe tumours: a continuous medium to describe the movement and the mechanical properties of the tissue, and a population dynamics approach to represent internal genetic inhomogeneity and instability of the tumour. In this way one can build models which cover several stages of tumour progression. In this paper we focus on describing transition from aerobic to purely glycolytic metabolism (the Warburg effect) in tumour cords. From the mathematical point of view this model leads to a free boundary problem where domains in contact are characterized by different sets of equations. Accurate stitching of the solution was possible with a modified ghost fluid method. Growth and death of the cells and uptake of the nutrients are related through ATP production and energy costs of the cellular processes. In the framework of the bi-population model this allowed to keep the number of model parameters relatively small.