Tumor growth has long been a target of investigation within the context of mathematical and computer modelling. The objective of this study is to propose and analyze a two-dimensional probabilistic cellular automata model to describe avascular solid tumor growth, taking into account both the competition between cancer cells and normal cells for nutrients and/or space and a time-dependent proliferation of cancer cells. Gompertzian growth, characteristic of some tumors, is described and some of the features of the time-spatial pattern of solid tumors, such as compact morphology with irregular borders, are captured. The parameter space is studied in order to analyze the occurrence of necrosis and the response to therapy. Our findings suggest that transitions exist between necrotic and non-necrotic phases (no-therapy cases), and between the states of cure and non-cure (therapy cases). To analyze cure, the control and order parameters are, respectively, the highest probability of cancer cell proliferation and the probability of the therapeutic effect on cancer cells. With respect to patterns, it is possible to observe the inner necrotic core and the effect of the therapy destroying the tumor from its outer borders inwards.
A nutrient-limited model for avascular cancer growth including cell proliferation, motility and death is presented. The model qualitatively reproduces commonly observed morphologies for primary tumors, and the simulated patterns are characterized by its gyration radius, total number of cancer cells, and number of cells on tumor periphery. These very distinct morphological patterns follow Gompertz growth curves, but exhibit different scaling laws for their surfaces. Also, the simulated tumors incorporate a spatial structure composed of a central necrotic core, an inner rim of quiescent cells an…
The comprehension of tumor growth is a intriguing subject for scientists. New researches has been constantly required to better understand the complexity of this phenomenon. In this paper, we pursue a physical description that account for some experimental facts involving avascular tumor growth. We have proposed an explanation of some phenomenological (macroscopic) aspects of tumor, as the spatial form and the way it growths, from a individual-level (microscopic) formulation. The model proposed here is based on a simple principle: competitive interaction between the cells dependent on their mu…
We are keenly interested in finding the limiting behaviors of total tumor size when tumor cells are subject to the periodic repetition of therapy and rest periods, called intermittent cyclic therapy. We hypothesize that each tumor cell can take either therapy-sensitive or therapy-tolerant phenotype, its phenotype transition is mainly driven by the presence or absence of environmental stress, and such a transition is reversible. Even though those aforementioned hypotheses make the model system simple, most of prior papers attempted to numerically find the optimal therapeutic scheduling that min…
We investigate a hybrid PDE/Monte Carlo technique for the variance reduced simulation of an agent-based multiscale model for tumor growth. The variance reduction is achieved by combining a simulation of the stochastic agent-based model on the microscopic scale with a deterministic solution of a simplified (coarse) partial differential equation (PDE) on the macroscopic scale as a control variable. We show that this technique is able to significantly reduce the variance with only the (limited) additional computational cost associated with the deterministic solution of the coarse PDE. We illustra…
A complete parametric analysis of dynamic regimes of a conceptual model of anti-tumor virus therapy is presented. The role and limitations of mass-action kinetics are discussed. A functional response, which is a function of the ratio of uninfected to infected tumor cells, is proposed to describe the spread of the virus infection in the tumor. One of the main mathematical features of ratio-dependent models is that the origin is a complicated equilibrium point whose characteristics crucially determine the main properties of the model. It is shown that, in a certain area of parameter values, the …
One of the mechanisms that ensure cancer robustness is tumor heterogeneity, and its effects on tumor cells dynamics have to be taken into account when studying cancer progression. There is no unifying theoretical framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity. Here we formulate a modeling approach that naturally takes stock of inherent cancer cell heterogeneity and illustrate it with a model of interaction between a tumor and an oncolytic virus. We show that several phenomena that are absent in homogeneous models, such as cancer recurrence,…
We model the interaction between the immune system and tumor cells including a time delay to simulate the time needed by the latter to develop a chemical and cell mediated response to the presence of the tumor. The results are compared with those of a previous paper, concluding that the delay introduces new instabilities in the system leading to an uncontrolable growth of the tumour. Then a cytokine based periodic immunotherapy treatment is included in the model and the effects of its dossage are studied for the case of a weak immune system and a growing tumour. We find the existence of metast…
Dual-functional nanoparticles, with the property of aggregation-induced emission and the capability of reactive oxygen species, were used to achieve passive/active targeting of tumor. Good contrast in in vivo imaging and obvious therapeutic efficiency were realized with a low dose of AIE nanoparticles as well as a low power density of light, resulting in negligible side effects.
The growth of many solid tumors has been found to be driven by chemo- and radiotherapy-resistant cancer stem cells (CSCs). A suitable therapeutic avenue in these cases may involve the use of a differentiating agent (DA) to force the differentiation of the CSCs and of conventional therapies to eliminate the remaining differentiated cancer cells (DCCs). To describe the effects of a DA that reprograms CSCs into DCCs, we adapt a differential equation model developed to investigate tumorspheres, which are assumed to consist of jointly evolving CSC and DCC populations. We analyze the mathematical pr…
Background: Promising results for mammary carcinoma treatment with photodynamic therapy (PDT) presuppose a careful selection of irradiation light wavelength. Methods: 4T1 tumors implanted in NOD-SCID mice were treated with Metvix-PDT under 625 nm, 660 nm and their combination light, for a fixed radiant exposure. The therapeutic outcome was assessed through Monte Carlo based computational simulations along with a preliminary in vivo study, where fluorescence, size and temperature measurements were conducted. Results: The light source combination protocol presents great potential, since it resul…
Cytotoxic chemotherapy is a common treatment for advanced prostate cancer. These tumors are also known to rely on angiogenesis, i.e., the growth of local microvasculature via chemical signaling produced by the tumor. Thus, several clinical studies have been investigating antiangiogenic therapy for advanced prostate cancer, either as monotherapy or combined with standard cytotoxic protocols. However, the complex genetic alterations promoting prostate cancer growth complicate the selection of the best chemotherapeutic approach for each patient's tumor. Here, we present a mathematical model of pr…
We present a mathematical model, based on ordinary differential equations, for the evolution of solid tumors and their response to treatment. Specifically the effects of a cytotoxic agent and a monoclonal antibody are included as control term in the equations. The variables considered here are: the number of cancerous cells sensitive to chemotherapy, the number of cancerous cells resistant to chemotherapy, the degree of angiogenesis and the average intensity of VEGF. The rules that govern the quantities mentioned above are based on a geometrical argument: we approximate the tumor mass as a sph…
In this work we propose and investigate a family of models, which admits as particular cases some well known mathematical models of tumor-immune system interaction, with the additional assumption that the influx of immune system cells may be a function of the number of cancer cells. Constant, periodic and impulsive therapies (as well as the non-perturbed system) are investigated both analytically for the general family and, by using the model by Kuznetsov et al. (V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson. Nonlinear dynamics of immunogenic tumors: Parameter estimation and…
The understanding of the macroscopic phenomenological models of the population growth at a microscopic level is important to predict the population behaviors emerged from the interactions between the individuals. In this work we consider the influence of the cell-cell interaction on the population growth rate $R$ in a tumor system, and show that, in most cases especially small proliferative probabilities, the regulative role of the interaction will be strengthened with the decline of the intrinsic proliferative probabilities. For the high replication rates of an individual and the cooperative …
Immunotherapies use components of the patient immune system to selectively target cancer cells. The use of CAR T cells to treat B-cell malignancies --leukaemias and lymphomas-- is one of the most successful examples, with many patients experiencing long-lasting complete responses to this therapy. This treatment works by extracting the patient's T cells and adding them the CAR group, which enables them to recognize and target cells carrying the antigen CD19+, that is expressed in these haematological tumors. Here we put forward a mathematical model describing the time response of leukaemias to …
A systematic understanding of the evolution and growth dynamics of invasive solid tumors in response to different chemotherapy strategies is crucial for the development of individually optimized oncotherapy. Here, we develop a hybrid three-dimensional (3D) computational model that integrates pharmacokinetic model, continuum diffusion-reaction model and discrete cell automaton model to investigate 3D invasive solid tumor growth in heterogeneous microenvironment under chemotherapy. Specifically, we consider the effects of heterogeneous environment on drug diffusion, tumor growth, invasion and th…
Recently, we have proposed a nutrient-limited model for the avascular growth of tumors including cell proliferation, motility and death \cite{jr}, that, qualitatively reproduces commonly observed morphologies for carcinomas {\it in situ}. In the present work, we analyze the effects of distinct chemotherapeutic strategies on the patterns, scaling and growth laws obtained for such nutrient-limited model. Two kinds of chemotherapeutic strategies were considered, namely, those that kill cancer cells and those that block cell mitosis but allows the cell to survive for some time. Depending on the ch…
Adaptive therapy is a dynamic cancer treatment protocol that updates (or "adapts") treatment decisions in anticipation of evolving tumor dynamics. This broad term encompasses many possible dynamic treatment protocols of patient-specific dose modulation or dose timing. Adaptive therapy maintains high levels of tumor burden to benefit from the competitive suppression of treatment-sensitive subpopulations on treatment-resistant subpopulations. This evolution-based approach to cancer treatment has been integrated into several ongoing or planned clinical trials, including treatment of metastatic ca…
In oncology, treating cancer with a beam of photons is a well established therapeutic technique, developed over 100 years, and today over 50% of cancer patients will undergo traditional X-ray radiotherapy. However, ionizing radiation therapy is not the only option, as the high-energy photons delivering their cell-killing radiation energy into cancerous tumor can lead to significant damage to healthy tissues surrounding the tumor, located throughout the beam's path. Therefore, in nowadays, advances in ionizing radiation therapy are competitive to non-ionizing ones, as for example the laser ligh…
In this paper we consider chemotherapy in a spatial model of tumor growth. The model, which is of reaction-diffusion type, takes into account the complex interactions between the tumor and surrounding stromal cells by including densities of endothelial cells and the extra-cellular matrix. When no treatment is applied the model reproduces the typical dynamics of early tumor growth. The initially avascular tumor reaches a diffusion limited size of the order of millimeters and initiates angiogenesis through the release of vascular endothelial growth factor (VEGF) secreted by hypoxic cells in the …
Melanoma is one of the most dangerous forms of cancer. The five-year survival rate is 98% if it is detected early. However, this rate plummets to 63% for regional disease and 17% when tumors have metastasized, that is, spread to distant sites. Furthermore, the incidence of melanoma has been rising by about 3% per year, whereas the incidence of cancers that are more common is decreasing. A handful of targeted therapies have recently become available that have finally shown real promise for treatment, but for reasons that remain unclear only a fraction of patients respond long term. These drugs …
Mechanical force is tightly connected to human health status, and the occurrence of disease can generally be ascribed to certain loss of force balance. However, the role of mechanical approaches in tumor therapy is largely neglected, while the currently available cancer prevention and treatment methods are generally either expensive or just cause too much side effect. In this article, we present a systematic interpretation on a promising strategy which was termed here as Mechanical Therapy for the first time based on the fact that mechanical force closely accompanies the whole life (growth and…
The goal of cancer immunotherapy is to boost a patient's immune response to a tumor. Yet, the design of an effective immunotherapy is complicated by various factors, including a potentially immunosuppressive tumor microenvironment, immune-modulating effects of conventional treatments, and therapy-related toxicities. These complexities can be incorporated into mathematical and computational models of cancer immunotherapy that can then be used to aid in rational therapy design. In this review, we survey modeling approaches under the umbrella of the major challenges facing immunotherapy developme…
A stochastic agent-based model for simulating tumor-immune dynamics and evaluating therapeutic strategies
School of Software, Tiangong University（天津工业大学软件学院）；School of Mathematical Sciences, Tiangong University（天津工业大学数学科学学院）；Center for Applied Mathematics, Tiangong University（天津工业大学应用数学中心）
We developed simulation methodology to assess eventual therapeutic efficiency of exogenous multiparametric changes in a four-component cellular system described by the system of ordinary differential equations. The method is numerically implemented to simulate the temporal behavior of a cellular system of multiple myeloma cells. The problem is conceived as an inverse optimization task where the alternative temporal changes of selected parameters of the ordinary differential equations represent candidate solutions and the objective function quantifies the goals of the therapy. The system under …
While the use of combination therapy is increasing in prevalence for cancer treatment, it is often difficult to predict the exact interactions between different treatment forms, and their synergistic/antagonistic effects on patient health and therapy outcome. In this research, a system of ordinary differential equations is constructed to model nonlinear dynamics between tumor cells, immune cells, and three forms of therapy: chemotherapy, immunotherapy, and radiotherapy. This model is then used to generate optimized combination therapy plans using optimal control theory. In-silico experiments a…
Many therapeutic advantages such as prolonged circulation in the blood, increased tumor drug uptake, enhanced therapeutic efficacy, and markedly reduced toxic side effects are provided by a carbon nanotube based chemotherapeutic formulation. In this system, doxorubicin (DOX) is loaded onto the sidewalls of functionalized single-walled carbon nanotubes by supramolecular pi-pi stacking.
The internal and external mechanical environment plays an important role in tumorogenesis. As a proxy of an avascular early state tumor, we use multicellular spheroids, a composite material made of cells, extracellular matrix and permeating fluid. We characterize its effective rheology at the timescale of minutes to hours by compressing the aggregates with osmotic shocks and modeling the experimental results with an active poroelastic material that reproduces the stress and strain distributions in the aggregate. The model also predicts how the emergent bulk modulus of the aggregate as well as …
We investigate a new dynamical system that describes tumor-host interaction. The equation that describes the untreated tumor growth is based on non-extensive statistical mechanics. Recently, this model has been shown to fit successfully exponential, Gompertz, logistic, and power-law tumor growths. We have been able to include as many hallmarks of cancer as possible. We study also the dynamic response of cancer under therapy. Using our model, we can make predictions about the different outcomes when we change the parameters, and/or the initial conditions. We can determine the importance of diff…
Oncolytic virotherapy - the use of viruses that specifically kill tumor cells - is an innovative and highly promising route for treating cancer. However, its therapeutic outcomes are mainly impaired by the host immune response to the viral infection. In the present work, we propose a multiscale mathematical model to study how the immune response interferes with the viral oncolytic activity. The model assumes that cytotoxic T cells can induce apoptosis in infected cancer cells and that free viruses can be inactivated by neutralizing antibodies or cleared at a constant rate by the innate immune …
Radiation Therapy (XRT) is one of the most common cancer treatment methods. In this paper, a new mathematical model is proposed for the population dynamics of heterogeneous tumor cells following external beam radiation treatment. According to the Target Theory, the tumor population is divided into m different subpopulations based on the diverse effects of ionizing radiation on human cells. A hybrid model con- sists of a system of differential equations with random variable coefficients representing the transition rates between subpopulations is proposed. This model is utilized to sim- ulate th…
Acquired resistance is one of the major barriers to successful cancer therapy. The development of resistance is commonly attributed to genetic heterogeneity. However, heterogeneity of drug penetration of the tumor microenvironment both on the microscopic level within solid tumors as well as on the macroscopic level across metastases may also contribute to acquired drug resistance. Here we use mathematical models to investigate the effect of drug heterogeneity on the probability of escape from treatment and time to resistance. Specifically we address scenarios with sufficiently efficient therap…
Two models of circulating tumor cell (CTC) dynamics have been proposed to explain the phenomenon of tumor 'self-seeding', whereby CTCs repopulate the primary tumor and accelerate growth: Primary Seeding, where cells from a primary tumor shed into the vasculature and return back to the primary themselves; and Secondary Seeding, where cells from the primary first metastasize in a secondary tissue and form microscopic secondary deposits, which then shed cells into the vasculature returning to the primary. These two models are difficult to distinguish experimentally, yet the differences between th…
Recent findings raise the possibility of PARP inhibitor therapy in colorectal cancers(CRCs). However, the extent of PARP-1 protein expression in clinical specimens of CRC is not known. Using immunohistochemistry we assessed PARP-1 protein expression in tissue microarrays of 151 CRCs and its association with the patient's age, sex, Astler-Coller stage, grade and site of the tumor. High PARP nuclear immunoreactivity was found in 68.2% (103/151) of all cases. In turn, 31.8% (48/151)of tumors showed low PARP expression, including 9 (6%) PARP-1 negative CRCs. There was a significant association of …
One of the most promising strategies to treat cancer is attacking it with viruses. Virus can kill tumor cells specifically or act as carriers that deliver normal genes into cancer cells. A model for virus therapy of cancer is investigated and some of its predictions are in agreement with results obtained from experimental tumors. Furthermore, the model reveals an oscillatory (periodic or aperiodic) response of tumor cells and virus populations which may difficult clinical prognosis. These results suggest the need for new \textit{in vivo} and \textit{in vitro} experiments aiming to detect this …
Patient-specific, mechanistic models of tumor growth incorporating artificial intelligence and big data
University of Pavia（帕维亚大学）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Oden Institute for Computational Engineering and Sciences（奥登计算工程与科学研究所）；Massachusetts General Hospital（麻省总医院）；Harvard Medical School（哈佛医学院）；Harvard University（哈佛大学）；Massachusetts Institute of Technology（麻省理工学院）；Dartmouth College（达特茅斯学院）；Geisel School of Medicine at Dartmouth（达特茅斯盖泽尔医学院）；Livestrong Cancer Institutes（利文斯顿癌症研究所）；Purdue University（普渡大学）；Weldon School of Biomedical Engineering（韦尔登生物医学工程学院）；University of Colorado Anschutz（科罗拉多大学安舒茨分校）
