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Singh A, Kumar N. Estimation of the injection criteria for magnetic hyperthermia therapy based on tumor morphology. Biomed Phys Eng Express 2024; 10:055017. [PMID: 39025085 DOI: 10.1088/2057-1976/ad64d8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/11/2024] [Accepted: 07/18/2024] [Indexed: 07/20/2024]
Abstract
Intratumoral multi-injection strategy enhances the efficacy of magnetic nanoparticle hyperthermia therapy (MNPH). In this study, criteria for the selection of injections and their location depending on the tumor shape/geometry are developed. The developed strategy is based on the thermal dosimetry results of different invasive 3D tumor models during MNPH simulation. MNPH simulations are conducted on physical tumor tissue models encased within healthy tissue. The tumor shapes are geometrically divided into a central tumor region containing maximum tumor volume and a peripheral tumor portion protruding in any random direction. The concepts of core and invasive radius are used to geometrically divide the tumor volume. Primary & secondary injections are used to inject MNP fluid into these respective tumor regions based on the invasiveness of the tumor. The optimization strategy is devised based on the zone of influence of primary & secondary injection. Results indicate that the zone of influence of secondary injection lies between 0.7 and 0.8 times the radial distance between the center of the tumor core and branch node point (extreme far endpoint on the invasive tumor surface). Additionally, the multi-injection strategy is more effective when the protrusion volume exceeds10%of the total volume. The proposed algorithm is used to devise multi-injection strategies for arbitrarily shaped tumors and will assist in pre-planning magnetic nanoparticle hyperthermia therapy.
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Affiliation(s)
- Amritpal Singh
- Department of Mechanical Engineering, Thapar Institute of Engineering and Technology, Patiala, India
| | - Neeraj Kumar
- Department of Mechanical Engineering, Thapar Institute of Engineering and Technology, Patiala, India
- Virginia Tech-TIET Center of Excellence in Emerging Materials, T I E T, Patiala, India
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2
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Blom E, Engblom S. Morphological Stability for in silico Models of Avascular Tumors. Bull Math Biol 2024; 86:75. [PMID: 38758501 DOI: 10.1007/s11538-024-01297-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2023] [Accepted: 04/16/2024] [Indexed: 05/18/2024]
Abstract
The landscape of computational modeling in cancer systems biology is diverse, offering a spectrum of models and frameworks, each with its own trade-offs and advantages. Ideally, models are meant to be useful in refining hypotheses, to sharpen experimental procedures and, in the longer run, even for applications in personalized medicine. One of the greatest challenges is to balance model realism and detail with experimental data to eventually produce useful data-driven models. We contribute to this quest by developing a transparent, highly parsimonious, first principle in silico model of a growing avascular tumor. We initially formulate the physiological considerations and the specific model within a stochastic cell-based framework. We next formulate a corresponding mean-field model using partial differential equations which is amenable to mathematical analysis. Despite a few notable differences between the two models, we are in this way able to successfully detail the impact of all parameters in the stability of the growth process and on the eventual tumor fate of the stochastic model. This facilitates the deduction of Bayesian priors for a given situation, but also provides important insights into the underlying mechanism of tumor growth and progression. Although the resulting model framework is relatively simple and transparent, it can still reproduce the full range of known emergent behavior. We identify a novel model instability arising from nutrient starvation and we also discuss additional insight concerning possible model additions and the effects of those. Thanks to the framework's flexibility, such additions can be readily included whenever the relevant data become available.
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Affiliation(s)
- Erik Blom
- Division of Scientific Computing, Department of Information Technology, Uppsala University, 751 05, Uppsala, Sweden
| | - Stefan Engblom
- Division of Scientific Computing, Department of Information Technology, Uppsala University, 751 05, Uppsala, Sweden.
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3
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Webb G, Zhao XE. Bifurcation analysis of critical values for wound closure outcomes in wound healing experiments. J Math Biol 2023; 86:66. [PMID: 37004561 DOI: 10.1007/s00285-023-01896-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/12/2022] [Revised: 03/02/2023] [Accepted: 03/07/2023] [Indexed: 04/04/2023]
Abstract
A nonlinear partial differential equation containing a nonlocal advection term and a diffusion term is analyzed to study wound closure outcomes in wound healing experiments. There is an extensive literature of similar models for wound healing experiments. In this paper we study the character of wound closure in these experiments in terms of the sensing radius of cells and the force of cell-cell adhesion. We prove a bifurcation result which differentiates uniform closure of the wound from nonuniform closure of the wound, based on a critical value [Formula: see text] of the force of cell-cell adhesion parameter [Formula: see text]. For [Formula: see text] the steady state solution [Formula: see text] of the model is stable and the wound closes uniformly. For [Formula: see text] the steady state solution [Formula: see text] of the model is unstable and the wound closes nonuniformly. We provide numerical simulations of the model to illustrate our results.
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Affiliation(s)
- Glenn Webb
- Mathematics Department, Vanderbilt University, Nashville, TN, 37240, USA.
| | - Xinyue Evelyn Zhao
- Mathematics Department, Vanderbilt University, Nashville, TN, 37240, USA
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4
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In-silico study of asymmetric remodeling of tumors in response to external biochemical stimuli. Sci Rep 2023; 13:941. [PMID: 36653410 PMCID: PMC9849277 DOI: 10.1038/s41598-022-26891-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2022] [Accepted: 12/21/2022] [Indexed: 01/19/2023] Open
Abstract
Among different hallmarks of cancer, understanding biomechanics of tumor growth and remodeling benefits the most from the theoretical framework of continuum mechanics. Tumor remodeling initiates when cancer cells seek new homeostasis in response to the microenvironmental stimuli. Cells within a growing tumor are capable to remodel their inter- and intra-connections and become more mobile to achieve a new homeostasis. This mobility enables the tumor to undergo large deformation. In this work, we studied the remodeling of homogeneous tumors, at their early stage of growth, in the context of continuum mechanics. We developed an evolution law for the remodeling-associated deformation which correlates the remodeling to a characteristic tensor of external stimuli. The asymmetric remodeling and the induced mechanical stresses were analyzed for different types of biochemical distributions. To experimentally investigate the model, we studied the remodeling of human glioblastoma (hGB) tumoroids in response to the gradient of nutrients. Using a tumoroid-on-a-chip platform, the degree of remodeling was estimated for the ellipsoidal tumoroids over time. It was observed that higher gradient of nutrients induces higher degree of ellipticity suggesting that the gradient of nutrient is a characteristic property of nutrient distribution that derives the remodeling. We also showed that remodeling gives rise to heterogeneity in cell distribution forming circumferentially aligned cells within the tumors. Compared to the existing studies on tumor growth, our work provides a biomechanical module that relates the remodeling to biochemical stimuli, and allows for large deformation. It also includes experimental component, a necessary but challenging step, that connects the theory and reality to evaluate the practicability of the model.
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5
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Zheng X, Zhao K, Jackson T, Lowengrub J. Tumor growth towards lower extracellular matrix conductivity regions under Darcy's Law and steady morphology. J Math Biol 2022; 85:5. [PMID: 35796898 PMCID: PMC9968407 DOI: 10.1007/s00285-022-01759-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/07/2021] [Revised: 02/27/2022] [Accepted: 04/16/2022] [Indexed: 10/17/2022]
Abstract
We study a classic Darcy's law model for tumor cell motion with inhomogeneous and isotropic conductivity. The tumor cells are assumed to be a constant density fluid flowing through porous extracellular matrix (ECM). The ECM is assumed to be rigid and motionless with constant porosity. One and two dimensional simulations show that the tumor mass grows from high to low conductivity regions when the tumor morphology is steady. In the one-dimensional case, we proved that when the tumor size is steady, the tumor grows towards lower conductivity regions. We conclude that this phenomenon is produced by the coupling of a special inward flow pattern in the steady tumor and Darcy's law which gives faster flow speed in higher conductivity regions.
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Affiliation(s)
- Xiaoming Zheng
- Department of Mathematics, Central Michigan University, Mount Pleasant, MI, 48858, USA.
| | - Kun Zhao
- Department of Mathematics, Tulane University, New Orleans, LA 70118, USA
| | - Trachette Jackson
- Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
| | - John Lowengrub
- Department of Mathematics, University of California, Irvine, CA 92697, USA
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6
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Webb G. The force of cell-cell adhesion in determining the outcome in a nonlocal advection diffusion model of wound healing. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2022; 19:8689-8704. [PMID: 35942731 DOI: 10.3934/mbe.2022403] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/15/2023]
Abstract
A model of wound healing is presented to investigate the connection of the force of cell-cell adhesion to the sensing radius of cells in their spatial environment. The model consists of a partial differential equation with nonlocal advection and diffusion terms, describing the movement of cells in a spatial environment. The model is applied to biological wound healing experiments to understand incomplete wound closure. The analysis demonstrates that for each value of the force of adhesion parameter, there is a critical value of the sensing radius above which complete wound healing does not occur.
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Affiliation(s)
- Glenn Webb
- Mathematics Department, Vanderbilt University, Nashville, TN, USA
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7
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Abstract
In this work, we studied the stability of radially symmetric growth in tumor spheroids using a reaction-diffusion model. In this model, nutrient concentration and internal pressure are local variables that implicitly relate the proliferation of cells to the growth of the tumor. The analytical solution of the governing model was presented in an orthonormal spherical harmonic basis. It was shown that the radially symmetric steady-state solution to the growth of tumor spheroids, under symmetric growth conditions, was unstable with respect to small asymmetric perturbations. Such perturbations excited the asymmetric modes of growth, which could grow in time and change the spherical configuration of the tumor. The number of such modes and their rates of growth depended on parameters such as surface tension, external energy and the rate of nutrient consumption. This analysis indicated that the spherical configuration of tumor spheroids, even under experimentally controlled symmetric growth conditions, were naturally unstable. This was confirmed by a comparison between the shapes of in vitro human glioblastoma (hGB) spheroids and the configuration of the first few asymmetric modes predicted by the model.
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8
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Mohammad Mirzaei N, Tatarova Z, Hao W, Changizi N, Asadpoure A, Zervantonakis IK, Hu Y, Chang YH, Shahriyari L. A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice. J Pers Med 2022; 12:807. [PMID: 35629230 PMCID: PMC9145520 DOI: 10.3390/jpm12050807] [Citation(s) in RCA: 8] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/05/2022] [Revised: 05/12/2022] [Accepted: 05/12/2022] [Indexed: 02/04/2023] Open
Abstract
The evolution of breast tumors greatly depends on the interaction network among different cell types, including immune cells and cancer cells in the tumor. This study takes advantage of newly collected rich spatio-temporal mouse data to develop a data-driven mathematical model of breast tumors that considers cells' location and key interactions in the tumor. The results show that cancer cells have a minor presence in the area with the most overall immune cells, and the number of activated immune cells in the tumor is depleted over time when there is no influx of immune cells. Interestingly, in the case of the influx of immune cells, the highest concentrations of both T cells and cancer cells are in the boundary of the tumor, as we use the Robin boundary condition to model the influx of immune cells. In other words, the influx of immune cells causes a dominant outward advection for cancer cells. We also investigate the effect of cells' diffusion and immune cells' influx rates in the dynamics of cells in the tumor micro-environment. Sensitivity analyses indicate that cancer cells and adipocytes' diffusion rates are the most sensitive parameters, followed by influx and diffusion rates of cytotoxic T cells, implying that targeting them is a possible treatment strategy for breast cancer.
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Affiliation(s)
- Navid Mohammad Mirzaei
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA; (N.M.M.); (Y.H.)
| | - Zuzana Tatarova
- Department of Radiology, Brigham & Women’s Hospital, Harvard Medical School, Boston, MA 02115, USA;
| | - Wenrui Hao
- Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA;
| | - Navid Changizi
- Department of Civil and Environmental Engineering, University of Massachusetts, Dartmouth, MA 02747, USA; (N.C.); (A.A.)
| | - Alireza Asadpoure
- Department of Civil and Environmental Engineering, University of Massachusetts, Dartmouth, MA 02747, USA; (N.C.); (A.A.)
| | - Ioannis K. Zervantonakis
- Department of Bioengineering, UPMC Hillman Cancer Center, University of Pittsburgh, Pittsburgh, PA 15219, USA;
| | - Yu Hu
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA; (N.M.M.); (Y.H.)
| | - Young Hwan Chang
- Department of Biomedical Engineering, Oregon Health & Science University, Portland, OR 97239, USA;
| | - Leili Shahriyari
- Department of Mathematics and Statistics, University of Massachusetts Amherst, Amherst, MA 01003, USA; (N.M.M.); (Y.H.)
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9
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Xie X. WELL-POSEDNESS OF A MATHEMATICAL MODEL OF DIABETIC ATHEROSCLEROSIS WITH ADVANCED GLYCATION END-PRODUCTS. APPLICABLE ANALYSIS 2022; 101:3989-4013. [PMID: 36188356 PMCID: PMC9524361 DOI: 10.1080/00036811.2022.2060210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/29/2021] [Accepted: 03/25/2022] [Indexed: 06/16/2023]
Abstract
Atherosclerosis is a leading cause of death worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 463 million people have diabetes, which increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. The pathophysiology of diabetic vascular disease is generally understood. Dyslipidemia with increased levels of atherogenic LDL, hyperglycemia, oxidative stress and increased inflammation are factors that increase the risk and accelerate development of atherosclerosis. In a recent paper [53], we have developed mathematical model that includes the effect of hyperglycemia and insulin resistance on plaque growth. In this paper, we propose a more comprehensive mathematical model for diabetic atherosclerosis which include more variables; in particular it includes the variable for Advanced Glycation End-Products (AGEs)concentration. Hyperglycemia trigger vascular damage by forming AGEs, which are not easily metabolized and may accelerate the progression of vascular disease in diabetic patients. The model is given by a system of partial differential equations with a free boundary. We also establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.
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Affiliation(s)
- Xuming Xie
- Department of Mathematics, Morgan State University, Baltimore, MD 21251
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10
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Xie X. WELL-POSEDNESS OF A MATHEMATICAL MODEL OF DIABETIC ATHEROSCLEROSIS. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2022; 505:125606. [PMID: 34483362 PMCID: PMC8415469 DOI: 10.1016/j.jmaa.2021.125606] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Atherosclerosis is a leading cause of death in the United States and worldwide; it emerges as a result of multiple dynamical cell processes including hemodynamics, endothelial damage, innate immunity and sterol biochemistry. Making matters worse, nearly 21 million Americans have diabetes, a disease where patients' cells cannot efficiently take in dietary sugar, causing it to build up in the blood. In part because diabetes increases atherosclerosis-related inflammation, diabetic patients are twice as likely to have a heart attack or stroke. Past work has shown that hyperglycemia and insulin resistance alter function of multiple cell types, including endothelium, smooth muscle cells and platelets, indicating the extent of vascular disarray in this disease. Although the pathophysiology of diabetic vascular disease is generally understood, there is no mathematical model to date that includes the effect of diabetes on plaque growth. In this paper, we propose a mathematical model for diabetic atherosclerosis; the model is given by a system of partial differential equations with a free boundary. We establish local existence and uniqueness of solution to the model. The methodology is to use Hanzawa transformation to reduce the free boundary to a fixed boundary and reduce the system of partial differential equations to an abstract evolution equation in Banach spaces, and apply the theory of analytic semigroup.
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Affiliation(s)
- Xuming Xie
- Department of Mathematics, Morgan State University, Baltimore, MD 21251
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11
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Improving cancer treatments via dynamical biophysical models. Phys Life Rev 2021; 39:1-48. [PMID: 34688561 DOI: 10.1016/j.plrev.2021.10.001] [Citation(s) in RCA: 14] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/16/2021] [Accepted: 10/13/2021] [Indexed: 12/17/2022]
Abstract
Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.
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12
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In-Silico Modeling of Tumor Spheroid Formation and Growth. MICROMACHINES 2021; 12:mi12070749. [PMID: 34202262 PMCID: PMC8303756 DOI: 10.3390/mi12070749] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/03/2021] [Revised: 06/21/2021] [Accepted: 06/22/2021] [Indexed: 12/20/2022]
Abstract
Mathematical modeling has significant potential for understanding of biological models of cancer and to accelerate the progress in cross-disciplinary approaches of cancer treatment. In mathematical biology, solid tumor spheroids are often studied as preliminary in vitro models of avascular tumors. The size of spheroids and their cell number are easy to track, making them a simple in vitro model to investigate tumor behavior, quantitatively. The growth of solid tumors is comprised of three main stages: transient formation, monotonic growth and a plateau phase. The last two stages are extensively studied. However, the initial transient formation phase is typically missing from the literature. This stage is important in the early dynamics of growth, formation of clonal sub-populations, etc. In the current work, this transient formation is modeled by a reaction–diffusion partial differential equation (PDE) for cell concentration, coupled with an ordinary differential equation (ODE) for the spheroid radius. Analytical and numerical solutions of the coupled equations were obtained for the change in the radius of tumor spheroids over time. Human glioblastoma (hGB) cancer cells (U251 and U87) were spheroid cultured to validate the model prediction. Results of this study provide insight into the mechanism of development of solid tumors at their early stage of formation.
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13
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Macnamara CK. Biomechanical modelling of cancer: Agent‐based force‐based models of solid tumours within the context of the tumour microenvironment. COMPUTATIONAL AND SYSTEMS ONCOLOGY 2021. [DOI: 10.1002/cso2.1018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/11/2022] Open
Affiliation(s)
- Cicely K. Macnamara
- School of Mathematics and Statistics Mathematical Institute University of St Andrews St Andrews Fife UK
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14
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Voss-Böhme A, Gerisch A. Multi-Scale Analysis of Contact-Dependent Interaction in Tissue Aggregation and Invasion. SYSTEMS MEDICINE 2021. [DOI: 10.1016/b978-0-12-801238-3.11449-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/01/2022] Open
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15
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Indirect Contributions to Tumor Dynamics in the First Stage of the Avascular Phase. Symmetry (Basel) 2020. [DOI: 10.3390/sym12091546] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022] Open
Abstract
A continuum model for tumor invasion in a two-dimensional spatial domain based on the interaction of the urokinase plasminogen activation system with a model for cancer cell dynamics is proposed. The arising system of partial differential equations is numerically solved using the finite element method. We simulated a portion of biological tissue imposing no flux boundary conditions. We monitored the cancer cell dynamics, as well the degradation of an extra cellular matrix representative, vitronectin, and the evolution of a specific degrading enzyme, plasmin, inside the biological tissue. The computations were parameterized as a function of the indirect cell proliferation induced by a plasminogen activator inhibitor binding to vitronectin and of the indirect plasmin deactivation due to the plasminogen activator inhibitor binding to the urokinase plasminogen activator. Their role during the cancer dynamical evolution was identified, together with a possible marker helping the mapping of the cancer invasive front. Our results indicate that indirect cancer cell proliferation biases the speed of the tumor invasive front as well as the heterogeneity of the cancer cell clustering and networking, as it ultimately acts on the proteolytic activity supporting cancer formation. Because of the initial conditions imposed, the numerical solutions of the model show a symmetrical dynamical evolution of heterogeneities inside the simulated domain. Moreover, an increase of up to about 12% in the invasion speed was observed, increasing the rate of indirect cancer cell proliferation, while increasing the plasmin deactivation rate inhibits heterogeneities and networking. As cancer cell proliferation causes vitronectin consumption and plasmin formation, the intensities of the concentration maps of both vitronectin and plasmin are superimposable to the cancer cell concentration maps. The qualitative imprinting that cancer cells leave on the extra cellular matrix during the time evolution as well their activity area is identified, framing the numerical results in the context of a methodology aimed at diagnostic and therapeutic improvement.
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16
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Shuttleworth R, Trucu D. Cell-Scale Degradation of Peritumoural Extracellular Matrix Fibre Network and Its Role Within Tissue-Scale Cancer Invasion. Bull Math Biol 2020; 82:65. [PMID: 32458057 PMCID: PMC7250813 DOI: 10.1007/s11538-020-00732-z] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/27/2019] [Accepted: 04/08/2020] [Indexed: 12/14/2022]
Abstract
Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. During the complex interaction between cancer cell population and the extracellular matrix (ECM), of key importance is the role played by both bulk two-scale dynamics of ECM fibres within collective movement of the tumour cells and the multiscale leading edge dynamics driven by proteolytic activity of the matrix-degrading enzymes (MDEs) that are secreted by the cancer cells. As these two multiscale subsystems share and contribute to the same tumour macro-dynamics, in this work we develop further the model introduced in Shuttleworth and Trucu (Bull Math Biol 81:2176–2219, 2019. 10.1007/s11538-019-00598-w) by exploring a new aspect of their interaction that occurs at the cell scale. Specifically, here we will focus on understanding the cell-scale cross talk between the micro-scale parts of these two multiscale subsystems which get to interact directly in the peritumoural region, with immediate consequences both for MDE micro-dynamics occurring at the leading edge of the tumour and for the cell-scale rearrangement of the naturally oriented ECM fibres in the peritumoural region, ultimately influencing the way tumour progresses in the surrounding tissue. To that end, we will propose a new modelling that captures the ECM fibres degradation not only at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour.
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Affiliation(s)
- Robyn Shuttleworth
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN Scotland, UK
| | - Dumitru Trucu
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN Scotland, UK
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17
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Moreira-Soares M, Cunha SP, Bordin JR, Travasso RDM. Adhesion modulates cell morphology and migration within dense fibrous networks. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2020; 32:314001. [PMID: 32378515 DOI: 10.1088/1361-648x/ab7c17] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/11/2019] [Accepted: 03/03/2020] [Indexed: 06/11/2023]
Abstract
One of the most fundamental abilities required for the sustainability of complex life forms is active cell migration, since it is essential in diverse processes from morphogenesis to leukocyte chemotaxis in immune response. The movement of a cell is the result of intricate mechanisms, that involve the coordination between mechanical forces, biochemical regulatory pathways and environmental cues. In particular, epithelial cancer cells have to employ mechanical strategies in order to migrate through the tissue's basement membrane and infiltrate the bloodstream during the invasion stage of metastasis. In this work we explore how mechanical interactions such as spatial restriction and adhesion affect migration of a self-propelled droplet in dense fibrous media. We have performed a systematic analysis using a phase-field model and we propose a novel approach to simulate cell migration with dissipative particle dynamics modelling. With this purpose we have measured in our simulation the cell's velocity and quantified its morphology as a function of the fibre density and of its adhesiveness to the matrix fibres. Furthermore, we have compared our results to a previousin vitromigration assay of fibrosarcoma cells in fibrous matrices. The results show good agreement between the two methodologies and experiments in the literature, which indicates that these minimalist descriptions are able to capture the main features of the system. Our results indicate that adhesiveness is critical for cell migration, by modulating cell morphology in crowded environments and by enhancing cell velocity. In addition, our analysis suggests that matrix metalloproteinases (MMPs) play an important role as adhesiveness modulators. We propose that new assays should be carried out to address the role of adhesion and the effect of different MMPs in cell migration under confined conditions.
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Affiliation(s)
| | - Susana P Cunha
- CQC, Department of Chemistry, University of Coimbra, Rua Larga, 3004-535 Coimbra, Portugal
| | - José Rafael Bordin
- Department of Physics, Institute of Physics and Mathematics, Federal University of Pelotas, Rua dos Ipês, Capão do Leão, RS, 96050-500, Brazil
| | - Rui D M Travasso
- CFisUC, Department of Physics, University of Coimbra, Rua Larga, 3004-516 Coimbra, Portugal
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18
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Strobl MAR, Krause AL, Damaghi M, Gillies R, Anderson ARA, Maini PK. Mix and Match: Phenotypic Coexistence as a Key Facilitator of Cancer Invasion. Bull Math Biol 2020; 82:15. [PMID: 31953602 PMCID: PMC6968991 DOI: 10.1007/s11538-019-00675-0] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/28/2019] [Accepted: 12/03/2019] [Indexed: 01/10/2023]
Abstract
Invasion of healthy tissue is a defining feature of malignant tumours. Traditionally, invasion is thought to be driven by cells that have acquired all the necessary traits to overcome the range of biological and physical defences employed by the body. However, in light of the ever-increasing evidence for geno- and phenotypic intra-tumour heterogeneity, an alternative hypothesis presents itself: could invasion be driven by a collection of cells with distinct traits that together facilitate the invasion process? In this paper, we use a mathematical model to assess the feasibility of this hypothesis in the context of acid-mediated invasion. We assume tumour expansion is obstructed by stroma which inhibits growth and extra-cellular matrix (ECM) which blocks cancer cell movement. Further, we assume that there are two types of cancer cells: (i) a glycolytic phenotype which produces acid that kills stromal cells and (ii) a matrix-degrading phenotype that locally remodels the ECM. We extend the Gatenby-Gawlinski reaction-diffusion model to derive a system of five coupled reaction-diffusion equations to describe the resulting invasion process. We characterise the spatially homogeneous steady states and carry out a simulation study in one spatial dimension to determine how the tumour develops as we vary the strength of competition between the two phenotypes. We find that overall tumour growth is most extensive when both cell types can stably coexist, since this allows the cells to locally mix and benefit most from the combination of traits. In contrast, when inter-species competition exceeds intra-species competition the populations spatially separate and invasion arrests either: (i) rapidly (matrix-degraders dominate) or (ii) slowly (acid-producers dominate). Overall, our work demonstrates that the spatial and ecological relationship between a heterogeneous population of tumour cells is a key factor in determining their ability to cooperate. Specifically, we predict that tumours in which different phenotypes coexist stably are more invasive than tumours in which phenotypes are spatially separated.
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Affiliation(s)
- Maximilian A. R. Strobl
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, OX2 6GG Oxford, UK
- Department of Integrated Mathematical Oncology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Andrew L. Krause
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, OX2 6GG Oxford, UK
| | - Mehdi Damaghi
- Department of Cancer Physiology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Robert Gillies
- Department of Cancer Physiology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Alexander R. A. Anderson
- Department of Integrated Mathematical Oncology, Moffitt Cancer Center, Magnolia Drive, Tampa, 12902 USA
| | - Philip K. Maini
- Wolfson Centre for Mathematical Biology, Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, OX2 6GG Oxford, UK
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Pirastehzad A, Taghizadeh A, Jamshidi AA. The formation of cancer stem cells in EMT6/Ro tumor: Hybrid modeling within its micro-environment. INFORMATICS IN MEDICINE UNLOCKED 2020. [DOI: 10.1016/j.imu.2019.100247] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022] Open
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21
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An analytical poroelastic model of a spherical tumor embedded in normal tissue under creep compression. J Biomech 2019; 89:48-56. [DOI: 10.1016/j.jbiomech.2019.04.009] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/16/2018] [Revised: 04/05/2019] [Accepted: 04/07/2019] [Indexed: 11/22/2022]
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22
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Alzahrani T, Eftimie R, Trucu D. Multiscale modelling of cancer response to oncolytic viral therapy. Math Biosci 2019; 310:76-95. [DOI: 10.1016/j.mbs.2018.12.018] [Citation(s) in RCA: 28] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2018] [Revised: 12/29/2018] [Accepted: 12/29/2018] [Indexed: 12/29/2022]
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Pham K, Turian E, Liu K, Li S, Lowengrub J. Nonlinear studies of tumor morphological stability using a two-fluid flow model. J Math Biol 2018; 77:671-709. [PMID: 29546457 DOI: 10.1007/s00285-018-1212-3] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2016] [Revised: 01/31/2018] [Indexed: 01/08/2023]
Abstract
We consider the nonlinear dynamics of an avascular tumor at the tissue scale using a two-fluid flow Stokes model, where the viscosity of the tumor and host microenvironment may be different. The viscosities reflect the combined properties of cell and extracellular matrix mixtures. We perform a linear morphological stability analysis of the tumors, and we investigate the role of nonlinearity using boundary-integral simulations in two dimensions. The tumor is non-necrotic, although cell death may occur through apoptosis. We demonstrate that tumor evolution is regulated by a reduced set of nondimensional parameters that characterize apoptosis, cell-cell/cell-extracellular matrix adhesion, vascularization and the ratio of tumor and host viscosities. A novel reformulation of the equations enables the use of standard boundary integral techniques to solve the equations numerically. Nonlinear simulation results are consistent with linear predictions for nearly circular tumors. As perturbations develop and grow, the linear and nonlinear results deviate and linear theory tends to underpredict the growth of perturbations. Simulations reveal two basic types of tumor shapes, depending on the viscosities of the tumor and microenvironment. When the tumor is more viscous than its environment, the tumors tend to develop invasive fingers and a branched-like structure. As the relative ratio of the tumor and host viscosities decreases, the tumors tend to grow with a more compact shape and develop complex invaginations of healthy regions that may become encapsulated in the tumor interior. Although our model utilizes a simplified description of the tumor and host biomechanics, our results are consistent with experiments in a variety of tumor types that suggest that there is a positive correlation between tumor stiffness and tumor aggressiveness.
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Affiliation(s)
- Kara Pham
- Department of Mathematics, University of California at Irvine, Irvine, CA, 92697-3875, USA
- Department of Mathematics, Fullerton College, Fullerton, CA, 92832, USA
| | - Emma Turian
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA
- Department of Mathematics, Northeastern Illinois University, Chicago, IL, 60625, USA
| | - Kai Liu
- Department of Mathematics, University of California at Irvine, Irvine, CA, 92697-3875, USA
| | - Shuwang Li
- Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, 60616, USA.
| | - John Lowengrub
- Departments of Mathematics and Biomedical Engineering, Center for Complex Biological Systems, Chao Family Comprehensive Cancer Center, University of California at Irvine, Irvine, CA, 92697-3875, USA.
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Islam MT, Reddy JN, Righetti R. An analytical poroelastic model of a non-homogeneous medium under creep compression for ultrasound poroelastography applications - Part II. J Biomech Eng 2018; 141:2686531. [PMID: 30029209 DOI: 10.1115/1.4040604] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2017] [Indexed: 11/08/2022]
Abstract
An analytical theory for the unconfined creep behavior of a cylindrical inclusion (simulating a soft tissue tumor) embedded in a cylindrical background sample (simulating normal tissue) is presented and analyzed in this paper. Both the inclusion and the background are considered as fluid-filled, porous materials, each of them being characterized by a set of mechanical parameters. Specifically, in this derivation, the inclusion is assumed to have significantly higher interstitial permeability than the background. The formulations of the effective Poisson's ratio (EPR) and fluid pressure in the inclusion and in the background are derived for the case of a sample subjected to a creep compression. The developed analytical expressions are validated using finite element models (FEM). Statistical comparison between the results obtained from the developed model and the results from FEM demonstrates accuracy of the proposed theoretical model higher than 99.4%. The model presented in this paper complements the one reported in the companion paper (Part I), which refers to the case of an inclusion having less interstitial permeability than the background.
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Affiliation(s)
- Md Tauhidul Islam
- Graduate Research Assistant, Ultrasound and Elasticity Imaging Laboratory, Department of Electrical & Computer Engineering, Texas A&M University, College Station, Texas, USA-77840
| | - J N Reddy
- Professor, Department of Mechanical Engineering, Texas A&M University, College Station, Texas, USA-77840
| | - Raffaella Righetti
- Associate Professor, Department of Electrical & Computer Engineering, Texas A&M University, College Station, Texas, USA-77840
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Islam MT, Chaudhry A, Unnikrishnan G, Reddy JN, Righetti R. An analytical poroelastic model for ultrasound elastography imaging of tumors. ACTA ACUST UNITED AC 2018; 63:025031. [DOI: 10.1088/1361-6560/aa9631] [Citation(s) in RCA: 13] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/16/2022]
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Bitsouni V, Trucu D, Chaplain MAJ, Eftimie R. Aggregation and travelling wave dynamics in a two-population model of cancer cell growth and invasion. MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA 2018; 35:541-577. [DOI: 10.1093/imammb/dqx019] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2016] [Accepted: 11/14/2017] [Indexed: 12/25/2022]
Affiliation(s)
- Vasiliki Bitsouni
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK
| | - Dumitru Trucu
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK
| | - Mark A J Chaplain
- School of Mathematics and Statistics, Mathematical Institute (MI), North Haugh
- University of St Andrews, St Andrews, KY16 9SS, Scotland, UK
| | - Raluca Eftimie
- Division of Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK
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Nargis NN, Aldredge RC, Guy RD. The influence of soluble fragments of extracellular matrix (ECM) on tumor growth and morphology. Math Biosci 2017; 296:1-16. [PMID: 29208360 DOI: 10.1016/j.mbs.2017.11.014] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/08/2016] [Revised: 11/22/2017] [Accepted: 11/30/2017] [Indexed: 01/27/2023]
Abstract
A major challenge in matrix-metalloproteinase (MMP) target validation and MMP-inhibitor-drug development for anti-cancer clinical trials is to better understand their complex roles (often competing with each other) in tumor progression. While there is extensive research on the growth-promoting effects of MMPs, the growth-inhibiting effects of MMPs has not been investigated thoroughly. So we develop a continuum model of tumor growth and invasion including chemotaxis and haptotaxis in order to examine the complex interaction between the tumor and its host microenvironment and to explore the inhibiting influence of the gradients of soluble fragments of extracellular matrix (ECM) density on tumor growth and morphology. Previously, it was shown both computationally (in one spatial dimension) and experimentally that the chemotactic pull due to soluble ECM gradients is anti-invasive, contrary to the traditional view of the role of chemotaxis in malignant invasion [1]. With two-dimensional numerical simulation and using a level set based tumor-host interface capturing method, we examine the effects of chemotaxis on the progression and morphology of a tumor growing in nutrient-rich and nutrient-poor microenvironments which was not investigated before. In particular we examine how the geometry of the growing tumor is affected when placed in different environments. We also investigate the effects of varying ECM degradation rate, the production rate of matrix degrading enzymes (MDE), and the conversion of ECM into soluble ECM. We find that chemotaxis due to ECM-fragment gradients strongly influences tumor growth and morphology, and that the instabilities caused by tumor cell proliferation and haptotactic movements can be prevented if chemotaxis is sufficiently strong. The influence of chemotaxis and the above factors on tumor growth and morphology are found to be more prominent in nutrient-poor environments than in nutrient-rich environments. So we extend our investigations of these antinvasive chemotactic influences by examining the effects of cell-cell and cell-ECM adhesion and low proliferation rate for tumors growing in low-nutrient environments. We find that as the extent of chemotaxis increases, the effects of adhesion on tumor growth and shape become negligible. Under conditions of low cell mitosis, chemotaxis may cause the tumor to shrink, as the extent of chemotaxis increases. Both stable and unstable tumor shrinkage are predicted by our model. Unexpectedly, in some cases chemotaxis may contribute toward developing instability where haptotaxis alone induces stable growth.
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Affiliation(s)
- Nurun N Nargis
- Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA
| | - Ralph C Aldredge
- Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA.
| | - Robert D Guy
- Department of Mathematics, University of California, Davis, CA 95616, USA
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González MM, Joa JAG, Cabrales LEB, Pupo AEB, Schneider B, Kondakci S, Ciria HMC, Reyes JB, Jarque MV, Mateus MAO, González TR, Brooks SCA, Cáceres JLH, González GVS. Is cancer a pure growth curve or does it follow a kinetics of dynamical structural transformation? BMC Cancer 2017; 17:174. [PMID: 28270135 PMCID: PMC5339962 DOI: 10.1186/s12885-017-3159-y] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/12/2016] [Accepted: 02/24/2017] [Indexed: 11/10/2022] Open
Abstract
BACKGROUND Unperturbed tumor growth kinetics is one of the more studied cancer topics; however, it is poorly understood. Mathematical modeling is a useful tool to elucidate new mechanisms involved in tumor growth kinetics, which can be relevant to understand cancer genesis and select the most suitable treatment. METHODS The classical Kolmogorov-Johnson-Mehl-Avrami as well as the modified Kolmogorov-Johnson-Mehl-Avrami models to describe unperturbed fibrosarcoma Sa-37 tumor growth are used and compared with the Gompertz modified and Logistic models. Viable tumor cells (1×105) are inoculated to 28 BALB/c male mice. RESULTS Modified Gompertz, Logistic, Kolmogorov-Johnson-Mehl-Avrami classical and modified Kolmogorov-Johnson-Mehl-Avrami models fit well to the experimental data and agree with one another. A jump in the time behaviors of the instantaneous slopes of classical and modified Kolmogorov-Johnson-Mehl-Avrami models and high values of these instantaneous slopes at very early stages of tumor growth kinetics are observed. CONCLUSIONS The modified Kolmogorov-Johnson-Mehl-Avrami equation can be used to describe unperturbed fibrosarcoma Sa-37 tumor growth. It reveals that diffusion-controlled nucleation/growth and impingement mechanisms are involved in tumor growth kinetics. On the other hand, tumor development kinetics reveals dynamical structural transformations rather than a pure growth curve. Tumor fractal property prevails during entire TGK.
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Affiliation(s)
- Maraelys Morales González
- Pharmacy Department, Oriente University, Natural Science Faculty, Patricio Lumumba Street, Santiago de Cuba, 90500, Cuba
| | - Javier Antonio González Joa
- National Center of Seismology Research, Street 7 # 2 between L and M, Terraza, Santiago de Cuba, 90400, Cuba
| | - Luis Enrique Bergues Cabrales
- Research and Innovation Department, Oriente University, National Center of Applied Electromagnetism, Ave. Las Américas, Santiago de Cuba, 90400, Cuba.
| | - Ana Elisa Bergues Pupo
- Physics Department, Oriente University, Natural Sciences Faculty, Patricio Lumumba Street, Santiago de Cuba, 90500, Cuba
| | - Baruch Schneider
- Faculty of Sciences and Literature, Department of Mathematics, Izmir University of Economics, Izmir, 353300, Turkey
| | - Suleyman Kondakci
- Faculty of Engineering and Computer Sciences, Department of Computer Engineering, Izmir University of Economics, Izmir, 353300, Turkey
| | - Héctor Manuel Camué Ciria
- Research and Innovation Department, Oriente University, National Center of Applied Electromagnetism, Ave. Las Américas, Santiago de Cuba, 90400, Cuba
| | - Juan Bory Reyes
- ESIME-Zacatenco, Instituto Politécnico Nacional, México, DF, 07738, Mexico
| | | | | | - Tamara Rubio González
- Dirección Municipal de Salud Pública, Servicio de Genética, Santiago de Cuba, 90500, Cuba
| | | | - José Luis Hernández Cáceres
- Facultad de Ciencias Médicas "Diez de Octubre", Universidad de Ciencias Médicas de La Habana, Avenida Independencia No. 8126, Esquina a Calle 100, Boyeros, La Habana, Cuba
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A Multiscale Mathematical Model of Tumour Invasive Growth. Bull Math Biol 2017; 79:389-429. [DOI: 10.1007/s11538-016-0237-2] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/27/2015] [Accepted: 12/02/2016] [Indexed: 10/20/2022]
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30
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Simmons A, Burrage PM, Nicolau DV, Lakhani SR, Burrage K. Environmental factors in breast cancer invasion: a mathematical modelling review. Pathology 2017; 49:172-180. [PMID: 28081961 DOI: 10.1016/j.pathol.2016.11.004] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/11/2016] [Revised: 11/07/2016] [Accepted: 11/13/2016] [Indexed: 12/17/2022]
Abstract
This review presents a brief overview of breast cancer, focussing on its heterogeneity and the role of mathematical modelling and simulation in teasing apart the underlying biophysical processes. Following a brief overview of the main known pathophysiological features of ductal carcinoma, attention is paid to differential equation-based models (both deterministic and stochastic), agent-based modelling, multi-scale modelling, lattice-based models and image-driven modelling. A number of vignettes are presented where these modelling approaches have elucidated novel aspects of breast cancer dynamics, and we conclude by offering some perspectives on the role mathematical modelling can play in understanding breast cancer development, invasion and treatment therapies.
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Affiliation(s)
- Alex Simmons
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia
| | - Pamela M Burrage
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia
| | - Dan V Nicolau
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia; Mathematical Institute, University of Oxford, Oxford, United Kingdom; Molecular Sense Ltd, Oxford, United Kingdom
| | - Sunil R Lakhani
- The University of Queensland, Centre for Clinical Research and School of Medicine and Pathology Queensland, The Royal Brisbane and Women's Hospital, Brisbane, Qld, Australia
| | - Kevin Burrage
- School of Mathematical Sciences, and ARC Centre of Excellence for Mathematical and Statistical Frontiers, Queensland University of Technology, Gardens Point, Brisbane, Qld, Australia; Department of Computer Science, University of Oxford, United Kingdom.
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31
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Lorenzi T, Lorz A, Perthame B. On interfaces between cell populations with different mobilities. ACTA ACUST UNITED AC 2017. [DOI: 10.3934/krm.2017012] [Citation(s) in RCA: 20] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Giverso C, Ciarletta P. On the morphological stability of multicellular tumour spheroids growing in porous media. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2016; 39:92. [PMID: 27726037 DOI: 10.1140/epje/i2016-16092-7] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/31/2016] [Accepted: 09/14/2016] [Indexed: 06/06/2023]
Abstract
Multicellular tumour spheroids (MCTSs) are extensively used as in vitro system models for investigating the avascular growth phase of solid tumours. In this work, we propose a continuous growth model of heterogeneous MCTSs within a porous material, taking into account a diffusing nutrient from the surrounding material directing both the proliferation rate and the mobility of tumour cells. At the time scale of interest, the MCTS behaves as an incompressible viscous fluid expanding inside a porous medium. The cell motion and proliferation rate are modelled using a non-convective chemotactic mass flux, driving the cell expansion in the direction of the external nutrients' source. At the early stages, the growth dynamics is derived by solving the quasi-stationary problem, obtaining an initial exponential growth followed by an almost linear regime, in accordance with experimental observations. We also perform a linear-stability analysis of the quasi-static solution in order to investigate the morphological stability of the radially symmetric growth pattern. We show that mechano-biological cues, as well as geometric effects related to the size of the MCTS subdomains with respect to the diffusion length of the nutrient, can drive a morphological transition to fingered structures, thus triggering the formation of complex shapes that might promote tumour invasiveness. The results also point out the formation of a retrograde flow in the MCTS close to the regions where protrusions form, that could describe the initial dynamics of metastasis detachment from the in vivo tumour mass. In conclusion, the results of the proposed model demonstrate that the integration of mathematical tools in biological research could be crucial for better understanding the tumour's ability to invade its host environment.
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Affiliation(s)
- Chiara Giverso
- Dipartimento di Matematica, MOX, Politecnico di Milano, Piazza Leonardo da Vinci, 32 - 20133, Milano, Italy
| | - Pasquale Ciarletta
- Dipartimento di Matematica, MOX, Politecnico di Milano, Piazza Leonardo da Vinci, 32 - 20133, Milano, Italy.
- UMR 7190, Institut Jean le Rond d'Alembert, CNRS and Sorbonne Universités, UPMC Univ Paris 06, 4 place Jussieu case 162, 75005, Paris, France.
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Lee HG, Kim Y, Kim J. Mathematical model and its fast numerical method for the tumor growth. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2015; 12:1173-1187. [PMID: 26775855 DOI: 10.3934/mbe.2015.12.1173] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hilliard equation in the original model. To numerically solve the new model, we apply a recently developed hybrid numerical method. We perform various numerical experiments. The computational results demonstrate that the new model is not only fast but also has a good feature such as distributing excess mass from the inside of tumor to its boundary regions.
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Affiliation(s)
- Hyun Geun Lee
- Institute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, South Korea.
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Ampatzoglou P, Dassios G, Hadjinicolaou M, Kourea HP, Vrahatis MN. A chemical energy approach of avascular tumor growth: multiscale modeling and qualitative results. SPRINGERPLUS 2015; 4:660. [PMID: 26558163 PMCID: PMC4630322 DOI: 10.1186/s40064-015-1417-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/10/2015] [Accepted: 10/09/2015] [Indexed: 11/10/2022]
Abstract
In the present manuscript we propose a lattice free multiscale model for avascular tumor growth that takes into account the biochemical environment, mitosis, necrosis, cellular signaling and cellular mechanics. This model extends analogous approaches by assuming a function that incorporates the biochemical energy level of the tumor cells and a mechanism that simulates the behavior of cancer stem cells. Numerical simulations of the model are used to investigate the morphology of the tumor at the avascular phase. The obtained results show similar characteristics with those observed in clinical data in the case of the Ductal Carcinoma In Situ (DCIS) of the breast.
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Affiliation(s)
- Pantelis Ampatzoglou
- School of Science and Technology, Hellenic Open University, Sachtouri 11, 26222 Patras, Greece
| | - George Dassios
- Department of Chemical Engineering, University of Patras, Patras, Greece
| | - Maria Hadjinicolaou
- School of Science and Technology, Hellenic Open University, Sachtouri 11, 26222 Patras, Greece
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Painter KJ, Bloomfield JM, Sherratt JA, Gerisch A. A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations. Bull Math Biol 2015; 77:1132-65. [DOI: 10.1007/s11538-015-0080-x] [Citation(s) in RCA: 45] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2014] [Accepted: 04/01/2015] [Indexed: 01/31/2023]
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Rieger H, Welter M. Integrative models of vascular remodeling during tumor growth. WILEY INTERDISCIPLINARY REVIEWS. SYSTEMS BIOLOGY AND MEDICINE 2015; 7:113-29. [PMID: 25808551 PMCID: PMC4406149 DOI: 10.1002/wsbm.1295] [Citation(s) in RCA: 44] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/27/2014] [Revised: 02/05/2015] [Accepted: 02/19/2015] [Indexed: 02/02/2023]
Abstract
UNLABELLED Malignant solid tumors recruit the blood vessel network of the host tissue for nutrient supply, continuous growth, and gain of metastatic potential. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature that is in many respects different from the hierarchically organized arterio-venous blood vessel network of the host tissues. Integrative models based on detailed experimental data and physical laws implement in silico the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. In this review, we give an overview over the current status of integrative models describing tumor growth, vascular remodeling, blood and interstitial fluid flow, drug delivery, and concomitant transformations of the microenvironment. WIREs Syst Biol Med 2015, 7:113-129. doi: 10.1002/wsbm.1295 For further resources related to this article, please visit the WIREs website. CONFLICT OF INTEREST The authors have declared no conflicts of interest for this article.
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Affiliation(s)
- Heiko Rieger
- Department of Theoretical Physics, Saarland UniversitySaarbrücken, Germany
| | - Michael Welter
- Department of Theoretical Physics, Saarland UniversitySaarbrücken, Germany
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Dervaux J, Magniez JC, Libchaber A. On growth and form of Bacillus subtilis biofilms. Interface Focus 2014; 4:20130051. [PMID: 25485075 DOI: 10.1098/rsfs.2013.0051] [Citation(s) in RCA: 41] [Impact Index Per Article: 4.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
A general feature of mature biofilms is their highly heterogeneous architecture that partitions the microbial city into sectors with specific micro-environments. To understand how this heterogeneity arises, we have investigated the formation of a microbial community of the model organism Bacillus subtilis. We first show that the growth of macroscopic colonies is inhibited by the accumulation of ammoniacal by-products. By constraining biofilms to grow approximately as two-dimensional layers, we then find that the bacteria which differentiate to produce extracellular polymeric substances form tightly packed bacterial chains. In addition to the process of cellular chaining, the biomass stickiness also strongly hinders the reorganization of cells within the biofilm. Based on these observations, we then write a biomechanical model for the growth of the biofilm where the cell density is constant and the physical mechanism responsible for the spreading of the biomass is the pressure generated by the division of the bacteria. Besides reproducing the velocity field of the biomass across the biofilm, the model predicts that, although bacteria divide everywhere in the biofilm, fluctuations in the growth rates of the bacteria lead to a coarsening of the growing bacterial layer. This process of kinetic roughening ultimately leads to the formation of a rough biofilm surface exhibiting self-similar properties. Experimental measurements of the biofilm texture confirm these predictions.
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Affiliation(s)
- Julien Dervaux
- Center for Studies in Physics and Biology , The Rockefeller University , New York, NY 10065 , USA
| | | | - Albert Libchaber
- Center for Studies in Physics and Biology , The Rockefeller University , New York, NY 10065 , USA
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Domschke P, Trucu D, Gerisch A, A. J. Chaplain M. Mathematical modelling of cancer invasion: Implications of cell adhesion variability for tumour infiltrative growth patterns. J Theor Biol 2014; 361:41-60. [DOI: 10.1016/j.jtbi.2014.07.010] [Citation(s) in RCA: 61] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2014] [Revised: 07/05/2014] [Accepted: 07/07/2014] [Indexed: 10/25/2022]
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Filipovic N, Djukic T, Saveljic I, Milenkovic P, Jovicic G, Djuric M. Modeling of liver metastatic disease with applied drug therapy. COMPUTER METHODS AND PROGRAMS IN BIOMEDICINE 2014; 115:162-170. [PMID: 24831076 DOI: 10.1016/j.cmpb.2014.04.013] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/21/2013] [Revised: 02/18/2014] [Accepted: 04/15/2014] [Indexed: 06/03/2023]
Abstract
Colorectal carcinoma is acknowledged as the second leading cause of total cancer-related death in the European Region. The majority of deaths related to colorectal carcinoma are connected with liver metastatic disease. Approximately, in 25% of all patients, liver metastatic disease is diagnosed at the same time as the primary diagnosis, while up to a quarter of others would develop liver metastases in the course of the illness. In this study, we developed reaction-diffusion model and analyzed the effect of drug therapy on liver metastatic disease for a specific patient. Tumor volumes in specific time points were obtained using CT scan images. The nonlinear function for cell proliferation rate as well as data about clinically applied drug therapy was included in the model. Fitting procedure was used for parameter estimation. Good agreement of numerical and experimental results shows the feasibility and efficacy of the proposed system.
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Affiliation(s)
- Nenad Filipovic
- Faculty of Engineering, University of Kragujevac, 34000, Serbia; BioIRC R&D Bioengineering Center, Kragujevac, 34000, Serbia; Harvard University, Boston, USA.
| | - Tijana Djukic
- Faculty of Engineering, University of Kragujevac, 34000, Serbia; BioIRC R&D Bioengineering Center, Kragujevac, 34000, Serbia.
| | - Igor Saveljic
- BioIRC R&D Bioengineering Center, Kragujevac, 34000, Serbia
| | - Petar Milenkovic
- Institute for Oncology and Radiology of Serbia, 11000 Belgrade, Serbia; Laboratory for Anthropology, Institute of Anatomy, School of Medicine, University of Belgrade, 11000 Belgrade, Serbia.
| | - Gordana Jovicic
- Faculty of Engineering, University of Kragujevac, 34000, Serbia
| | - Marija Djuric
- Laboratory for Anthropology, Institute of Anatomy, School of Medicine, University of Belgrade, 11000 Belgrade, Serbia.
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Hao W, Friedman A. The LDL-HDL profile determines the risk of atherosclerosis: a mathematical model. PLoS One 2014; 9:e90497. [PMID: 24621857 PMCID: PMC3951264 DOI: 10.1371/journal.pone.0090497] [Citation(s) in RCA: 104] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2013] [Accepted: 02/03/2014] [Indexed: 12/23/2022] Open
Abstract
Atherosclerosis, the leading death in the United State, is a disease in which a plaque builds up inside the arteries. As the plaque continues to grow, the shear force of the blood flow through the decreasing cross section of the lumen increases. This force may eventually cause rupture of the plaque, resulting in the formation of thrombus, and possibly heart attack. It has long been recognized that the formation of a plaque relates to the cholesterol concentration in the blood. For example, individuals with LDL above 190 mg/dL and HDL below 40 mg/dL are at high risk, while individuals with LDL below 100 mg/dL and HDL above 50 mg/dL are at no risk. In this paper, we developed a mathematical model of the formation of a plaque, which includes the following key variables: LDL and HDL, free radicals and oxidized LDL, MMP and TIMP, cytockines: MCP-1, IFN-γ, IL-12 and PDGF, and cells: macrophages, foam cells, T cells and smooth muscle cells. The model is given by a system of partial differential equations with in evolving plaque. Simulations of the model show how the combination of the concentrations of LDL and HDL in the blood determine whether a plaque will grow or disappear. More precisely, we create a map, showing the risk of plaque development for any pair of values (LDL,HDL).
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Affiliation(s)
- Wenrui Hao
- Mathematical Biosciences Institute, The Ohio State University, Columbus, Ohio, United States of America
| | - Avner Friedman
- Mathematical Biosciences Institute & Department of Mathematics, The Ohio State University, Columbus, Ohio, United States of America
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41
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Al-Mamun M, Brown L, Hossain M, Fall C, Wagstaff L, Bass R. A hybrid computational model for the effects of maspin on cancer cell dynamics. J Theor Biol 2013; 337:150-60. [DOI: 10.1016/j.jtbi.2013.08.016] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/20/2013] [Revised: 08/07/2013] [Accepted: 08/15/2013] [Indexed: 01/01/2023]
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DuBois C, Farnham J, Aaron E, Radunskaya A. A multiple time-scale computational model of a tumor and its micro environment. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2013; 10:121-150. [PMID: 23311365 DOI: 10.3934/mbe.2013.10.121] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/01/2023]
Abstract
Experimental evidence suggests that a tumor's environment may be critical to designing successful therapeutic protocols: Modeling interactions between a tumor and its environment could improve our understanding of tumor growth and inform approaches to treatment. This paper describes an efficient, flexible, hybrid cellular automaton-based implementation of numerical solutions to multiple time-scale reaction-diffusion equations, applied to a model of tumor proliferation. The growth and maintenance of cells in our simulation depend on the rate of cellular energy (ATP) metabolized from nearby nutrients such as glucose and oxygen. Nutrient consumption rates are functions of local pH as well as local concentrations of oxygen and other fuels. The diffusion of these nutrients is modeled using a novel variation of random-walk techniques. Furthermore, we detail the effects of three boundary update rules on simulations, describing their effects on computational efficiency and biological realism. Qualitative and quantitative results from simulations provide insight on how tumor growth is affected by various environmental changes such as micro-vessel density or lower pH, both of high interest in current cancer research.
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Affiliation(s)
- Christopher DuBois
- University of California, Irvine, Dept. of Statistics, School of Information and Computer Science, 3019 Bren Hall, Irvine, CA 92617-5100, USA.
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Sciumè G, Shelton S, Gray WG, Miller CT, Hussain F, Ferrari M, Decuzzi P, Schrefler BA. A multiphase model for three-dimensional tumor growth. NEW JOURNAL OF PHYSICS 2013; 15:015005. [PMID: 24554920 PMCID: PMC3926362 DOI: 10.1088/1367-2630/15/1/015005] [Citation(s) in RCA: 56] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/03/2023]
Abstract
Several mathematical formulations have analyzed the time-dependent behaviour of a tumor mass. However, most of these propose simplifications that compromise the physical soundness of the model. Here, multiphase porous media mechanics is extended to model tumor evolution, using governing equations obtained via the Thermodynamically Constrained Averaging Theory (TCAT). A tumor mass is treated as a multiphase medium composed of an extracellular matrix (ECM); tumor cells (TC), which may become necrotic depending on the nutrient concentration and tumor phase pressure; healthy cells (HC); and an interstitial fluid (IF) for the transport of nutrients. The equations are solved by a Finite Element method to predict the growth rate of the tumor mass as a function of the initial tumor-to-healthy cell density ratio, nutrient concentration, mechanical strain, cell adhesion and geometry. Results are shown for three cases of practical biological interest such as multicellular tumor spheroids (MTS) and tumor cords. First, the model is validated by experimental data for time-dependent growth of an MTS in a culture medium. The tumor growth pattern follows a biphasic behaviour: initially, the rapidly growing tumor cells tend to saturate the volume available without any significant increase in overall tumor size; then, a classical Gompertzian pattern is observed for the MTS radius variation with time. A core with necrotic cells appears for tumor sizes larger than 150 μm, surrounded by a shell of viable tumor cells whose thickness stays almost constant with time. A formula to estimate the size of the necrotic core is proposed. In the second case, the MTS is confined within a healthy tissue. The growth rate is reduced, as compared to the first case - mostly due to the relative adhesion of the tumor and healthy cells to the ECM, and the less favourable transport of nutrients. In particular, for tumor cells adhering less avidly to the ECM, the healthy tissue is progressively displaced as the malignant mass grows, whereas tumor cell infiltration is predicted for the opposite condition. Interestingly, the infiltration potential of the tumor mass is mostly driven by the relative cell adhesion to the ECM. In the third case, a tumor cord model is analyzed where the malignant cells grow around microvessels in a 3D geometry. It is shown that tumor cells tend to migrate among adjacent vessels seeking new oxygen and nutrient. This model can predict and optimize the efficacy of anticancer therapeutic strategies. It can be further developed to answer questions on tumor biophysics, related to the effects of ECM stiffness and cell adhesion on tumor cell proliferation.
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Affiliation(s)
- G Sciumè
- Department of Civil, Environmental and Architectural Engineering, University of Padua, Italy
- Laboratoire de Mécanique et Technologie, Ecole Normale Supérieure de Cachan, France
| | - S Shelton
- Department of Environmental Sciences and Engineering, University of North Carolina at Chapel Hill, USA
| | - WG Gray
- Department of Environmental Sciences and Engineering, University of North Carolina at Chapel Hill, USA
| | - CT Miller
- Department of Environmental Sciences and Engineering, University of North Carolina at Chapel Hill, USA
| | - F Hussain
- Department of Mechanical Engineering, University of Houston, USA
- Department of Nanomedicine, The Methodist Hospital Research Institute, Houston, USA
| | - M Ferrari
- Department of Nanomedicine, The Methodist Hospital Research Institute, Houston, USA
- Department of Medicine, Weill Cornell Medical College of Cornell University, New York, USA
| | - P Decuzzi
- Department of Nanomedicine, The Methodist Hospital Research Institute, Houston, USA
- Department of Translational Imaging, The Methodist Hospital Research Institute, Houston, USA
| | - BA Schrefler
- Department of Civil, Environmental and Architectural Engineering, University of Padua, Italy
- Department of Nanomedicine, The Methodist Hospital Research Institute, Houston, USA
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Guebel DV, Schmitz U, Wolkenhauer O, Vera J. Analysis of cell adhesion during early stages of colon cancer based on an extended multi-valued logic approach. MOLECULAR BIOSYSTEMS 2012; 8:1230-42. [PMID: 22298312 DOI: 10.1039/c2mb05277f] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Subscribe] [Scholar Register] [Indexed: 12/13/2022]
Abstract
Cell adhesion in the normal colon is typically associated with differentiated cells, whereas in cancerous colon it is associated with advanced tumors. For advanced tumors growing evidence supports the existence of stem-like cells that have originated from transdifferentiation. Because stem cells can also be transformed in their own niche, at the base of the Lieberkühn's crypts, we conjectured that cell adhesion can also be critical in early tumorigenesis. To assess this hypothesis we built an annotated, multi-valued logic model addressing cell adhesion of normal and tumorigenic stem cells in the human colon. The model accounts for (i) events involving intercellular adhesion structures, (ii) interactions involving cytoskeleton-related structures, (iii) compartmental distribution of α/β/γ/δ-catenins, and (iv) variations in critical cell adhesion regulators (e.g., ILK, FAK, IQGAP, SNAIL, Caveolin). We developed a method that can deal with graded multiple inhibitions, something which is not possible with conventional logical approaches. The model comprises 315 species (including 26 genes), interconnected by 269 reactions. Simulations of the model covered six scenarios, which considered two types of colonic cells (stem vs. differentiated cells), under three conditions (normal, stressed and tumor). Each condition results from the combination of 92 inputs. We compared our multi-valued logic approach with the conventional Boolean approach for one specific example and validated the predictions against published data. Our analysis suggests that stem cells in their niche synthesize high levels of cytoplasmatic E-cadherin and CdhEP(Ser684,686,692), even under normal-mitogenic stimulus or tumorigenic conditions. Under these conditions, E-cadherin would be incorporated into the plasmatic membrane, but only as a non-adhesive CdhE_β-catenin_IQGAP complex. Under stress conditions, however, this complex could be displaced, yielding adhesive CdhE_β-catenin((cis/trans)) complexes. In the three scenarios tested with stem cells, desmosomes or tight junctions were not assembled. Other model predictions include expected levels of the nuclear complex β-catenin_TCF4 and the anti-apoptotic protein Survivin for both normal and tumorigenic colonic stem cells.
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Affiliation(s)
- Daniel V Guebel
- Department of Systems Biology and Bioinformatics, University of Rostock, 18051 Rostock, Germany.
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Choe SC, Zhao G, Zhao Z, Rosenblatt JD, Cho HM, Shin SU, Johnson NF. Model for in vivo progression of tumors based on co-evolving cell population and vasculature. Sci Rep 2011; 1:31. [PMID: 22355550 PMCID: PMC3216518 DOI: 10.1038/srep00031] [Citation(s) in RCA: 15] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/03/2011] [Accepted: 06/16/2011] [Indexed: 01/30/2023] Open
Abstract
With countless biological details emerging from cancer experiments, there is a growing need for minimal mathematical models which simultaneously advance our understanding of single tumors and metastasis, provide patient-personalized predictions, whilst avoiding excessive hard-to-measure input parameters which complicate simulation, analysis and interpretation. Here we present a model built around a co-evolving resource network and cell population, yielding good agreement with primary tumors in a murine mammary cell line EMT6-HER2 model in BALB/c mice and with clinical metastasis data. Seeding data about the tumor and its vasculature from in vivo images, our model predicts corridors of future tumor growth behavior and intervention response. A scaling relation enables the estimation of a tumor's most likely evolution and pinpoints specific target sites to control growth. Our findings suggest that the clinically separate phenomena of individual tumor growth and metastasis can be viewed as mathematical copies of each other differentiated only by network structure.
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Affiliation(s)
- Sehyo C Choe
- Division of Theoretical Bioinformatics (B080), German Cancer Research Center (DKFZ), Im Neuenheimer Feld 580, 69120 Heidelberg, Germany; Department of Bioinformatics and Functional Genomics, Institute of Pharmacy and Molecular Biotechnology (IPMB) and Bioquant, Heidelberg University, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany.
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Buzea CG, Agop M, Moraru E, Stana BA, Gîrţu M, Iancu D. Some implications of Scale Relativity theory in avascular stages of growth of solid tumors in the presence of an immune system response. J Theor Biol 2011; 282:52-64. [PMID: 21600219 DOI: 10.1016/j.jtbi.2011.05.001] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/12/2011] [Revised: 04/18/2011] [Accepted: 05/03/2011] [Indexed: 11/24/2022]
Abstract
We present a traveling-wave analysis of a reduced mathematical model describing the growth of a solid tumor in the presence of an immune system response in the framework of Scale Relativity theory. Attention is focused upon the attack of tumor cells by tumor-infiltrating cytotoxic lymphocytes (TICLs), in a small multicellular tumor, without necrosis and at some stage prior to (tumor-induced) angiogenesis. For a particular choice of parameters, the underlying system of partial differential equations is able to simulate the well-documented phenomenon of cancer dormancy and propagation of a perturbation in the tumor cell concentration by cnoidal modes, by depicting spatially heterogeneous tumor cell distributions that are characterized by a relatively small total number of tumor cells. This behavior is consistent with several immunomorphological investigations. Moreover, the alteration of certain parameters of the model is enough to induce soliton like modes and soliton packets into the system, which in turn result in tumor invasion in the form of a standard traveling wave. In the same framework of Scale Relativity theory, a very important feature of malignant tumors also results, that even in avascular stages they might propagate and invade healthy tissues, by means of a diffusion on a Newtonian fluid.
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Affiliation(s)
- C Gh Buzea
- National Institute of Research and Development for Technical Physics, D. Mangeron 47, Iaşi 700050, Romania.
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Geris L, Ashbourn JMA, Clarke T. Continuum-level modelling of cellular adhesion and matrix production in aggregates. Comput Methods Biomech Biomed Engin 2011; 14:403-10. [PMID: 21516526 DOI: 10.1080/10255842.2010.539209] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
Abstract
Key regulators in tissue-engineering processes such as cell culture and cellular organisation are the cell-cell and cell-matrix interactions. As mathematical models are increasingly applied to investigate biological phenomena in the biomedical field, it is important, for some applications, that these models incorporate an adequate description of cell adhesion. This study describes the development of a continuum model that represents a cell-in-gel culture system used in bone-tissue engineering, namely that of a cell aggregate embedded in a hydrogel. Cell adhesion is modelled through the use of non-local (integral) terms in the partial differential equations. The simulation results demonstrate that the effects of cell-cell and cell-matrix adhesion are particularly important for the survival and growth of the cell population and the production of extracellular matrix by the cells, concurring with experimental observations in the literature.
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Affiliation(s)
- Liesbet Geris
- Biomechanics Research Unit, Universite de Liege, Liege, Belgium.
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48
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Computational analysis of matrix-degrading enzymes in cancer-cell invasion. BIOCHIP JOURNAL 2011. [DOI: 10.1007/s13206-011-5103-6] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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49
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Mahmood MS, Mahmood S, Dobrota D. Formulation and numerical simulations of a continuum model of avascular tumor growth. Math Biosci 2011; 231:159-71. [PMID: 21396381 DOI: 10.1016/j.mbs.2011.03.002] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/17/2010] [Revised: 02/22/2011] [Accepted: 03/01/2011] [Indexed: 11/29/2022]
Abstract
In this paper we present a continuum mathematical model for a multicellular spheroid that mimics the micro-environment within avascular tumor growth. The model consists of a coupled system of non-linear convection-diffusion-reaction equations. This system is solved using a previously developed conservative Galerkin characteristics method. In the model considered, there are three cell types: the proliferative cells, the quiescent non-dividing cells which stay in the G(0) phase of the cell cycle and the necrotic cells. The model includes viable cell diffusion, diffusion of cellular material and the removal of necrotic cells. We assume that the nutrients diffuse passively and are consumed by the proliferative and quiescent tumor cells depending on the availability of resources (oxygen, glucose, etc.). The numerical simulations are performed using different sets of parameters, including biologically realistic ones, to explore the effects of each of these model parameters on reaching the steady state. The present results, taken together with those reported earlier, indicate that the removal of necrotic cells and the diffusion of cellular material have significant effects on the steady state, reflecting growth saturation, the number of viable cells, and the spheroid size.
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Affiliation(s)
- Mohammed Shuker Mahmood
- Department of Applied Mathematics, Mechanical Engineering Faculty, University of Žilina, 010 26 Žilina, Slovakia.
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Majumder D, Mukherjee A. A passage through systems biology to systems medicine: adoption of middle-out rational approaches towards the understanding of therapeutic outcomes in cancer. Analyst 2011; 136:663-78. [DOI: 10.1039/c0an00746c] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/03/2023]
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