Conservation Laws in Cancer Modeling

We review mathematical models of tumor growth based on conservation laws in the full system of cells and interstitial liquid. First we deal with tumor cords evolving in axisymmetric geometry, where cells motion is simply passive and compatible with the saturation condition. The model is characterized by the presence of free boundaries with constraints driving the free boundary conditions, which in our opinion are particularly important, especially in the presence of treatments. Then a tumor spheroid is considered in the framework of the so-called two-fluid scheme. In a multicellular spheroid, on the appearance of a fully degraded necrotic core, the analysis of mechanical stresses becomes necessary to determine the motion via momentum balance, requiring the specification of the constitutive law for the “cell fluid.” We have chosen a Bingham-type law that presents considerable difficulties because of the presence of a yield stress, particularly with reference to the determination of an asymptotic configuration. Finally, we report some recent PDE-based models addressing complex processes in multicomponent tumors, more oriented to clinical practice.

[1]  Alfio Quarteroni,et al.  Complex Systems in Biomedicine , 2006 .

[2]  Philip K. Maini,et al.  A general reaction–diffusion model of acidity in cancer invasion , 2013, Journal of Mathematical Biology.

[3]  A. Fasano.  Glucose Metabolism in Multicellular Spheroids, ATP Production and Effects of Acidity , 2012 .

[4]  S. McDougall,et al.  Mathematical modeling of tumor-induced angiogenesis. , 2006, Annual review of biomedical engineering.

[5]  Antonio Fasano,et al.  Slow and fast invasion waves in a model of acid-mediated tumour growth. , 2009, Mathematical biosciences.

[6]  A. Bertuzzi,et al.  A MATHEMATICAL MODEL FOR TUMOR CORDS INCORPORATING THE FLOW OF INTERSTITIAL FLUID , 2005 .

[7]  Lei Xu,et al.  Normalization of the vasculature for treatment of cancer and other diseases. , 2011, Physiological reviews.

[8]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[9]  D. Hirst,et al.  TUMOUR CELL PROLIFERATION IN RELATION TO THE VASCULATURE , 1979, Cell and tissue kinetics.

[10]  L. Preziosi,et al.  Cell adhesion mechanisms and stress relaxation in the mechanics of tumours , 2009, Biomechanics and modeling in mechanobiology.

[11]  A. Bertuzzi,et al.  Modelling the evolution of a tumoural multicellular spheroid as a two-fluid Bingham-like system , 2013 .

[12]  Antonio Fasano,et al.  Regression and regrowth of tumour cords following single-dose anticancer treatment , 2003, Bulletin of mathematical biology.

[13]  A. Bertuzzi,et al.  Interstitial pressure and extracellular fluid motion in tumor cords. , 2005, Mathematical biosciences and engineering : MBE.

[14]  D. Ribatti,et al.  The discovery of angiogenic factors: a historical review. , 2000, General pharmacology.

[15]  H. Frieboes,et al.  Nonlinear modelling of cancer: bridging the gap between cells and tumours , 2010, Nonlinearity.

[16]  W. Looney,et al.  DYNAMIC HISTOLOGY OF A RAT HEPATOMA AND THE RESPONSE TO 5‐FLUOROURACIL * , 1980, Cell and tissue kinetics.

[17]  Yijie Wang,et al.  Stabilization of HIF-2α Induces sVEGFR-1 Production from Tumor-Associated Macrophages and Decreases Tumor Growth in a Murine Melanoma Model , 2012, The Journal of Immunology.

[18]  G. Nuovo,et al.  Granulocyte macrophage colony-stimulating factor inhibits breast cancer growth and metastasis by invoking an anti-angiogenic program in tumor-educated macrophages. , 2009, Cancer research.

[19]  C. Bruni,et al.  Optimal solution for a cancer radiotherapy problem , 2013, Journal of mathematical biology.

[20]  A. Fasano,et al.  Investigating the steady state of multicellular spheroids by revisiting the two-fluid model. , 2011, Mathematical biosciences and engineering : MBE.

[21]  Antonio Fasano,et al.  Mathematical modelling of tumour growth and treatment , 2006 .

[22]  Helen M. Byrne,et al.  A two-phase model of solid tumour growth , 2003, Appl. Math. Lett..

[23]  J. Murray,et al.  Virtual and real brain tumors: using mathematical modeling to quantify glioma growth and invasion , 2003, Journal of the Neurological Sciences.

[24]  Avner Friedman,et al.  A hierarchy of cancer models and their mathematical challenges , 2003 .

[25]  Nicola Bellomo,et al.  On the foundations of cancer modelling: Selected topics, speculations, and perspectives , 2008 .

[26]  J. Moore,et al.  Tumour cords in 52 human bronchial and cervical squamous cell carcinomas: inferences for their cellular kinetics and radiobiology. , 1985, British Journal of Cancer.

[27]  Jason M. Graham,et al.  A spatial model of tumor-host interaction: application of chemotherapy. , 2008, Mathematical biosciences and engineering : MBE.

[28]  A. Fasano,et al.  The Steady State of Multicellular Tumour Spheroids: A Modelling Challenge , 2011 .

[29]  J. Freyer,et al.  Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply. , 1986, Cancer research.

[30]  E. T. Gawlinski,et al.  A reaction-diffusion model of cancer invasion. , 1996, Cancer research.

[31]  Luigi Preziosi,et al.  A Multiphase Model of Tumour and Tissue Growth Including Cell Adhesion and Plastic Re-organisation , 2011 .

[32]  B. Sleeman,et al.  Mathematical modeling of the onset of capillary formation initiating angiogenesis , 2001, Journal of mathematical biology.

[33]  J. Folkman,et al.  ISOLATION OF A TUMOR FACTOR RESPONSIBLE FOR ANGIOGENESIS , 1971, The Journal of experimental medicine.

[34]  L. Preziosi,et al.  Modelling Solid Tumor Growth Using the Theory of Mixtures , 2001, Mathematical medicine and biology : a journal of the IMA.

[35]  V. Poghosyan,et al.  Numerical study of the correspondence between the dissipative and fixed-energy Abelian sandpile models. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  V. Shelukhin,et al.  Generalized solutions to the equations of compressible Bingham flows , 1999 .

[37]  Antonio Fasano,et al.  A Free Boundary Problem with Unilateral Constraints Describing the Evolution of a Tumor Cord Under the Influence of Cell Killing Agents , 2005, SIAM J. Math. Anal..

[38]  Juan Belmonte-Beitia,et al.  Bright solitary waves in malignant gliomas. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Avner Friedman,et al.  Hypoxia Inducible Factors-Mediated Inhibition of Cancer by GM-CSF: A Mathematical Model , 2012, Bulletin of mathematical biology.

[40]  Alberto Gandolfi,et al.  Chemotherapy of vascularised tumours: role of vessel density and the effect of vascular "pruning". , 2010, Journal of theoretical biology.

[41]  John R. King,et al.  Vanishing beyond all orders: Stokes lines in a water-wave model equation , 2009 .

[42]  J. Folkman,et al.  SELF-REGULATION OF GROWTH IN THREE DIMENSIONS , 1973, The Journal of experimental medicine.

[43]  Yiider Tseng,et al.  Single-molecule analysis of cadherin-mediated cell-cell adhesion , 2006, Journal of Cell Science.

[44]  Antonio Fasano,et al.  Dynamics of tumour cords following changes in oxygen availability: A model including a delayed exit from quiescence , 2005, Math. Comput. Model..

[45]  M. Neeman,et al.  Self-diffusion of water in multicellular spheroids measured by magnetic resonance microimaging. , 1991, Cancer research.

[46]  H. Greenspan.  Models for the Growth of a Solid Tumor by Diffusion , 1972 .

[47]  P. Hahnfeldt,et al.  Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. , 1999, Cancer research.

[48]  S. Ghosal,et al.  Electromigration Dispersion in Capillary Electrophoresis , 2012, Bulletin of mathematical biology.

[49]  Avner Friedman,et al.  Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy , 2013 .

[50]  A. Bertuzzia,et al.  Reoxygenation and Split-Dose Response to Radiation in a Tumour Model with Krogh-Type Vascular Geometry , 2008 .

[51]  Urszula Ledzewicz,et al.  Mathematical methods and models in biomedicine , 2013 .

[52]  A. Tosin,et al.  Mathematical model of tumour cord growth along the source of nutrient , 2007 .

[53]  Kumbakonam R. Rajagopal,et al.  Mechanics of Mixtures , 1995 .

[54]  C. Verdier,et al.  Fractal approach to the rheology of concentrated cell suspensions. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[55]  P. Maini,et al.  A cellular automaton model for tumour growth in inhomogeneous environment. , 2003, Journal of theoretical biology.

[56]  A. Brú,et al.  The universal dynamics of tumor growth. , 2003, Biophysical journal.

[57]  Alexander R. A. Anderson,et al.  Mathematical modelling of flow in 2D and 3D vascular networks: Applications to anti-angiogenic and chemotherapeutic drug strategies , 2005, Math. Comput. Model..

[58]  Luigi Preziosi,et al.  Cancer Modelling and Simulation , 2003 .

[59]  C. Please,et al.  Tumour dynamics and necrosis: surface tension and stability. , 2001, IMA journal of mathematics applied in medicine and biology.

[60]  Philip Hahnfeldt,et al.  The Tumor Growth Paradox and Immune System-Mediated Selection for Cancer Stem Cells , 2013, Bulletin of mathematical biology.

[61]  C. Marsh,et al.  Hypoxia-Inducible Factor-2α Regulates GM-CSF–Derived Soluble Vascular Endothelial Growth Factor Receptor 1 Production from Macrophages and Inhibits Tumor Growth and Angiogenesis , 2011, The Journal of Immunology.

[62]  Rakesh K. Jain,et al.  Normalizing tumor vasculature with anti-angiogenic therapy: A new paradigm for combination therapy , 2001, Nature Medicine.

[63]  A. Bertuzzi,et al.  Interstitial Pressure And Fluid Motion In Tumor Cords , 2005 .

[64]  Urszula Ledzewicz,et al.  Optimal response to chemotherapy for a mathematical model of tumor–immune dynamics , 2012, Journal of mathematical biology.

[65]  Miguel A. Herrero,et al.  Reaction-diffusion systems: a mathematical biology approach , 2003 .

[66]  D L S McElwain,et al.  A history of the study of solid tumour growth: The contribution of mathematical modelling , 2004, Bulletin of mathematical biology.

[67]  A. Bertuzzi,et al.  Cell resensitization after delivery of a cycle-specific anticancer drug and effect of dose splitting: learning from tumour cords. , 2007, Journal of theoretical biology.

[68]  L. Xing,et al.  Optimization of radiotherapy dose-time fractionation with consideration of tumor specific biology. , 2005, Medical physics.

[69]  M. Chaplain,et al.  Continuous and discrete mathematical models of tumor-induced angiogenesis , 1998, Bulletin of mathematical biology.

[70]  Alexander R A Anderson,et al.  Quantifying the Role of Angiogenesis in Malignant Progression of Gliomas: in Silico Modeling Integrates Imaging and Histology Nih Public Access Author Manuscript Introduction , 2011 .