Abstract
A diffusion model of tumor growth, vascularization and necrosis is used to analyze experimental data describing the temporal changes in tumor cell and blood vessel radial distributions in a host-tissue field transplanted with a fibrosarcoma. The experimental results showed a peak density of vessels occurring at the advancing migration front of the tumor and a decline in the vessel surface area at the tumor center with time. The peak density of tumor cells shifts away from the tumor center with time. These dynamic changes can be explained by a mathematical model which views the process as one of diffusion and proliferation in time and space. Coupled diffusion equations with nonlinear source and sink terms describe the proliferation, death, and migration of tumor cells and vascular surface area. The concept of an angiogenic factor elaborated by tumor cells is incorporated.
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Literature
Algire, G. H. and H. W. Chalkley. 1946. “Vascular Reactions of Normal and Malignant TissueIn Vivo—I. Vascular Reactions of Mice to Wounds and to Normal and Neoplastic Transplants”J. Natn. Cancer Inst,6, 73–85.
Burton, A. C. 1966. “Rate of Growth of Solid Tumors as a Problem of Diffusion”Growth,30, 157–176.
Carnahan, B., H. A. Luther and J. O. Wilkes. 1969.Applied Numerical Methods. New York: John Wiley and Sons.
Carslaw, H. S. and J. C. Jaeger. 1956.Conduction of Heat in Solids, pp. 230–240. Oxford: Clarendon Press.
Eddy, H. A., and G. W. Casarett. 1973. “Development of the Vascular System in the Hamster Malignant Neurolemmoma”Microvascular Res.,6, 63–82.
Folkman, J., E. Merler, C. Abernathy, and G. Williams. 1971. “Isolation of a Factor Responsible for Tumor Angiogenesis”J. Exp. Med.,133, 275–288.
Florey, H. W. 1970.General Pathology, pp. 656–660. Philadelphia and London: W. B. Saunders Company.
Liotta, L., J. Kleinerman and G. Saidel. 1974. “Quantitative Relationships of Intravascular Tumor Cells, Tumor Vessels, and Pulmonary Metastases Following Tumor Implantation”Cancer Res.,34, 997–1004.
Saidel, G. M., L. A. Liotta, and J. Kleinerman. 1976. “System Dynamics of a Metastatic Process from an Implanted Tumor”J. Theor. Biol.,56, 417–434.
Tannock, I. F.. 1968. “The Relation Between Cell Proliferation and the Vascular System in Mouse Mammary Tumor”.Br. J. Cancer,22, 258–273.
— 1970. “Population Kinetics of Carcinoma Cells, Capillary Endothelial Cells, and Fibroblasts in a Transplanted Mouse Mammary Tumor”Cancer Res.,30, 2470–2476.
— and S. Hayashi. 1972. “The Proliferation of Capillary Endothelial Cells”Cancer Res.,32, 77–82.
— and G. G. Steel. 1969. “Quantitative Techniques for Study of the Anatomy and Function of Small Blood Vessels”J. Natn. Cancer Inst.,42, 771–782.
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Liotta, L.A., Saidel, G.M. & Kleinerman, J. Diffusion model of tumor vascularization and growth. Bltn Mathcal Biology 39, 117–128 (1977). https://doi.org/10.1007/BF02460686
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DOI: https://doi.org/10.1007/BF02460686