Skip to main content
Article thumbnail
Location of Repository

A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis

By L. Olsen, J. A. Sherratt, P. K. Maini and F. Arnold

Abstract

Angiogenesis, the process by which new blood capillaries grow into a tissue from surrounding parent vessels, is a key event in dermal wound healing, malignant-tumour growth, and other pathologic conditions. In wound healing, new capillaries deliver vital metabolites such as amino acids and oxygen to the cells in the wound which are involved in a complex sequence of repair processes. The key cellular constituents of these new capillaries are endothelial cells: their interactions with soluble biochemical and insoluble extracellular matrix (ECM) proteins have been well documented recently, although the biological mechanisms underlying wound-healing angiogenesis are incompletely understood. Considerable recent research, including some continuum mathematical models, have focused on the interactions between endothelial cells and soluble regulators (such as growth factors). In this work, a similar modelling framework is used to investigate the roles of the insoluble ECM substrate, of which collagen is the predominant macromolecular protein. Our model consists of a partial differential equation for the endothelial-cell density (as a function of position and time) coupled to an ordinary differential equation for the ECM density. The ECM is assumed to regulate cell movement (both random and directed) and proliferation, whereas the cells synthesize and degrade the ECM. Analysis and numerical solutions of these equations highlights the roles of these processes in wound-healing angiogenesis. A nonstandard approximation analysis yields insight into the travel ling-wave structure of the system. The model is extended to two spatial dimensions (parallel and perpendicular to the plane of the skin), for which numerical simulations are presented. The model predicts that ECM-mediated random motility and cell proliferation are key processes which drive angiogenesis and that the details of the functional dependence of these processes on the ECM density, together with the rate of ECM remodelling, determine the qualitative nature of the angiogenic response. These predictions are experimentally testable, and they may lead towards a greater understanding of the biological mechanisms involved in wound-healing angiogenesis

Topics: Biology and other natural sciences
Year: 1997
DOI identifier: 10.1093/imammb
OAI identifier: oai:generic.eprints.org:441/core69

Suggested articles

Citations

  1. (1996). A mathematical model for fibro-proliferative wound healing disorders.
  2. (1991). A mathematical model for the diffusion of tumour angiogenesis factor into the surrounding host tissue.
  3. (1995). A mechanochemical model for adult dermal A
  4. (1991). A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor.
  5. (1996). A model of woundhealing angiogenesis in soft tissue.
  6. (1947). A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type.
  7. (1993). A stochastic model for adhesion-mediated cell random motility and haptokinesis.
  8. (1997). Aggregation, blowup and collapse: The ABCs of taxis in reinforced random walks.
  9. (1988). Angiogenesis. In: The Molecular and Cellular Biology of Wound Repair
  10. (1992). Angiogenesis. In: Wound Healing: Biochemical and Clinical Aspects
  11. (1995). Between molecules and morphology. Extracellular matrix and creation of vascular form.
  12. (1989). Biochemical consequences of mechanical forces generated by distension and distortion.
  13. (1993). Biology of dermal wound repair.
  14. (1995). Cellular pattern formation during dictyostelium aggregation.
  15. (1993). Cellular tensegrity: Defining new rules of biological design that govern the cytoskeleton./. Cell Sci.
  16. (1994). Chemotaxis and chemokinesis in eukaryotic cells: The Keller-Segel equations as an approximation to a detailed model.
  17. (1984). Collagen biosynthesis by cells in a tissue equivalent matrix in vitro.
  18. (1992). Comparison of the effects of moist and dry conditions on the process of angiogenesis during dermal repair.
  19. (1982). Connective tissue morphogenesis by fibroblast traction. I: Tissue culture observations.
  20. (1993). Endothelial cells exhibiting angiogenesis in vitro proliferate in response to TGF01.J. Cell Biochem.
  21. (1982). Fibronectin involvement in granulation tissue and wound healing in rabbits.
  22. (1988). Fibronectin: A primitive matrix. In: The Molecular and Cellular Biology of Wound Repair
  23. (1985). Growth inhibition of human fibroblasts by reconstituted collagen fibrils.
  24. (1989). How does the ECM control capillary morphogenesis?
  25. (1988). Inhibition of angiogenesis through modulation of collagen metabolism.
  26. (1989). Mathematical Biology.
  27. (1994). Mathematical modelling of corneal epithelial wound healing.
  28. (1996). Mathematical modelling of wound healing and tumour growth—2 sides of the same coin.
  29. (1995). Mathematical models for tumour angiogenesis— Numerical simulations and nonlinear wave analysis.
  30. (1988). Mathematical Models in Biology.
  31. (1980). Mathematical Models in Molecular and Cellular Biology.
  32. (1992). Mathematical models of wound healing in embryonic and adult epidermis.
  33. (1988). Mechanisms of parenchymal cell migration into wounds. In: The Molecular and Cellular Biology of Wound Repair
  34. (1989). Mechanochemical switching between growth and differentiation during fibroblast growth factor stimulated angiogenesis in vitro—Role of extracellular matrix.
  35. (1994). Numerical Solution of Partial Differential Equations.
  36. (1992). On explosions of solutions to a system of partial differential equations modelling chemotaxis.
  37. (1995). Order and disorder in the temporal organ of wound repair. Wound Repair Regen.
  38. (1988). Overview and general considerations of wound repair. In: The Molecular and Cellular Biology of Wound Repair
  39. (1993). Platelet-derived growth factor indirectly stimulates angiogenesis in vitro.
  40. (1995). Prognostic and predictive value of the determination of tumour angiogenesis in primary solid tumours.
  41. (1997). Regulation of dermal wound angiogenesis—Role of vEGF.
  42. (1985). Regulation of proliferation of bovine aortic endothelial cells, smooth muscle cells and adventitial fibroblasts in collagen lattices.
  43. (1982). Role of platelets and fibrin in the healing sequence—An in vivo study of angiogenesis and collagen synthesis.
  44. (1980). Sequential appearance of fibronectin and collagen in experimental granulation tissue.
  45. (1993). Sequential changes in histological pattern and extracellular matrix during the healing of chronic venous leg ulcers. / Cell. Biochem S17E,
  46. (1996). Subcutaneous tissue 280 L. OLSEN ET AL. oxygen pressure—A reliable index of peripheral perfusion in humans after injury. / Trauma 40,S116-S122.
  47. (1970). The initiation of slime mold aggregation viewed as an instability.
  48. (1989). The Molecular Biology of the Cell.
  49. (1992). The skin. In: Wound Healing: Biochemical and Clinical Aspects
  50. (1937). The wave of advance of advantageous genes.
  51. (1991). Tissue oxygenation, anemia, and perfusion in relation to wound healing in surgical patients.
  52. (1984). Wound Repair.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.