微血管是人體內物質交換場所，透過微血管運輸血液、氧氣及養份供給身體器官與組織，代謝出體內有害物質，若可以在了解微血管的形成，對於疾病治療與防止人造器官與組織壞死有很大的幫助。
本文建立數學模型分析體外培養內皮細胞，細胞自發性移動而形成微血管的網狀結構，將影響微血管網狀分佈的因素加入模型中，此數學模型考慮阿米巴型細胞密度、間質型細胞密度、旁泌性血管內皮生長因子濃度、基質密度及基質位移。模型假設阿米巴型細胞與間質型細胞在基質上的因貼附力的不同，所以有不同的遷徙行為，血管內皮生長因子為旁泌性包含在生物膠內，並會釋出與細胞外基質結合，細胞移動受到基質趨觸性及血管內皮生長因子趨化性的的影響與基質受到細胞牽引力的位移等作用，使得細胞逐漸移動而構成網狀微血管。為了分辨及解析微血管網狀結構的形成因素，本文分別建立僅有趨化性影響細胞移動的化學模型，基質趨觸性影響細胞移動的力學模型，與趨化性、趨觸性皆影響細胞移動的化學力學模型。再藉由線性穩定性分析方式探討不同模型及參數對於網狀形態產生的影響，並將參數分類為穩定和不穩定參數，調控參數探討細胞聚落分佈型態的力學機制。

Capillary vessels function as an exchange network, which bring in oxygen and nutrients and take away the metabolites for the organs and tissues in the human body. Understanding the formation of capillary networks is helpful to the treatment of cancer and developing engineered tissues of large size.
This study developed a mathematical model for describing the plexus formation by endothelial cells cultivated in vitro. The model variables include the cell densities of the amoeboid-type angioblasts and the mesenchymal-type endothelial cells, the concentration of vascular endothelial growth factor (VEGF) embedded in and released from the substrate, and the substrate density and displacement. The amoeboid cells seeded onto the substrate gradually differentiate into the mesenchymal cells. These two cell types have different attachment behaviors to the substrate and migrate with different speeds. Due to the traction by the mesenchymal cells, the substrate is displaced and in turn drags the cells with it. The motion of the amoeboid cells is affected by the directed migration to the chemotaxis of VEGF. The mesenchymal cells migrate not only to the chemotaxis but also to the haptotaxis induced by the variation of the substrate density. The interactions of these factors ultimately determine the distribution of the cells. In order to carefully distinguish the effects of the possible factors, this study set up three models: the first called the chemical model considered only the chemotaxis and assumed a uniform substrate; the second termed the mechanical model consider only the interaction between the cells and the substarte displacement and assumed there was no VEGF being released from the substrate; and the final one referred to as the mechano-chemical model included both VEGF chemotaxis and heptotaxis between the cells and substrate. The model parameters were investigated by linear stability analysis. The mechanisms that underlie the plexus formation of the endothelial cells were explained via the parameter analysis.

Ambrosi D, Bussolino F and Preziosi L. 2005. A review of vasculogenesis models. Journal of Theoretical Medicine 6(1): 1-19.
Balaban NQ, Schwarz US, Riveline D, Goichberg P, Tzur G, Sabanay I, Mahalu D, Safran S, Bershadsky A, Addadi L and Geiger B. 2001. Force and focal adhesion assembly: a close relationship studied using elastic micropatterned substrates. Nature cell biology 3:466-472.
Barocas VH, Moon AG and Tranquillo RT. 1995. The fibroblastpopulated collagen microsphere assay of cell traction force-Part 2: measurement of the cell traction parameter.Journal of biomechanical engineering 117:161–170.
BD Biosciences, 2011. BD Matrigel matrix Frequently Asked Questions.
Benkherourou M, Gumery PY, Tranqui L and Tracqui P. 2000. Quantification and macroscopic modeling of the nonlinear viscoelastic behavior of strained gels with varying fibrin concentrations. IEEE Transactions on biomedical engineering 47: 1465-1475.
Ferrenq I, Tranqui L, Vailhe B, Gumery, PY and Tracqui P. 1997. Modelling biological gel contraction by cells: mechanocellular formulation and cell traction force quantification. Acta biotheoretica 45:267-293.
Folkman J and Haudenschild C. 1980. Angiogenesis in vitro. Nature 288:551–556.
Folkman J. 2001. Angiogenesis-dependent diseases. Semin Oncol 28:536-542.
Friedl P and Wolf K. 2003. Tumour-cell invasion and migration: diversity and escape mechanisms. Nature Reviews Cancer, 3(5): 362-374.
Holmes MJ and Sleeman BD. 2000. A mathematical model of tumour angiogenesis incorporating cellular traction and viscoelastic effects. Jourmal of theoretical biology 202:95-112.
Ishaug-Riley SL, Okun LE, Prado G, Applegate MA and Ratcliffec A. 1999. Human articular chondrocyte adhesion and proliferation on synthetic biodegradable polymer films. Biomaterials 20:2245-2256.
Köhn-Luque A, de Back W, Starruß J, Mattiotti A, Deutsch A, Pérez-Pomares JM, and Herrero MA. 2011. Early embryonic vascular patterning by matrix-mediated paracrine signalling: a mathematical model study. PloS one 6(9): e24175.
Kubota Y, Kleinman Martin GR and Lawley TJ. 1988. Role of laminin and basement membrane in the morphological differentiation of human endothelial cells into capillary-like structures. The journal of cell biology 107:1589–1598.
Langer R and Vacanti JP. 1993. Tissue engineering. Science 260:920-926.
Manoussaki D, Lubkin SR, Vernon R and Murray JD. 1996. A mechanical model for the formation of vascular networks in vitro. Acta biotheoretica 44:271–282.
Moon AG and Tranquillo RT. 1993. Fibroblast-populated collagen microsphere assay of cell traction force: Part 1. continuum model. AIChE Journal 39:163-175.
Murray JD and Oster GF. 1984. Cell traction models for generating pattern and form in morphogenesis. Journal of mathematical biology 19:265–279.
Namy P, Ohayon J and Tracqui P. 2004. Critical conditions for pattern formation and in vitro tubulogenesis driven by cellular traction fields. Journal of theoretical biology 227:103-120.
Nico B, Vacca A, De Giorgis M, Roncali L, and Ribatti D. 2001. Vascular endothelial growth factor and vascular endothelial growth factor receptor-2 expression in the chick embryo area vasculosa. The Histochemical Journal,33(5):283-286.
Preziosi L and Astanin S. 2006. Modelling the formation of capillaries. Complex systems in biomedicine. Springer Milan 109-145.
Scherer GW, Hdach H and Phalippou J. 1991. Thermal expansion of gels: a novel method for measuring permeability. Journal of Non-crystalline solid 130: 157-170.
Serini G, Ambrosi D, Giraudo E, Gamba A, Preziosi L, and Bussolino F. 2003. Modeling the early stages of vascular network assembly. The EMBO Journal 22(8): 1771-1779.
Tosin A, Ambrosi D, and Preziosi L. 2006. Mechanics and chemotaxis in the morphogenesis of vascular networks. Bulletin of mathematical biology 68(7): 1819-1836.
Tranqui L and Tracqui P. 2000. Mechanical signalling and angiogenesis. The integration of cell-extracellular matrix couplings. Comptes rendus de l'Académie des Sciences 323:31-47.
Vailhé B, Ronot X, Tracqui P, Usson Y and Tranqui L. 1997. In vitro angiogenesis is modulated by the mechanical properties of fibrin and is related to αvβ3 integrin localisation. In vitro cellular & developmental biology 33:763-773.
Xu K and Cleaver O. 2011. Tubulogenesis during blood vessel formation. Seminars in Cell & Developmental Biology 22:993– 1004.
Zhang WJ, Liu W and Cao Y. 2007. Tissue engineering of blood vessel. Journal of Cellular and Molecular Medicine 11:945-957.
吳思穎，2012，體外培養內皮細胞形成毛細血管結構前期之穩定性分析及模擬，中央大學機械工程學系碩士論文