MBI Publications

MBI Publications for Ischemia (2)

  • E. Green, N. Ovenden and F. Smith
    Flow in a multi-branching channel with compliant walls
    Journal of Engineering MathematicsVol. 64 No. 4 (2009) pp. 353-365


    The problem of fluid flow in a compliant-walled channel which branches into two or more daughters is considered with the aim of understanding blood flow through arterio-venous malformations (AVMs) in the brain. The outer walls of the channel are assumed for definiteness to behave as spring-back plates, whilst the divider is taken as rigid. The fluid is assumed to be incompressible and inviscid. When the Strouhal number is small (as occurs in practice in the brain), there are two main axial length scales, one much longer than the vessel width and the other comparable with the vessel width. Also, in the case of small wall displacements, one can analyse the local flow-structure interaction problem using a complex variable method. The flow shows markedly different qualitative features downstream of the branching, depending on the wall stiffness. Keywords Branching channel ? Compliant walls ? Inviscid fluid flow ? Matching A. Matzavinos, C.Y. Kao, J.E.F. Green, A. Sutradhar, M. Miller and A. Friedman, Modelling oxygen transport in surgical tissue transfer, Proceedings of the National Academy of Sciences, 106 (29) p12091-12096, 2009 http://www.pnas.org/content/106/29/12091.abstract Abstract Reconstructive microsurgery is a clinical technique used to transfer large amounts of a patient's tissue from one location used to another in order to restore physical deformities caused by trauma, tumors, or congenital abnormalities. The trend in this field is to transfer tissue using increasingly smaller blood vessels, which decreases problems associated with tissue harvest but increases the possibility that blood supply to the transferred tissue may not be adequate for healing. It would thus be helpful to surgeons to understand the relationship between the tissue volume and blood vessel diameter to ensure success in these operations. As a first step towards addressing this question, we present a simple mathematical model that might be used to predict successful tissue transfer based on blood vessel diameter, tissue volume, and oxygen delivery.
  • C. Xue, A. Friedman and C. Sen
    A mathematical model of ischemic cutaneous wounds
    PNASVol. 106 No. 39 (2009) pp. 16782-16787


    Chronic wounds represent a major public health problem affecting 6.5 million people in the United States. Ischemia, primarily caused by peripheral artery diseases, represents a major complicating factor in cutaneous wound healing. In this work, we sought to develop a mathematical model of ischemic dermal wounds. The model consists of a coupled system of partial differential equations in the partially healed region, with the wound boundary as a free boundary. The extracellular matrix (ECM) is assumed to be viscoelastic, and the free boundary moves with the velocity of the ECM at the boundary. The model equations involve the concentration of oxygen, PDGF and VEGF, the densities of macrophages, fibroblasts, capillary tips and sprouts, and the density and velocity of the ECM. Simulations of the model demonstrate how ischemic conditions may limit macrophage recruitment to the wound-site and impair wound closure. The results are in general agreement with experimental findings.

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