1Physiology 472/572 - 2011 - Quantitative modeling of biological systems Lecture 17: Blood flow in the circulatory system Distribution of parameters in the circulatory system • immense number of vessels, branched network • orders of magnitude ranges of length and diameter • veins/venules larger than corresponding arteries/arterioles - most volume is in veins - by Poiseuille's law, most resistance is in arterioles Resistance to blood flow • driving pressure divided by flow rate • for systemic circulation, TPR = (MAP - CVP) / CO - TPR = total peripheral resistance - MAP = mean arterial pressure - CVP = central venous pressure2- CO = cardiac output (volume/time) • for single tube, 4dL128QpRμπ=Δ= • for multiple (N) tubes in parallel, 4dLN128NQpRμπ=Δ= • on graph, d goes down 3-4 orders of magnitude, L goes down 3 orders, N goes up 10 orders • resistances at different levels are comparable, but weighted towards arterioles - most resistance is there Artery walls • components - elastic (elastin, collagen), smooth muscle (contractile) • function is convective transport • must absorb pulsatile pressure generated by heart Wave propagation in arteries - simplified analysis • neglect viscosity, velocity u(x,t) is uniform over cross-section • assume tube radius depends on pressure - a = a(p) • assume small radius changes, assume small fluid velocity Conservation of mass - A = πa2 is cross-section area30)Au(xtAsoand]Au[]Au[Adxtdxxx=∂∂+∂∂−=∂∂+ so 0xaau2xuataa22=∂∂π+∂∂π+∂∂π Conservation of momentum0xpρ1xuutui.e.xp)xuutuρ( =∂∂+∂∂+∂∂∂∂−=∂∂+∂∂ Derivation of wave equation • assume small radius changes: a = a0 + a1(x,t) where a1 is small • assume u is small • linearize - i.e., neglect all products of small quantities • define compliance γ = 2 da/dp • conservation of mass gives0xuatpγi.e.,0xu2ata00=∂∂+∂∂=∂∂+∂∂ • conservation of momentum gives 0xpρ1tu=∂∂+∂∂ • eliminate u to get wave equation0xpctpi.e.0xpρatpγ2222222022=∂∂−∂∂=∂∂−∂∂ where c = (a0/γρ)1/2 is the wave speed Solutions of wave equation • general solution is p = f(x - ct) + g(x + ct) where f and g are arbitrary functions • check by substituting in wave equation • represents superposition of waves travelling to the right and to the left • in arteries, the dominant wave travels away from the heart, but there is also a reflected component due to non-uniformities (branches, taper, etc.)4Wave speed c • mechanics of a stretching thin-walled tube implies that γ = 2a02/(Eh) where E is Young's modulus and h is wall thickness • wave speed is c = (Eh/2ρa0)1/2 - relationship discovered in 1808 (Thomas Young), called the Moens-Korteweg formula • numerical values (canine aorta - Caro et al., 1974) - E = 4.8 × 105 newton/m2 - h/2a0 = 0.07 - ρ = 103 kg/m3 • gives c = 5.8 m/s • actual is about 5-10 m/s throughout arteries • can feel pulse in wrist and neck simultaneously - 0.1 s delay at wrist Propagation of the arterial pressure pulse • many factors contribute - nonlinear elasticity and viscoelasticity of walls - finite amplitude of pulse - blood velocity not small relative to pulse velocity - arteries are branched and tapered - reflections - blood viscosity • pressure pulse shape changes with propagation, becomes
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