1Physiology 472/572 - 2009 - Quantitative modeling of biological systems Lecture 11: Oxygen transport Oxygen solubility in tissue • consider a gas in solution at equilibrium with the gaseous phase • Henry's law for solubility of a gas: c = αP where c is the concentration in solution, P is the partial pressure in the gas phase and α is the solubility • solubility of oxygen in water at 37C ≈ 3 × 10-5 cm3O2/cm3/mmHg • solubility in tissue is close to that in water • partial pressure of oxygen in air at sea level is 20% of 760 mmHg or 152 mmHg • typical partial pressure of oxygen in arterial blood is 100 mmHg Oxygen diffusion in tissue - diffusion distance • diffusivity of oxygen in tissue, D ≈ 1.5 × 10-5 cm3/s • oxygen consumption in heavily working muscle: 35 cm3O2/100cm3/min • consider steady-state one-dimensional diffusion: r cDtc2+∇=∂∂ • then 0 = Dα d2P/dx2 − g where g = 35/60/100 = 0.005833 cm3O2/cm3/s • suppose that at the diffusion distance L, the oxygen is all used up • boundary conditions P = P0 at x = 0, P = dP/dx = 0 at x = L • from differential equation, P = A + Bx + gx2/(2Dα) • from boundary conditions, A = P0, B = - gL/(Dα), A + BL + gL2/(2Dα) = 0 • diffusion distance with P0 = 100 mmHg, L = (2DαP0/g)1/2 = 3.9 × 10-3 cm = 39 μm • oxygen must be brought by convection within this distance of every point in the tissue2Convective transport of oxygen • suppose a runner consumes 3 L/min of oxygen, which must be transported by convection • suppose that the circulating fluid is water instead of blood • if 'blood' is fully equilibrated with air, then c = αP = 4.56 × 10-3 cm3O2/cm3 • rate of convective oxygen delivery is QC where Q is the volume flow rate • to supply oxygen demand, QC > 3 L/min, so Q > 660 L/min Red blood cells and hemoglobin • blood contains 55-60% plasma, 40-45% erythrocytes (red blood cells) • also leukocytes (white blood cells 7-22 μm, various types) and platelets (2-4 μm) • red blood cell has a biconcave disk shape when unstressed • thin membrane - lipid bilayer with protein skeleton • contents - concentrated hemoglobin solution, no nucleus in mammals • each hemoglobin molecule can bind up to 4 oxygen molecules • define saturation S as fraction of occupied binding sites • Hill equation (empirical relationship): n50nnPPPS+= • for human blood, P50 = 26 mmHg, n = 2.7 • oxygen carrying capacity 0.5 cm3O2/cm3 RBC, 0.2 cm3O2/cm3 blood at 40% hematocrit • to deliver 3 L/min requires 15 L/min of blood, a more reasonable rate3Krogh cylinder model for oxygen delivery to skeletal muscle • assumes each capillary supplies oxygen to a surrounding cylindrical region • assumes uniform oxygen consumption rate • neglects diffusion in axial direction • assumes steady-state conditions • assumes symmetry about cylinder axis • r cDtc2+∇=∂∂becomesgzPθPr1rPrrr1D022222−⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂+⎟⎠⎞⎜⎝⎛∂∂∂∂=α • αDgrPrrr1=⎟⎠⎞⎜⎝⎛∂∂∂∂with boundary conditions P(rc) = P0, P'(rt) = 0 • solution is ⎟⎟⎠⎞⎜⎜⎝⎛−−+=c2t2c20rrlnr2rr4DgPPα Limitations of Krogh model • assumes idealized geometry • neglects axial decline of oxygen levels • neglects intravascular resistance to oxygen diffusion • assumes constant oxygen consumption rate • neglects diffusive interactions with other
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