RENAL PHYSIOLOGY Florida State University Advanced Topics in Biomedical Mathematics MAP5932, Spring 2007 03/16/07 Brinda Pamulapati Goal 1 Background of the kidney 2 Glomerulus 3 Mathematical model of Glomerulus 4 Co-current and counter- current mechanism 5 Mathematical model of the co-current and counter-current mechanism Kidney Kidney and Nephron picture Glomerulus and Bowman's Capsule Mathematical model of the glomerular filter There are 3 pressures that effect the rate of glomerular filtration: 1 the pressure inside the glomerular capillaries that promote filteration(p1) 2 the pressure inside the Bowman's capsule that opposes filtration(p2) 3 the colloidal osmotic pressure of the plasma proteins inside the capillaries that opposes filteration(pi) Schematic diagram of the glomerular filtration(one dimentional) q2 Qi q1 Qe x=0 x=L Mathematical model of the glomerular filter(cont.) dq 1 = K (P " P + ! ) dx f 2 1 c P1 ,P2 = hydrostatic pressure ! c = osmotic pressure of the suspended protei ns and formed elements of the blood K f = capillary filteration rate where osmotic pressure is ! c = RTc Conservation Equation ciQi = cq1 ! c ciQi = q1 ( from osmosis RTc) RT ! c = ! c = RTciQi / q1 Qi ! c = ! i (where ! i = RTci ) q1 Mathematical Model of the Glomerulus dq1 = K (P " P + ! ) ................(20.1) dx f 2 1 c Qi ! c = ! i .......................................................(20.4) qi # Q $ e %! Q & Q ' " e +! ln & i ' =1% K L i f ..............(20.5) Qi & 1%! ' !Qi & ' ( ) Qe =efflux through the efferent arterioles L=length of the filter !=" i /(P1 # P2) Cocurrent and Countercurrent Mechanism What is Cocurrent and Countercurrent Mechanism Why Study about it ? The human kidney use countercurrent exchange to remove water from urine so the body can retain water that was used to move the nitrogenous waste products. Mathematical Model of the Cocurrent and Countercurrent Mechanism !C1 !C1 + q = d(C "C ) .....................(20.15) !t 1 !x 2 1 !C2 !C2 + q2 = d(C1 " C2 ) ..................(20.16) !t !x Mathematical Problem To Find: The outflow concentration Given : 1) The inflow concentration 2) The length of the exchange chamber 3) Flow velocities are known 0 0 Assume: Flows are in steady state C1 &C2 The input concentrations are COCURRENT MECHANISM C1 (L) 1+ "# 1$ " $!L 0 = + # e C1 1+ # 1+ # C 0 q d % 1 & 2 , 2 , 1 " = 0 # = ! = ' + ( C1 q1 q1 ) # * COUNTERCURRENT MECHANISM $ $!L C1 (L) $"# + (1$ # + "#)e = L 0 $ $! C1 e $ # C L C 0 q d % 1 & 2 ( ) 2 , 2 , 1 " = 0 = 0 # = $ ! = ' $ ( C1 C1 q1 q1 ) # * Conclusion Total transfer of solute is always more efficient with a countercurrent than with a cocurrent. Sources J.Keener, J.Sneyd, Mathematical Physiology http://en.wikipedia.org/wiki/Image:Kidneys_from_behind.jp g http://ocw.mit.edu/NR/rdonlyres/Health-Sciences-and Technology/HST-542JSpring-2004/BB83F266-3398- 4154-A81D- 758E76A74EB5/0/renal_physiology.pdf http://coe.fgcu.edu/faculty/greenep/kidney/index.html http://en.wikipedia.org/wiki/Countercurrent_exchange.