RENAL PHYSIOLOGY Florida State University Advanced Topics in - PowerPoint PPT Presentation

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 ) = � + � f 2 1 c dx P P , hydrostatic pressure = 1 2 osmotic pressure of the suspended protei ns and formed elements of the blood � = c K capillary filteration rate = f where osmotic pressure is RTc � = c

Conservation Equation c Q cq = i i 1 � c Q c q ( from osmosis RTc ) = � = i i 1 c RT RTc Q / q � = c i i 1 Q (where RTc ) i � = � � = c i i i q 1

Mathematical Model of the Glomerulus dq 1 K ( P P ) = � + � ................(20.1) f 2 1 c dx Q i � = � .......................................................(20.4) c i q i Q � � e � � � � Q Q � e ln i 1 K L i � � + � = � f Q 1 Q ..............(20.5) � � � � � i i � � � � Q =efflux through the efferent arterioles e L=length of the filter = /( P P ) � � � i 1 2

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 C C � � q d C ( C ) 1 1 + = � 1 2 1 .....................(20.15) t x � � C C � � 2 q 2 d C ( C ) + = � ..................(20.16) 2 1 2 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 Assume: Flows are in steady state 0 0 C & C 1 2 The input concentrations are

COCURRENT MECHANISM C L ( ) 1 1 + �� � � 1 L e � � = + � 0 C 1 1 + � + � 1 0 C q d 1 � � 2 , 2 , 1 � = � = � = + � � 0 C q q � � � 1 1 1 COUNTERCURRENT MECHANISM L � � C L 1 e ( ) ( ) � � �� + � � + �� 1 = 0 L C � � e � � � 1 C L 0 ( ) C q d 1 � � 2 2 , 2 , 1 � = = � = � � = � � � 0 0 C C q q � � � 1 1 1 1

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