RENAL PHYSIOLOGY

Florida State UniversityAdvanced Topics in Biomedical Mathematics

MAP5932, Spring 2007 03/16/07

Brinda Pamulapati

Goal

1 Background of the kidney2 Glomerulus3 Mathematical model of Glomerulus4 Co-current and counter- current mechanism5 Mathematical model of the co-current and

counter-current mechanism

Kidney

Kidney and Nephron picture

Glomerulus and Bowman's Capsule

Mathematical model of theglomerular filter

There are 3 pressures that effect the rate ofglomerular filtration:1 the pressure inside the glomerularcapillaries that promote filteration(p1)2 the pressure inside the Bowman's capsulethat opposes filtration(p2)3 the colloidal osmotic pressure of theplasma proteins inside the capillaries thatopposes filteration(pi)

Schematic diagram of theglomerular filtration(one dimentional)

q1

q2

QiQe

x=0 x=L

Mathematical model of theglomerular filter(cont.)

12 1( )f c

dqK P P

dx!= " +

cRTc! =

where osmotic pressure is

1 2

c

f

, hydrostatic pressure

osmotic pressure of the suspended protei ns and formed elements of the blood

K capillary filteration rate

P P

!

=

=

=

Conservation Equation

( osmosis )cfrom RTc! =

1

1

1

i

1

/

(where )

i i

ci i

c i i

ic i i

c Q cq

c Q qRT

RTcQ q

QRTc

q

!

!

! ! !

=

=

=

= =

Mathematical Model of theGlomerulus

12 1( )f c

dqK P P

dx!= " +

i

c i

i

Q

q! !=

................(20.1)

.......................................................(20.4)

ln 11

e

e i if

i i

Q

Q QK L

Q Q

!"

!! !

# $%& '

& '+ = %%& '

& '( )

..............(20.5)

i 1 2

=efflux through the efferent arterioles

L=length of the filter

= /( )

eQ

P P! " #

Cocurrent and CountercurrentMechanism

What is Cocurrent and Countercurrent Mechanism

Why Study about it ?The human kidney use countercurrent exchange to remove waterfrom urine so the body can retain water that was used to move thenitrogenous waste products.

Mathematical Model of theCocurrent and Countercurrent

Mechanism

1 11 2 1

2 22 1 2

( )

( )

C Cq d C C

t x

C Cq d C C

t x

! !+ = "

! !

! !+ = "

! !

.....................(20.15)

..................(20.16)

Mathematical Problem

To Find: The outflow concentrationGiven : 1) The inflow concentration

2) The length of the exchangechamber

3) Flow velocities are known

Assume: Flows are in steady state The input concentrations are

0 0

1 2&C C

( )

( ) ( )

( )

1

0

1

0

2 2

0

1 1 1

1

0

1

0

2 2 2

0 0

1 1 1 1

COCURRENT MECHANISM

1 1

1 1

1, , 1

COUNTERCURRENT MECHANISM

1

1, , 1

L

L

LC Le

C

C q d

C q q

C L e

C e

C L C q d

C C q q

!

!

!"# "#

# #

" # !#

"# # "#

#

" # !#

$

$

$

$

$

+ $= +

+ +

% &= = = +' (

) *

$ + $ +=

$

% &= = = $ = $' (

) *

Conclusion

Total transfer of solute is always moreefficient with a countercurrent than with a

cocurrent.

Sources

J.Keener, J.Sneyd, Mathematical Physiologyhttp://en.wikipedia.org/wiki/Image:Kidneys_from_behind.jp

ghttp://ocw.mit.edu/NR/rdonlyres/Health-Sciences-and Technology/HST-542JSpring-2004/BB83F266-3398-4154-A81D-

758E76A74EB5/0/renal_physiology.pdfhttp://coe.fgcu.edu/faculty/greenep/kidney/index.html

http://en.wikipedia.org/wiki/Countercurrent_exchange