Flow Through The Kidney - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 329

the two compartments of 1 mL/min in each direction. The initial concentration of salt within the
dialysate is 0 grams, and the initial concentration of salt within the blood is 150 g. The volume of
the blood compartment is 100 mL, and the volume of the dialysate compartment is 100 mL.
Determine the salt quantity as a function of time for both compartments of this system.

Solution
First, set up the differential equations and the initial conditions:
dc 1
dt 52

= =

1ml min

100 mL

1mL min

100 mL

5 grams
min

c 1 1 c 2 1

dc 2
dt 5

1ml min

100 mL

= 1mL min

100 mL

=

c 1 2 c 2

c 1 ð 0 Þ 5 150 g

c 2 ð Þ 5

0 0g

Using the Laplace Transform, this system becomes

5
s

sC 1 2 c 1 ð Þ 52 :

01 C 1 1

:

01 C 2 1 0 0 0

sC 2 2 c 2 ð 0 Þ 5 0 : 01 C 1 2 0 : 01 C 2

5
s 1

150 s 1 5

C 1 ð s 0 : 01 Þ 2 0 : 01 C 2 5 150

1 5

s

0 : 01 C 1 5 C 2 ð s 0 : 01 Þ

1

C 1 5 100 C 2 ð s 0 : 01 Þ

1

150 s 5

1

½ 100 C 2 ð s 0 : 01 Þ ð s 0 : 01 Þ 2 0 : 01 C 2 5

1 1

s

150 s 5

1

s 2 100 C 2 ð 0 : 02 s Þ 5

1

s

1 : 5 s 1 0 : 05

Þ 5 2

5 2 5

s 2 1

: 5

C 2 5 1

s 2 ð s 0 : 02 s s 0 : 02

1 1

150 s 2

6 : 5 s 0 : 05 200

s

2 5

s 2 2

: 50

1 1

C 1 5

5 2

s 2 ð s 0 : 02 Þ s 0 : 02

1 1

50 e 2 0 : 02 t

c 1 ð t Þ 5 200 2 : 5 t

2 2

5 e 2 0 : 02 t c 2 ð t Þ 52 5 1 2 : 5 t 1

12.7 DISEASE CONDITIONS

12.7.1 Renal Calculi
Renal calculi are commonly known as kidney stones. Most of the time, calculi appear in
the kidneys due to urine becoming supersaturated with salts. These salts become