Mathematical Biology: Modeling the Renal System - PowerPoint Presentation

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Mathematical Biology:Modeling the Renal System

Tori

Turkington

Cinzia

Ballantyne

Jordan

Mattheisen

Caroline Martel

The Kidney

R

emoves

waste products from the bodyMaintains homeostasis:electrolyte levelsacid/base/balancesalt/water balance

The Nephron

R

esponsible

for filtering water, wastes, and Na/KRedistributes substancesEach kidney has about 1 million nephrons

The Nephron

Dynamics of Sodium and Water Transport Along Renal Tubules

=

distance along the tubule

=

concentration of the sodium ion inside the tubule at position x

=

flow of water (volume per unit time) along tubule at x in the direction of increasing x

=

outward transport of water (volume per unit time per length of tubule) across the walls of the tubule

=

outward transport of sodium ions (number of ions per unit time per length of tubule) across the walls of the tubule

=

distance along the tubule=concentration of the sodium ion inside the tubule at position x=flow of water (volume per unit time) along tubule at x in the direction of increasing x=outward transport of water (volume per unit time per length of tubule) across the walls of the tubule=outward transport of sodium ions (number of ions per unit time per length of tubule) across the walls of the tubule

Flow through the tubule is steadyF

low

into the segment must equal the sum of the flow out of the segment and the flux of water out through the walls of the

tubule

(2.1)

(2.1)

D

ifferentiate

with respect to

(2.2)

(2.2)

(2.3)(2.3)

Now, let sodium concentration in the tubule be

S

odium

ion per unit time transported along the tubule by the flow past the point is

(2.4)

(2.4)

Loop of Henle

=

concentration of sodium

ion

=

water

flow in the

tubules

= external

concentration of sodium

ionLet the flow be positive in the descending limb and negative in the ascending limb. 

Assumptions:1.

the walls

of the descending limb are permeable to water but not to sodium

ions

(3.3)

(3.3)

(3.1)

(3.2)

(3.1)

(3.2)

permeability to water is

>>> that

the flux makes the internal and external concentrations of sodium ion equal2. sodium ions are pumped from the ascending limb at a steady rate per unit time ascending limb is

impermeable to

water

(3.4)

(3.5)

Assumptions:

3.

At the turn in the Loop

all sodium ions and water leaving the descending limb enter the ascending limb

(3.6)

(3.7)

(3.6)

(3.7)

4.

peritubular

capillaries pick up sodium ions and water locally

driving force for the reverse filtration due to oncotic pressureThe flux of sodium ions and the flow of water is related by the equation:(3.8)(3.8)

Differential equation for c(x):

From these we derive the following:

(3.9)

(3.10)

(3.9)

(3.10)

The ratio of fluxes is renamed as α

α determines the maximum

sodium ion concentrating ability of the nephron

through

(3.16)

(3.16)

(3.17)

(3.17)

(3.15)

(3.15)

Considering the ascending limb:

(3.19)

(3.19)

Rearranging the equation 3.5 we get 3.19

It follows that

(3.20)

(3.20)

Because

=

)

(3.21)

(3.21)

(3.22)

(3.22)

(3.23)

(3.23)

Juxtaglomerular Apparatus Physiology

=

concentration

of sodium in the filtrate as it leaves the afferent arteriole

=

concentration of sodium sought by the juxtaglomerular apparatus

=

concentration

of sodium in the filtrate as it leaves the afferent arteriole

=

concentration of sodium sought by the juxtaglomerular apparatus

Define:Marieb, E. (2006). Essentials of human anatomy & physiology (8th ed.). San Francisco: Pearson/Benjamin Cummings.

Juxtaglomerular Apparatus

(4.1)

(4.2)

(4.1)

(4.2)

Assume the following:

(4.3)

(4.4)

(4.5)

(4.3)

(4.4)

(4.5)

Recall:

(3.23)

Recall:

(3.23)

Therefore:

Juxtaglomerular Apparatus

(4.8)

(4.9)

(4.10)

(4.11)

(4.8)

(4.9)

(4.10)

(4.11)

(4.6)

(4.7)

(4.6)

(4.7)

For small values of a:

Recall:

(4.3)

Recall:

(4.3)

Now we re-write previous section’s conclusions with the parameter c*

Juxtaglomerular Apparatus99%:1%

reabsorption: error in reabsorption ratio

http

://www.secretlyhealthy.com/wp-content/uploads/2015/06/Dehydration.png

The Distal Tubule and Collecting duct

(

Moyes

, Schulte, 2015)

Function of the Distal Tubule and Collecting ductControls concentration of solutes in urine

Process aided by ADH (antidiuretic hormone)

ADH allows distal tubules and collecting duct to be permeable to water

(Andrew Biology, 2011)

Flux of Na+ and fluid

When ADH is present Na

+

flux at the end of the collecting loop is modeled by:

When ADH is present fluid flux can be modeled by:

(McGraw Hill, 2014)

Citations

Hoppensteadt

, F., &

Peskin, C. (2002). Modeling and simulation in medicine and the life sciences(2nd ed.). New York: Springer.

Labeled

Diagram of Nephron. (2014, April 4). Retrieved December 11, 2015, from http://

www.buzzle.com/articles/labeled-diagram-of-nephron.html

Marieb

, E. (2006). Essentials of human anatomy & physiology (8th ed.). San Francisco: Pearson/Benjamin

Cummings.

Secretly Healthy. (2015, June 11). Why Dehydration is Making You Fat. Retrieved from http

://www.secretlyhealthy.com/wp-content/uploads/2015/06/Dehydration.png