Renal Stone Model
System Dynamics Approach
Mohammad T. Mojtahedzadeh Mohammad Kazemeynee Hamideh Azizkhan
Inst. for Research in Planning Yazd Medical Tehran Medical
and Development, University University
P. O. Box 19395/4647, Yazd, Iran Tehran, Iran
Tehran, Iran
Abstract
A clearly defined etiology for urinary calculi has not yet been established. In this
paper, a primary System Dynamics model is presented to a better understanding the
process of renal stone formation. The paper attempts to explain why in most patients
who have recurrent formation of calculi, urinary excretion of calcium is normal, and
why can not be over emphasized on hydration in the prevention of urinary calculi.
Introduction
Natural systems are made up of the organization of particles whose interactions
determine the properties and behavior of the system. Human mind with all its
wonderful powers can only assemble a few of numerous relations which exists
between the parts of a complex system, simultaneously. System Dynamics is a method
for simulating the behavior of the system using control theory and computer.
Up to now System Dynamics has been applied to several biomedical problems
(Brush 1985, Foster 1971). In this paper, a simple System Dynamics model is
presented to analyze the formation process of renal stone. The paper is divided into
five sections. The first section sets the dynamic problem, the second one presents the
construction of the model, response to treatment are analyzed at several point in
section three, finally, the conclusion comes in section four.
Dynamic Problem
The history of renal calculus returns back to almost 7000 years ago. It is a very
common global problem. The patient with renal calculi have been faced with broad
spectrum of complications.
Many attempts have been made to establish the etiology of this condition but the
mechanism of the calculus formation and development remains obscure. It is obvious
that there are many factors which are involved in the formation of calculi. But
probably the complicated effects of these factors together in a biological environment
make the proper diagnosis of this problem more difficult. Here are a summary of
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some important theoretical points in the calculus formation in the urinary system
which include:
1-Saturation and super saturation of some of substance in the urine eventually cause
the precipitation crystals.
2-Presence of certain factors in the urine which probably can play important roles
in stone formation.
3-Loss of substances in urine with inhibiting effects on stone formation.
4-Accumulation of different urinary crystal in the development of calculus.
5-The relationship of the time of the remaining primary nucleus or reduced rate of
urine flow which can increase the chance of enlargement of the stone.
6-The urine PH which has an effect in dissolving or precipitating the minerals.
80 to 90 percent of urinary calculi contain some of calcium salts. Therefore the
calcium is considered to be the major substance in the formation of urinary stone. The
calcium metabolism in the human body depends on its intestinal absorption and renal
excretion. There is also a relationship between the calcium metabolism and other
factors such as endocrine glands, vitamins and bone metabolism. The urinary stone
formation is also influenced by the amount water excretion by the kidneys which can
either dilute or concentrate the urine.
Despite the complicated: biological metabolism of the calcium and water in the
human body, it can be started in the beginning in a simple model’ then the other
physiological factors can be added to this model.
Model Construction
A simple dynamic model is presented to analyze the formation process of renal
stone. The dynamic model is made up of three sections; Water, Calcium, and Stone. The
water and calcium sections present their circulations in the body, while stone section
describes the formation process of stone in the kidney. The dynamic behavior of the
model is the result of interaction between these three sections. The response of the
important variables to treatment at several points i.e.: different intensity levels, are
simulated for a period of 48 hours.
A) The Water Section
Figure 1 shows the circular flow of water in the body. Let the amount of the intake
water to be MW, in the model the beginning is from when it enter the assimilation
part of digestive system. The assimilation is done in an hour (WA is the amount of
water which is being assimilated). After assimilation, the amount of water in the body
(W) increases, this increase results in the increase of the water excretion (TWL is
the amount of water which is excreted). So there is a relationship between the amount
of excreted water and the amount of water in the body. The water excretion decreases
the water level.
Body losses water in three ways: From the kidneys, through the skin, from the lungs.
Urine excretion is equivalent to its normal amount modified by the current water
level, by a multiplier that summarize the renal response to blood volume. The water
loss via the skin and lung is equivalent to their normal amount, modified by the
current water level.
The necessary amount of water for the body is determined by considering equilibrium
amount of water and a proportion of difference between the equilibrium amount of
water and the amount of water in the body. It decreases with an increase in the water
level. The necessary amount of water is taken within 3 hours.
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ere
figure 2-a: Circular Flow of Calcium in Body
calei —_—|
necessity -
: Cnn MON
fhe’, ———J
figure 2-b : Causal Loops of Calcium Circulation in Body
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ee
--> Information Flow
—— Material Flow
figure 2-a : Circular Flow of Calcium in Body
calciuay ¢<————
necessity -
lei! i
Gy) wee Cy akin
. :
. ——
alcium
ge
figure 2-b : Causal Loops of Calcium Circulation in Body
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B) The Calcium Section
Figure 2 shows the circular flow of calcium in the body which is very similar to
what was said about water. Assume the amount of calcium entered the body through the
food intake to be MCA, again the beginning is the assimilation part of digestive system.
Here the assimilation takes place in 4 hours (CAA is the amount of calcium which is
being assimilated). After assimilation, the amount of calcium in the body (CA)
increases and results in the increase of the calcium excretion (CAX). The rate of
calcium excretion is equivalent to its normal amount and change with the current
calcium level in the body, proportionally. The calcium excretion decreases the
calcium level.
The necessary amount of calcium in the body (DMCA) is determined by considering
the excreted amount of calcium and a proportion of deference between the
equilibrium amount of calcium (CAQ) and the current amount of calcium in the body
(CA). It decreases with an increase in the calcium level. The necessary amount of
calcium is taken within 6 hours.
C) The Stone Section
By considering the amount of urine and the amount of the excreted calcium per unit
of time, the urinary calcium is determined. Since 80 to 90 percent of renal stone
contain some of calcium composition, the amount of calcium in the urine has an
important effect on the formation of stone. When the urinary calcium is normal and
there is no stone in the kidney, there will be no concretion either. The size of the
stone is also an important factor in the continuance of the concretion. The calculus
decreases the threshold level of the stone formation, it's because the existence of stone
increases the chance of collision between the calcium particles. In order to decreases
this chance the urinary calcium should be reduced. The threshold level of the stone
formation is the amount calcium in the urine for which the probability of collision of
the particles and calculus is so small that can be ignored. Thus, if the urinary calcium
is equal to threshold level of stone formation the calcium particles will not join up
and the stone will not be enlarged. Otherwise, the particles will be collide and join up
and make the size of the stone larger and larger (Figure 3).
calculus
+
+ threshold
uninary
urinary caloium == =
masinans oatshut!
figure 3: Causal loop of renal stone foraation
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Model behavior
The existence of the stone in the kidney reduces the threshold level of stone
formation. The urinary calcium will be more than the threshold, if the calcium and
urine excretion both stayed unchanged. As a result the collision of calcium particles
with each other and with the stone will be more probable, and that results in the
enlargement of the stone. The enlargement of the stone reduces the threshold to less
than the former level, for a larger size of stone the more probable is the collision of
the particles.
Assuming the amount of calcium in the urine to be relatively constant, a decrease in
the threshold level results in an enlargement i in the size of the stone. (See the curve
"A" in figure 4).
Selection of the treatment
The amount of the calcium in the urine should be reduced in order to prevent the
continuation of the enlargement of the stone. This can be done either by increasing the
amount of the excreted urine per unit of time or by decreasing the amount of the
excreted calcium per unit of time or both together. x
“B" and "C" curves in figure 5 show the effect of 5 and 10 percent/increase in water
intake on the urine excretion. The drinking begins at time 0 and continues for 48
hours.
The "B" curve of figure 6 shows the effect of one percent decrease in calcium intake
on the calcium excretion. This decrease at time 0 for 48 hours slowly decreases the
amount of the excreted calcium.
The "B" curve in figure 7 shows the urinary calcium when the water intake is
increased by 5 percent and the calcium intake is reduced by one percent. "C” shows
the urinary calcium when there is an increase of 10 percent in the drinking water
and a decrease of one percent in the calcium intake.
The "B" and "C" curves of figure 4 show the mass of the formed stone in the same
interval when there is an increase of 5 percent and an increase of 10 percent in the
drinking water and decrease of one percent in the calcium intake.
As can be seen in figure 4, the formation of stone may be slowed down by increasing
the drinking amount of water intense so that the curve of the formed mass of stone is
shifted from "A" to "B" and form "B” to "C" within 48 hours.
The formation of the stone can further be slowed down by an additional increase in the
amount of water along with an additional decrease in the amount of calcium.
"D" and "E" curves in figure 5 show the effect of an increase of 20 percent and an
increase of 40 percent in the drinking water on the urine excretion. As the graph
shows the increase of the drinking water causes an oscillation in the excreted amount
of urine. The reason for this oscillation, is that both the capacity of the urine
excretory system and body's ability to keep more than equilibrium amount of water
for along time are limited.
The "C" curve of figure 6 shows the effect of a decrease of 5 percent in the taking
calcium on the calcium excretion. The decrease in the taking calcium causes a smooth
oscillation in the calcium excretion. The reason is that the body's ability to cope with
an amount of calcium less than equilibrium amount is limited.
The "D" and "E" curves in figure 7 show the urinary calcium when there is an
increase of 20 percent and an increase of 40 percent in the drinking water along with
a decrease of 5 percent in the taking calcium.
|
|
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i
it
Figure 4:
—--—---— Renal calculi(mg): ox Increase in Water Intake & OX Decrease in Calcium Intake
Renal calculi(mg): 5% Increase in Water Intake & 1% Decrease in Calcium Intake
————-— Renal calculi(mg): 10% Increase in Water Intake & 1% Decrease in Calcium Intake
&
&
seetee Renal calculi(mg): 20% Increase in Water Intake & 5% Decrease in Calcium Intake
Renal calculi(mg): 40X Increase in Water Intake & 5% Decrease in Calcium Intake
76. 7
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67.5) 5 pte SE
ae
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65. ‘ eee]
62.5]
68.
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TIME
Figure 5:
Urine Excretion(mI/hour): 0% Increase in Water Intake
—— — — Urine Excretion(mi/hour): 5% Increase in Water Intake
+7 Urine Excretion(m1/hour): 10% Increase in Water Intake
Urine Excretion(mi/hour): 20% Increase in Water Intake
Urine Excretion(ml/hour): 40x Increase in Water Intake
67.5)
63.7:
bmp
—Z. 8. 18. 28. 38. 48.
TIME
wT
——
a _
“Ghee
nee
Figure 6:
Calcium Excretion(mg/hour): 0x Decrea:
in Calcium Intake
— — — Calcium Excretion(mg/hour, in Calcium Intake
-—-—— Calcium Excretion(mg/hour): in Calcium Intake
11.
11.25] Ax
=
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a
~
11.2] ~ EA
\ —ae a
\
ia?
14.15 >»
Py Le -----4
18. 28. 38. 48.
TIME
Figure 7:
Urinary Calcium: 0% Increase
Urinary Calcium: 5% Increase
Urinary Calcium: 10% Increase
" Urinary Calcium: 20% Increase
Urinary Calcium: 40% Increase
188.
in Water Intake & Ox Decrease in Calcium Intake
in Water Intake & 1% Decrease in Calcium Intake
in Water Intake & 1% Decrease in Calcium Intake
in Water Intake 4 5¥ Decrease in Calcium Intake
in Water Intake & 5% Decrease in Calcium Intake
172.5] y
w OX
. ,
165. ‘ S Torre
=o
S oe:
157. r
158.
38. 48.
TIME
Figure 8:
10% Increase in Water Intake & OX Decrease in Calcium Intake
—-- —---— Renal calculi(mg): Adjustment Time of Water Intake = 1
—_—_-__——. Renal calculi(mg): adjustment Time of Water Intake = 3 (Normal)
—————- Renal calculi(mg): Adjustment Time of Water Intake = 6
40x Increase in Water Intake & 5% Decrease in Calcium Intake
Renal calculi(mg): Adjustment Time of Water Intake = 1
Renal calculi(mg): Adjustment Time of Water Intake = 3 (Normal)
Renal calculi(mg): Adjustment Time of Water Intake = 6
TIME
Figure 9:
10% Increase in Water Intake & 0%, Decrease in Calcium Intake
—--— Urine Excretion(mi/h): Adjustment Time of Water Intake = 1
Urine Excretion(mi/h): Adjustment Time of Water Intake = 3 (Normal)
Urine Excretion(ml/h Adjustment Time of Water Intake = 6
40% Increase in Water Intake & 5% Decrease in Calcium Intake
urine Excretion(mi/h): Adjustment Time of Water Intake = 1
Urine Excretion(ml/h): Adjustment Time of Water Intake = 3 (Normal)
Urine Excretion(mI/h): Adjustment Time of Water Intake = 6
71.25) E ‘
6?. - Sere Pera
63.75]
68.
-2 8 18. 28. 38. 48.
TIME
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The "D" and “E” curves in figure 4 show the mass of the formed stone in each of those
conditions. The rate of concretion aggravates when there is a decrease in the urine
excretion.
When the increase in the water taken is accompanied with a decrease in the
adjustment time, the amplitude of the oscillation in urine excretion will decrease and
that will make the rate of concretion slower (See figures 8 and 9).
Conclusion
A preliminary model has been developed to analyze the formation process of renal
stone. Given its structure, the model gives reasonable response to multiple external
input, as treatment, including different maintenance amount of calcium and water.
The model shows, when the taking of water increases drastically the excreted amount
of urine oscillates. t may aggravate the rate of concretion. The reason for this
oscillation in urine excretion is that both the capacity of the urine excretory system
and the body's ability to keep more than equilibrium amount of water for a long period
of time are limited. Shortening the drinking intervals will decrease amplitude of
oscillation in urine excretion and will cause slower rate of concretion.
The model also show, how concretion in the sufferer of renal stone ‘May continue even
though the amount of calcium in their urine is less than its amount in the urine of
normal people. The calculus decrease the threshold level of stone formation. It's
because the existence of stone increases the chance of collision between the calcium
particles. The enlargement of stone will continue because the amount of calcium in the
suffere's urine is still more than its threshold level.
References
-Bush J. W, et. al., 1985, “fluid Therapy in Acute Large Area Burns, A System
Dynamics Model", System Dynamics Review 1: 20-41.
-Estrup C, 1973, "The Dynamics of Body Metabolism", System Dynamics Group,
Mass, MIT.
-Forrester, Jay W., 1968, Principles of Systems, Cambridge: The MIT Press
-Foster, R. O., 1970, "The Dynamics of Blood Suger Regulation", MSc Thesis,
Cambridge, MIT.
-Netter, Frank, 1986, "Kidneys, Ureters, and Urinary Bladder", The CIBA
Collection of Medical lilustrations. CIBA.
-Richardson George P, Alexander L. Pugh Ill, 1981, Introduction to System
Dynamics Modeling with DYNAMO, Cambridge: The MIT Press.
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