THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 279
COMPUTER MODEL OF THE RENAL FILTER CONTROL SYSTEM
Roa, Laura
E.T.S.I.1., Universidad de Sevilla.
Cantero, A.
Becario de la C.A.1.C.Y.T.
Solis Galera, A.
Becario adscrito al Dpto de Cirujia Plastica y Quemados,
Ciudad Sanitaria Virgen del Rocio. Sevilla
Franco, A.
Dpto de Cirujia Plastica y Quemados. Ciudad Sanitaria Virgen
del Rocio. Sevilla.
SUMMARY
@ mathematical model and digital computer simulation of the
human renal filtration controls are herein developed.The
purpose of the model is to provide a method of analysing renal
filtration control hypotheses which cannot easily be tested in
an animal or human.
The method used in the construction of the model was system
dynamics.
We propose an original formulation for the numerous different
variables, eg. Bowman capsule pressure, glomerular
absorption, net filtration and other considered variables in
the model.
This model can simulate the dynamic, functions of variables
such as colloidal osmotic pressure, glomerular capillaries,
tubular filter,along with other variables of difficult
clinical determinants.
The model simulates disparate situations, such as the effects
of renal filtration variations of arterial pressure,
concentration of plasma proteins,
The results presented coincide with those of other authors.
INTRODUCTION
As is well known, the renal system regulates the longterm
distribution of the human bodily liquids. The importance of
the study of this regulatory mechanism is due as much to the
number of pathological causes which can give rise to a kidney
failure as to the critical clinical situations which are thus
created, bath of acute or chronic nature. Just as well- known
is the importance of renal filtration as one of the determi—
nant variables of diuresis, as is demonstrated in the ample
bibliography which exists on the subjet.
280 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
Amongst others, we can highlight the physiological study
carried out by Guyton (1974) giving an overall vision of the
regulating mechanisms which participate in renal filtration,
and also the exhaustive analysis carried out by Rose (1985)
from a physiopathological point of view. In all these
studies, the importance of two variables in the establishment
of renal filtration is emphasized : arterial pressure and the
concentration of plasma proteins, irrespective of whether
normal or pathological situations are being considered.
It is in this line of research into renal filtration which the
present study is included, but attempting to give a macrosco-
pic and dynamic vision of the different mechanisms which take
part in the regulation and interactions of renal filtration.
CONSTRUCTION OF THE MODEL
The techniques used in the construction of the model were
those of Dynamics of Systems (Forrester,1968; Goodman, 1974;
Aracil, 1984...).
The model was developed in two stages. The first stage dealt
with the control mechanisms which regulate the net renal fil-
tration. For this purpose the functioning of the renal system
has been considered as that of a nephrom. Figure 1 shows the
causal diagram corresponding to this stage.
To quantify the relations established in this first stage of
the model, the experimental data determined by other authors
has been used (Guyton,1976; Ganong,1982..). The conclusions
reached were the follawin
i- The pressure in the Bowman’s Capsule is considered as a
function of the net glomerular filtration at the afferent ar—
terial level, and the reabsorption at the efferent arterial
level.
2- The interstitial glomerular pressures are not taken into
account.
With this first version of the model, it is possible to ana—
lyse the behaviour of the net glomerular filtration (GNF) when
the variables of arterial pressure (AP) and the con-
centration of plasma proteins (PPC) are disturbed in iso-
lation.
However this type of alteration would not have a great phy-
siological significance since isolated alterations in the PPC
and AP variables are going to produce certain reacti-
ons in the regulatory mechanisms of AP which do not permit
an alteration in one of them whilst the other remains cons-
tant, given the relations between these variables. For this
same reason, neither would we be able to disturb simulta~
neously both variables fixing a priori certain percentage va~
riations.
In the second phase of the model the regulatary mechanisms of
the variable AP are added to the existing structure.
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 281
At this stage, the exchange of fluids between the different
compartments ( extra and intracellular) are studied when so-
me disturbance modifies the variables that we have named “net
liquid input", defined as the difference between the input and
output liquids by the mouth and the remaining vias (except the
kidneys) respectively, and the "filled factor", which permits
us to evaluate the alteration in the body fluids after an in-
fusion of any solution into the circulating fluid.
The regulatory mechanics which have been considered in this
stage are the following: the kidney’s long-term requlation of
arterial pressure, transcapillary fluid exchange, circulatory
adaptability, the limphatic system and the regulation of the
cardiac output by peripheral resistence.
In addition, the baroreceptor reflex system must be included
to show the influence of sympathetic tone on the cardiac out—
put, peripheral resistences, and the kidneys to alter urinary
output and mean circulatory pressure
Figure 2 show the causal diagram of the transcapillary fluid
exchange. Similar causal diagrams have been elaborated for
the remaining control mechanisms.
The state variables which determine the behaviour of the model
are the plasma, free interstitial and intracellular volume.
the interstitial and plasma protein, the extracellular and the
intracellular electrolytes.
Figure 3 show the Forrester diagram of the interstitial
compartment , similar Forrester diagrams have been elaborated
for the remaining causal diagrams.
The considered supositions for the variables quantification
wich takes places in this stage have similarity with the most
of the macroscopic hemodinamic studyes (Rudinger, 1966:
Warner, 1969 ; Leonard, 1975 3 Thain, 1967 ; Tanaka, 1979 3
Abbrecht, 1980 ; Kakinchi, 1981 : Roa , 1982 ; Venkatachalam ,
1978...).
In this form, a new version model can be perfomed which show
the evolution of renal filtred under the influence of the
long-intermediate and short-term control variables.
The function CVI have been incorporated to the program for the
simulation of the model. This function is an interpolacion
function used to determine the pressure of the right auricle
from the values of cardiac output (Pickering, 1969). The
control variables were incorporated into the model by CLIP
function which facilitates the accesibility of the model to
experts in the renal system even if they are not experts in
system dynamics or computer.
ANALYSIS OF SENSIBILITY
The singular perturbations methods has been used to analyse
the sensibility of the more significative state variables when
282 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
the following parameters are perturbed: Initial frecuency of
baroreceptor pulse (IFBF) and mean circulatory normal pressure
CHEN)
The results of this analysis show that the model is not
practically altered by perturbations of the parameter IFBF.
The effects of varying the parameter MCNP are negligible for
perturbations less that 10 per cent of the normal values of
this parameter and when the simulation time is short (less 10
minutes).
This methad show that the more significative state variables
return to near normal values at the end of the simulation when
a changes occurs in the interstitial or intracellular volume
within physiological limits.
A more detailed analysis of model sensibility is made by
Mantecarlo’s procedure.
The effect of change in the two parameters ‘MCNF (mean circu-
latory normal pressure) and initial values of the most sig-
nificant stated variables (plasma interstitial and intrace—
llular volume) have been analysed. Variations of 5% in both,
MCNP and initial values were produced in the model for va-
rious simulated times, TS, (TS=10, 180, and 340 minutes). The
results have shown that interstitial and intercellular volume
return to near normal values for TS= 180 minutes, while the
plasma volume returns for TS= 360 minutes
In conclusion, these results show that the model depends on
its own structure but is not practically altered by the par—
ticular values of the initial conditions and model parameters
when physiolagical conditions are considered.
SIMULATION RESULTS
In order to prove the validity and use of the constructed mo~
del different experiences have been simulated. The results are
presented of simulating the known experience of increasing the
net liquid intake from its normal value of 1 ml/min, to 3.5
ml/min. in an isotonic solution with the body fluids during a
period of six days.
Figure 4 shows the behaviour of the variable AP As can be
observed, this variable increases during the first two days
and later decreases and establishes itself at a value of
approximately 106 mmHg which allows it to eliminate by diure-
sis the excess liquid administered.
Figure 5 represents the behaviour of PPC,showing how while the
variable AP increases, the concentration of plasma pro- teins
decreases, and, after the second day, begins to recover
establishing it self a value lower than the normal.
The behaviour of these variables is justified if one observes
the evolution of the variables plasma volume (PV) Fig 6, — and
interstitial volume (IV) Fig 7. Very noticeable is the sharp
THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 283
increase of PV during the first day which later decreases ve—
ry slowly in spite of the fact that the disturbance is main—
tained. On the contrary, the increase of IV is slower due ta
the fact that the regulatory system of transference of li-
quids between the plasma and interstitial compartments is a
short-term mechanism.
Figure 8 represents the behaviour of the cardiac output va—
riable (CG) highlighting the fact that in the first two days
its evolution is parallel to that of the AP, and that after
the second day it decreases until approximately the fourth day
on which it reaches practically its normal value in spite of
the increase in the plasma volume. These results are in a~
greement with other reports in the references (Guyton et al
1973, Guyton 1976).
Figure 9 shows the dynamic evolution of the net filtration
variable (NGF). The parallel between the behaviour of this
variable and that of the arterial pressure is noticeable, a
result coinciding with those of other researchers which de-
monstrates the importance of the arterial pressure variable in
the establishment of the net filtration and the favorable
effect which the behaviour of the PPC exercises on the beha-,
viour of the NGF. :
CONCLUSIONS
Using the system dynamic approach a model has been construc
ted which is capable of predicting an integrated response fram
all the elements involved in the renal filtration.
The development of this model means, in our opinion, that it
is possible to make the following contributions:
A contribution to the physiopathological study of renal fil—
tration, from a macroscopic perspective which allows analysis
of the interactions between the different regulatory mecha—
nisms considered.
The knowledge through simulations of the qualitative and
quantitative behaviour of different variables difficult to
determine in daily clinical practice.
The possibilty of analysing the effects of different solu-~
tions on filtration.
The beginning of a line of research that will enable us, on
adding to the present model the mechanisms of tubular control
to make new contributios to the study of renal physiopa—
thology. *
Also worthy of emphasis is the general accessibilty of the
model to experts in the renal system, even if -they are not
expert in system dynamics or computer.
284 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
REFERENCES
@bbrecht, P.H. (1980). “Regulation of extracellular fluid
volume and osmolality" Annals of Biomedical Eng. ,vol.®8 pp
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Aracil,d (1984)."Intraduction ala dynamique des systémes"
Presses Universitaires de Lyon.
Forrester,J W (1968). “Principles of Systems". _Wringt-Allen
Press.
Ganong, F.W. (1982). "Fisiologia Medica", @ Ediccién. El Ma~
Aual Moderno S.A., Mexico D.F.
Guyton, A.C.,Jones, C.E.,Coleman, T.6. (1973). “Circulatory
Physiology: Cardiac output and its regulation" , W.B.
Sanders Co, pp. 298-300.
Guyton, A.C. (1976). “Textbook of medical physiology", W.B.
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Goodman, M.P. (1974). "Study notes in system dynamics"
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Kakiuchi,Y.; Nakajima,S.; Arai, T.3 Horimoto,M.3 Kikuchi,Y.+
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by changes in plasma oncotic pressure of anasthetized
dog". Japanesse journal of physiology.
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Leonard, J.1.; Abbrecht,P.H. (1973). "Dynamics of plasma-in-
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fusion in dogs". Circulation research,vol XXXIII
pp. 735-748.
Madden, A., De Deen, W. and Breener, B.M. (1974 )"Dynamics of
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Rose,B.D. (1985)."Fisiopatolagia de las enfermedades renales".
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Warner, H.R. and Russell, R.0., (1969). “Effect of combined
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THIS MATERIAL IS BASED ON WORK SUPPORTED BY THE C.A.I.C.Y.T.
UNDER GRANT 1078/84.
ty anrertan PRESSURE
GLOMERULAR PRESSURE
AFFERENT —_ GLOMERULAR PRESSURE
CAPILLARY EFFERENT
+|
GLOMERULAR FILTRATE
GLOMERULAR ABSORPTION
BOWMAN“S:
PLASMA COLLOID CAPSULE pear:
OSMOTIC PRESSURE PRESSURE ErOn.
+ FLOW
PLASMA PROTEIN
CONCENTRATION
NET GLOMERULAR RENAL
FILTRATE PLASMA
FLOW
+
TOTAL PROTEINS IN £ conorpar osworrc PRESSURE -
GLOMERULAR CAPILLARIES IN GLOMERULAR CAPILLARIES +
PLASMA FLOW
IN GLOMERULAR
CAPILLARIES
Fig. 1 : Causal Diagram of glomerular filtrate
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THE 1980 INTERNA TONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY, SEVILLA, OCTOBER, 1986
4 INTERSTITIAL
COLLOID osMoTIC
‘PRESSURE,
SL ARTERIAL,
PRESSURE
PLASMA COLLOID osMorIc
PRESSURE,
+
PROTEIN CONCENTRATION PLASMA PROTEIN:
/ a ‘CONCENTRATION
+
PROTEIN
FLOW oS
i >
‘TOTAL PLASMA
TOTAL INTERSTITIAL
AL ete (PRoTamN
Figure 2 : Causal diagram of transcapillary fluids exchange.
or
288 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
Figure 3. Forrester diagram of interstitial compartment
THE 1966 INTERNA TIUNAL CONFERENGE UP TE STOTEM UINAMILG OUUIETY. SEVILLA, UUIUBEN, 1900 2OU
110
PA
cy
Es
B 105
<
a
100
1 2 3 4 5 6
TIME (day)
Figure 4. Simulation results
for mean arterial
pressure (PA)
0.100
4a
Gi
30-075
ae CPP
a
3)
0.050
1 2 3 4 5 6
TIME (day)
Figure 5. Simulation results
for plasmatic pro-
teins concentration (CPP)
290 THE 1986 INTERNATIONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986
5.0
ve
a
4.0
§
3.0
1 2°03 4 «5 6
TIME (day)
Figure 6. Simulation results
for plasmatic volu
me (VP)
13.5
13.0
3g
g
2 VLINT
Bias
12.0
1 2 3 4 °&«5 6
TIME (day)
Figure 7. Simulation results
for interstitial
fluid volume (VLINT)
THE 1900 INTERNA TONAL CONFERENCE OF THE SYSTEM DINAMICS SOCIETY. SEVILLA, OCTOBER, 1986 291
5.50
6c (1/min)
w
nu
a
Gc
5.00
1 2 3 4 5 6
TIME (day)
Figure 8. Simulation results
for cardiac output (Gc)
160
£GN
c
4
E
£
3
zg
~ 140
z
6
&
120
Ltt
4 5 6
TIME (day)
Pigure 9. Simulation results
for net glomerular
filtrate (FGN)
