A Dynamic Simulation Model for Long-Term Hypertension
Progression
Firat Incioglu
M.S. Graduate from Industrial Engineering Department
Bogazici University
Saral Sitesi D Blok 19 Gayrettepe 34349, Istanbul, Turkey
Telephone: +90 (532) 321-7850
E-mail: firat.incioglu@ gmx.net
1 ABSTRACT
Dynamics of blood pressure over the life span of a human being demonstrates a
growth path. The most significant theories which try to explain dynamics of blood
pressure adopt a kidney-dependent approach. Structural reductions in the size of renal
arterioles (vascular remodeling) and loss of nephrons are considered to be primarily
responsible for the progressive increase in blood pressure. A dynamic simulation model
is constructed to realistically reproduce the long-term progression of blood pressure in
healthy and in hypertensive people. It focuses on systemic interactions that result in
vascular remodeling of renal arterioles and in loss of nephrons. These hypertensive
mechanisms are integrated with fluid volume and blood pressure control mechanisms
which are aimed to achieve homeostatis. This study suggests that progression of blood
pressure can suitably be modeled by conceptualizing the problem as a long-term
control of fluid excretion capacity. The reference behaviors for normal and hypertensive
people underline alternative pathways in blood pressure progression. Experiments with
the model demonstrate that management of the number of remodeled arterioles over
time should be an essential task in long-term blood pressure progression control.
Scenario runs with the simulation model help distinguish successful policies from the
ineffective interventions.
2 INTRODUCTION
Sustained elevation in blood pressure (BP) above an average mean arterial
pressure of 100 mmHg is considered as hypertension. Most cases of hypertension (95
per cent of cases) are estimated to be of an unknown cause, i.e. there is not a specific
clinically identifiable cause which is responsible for chronic elevation of BP. This type
of hypertension is called essential or primary (since hypertension is not secondary to
another condition, such as renal artery stenosis, obesity, stress or pregnancy). The
increase in blood pressure over the life span of a human being is an inevitable result of
the aging process. As Arthur Guyton, the pioneer in applying control systems to
physiology, wrote, “presumably, if all persons lived to infinite age, everyone might
eventually develop a consuming state of hypertension..” (Guyton, 1980, p. 471). The
natural progression of BP to elevated levels does not pose a great threat to most people
as they die of other causes prior to experiencing lethal consequences of hypertension.
However, the accelerated growth behavior of BP observed in essential hypertension is a
big threat to young and middle-aged people as it leads to target organ damage in vital
organs such as heart, kidneys and brain.
The measured BP of a human being is the end result of interactions of a number
of physiological control systems. Although BP is in a constant state of adjustment via
these systems to respond to the current needs of the body, the long-term level of BP is
proportional to the amount of water in the body. Thus, as Guyton convincingly
demonstrated in his system analysis work, the long-term level of BP is to a great extent
determined by kidneys’ ability to excrete salt and water from the body (Guyton, 1980).
There is consensus in the field of hypertension that progression of BP over time
is caused by structural changes in the kidneys which affect kidneys’ salt and water
excretion. These changes can best be understood by focusing on the nephron, the self-
sufficient functional unit of the kidney. Each nephron undergoes structural changes that
affect its excretion capacity. The most significant changes that take place in the nephron
are the remodeling of its afferent arteriole, which reduces the cross-section of the blood
vessel connecting the nephron to the circulatory system, and glomerularsclerosis,
scarring incurred at the glomerulus of the nephron. Both of these changes lead to
decreased filtration and excretion from the nephron. The loss in excretion due to
structural changes is compensated by healthy nephrons which can increase their
excretion by adaptive functional (i.e. non-structural) increases in the cross-section of
their afferent arterioles. The compensation takes place over renin-angiotensin system by
adjusting the amount of renin secreted from each healthy nephron. As long as the loss of
excretion capacity can be compensated by the adaptive responses of healthy nephrons,
long-term level of BP can be maintained at normal levels.
What causes these structural changes is still not exactly established, but in
general there seems to be consensus that hypertension itself may be responsible (Lever
and Harrap, p.70 in Kaplan, 1998). In the case of glomerularsclerosis, the glomerulus is
scarred as a result of natural aging process and/or increased BP sustained over long
periods. Remodeling of arterioles is also believed to be caused by elevated BP.
However, recent work demonstrates that remodeling is caused by frequent or persistent
constriction of the afferent arteriole independent of BP (Johnson et al., 2005). In
infusion experiments it was shown that a variety of vasoactive substances can lead to
remodeling when their elevated levels are sustained over time.(Johnson et al., 2005).
The renin-angiotensin system (RAS) which is responsible for functional regulation of
arterioles by controling the concentration of such a vasocative substance, angiotensin II,
is most likely to be involved in mediating remodeling (Schiffrin, 2004; Johnson et al.,
2005).
In medical literature essential hypertension is characterized with remodeling of
renal arterioles (Sealey et al., 1988). Yet, the increase in BP of a typical essential
hypertensive case is not caused by healthy nephrons’ insuffcient excretion capacity to
compensate for decreased excretion of remodeled nephrons. It is rather caused by
remodeling’s effects on angiotensin II (Ang II) levels which interfere with healthy
nephrons’ ability to increase their excretion. The interference takes place over the renin-
angiotensin system in the following way. Each nephron regulates its renin secretion in
order to achieve a concentration of Ang II which is consistent with the nephron’s own
Ang II need. However, each nephron consumes an amount of Ang II proportional to
global Ang II concentration, which is determined by the cumulative renin secretion of
all nephrons. In a kidney that has a non-uniform composition with normal and
remodeled nephrons, normal nephrons suppress their renin secretion, whereas
remodeled nephrons secrete high amounts of renin. Thus, in a person with such kidneys,
the long-term global renin may settle to a high level that inhibits normal nephrons’
capability to achieve compensatory increase in their excretion. The sustained elevations
in BP of essential hypertensive people are believed to be the result of such improper
global renin levels interfering with adaptive functional changes of normal nephrons.
(Sealey et al.,1988). Whether global renin reaches such disruptive levels depends on
proliferation of remodeling in the kidney and on the severity of remodeling of each
arteriole since renin secretion per remodeled nephron is inversely proportional with its
remodeling grade.
3 KEY FEEDBACK RELATIONSHIPS OF BP PROGRESSION
Recent work which suggests that remodeling develops due to frequent
vasoconstriction provides a possible alternative explanation for accelerated progression
of BP in essential hypertensive people. High levels of global renin in people who have
kidneys with non-uniform nephrons may intiate further remodeling of healthy nephrons.
In other words, once the initial injury to a significant number of arterioles is acquired,
further injury may be reinforced over renin-angiotensin system (RAS). A reinforcing
mechanism involving RAS was previously proposed by Guyton as an attempt to explain
fast progressing hypertension, malignant hypertension (Guyton and Coleman, 1969).
However, their model focused on a short-term positive feedback between renin and
functional constriction of arterioles, but not on the structural changes caused in the
arterioles. Unable to maintain high levels of renin in a uniform kidney model Guyton
commented that such a positive feedback was unlikely to be the primary cause of
progression of BP (Guyton, 1980). However, he left the door open to the possibility that
once structural changes occured such a positive feedback could be effective. This study
explores the possibility of such a positive feedback in essential hypertension.
The proposed mechanism involves nephrons coupled with remodeled arterioles,
denoted as remodeled nephrons, which secrete very high amounts of renin.
Unsuppressed secretion of renin leads to high Ang II levels in the blood. Over time, the
state of vasoconstriction caused by higher than normal Ang II levels leads to remodeling
of healthy nephrons (Schiffrin, 2004; Johnson et al., 2005). Since renin secretion from
remodeled nephrons is higher, the proliferation of remodeled nephrons increases global
Ang II levels further (Please refer to (+) remodeling loop in Figure 3.1)
In the other positive feedback mechanism, healthy nephrons die due to the natural
aging process. As a result water excretion capacity of kidneys decreases. As fluid
volume increases, the pressure exerted upon remaining nephrons increases. Overload of
pressure makes nephrons more susceptible to experience injuries and become obsolete.
Hence, this positive feedback loops results in the reduction of number of alive nephrons.
(Brenner and Chertow, 1994; please refer to (+) nephron loss loop in Figure 3.1)
Any changes that affect long-term control of excretion are compensated by
functional regulation of renal arteriole over the renin-angiotensin system. The goal of
this system is to maintain fluid volume at its normal physiological level. (Guyton and
Hall, 2000; Laragh, 2002; please refer to (-) FV-RAS loop in Figure 3.1).
The model integrates the main long-term mechanisms involved in the
progression and control of BP over the life span of a human being. The goal of this
study is to demonstrate the differences between the dynamics of key variables of normal
and essential hypertensive subjects. The study is also intended to facilitate
experimentation for policies which can slown down the dangerous progression of BP in
essential hypertensive subjects.
pal od <r
pe ~ Blo ex
ee Pressure ra MK
Glomenulat* 4 y
- Pressure / We Total Renin
A Ange pese Secretion RM,
: Fluid 7 .
) Ww aerlnake Fyne / (
/ ~ i | ( remot A \
SO { 4 Fy-Ras| Ria \ gt
| ib \ \ "/ Angiotensin \ / |
" [ 4Nephron Loss _ Water —/ 0 NY |
Glomerular \ } Excetion ( /
sclerosis ae Ze { \s /
\ \ Renal Arteriole
\ \ ‘Vasoconstriction Remodeled
_ \ / Nephrons (R.M.)
q Ne jf See a
GER / +
. ak at
™ A
“* Normal ss Se.
Nephrons ———~ ee See?
Figure 3.1.Integrated CLD of key feedback loops in the model
4 MODEL DESCRIPTION
The model is composed of three main stocks: Fluid Volume, Normal Arterioles and
Remodeled Arterioles. The latter two are also denoted as normal and remodeled
nephrons, respectively. Fluid Volume is the main determinant of long-term blood
pressure. The model can be analyzed in two sectors, Fluid Volume and Nephron.
Normal physiological values from medical textbooks are used for normal values of
constants (Guyton and Hall, 2000). The model represents a human being with two
million nephrons who has not experienced any loss of nephrons due to aging.
4.1 Fluid Volume (FV) Sector
This sector has a single stock, Fluid Volume. Its inflow and outflow are the
weekly water intake into the body and water excretion from the kidneys, respectively.
Evaporation is considered to be negligible and water intake is taken as an exogenous
variable in the model.
Normal filtration and excretion of each nephron is proportional to the amount of
water in the body. For each nephron type, FV and water excretion are involved in a
negative feedback loop over normal single nephron glomerular filtration rate (normal
snGFR), which corresponds to filtration from a nephron for the current level of FV. This
loop ensures that there be zero filtration when there is no water in the body.
FV is also involved in the goal-seeking FV-RAS loop of normal nephrons. The
goal of this structure is to maintain FV at its normal physiological level despite changes
in the number and type of nephrons as well as changes in water intake. Normal snGFR
is compared to required snGFR of each normal nephron, single nephron filtration
necessary to maintain FV at its normal level.
Renin-Angiotensin system is instrumental in adjusting actual single nephron
filtration, Actual snGFR, to required single nephron filtration, Required snGFR, because
of its role in the regulation of renal arteriole resistance. Glomerular filtration rate of
each nephron is proportional to the size and inversely proportional to the resistance of
its renal arterioles. Renal arteriole resistance can be adjusted by changing the amount
angiotensin II consumed by the arteriole. Renin secretion of each nephron is determined
by the nephron’s own required Ang II consumption necessary to achieve the required
arteriole resistance (Sealey et al., 1988). Based on this notion the model uses a sequence
of equations that retums an actual renin degeneration rate per nephron for a given ratio
of the required snGFR to normal snGFR. As part of this formulation there is a pair of
graphical functions, required single nephron renin secretion over normal single nephron
renin secretion (Required sn Ren Sec over normal snRen sec), and effect of renin per
nephron on snGFR (E of RpN on snGFR N.N.) which are each other’s inverse functions
(Figure 4.1. and Figure 4.2). In the case of a single normal nephron population the
output of the first function becomes the input of the second function as long as required
snGFR is attainable within the maximum and minimum physiological limits of renin
secretion. Thus, formulation pair ensures that indicated snGFR after adjustment by
arteriole resistance equals the Required snGFR.
ir Gutput — z Ym
046 6 “ c=
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oes6 [2697 H
ora [7.835
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1.088 0.8318
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frase Joan | Ab
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New 2 i Q bl
=a |
_Immport Vals_| x -min:|0 _z|xe1.387 yo2.421 Xmax|4 + |_ Reset Sealine
Figure 4.1. Required single nephron renin secretion over normal single nephron renin
secretion as a function of required to normal snGFR ratio
Required single nephron renin secretion over normal single nephron renin
secretion is checked against the minimum and maximum single nephron renin
contribution factors. The result is then multiplied with normal renin contribution per
normal nephron to give adjusted required single nephron renin secretion.
Total required renin secretion by normal nephrons is calculated by multiplying
adjusted required single nephron renin secretion by total alive nephrons. In the presence
of two nephron subpopulations, normal nephrons adjust their total renin secretion by
taking into account the total renin secretion from remodeled nephrons. The renin
contribution from each nephron group is added together to give Plasma Renin. Since
renin is produced locally but global Ang II is consumed by each nephron, the weighted
average for renin degeneration rate per nephron, Renin per Nephron, is calculated by
dividing Plasma Renin to the total number of alive nephrons.
fT Uutput
(0.02 4 =
foorsae [3.281
lor74a [24g1
fozes4 [2105
(0.6667 :
0.8318 [1.088 ee ae aaa ia
1 1 i
ier [ost2s a ore
fies [ove Teen
New a H : |
—
Import Vals | _x-min:|0 sr lxe01468 93.93 Xmax|6 >| Reset Scaling |
Figure 4.2 Effect of renin per nephron on single nephron glomerular filtration rate
Effect of renin per nephron on snGFR = f (Renin per nephron / normal renin per
nephron)
Resistance Adjusted Indicated snGFR = normal snGFR * Effect of renin per nephron on
snGFR
Resistance adjusted indicated snGFR represents the glomerular filtration rate of
anormal nephron after the indicated adjustment by renal arteriole resistance. Its value is
checked against the maximum physiologically possible snGFR, Max snGFR capacity
(Figure 4.3).
Actual snGFR = Max snGFR capacity * Effect of Capacity on snGFR
Effect of Capacity on snGFR = f (Resistance Adjusted Indicated snGFR / Max snGFR
capacity)
ie va = Sane
Lg a aaa 1 ¥
Ca e—
O7 O7 dl
fo7es_[a77e3 K
0.8807 |0.8509 Z
1.018 0.9167 #
1.141 [0.9693 fe
13 1 Zi
=a /
mee 4 ¥
= ff “nin:
New Z e =]
a
Import Vals | Xmin{0 | x=0.485 v=1.018 Xmaxl1.3 _~| Reset Sealing
Figure 4.3 Effect of maximum renal arteriole capacity on snGFR
Single nephron water excretion, sn Excretion, for each nephron type will be
calculated from Actual snGFR by multiplication with a constant that represents the
normal physiological ratio between water excretion and glomenular filtration rate. Total
Water Excretion is equal to the sum of water excretion from all nephrons.
The variable representing measured blood pressure, BP Output, is not involved
in any of the feedback loops of the model. In reality, BP equals cardiac output times
total peripheral resistance. BP output is formulated as a multiplicative effect formulation
with FV over Target FV and effect of renin on BP. Effect of Renin BP is an s-shaped
function which has a positive relationship with Plasma Renin over Normal Renin. This
function represents the effect of Ang II on the resistance of other non-renal vessels in
the body. However, effects of Ang II on peripheral resistance is allowed to affect BP
only minorly, since long-term BP is to a great extent determined by the level of FV
(Guyton, 1980)
BP Output = Normal Set BP* FV/Target FV*Effect of Renin on BP
Effect of Renin on BP= f (Plasma Renin/Normal Renin)
Yeni
| fos >
io fis — =
New
a |
Import Vals | x -min:|0 x} x4.373 y=1.204 Xmax|10 _»]_Reset Sealing
Figure 4.4.Effect of plasma renin on the resistance of non-renal vessels
'
4.2 Nephron Sector
Human beings are born with a fixed number of nephrons (approximately 2 million
nephrons); nephrons do not regenerate after birth.
Nephron Sector is comprised of two stocks, Normal Arterioles and Remodeled
Arterioles. In reality, various grades of remodeling would be present in a kidney.
However, in this model remodeled nephrons are classified only with a single grade.
Remodeled nephrons are considered to be incapable of regulating their renin secretion
and excretion, whereas normal nephrons can adapt their excretion over FV-RAS loop.
Thus, arteriole resistance and renin secretion is assumed to be constant for all remodeled
nephrons. To account for excretion by remodeled nephrons, required total GFR by
normal nephrons is set equal to the difference of required total GFR and total normal
GFR from remodeled nephrons.
The presence of remodeled arterioles makes it possible for the positive
remodeling feedback to be initiated. Renin per nephron is the variable responsible for
initiating conversion of normal arterioles to remodeled arterioles. If Renin per Nephron
rises above the threshold level, which correponds to normal physiological renin
degeneration rate of normal arterioles, remodeling of normal arterioles will be initiated
(Figure 4.5). There is a delay between functional constriction of a normal arteriole and
its conversion to a remodeled arteriole. This delay is reflected by the Average
Remodeling Stimuli N to M stock. This stock accumulates past remodeling stimuli and
updates Arteriole Conversion rate with an average delay of two weeks.
Arteriole Conversion = Normal Arterioles * max conversion fraction from normal to
remodeled nephons * Average Remodeling Stimuli for
conversion
Change in Remodeling Stimuli = (Effect of renin per nephron on functional resistance -
Average remodeling stimuli for conversion ) /
remodeling delay
Soma:
o
nT
THe
Hi
ii
1
i
I I
Import Vals | %-min:|0 vl e1.124 v2.07 Xmaxl21 —-»| Reset Scali
Figure 4.5. Effect of renin per nephron on functional afferent resistance
The effect function ensures that remodeling will only be initiated above the
normal physiological level of constriction. When Renin per Nephron is approximately
twice as great as the remodeling threshold RpN, this function retums the maximum
possible remodeling stimuli which corresponds to maximum arteriole conversion rate.
Max conversion fraction from normal to remodeled nephrons represents the
fraction of normal nephrons per week that would become remodeled under maximum
remodeling stimuli. In animal experiments, significant remodeling was initiated in a
couple of weeks under extremely high Ang II infusion (Franco et al., 2001). However,
there is a significant difference between Ang II infusion experiments and normal
physiological conditions. Even the highest Ang II levels under normal physiological
conditions for spontaneously hypertensive rat (SHR), which characterizes essential-
hypertensive subjects, are much lower than Ang II levels attained in infusion
experiments. Since there are no experiments on human subjects, max conversion
fraction N to M is a highly uncertain parameter.
The nephron stocks also decrease through their loss rates. For each nephron type
there is a natural death fraction and a pair of effect functions which represent the effects
of glomerular pressure on death rate. Actual snGFR of each nephron represents the
glomerular pressure exerted upon the glomerulus. If Actual snGFR is above its normal
physiological level, the normal loss rate is multiplied by a monotonically increasing
function of Actual snGFR (Figure 4.6). On the other hand, Effect of low blood flow on
nephron loss rate represents the detrimental effects of low glomerular pressures on
nephrons (Figure 4.7). Although these effect functions are formulated for both stocks,
only Effect of high blood flow on nephron loss is applicable for normal nephrons for
actual operating ranges of the model. On the other hand, remodeled nephrons mostly
loose their nephrons because of low actual snGFR (Figure 4.7)
Nephron loss rate = Arterioles * normal nephron loss fraction * Effect of high blood
flow on nephron loss rate * Effect of low blood flow on nephron
loss rate
Effect of high blood flow on nephron loss rate =f (Actual snGFR /Normal snGFR)
ig utput vinden
La 1 S H a x
a :
na
a '
| EL H
| Oa bs : Me
i706 |i1sa : es
fioss (7.333 ; wt
2339 (fts79 yn ee eee
267 [1.942 H
305s 2211 =| ‘mine
New 0 >
a |
Import Vals X-min:|0 x) x=4.596 y=3. 333 Xmax|45 | Reset Scaling
Figure 4.6. Effect of high blood flow on nephron loss rate of normal nephrons
Effect of low blood flow on nephron loss rate =f (Actual snGFR / Normal snGFR)
Input Output
0 3 z
[o.08257
2.934
Yeman
Import Vals a jo cx}0.4312 ——y=0,08579 Xmaxe|1 _x| Reset Seal
Figure 4.7. Effect of low blood flow on nephron loss of remodeled nephrons
Normal nephron loss fraction represents the average lifespan of a normal
nephron. In normal subjects, under healthy conditions initial nephron number is to a
great extent preserved up until 4th decade of life. After 30’s people start loosing their
nephrons at a rate of approximately 1 per cent per year (Guyton and Hall, 2000). Thus,
normal nephron loss fraction is set to 0.0005 / week. On the other hand, remodeled
nephrons have a higher normal physiological death rate. The difference is due to the fact
that remodeled arterioles cannot achieve effective functional regulation that responds to
short-term variations in glomerular pressure. As a result, remodeled nephrons are
considered to be more vulnerable to high and low pressures (Johnson et al., 2005).
Remodeled nephron loss fraction is set to two times of normal nephron loss fraction
(0.001/week). The uncertainty regarding these parameters will be explored in scenario
analysis section.
10
E of Reninon
BP
7 Required/normal , Required snGFR
7 see ae. - SCHRNINE aks ‘Required Total
_/~contibution per lie exbtei’Sec, <_ GFR NN.
meals Required snRen’S 4
over normal snRen Sec
in fisted RE
minsn ren sec Adjtisted Réquired
fraction IV.N. oN Rat Soc
Plasma Kerin
Required by N.N nomnal renin
contribution per
RM
‘Additional Desiréd
Excretion
marten)
3
Excretion FV/Target,
FV
Required tofal Renin __ waterinake
nis Contibution
: > Total Water
E of Max Capacity Excreyog (yea)
M.
on snGFRR
sn Excretion /
. Exerc ws Eyer
AajSte Plasma NN
rae / /
¢ / max snGFR /
bil oe ( capacity NIN.
Nephion ep oeayy | YS
\ SAGER NN. \ min snGFR
= Ffsistanct Adjusted | Sey
Indicated snGFR N.N.
~E of Max Capacity - Eifect of Low Blood”
remodeli +
threshold RpN- Effect of RpN on onsnGFR NWN, jow on Nephron Loss
Functional Afferent Rate RM.
Resistance N.N. f (® a
Average Effect of Low Blood
jangeim® ——] _ Remodeling nae ‘on Nephron Loss
“remodeling N to M_ | Stimuli N to M | Rae Arteriole RE
“& conversion
Effect of High Blood
remodeling delay a
NtoM mner anit Effect of High Blood = 4} Flow on Neptton Loss .
TimitN-N.—— "Flow on Nepthon Loss max RateR.M, Tembdeled nephron
ate convertion recated ment
- aheaW i oss fracto
Noa St max shGFR
ane, Agengle TimitM.
normal nephron. ‘Nephrons
fos fection Nephron
Figure 4.8. Stock-Flow Diagram
5 VALIDATION AND ANALYSIS OF THE MODEL
Verification and validation was carried out on the model using the standard procedures
(Barlas, 1996). Structure-oriented behavior tests confirmed that each key loop, FV-RAS,
remodeling and nephron loss operate as expected. Due to lack of availability of real data on
progression of blood pressure over the course of years in either normal subjects or essential
hypertensive categories, behavioral validation could only be conducted in comparison to
described behavior in medical texts (Guyton and Hall, 2000). The base behavior for normal
people also serves for the behavior validation test of the model.
Base cases for normal, potential-hypertensive and essential-hypertensive subjects are
demonstrated to characterize different paths of progression. The difference between normal
and hypertensive subjects arises from the initial presence of remodeled nephrons. In real life,
the presence of an initial remodeled subpopulation may be the result of a short-term injury or
it may be congenital (Johnson et al, 2005). Potential-hypertensives differ from essential-
hypertensives only by the lower renin secretion from their remodeled nephrons.
Table 5-1. Initial conditions and parameters of base cases
a a normal Renin
Different Base Cases: rer al aerate Contribution per
Remodeled Nephron
1.4* normal Renin
Normal Subjects 2000000 0 Contribution per
Normal Nephron
1.4* normal Renin
Potential-Hypertensive 1600000 400000 Contribution per
Normal Nephron
ra
Essential-Hypertensive 1600000 400000 ce
5.1 Base Behavior for Normal Subjects
People after 4th decade of life are estimated to loose 10 per cent of their nephrons in
every ten years because of aging of nephrons and other conditions such as benign
nephrosclerosis (Guyton and Hall, 2000). The first run of normal subjects (NN ref) represents
a similar scenario. The subject looses about 65 per cent of nephrons over 1730 weeks (35
years) (Figure 5.2). Interestingly, fluid does not start accumulating in the body until after
week 1500 (30 years) when nephron number is reduced by almost 60 per cent (Figure 5.2 and
Figure 5.3). This is consistent with real examples since people who have lost as much as 70
per cent of nephrons can maintain normal excretion of water (Guyton and Hall, 2000). The
subject in the first run demonstrates a normal subject who develops hypertension over 35
years as a result of significant nephron loss. The subject in the second run (NNref2) has a
smaller normal nephron loss fraction and represents an alternative, a slower progression of
nephron loss (Figure 5.2). The dynamics seen in normal subjects will be analyzed on the first
tun since the second subject does not experience any rise in BP within the time frame of
simulation.
BP Output Nommal Arterioles
203.11 2M
176.10 ‘E67:
1.342M
149.10
1.013M
122.09
684,758
95.08 Fepo_5 0 173° 346 «6519 «692 865 1038 1211 1384 1557 1730
0 173 346 519 692 865 1038 1211 1384 1557 1730 Hime ee’)
Time (Week) Nommal Arterioles : NNref
BP Output: NNref Normal Arterioles : NNref2
BP Output: NNref2
aug 126.15
23.53 115.98
Fi 5.2. D ics of Normal Arteriol
Figure 5.1. Dynamics of blood pressure ‘gue Yynamics of Norm enoles
The stable dynamics of FV up until week 1500 is due to the fact that remaining normal
nephrons increase their excretion to compensate for excretion lost by each dying nephron
(Figure 5.3). Only when surviving nephrons approach their maximum filtration capacity their
compensation becomes imperfect and actual snGFR starts falling below required snGFR
(Figure 5.7). Once maximum snGFR capacity (0.25 ml/day) is approached, BP rises to
hypertensive levels within 200 to 300 weeks (4-6 years) as demonstrated by the behavior of
BP after week 1500, after 55 years of age (Figure 5.4).
Fluid Volume BP Output
136.32
a
19.26 105.81
ery
15 : 95.64 eee el
173° 346 «519 692) 865) :1038 «1211 1384 1557 1730 1250 1300 1350 1400 1450 1500 1550 1600 1650 1700
Time (Week) ‘Time (Week)
Fluid Volume : NNref is BP Output : NNref
Fluid Volume : NNref2 x BP Output : NNref2
Figure 5.3. Dynamics of fluid volume Figure 5.4. Progression dynamics of
blood pressure
Nephrons respond to the need for higher snGFR by decreasing their renin secretion
which allows them to increase their actual snGFR above their normal snGFR (Figure 5.5,
Figure 5.6, Figure 5.7) A key observation is that there is an exact match between required
single nephron renin secretion and renin per nephron (Figure 5.5 and Figure 5.6). The tight
regulation of required and actual renin is not always the case as it will be demonstrated in the
run for essential hypertension.
Required snRen Sec over normal snRen Sec
RpN over NRpN
0.1194
0 173 346 519 692 865 1038 1211 1384 1557 1730
Time (Week)
Required snRen Sec over normal snRen Sec : NNref
Required snRen Sec over normal snRen Sec : NNref2
Figure 5.5. Dynamics of required single
nephron renin secretion
0 173 346 519 692 865 1038 1211 1384 1557 1730
‘Time (Week)
RpN over NRpN : NNref
RpN over NRpN : NNref2 —2
Figure 5.6. Dynamics of renin per nephron
Dynamics in normal subjects is characterized by initially stable, well-controlled BP
and declining renin per nephron and compensation by remaining nephrons. Only after
significant nephron loss does the water excretion fall below water intake (Figure 5.8.)
snGFR
0.5618
0.3259
0.09 _/J
0 346 692 1038 1384 1730
Time (Week)
“Actual snGFR per N.N." : NNref mil/day
“Required snGFR N.N." : NNref mil/day
nomal snGFR : NNref 3 + 3 3 ml/day
Figure 5.7. Comparative Dynamics of snGFR
5.2 Base Behavior for Potential-Hypertensives
Water Flow
10.08
9.833
9.586
0 346 692 1038 1384 1730
Time (Week)
water intake : NNref +——t—-2—z_ L/Week
water excretion: NNref ——2—2—_ L/Week
Figure 5.8. Water balance
The BP dynamics of potential-hypertensive subjects is primarily driven by nephron loss
from normal and remodeled nephrons. Reduction in filtration capacity can be compensated by
functional vasodilation of normal arterioles similar to the case of normal subjects. In order to
increase their snGFR, remaining nephrons reduce their own renin contribution. The dynamics
of key variables are similar to the case of normal subjects. FV and BP are maintained near
target values for long periods of time. The only difference from normal subjects is the
coexistence of a decreasing remodeled subpopulation (Figure 5.11). Dynamics of the
reference case of normal subjects (NNref) will be presented along with the dynamics of
potential-hypertensives to demonstrate these differences.
BP Output Nommal Arterioles
2M
1.669 M
1,338 M
121.26 1.007M
a — 265 530. 795 1060 1325 ae
5
Time (Week) 0 265, —— wis 1060 1325
BP Output: NNref Nonmal Arterioles : NNref
BP Output : Potential hypertension case ‘Normal Arterioles : Potential hypertension case
Figure 5.9. Dynamics of blood pressure- Figure 5.10. Dynamics of normal nephrons-
potential hypertension potential hypertension
Blood pressure remains around its normal level for up until week 1060, 20 years
(Figure 5.9). The dynamics of BP in potential-hypertensives are indistinguishable from
normal subject for the first 20 years. However, since potential-hypertensives need to
compensate for greater loss of filtration than normal subjects, their renin per nephron are
lower throughout the simulation (Figure 5.12)
400,000
300,000
200,000
100,000
0
Remodeled Auterioles : NNref
Remodeled A terioles : Potential hypertension case
Remodeled Arterioles RpN over NRpN
1
0.794
0.5589
0.3384
0.1179
0 265 530 795 1060 1325 0 265 530 795 1060 1325
Time (Week) Time (Week)
RpN overNRpN : NNref
RDN overNRpN : Potential hypertension case
Figure 5.11. Dynamics of remodeled Figure 5.12. Dynamics of renin per nephron-
nephrons-potential hypertension potential hypertension
In potential-hypertensive subjects, actual snGFR can exactly match required snGFR up until
week 1060 (Figure 5.13). Even though remodeled nephrons secrete higher amounts of renin,
this poses no problem for normal arterioles’ functional adjustment. The high renin
contribution from remodeled nephrons can be counterbalanced by reductions in normal
nephrons’ own renin contribution. However, when renin secretion by remodeled nephrons is
very high, even total elimination of normal nephrons’ renin secretion may not be sufficient to
increase Actual snGFR to match the required snGFR as it is the case in essential hypertension.
The dynamics key variables and water balance are similar to the case of normal subjects.
Nevertheless, subjects with significant remodeled nephron subpopulation are classified as
“potential” hypertensive, because their excretion capacity will be approached earlier in life.
snGFR
Water Flow
0.5547
10.08
0.4385
0.3223, 9.852
0.2061
9.625
ae ee 0 265 530 795 1060 1325
Time (Week) ° ° Time (Week)
"Actual snGFR per N.N." : Potential hypertension case. —2—2—2—2—+.
"Required snGFR N.N." : Potential hypertension case. —2—2—2—2—2—
normal snGFR : Potential hypertension case —3—3—3—3—3—3 3
Figure 5.13. Comparative dynamics of
snGFR-potential hypertension
water intake : Potential hypertension case _L/Week
water excretion : Potential hypertension cask/W eek
Figure 5.14 Water Balance- potential
hypertension
5.3 Base Behavior for Essential- Hypertensives
Although the number of remodeled nephrons is the same for potential- and essential
hypertension case, in essential hypetension the renin secretion from remodeled nephrons is
high enough that FV-RAS control mechanism of normal nephrons is distorted at the
beginning of simulation. The dynamics of key variables significantly differ between the two
cases, because the initial levels of renin instigate the positive remodeling loop.
The BP of essential-hypertensive patient (run1) rises above the lethal level of 200
mmHg within 160 weeks (Figure 5.15). This case demonstrates a hypertension patient who
starts in the incipient stages of essential hypertension and is treated with any medication.
BP Output
0 20 40 60 80 100 120 140 160 180 200
‘Time (Week)
BP Output : Essential hypertension case. 3333-2
BP Output : Potential hypertension case. 22222 222
Figure 5.15. Dynamics of blood pressure- essential hypertension
‘Nonmal Arterioles Remodeled A terioles
16M 968,742
1.388 M 794,388
1.176 M 620,033
965,050 445,678
753,400 271,324
0 20° 40° 6080100120 140 160 180 200 0 20 40 6 80 100 120 140 160 180 200
Time (Week) Time (Week)
Nomnal Atterioles : Essential hypertension case
Nomnal Arterioles : Potential hypertension case
Remodeled A iterioles : Essential hypertension case. —3—3—+ 23+
Remodeled A terioles: Potential hypertension case. —2—2—2—2—2—
Figure 5.16. Dynamics of Normal Figure 5.17. Dynamics of Remodeled
Arterioles-essential hypertension Arterioles- essential hypertension
The precipitous fall in Normal Arterioles does not result from loss of nephrons as it
does in the potential hypertension case. It results predominantly from conversion of normal
arterioles to remodeled arterioles (Figure 5.16, Figure 5.17 and Figure 5.18). Dynamics of the
base case of essential hypertension is dominated by remodeling rather than nephron loss as
demonstrated by the contrast between total conversions and deaths from normal nephrons
(Figure 5.18 vs. Figure 5.19).
Converted Arteroles Nommal Arteriole Dead Nephrons
719,935 155,887
Al
539,951 116915 aa
359,967 77,943
179,983 38,971
0 LETT | 0
ge 20 AO Be, 1D 120 a0 HO ABO” 200, 0 2 40 60 8 100 120 140 160 180 200
Time (Week) Time (Week)
Converted Arterioles : Essential hypertension case. —i—a—+—a—2—+
Normal Arteriole Dead Nephrons : Essential hypertension case. —t—3— +
Normal Arteriole Dead Nephrons : Potential hypertension case
Figure 5.18. Dynamics of Converted Figure 5.19. Dynamics of normal nephron
Arterioles-essential hypertension deaths-essential hypertension
Converted Arteries : Potential hypertension case
The increase in the number of remodeled arterioles drives up plasma renin levels and
renin per nephron (RpN) (Figure 5.20 and Figure 5.21). The increase in RpN means that
normal arterioles will be constricted more than their normal states. Thus, more blood flow
would be necessary to achieve the same amount of water excretion. The discrepancy in goal-
seeking FV-RAS control mechanism of normal nephrons is best demonstrated by the
difference between the Actual snGFR, runi, and Required snGFR, run2 (Figure 5.23).
Throughout the simulation, Actual snGFR is below Required snGFR in contrast to the
previous two base cases.
% Plasma Renin over NPR
0 2 40 60 80 100 120 140 160 180 200
Time (Weeks)
" Plasma Renin over NPR" : Essential hypertension case. —-———— +
“4 Plasma Renin over NPR" : Potential hypertension case. —2—2—2—2—
Figure 5.20. Dynamics of plasma renin-
essential hypertension
RpN over NRpN
3.172
2.530
1.888
1.246
0.6043
o 20 «40 «660 680 ©6100 120 140 160 180 200
Time (Week)
RpN over NRPpN : Essential hypertension case
RpN over NRPN : Potential hypertension case
Figure 5.21. Dynamics of renin per
nephron-essential hypertension
Actual snGFR must equal required snGFR to bring FV back to its target level and
achieve excretion that equals intake. The persistent difference between required and actual
snGFR over weeks results in the gap between water intake and water excretion throughout the
life span of the subject. Note that the horizontal line in Figure 5.22, run 1, denotes the water
intake and that run 2 denotes the water excretion in essential hypertension case. The positive
water balance between the two flows results in accumulation of FV over time. On the other
hand, water excretion of potential hypertension case, run 3 perfectly matches water intake.
Water Flow
0 20 40 60 80 100 120 140 160 180 200
Time (Week)
water intake : Essential hypertension case —t——t—i-— 2-3. L/Week
water excretion : Essential hypertension case --—2—2—2—2 2 L/Week
water excretion : Potential hypertension case. —3—3—3—3—3— LWeek
Figure 5.22. Dynamics of Water Balance-
essential hypertension
5.4 Scenario and Policy Analysis
snGFR
0.4251
0.2619
0.0986 Thee eer
0 20 40 60 80 100 120 140 160
Time (Week)
180 200
normal snGFR : Essential ‘hypertension case. —3—3—_3—_3-—_3-
Figure 5.23. Dynamics of snGFR-essential
hypertension
A number of experiments were conducted on uncertain parameters in base cases. The
scenarios below demonstrate experiments on essential hypertension base case.
5.4.1 Low Normal Nephron Loss Fraction and Low Remodeled Nephron Loss Fraction
The base case of essential hypertension corresponds to normal values of nephron loss
fractions. In this experiment, nephron loss fraction for normal nephrons will be decreased to
20% of its normal value (from 0.0005 to 0.0001) and remodeled nephron loss fraction will be
decreased to half of its normal value (from 0.001 to 0.0005). This scenario is demonstrated in
tun 1 (Essential hypertension LL) in comparison to the essential hypertension case, run 2.
BP Output RpN over NRpN
0 2 40 60 «80 ©6100 120 140 160 180 200
‘Time (Week) 0 20 40 60 80 100 120 140 160 180 200
Time (Week)
BP Output : Essential hypertension LL
BP Output : Essential hypertension case
RpN over NRpN : Essential hypertension LL. —+—z—3—3 3 —z-—_
RpN over NRpN : Essential hypertension case. —2—2—2—2—2 2 —
Figure 5.24. Dynamics of blood pressure for — Figure 5.25. Dynamics of renin per nephron
low nephron loss fractions for low nephron loss fractions
The behavior of BP is not significantly different from the essential hypertension case
despite significant reductions in nephron loss fractions. Renin per nephron levels are slightly
higher as a result of higher remodeled population (Figure 5.25 and Figure 5.27). The number
of normal nephrons and remodeled nephrons are higher, consistent with expectations, but the
general behavior pattern of BP in essential hypertension does not change by reductions in
nephron loss fractions.
Normal Arterioles Remodeled Arterioles
16M 1.093 M
1.388 M 919,761
1.176M 746,336
965,050 572,911
753,400 399,486
020406080 100 120-140-160 180200 0 2 «40 «660 «680 «(100 +120 140 160 180 200
Time (Week) Time (Week)
Remodeled Arterioles : Essential hypertension LL. ——3—3—3—+—3_-
Nora) Seiles Entel re aeee LL Remodeled Arterioks : Essential hypertension case. ——2—2>—2—> > —
Noma Arterioles : Essential hypertension case
Figure 5.26. Dynamics of normal arterioles Figure 5.27. Dynamics of remodeled
for low nephron loss fractions arterioles for low nephron loss fractions
5.4.2 150 per cent Increase in Remodeled Nephron Loss Fraction only
In this scenario, the normal nephron loss fraction is kept at its normal level of 0.0005,
whereas remodeled nephron loss fraction is increased up to 2.5 times of its normal value
(from 0.001 to 0.0025). This change has a very significant impact on the behavior. Higher
loss from remodeled nephrons leads to normalization of BP. The behavior of BP in the
reference essential hypertension case and in the case with reduction in both of nephron loss
19
232.91
198.46
164.00
129.54
95.09
BPO}
fractions (see 5.4.1) are in stark contrast to BP of this run (Figure 5.29). In both of those
cases, the subject reaches lethally high levels of blood pressure within 160 weeks. In the
current scenario, BP decreases from its initially high levels and demonstrates a stable
behavior around its normal value of 100 between week 500 and 800 (Figure 5.28). The
progression of blood pressure to dangerous levels takes place over a longer period of time.
The subject does not experience any significant increase in BP up until week 1000.
BP Output BP Output
244.60
213.14
181.68
150.22
118.76
0 2 40 60 80 100 120 140
Time (Week)
160 180
0 130 «260 «390 = 520650
Time (Week)
780 910 1040 1170
jutput : Essential hypertension nh.
Figure 5.28. Dynamics of blood pressure -
u ; Figure 5.29. Comparative dynamics of
high remodeled nephron loss fraction
blood pressure; early phase
BP Output: Essential hypertension nh #4243 34-4
BP Output: Essential hypertension case. 22222222
RpN over NRpN snGFR
1.141 0.6502
05102
0.8847
0.3701
0.6284
0.2300
akan 0.0900 a oA
0 130 260 390 520 650 780 910 1040 1170
0.1159 Time (Week)
0 130 260 390 520 650 780 910 1040 1170 “Actual snGFR per N.N." : Essential hypertension ah
Time (Week)
RpN over NRpN : Essential hypertension nh. —3—3—3—3 33
Figure 5.30. Dynamics of renin per nephron -
high remodeled nephron loss fraction
“Required snGFR N.N." ;: Essential hypertension nh. —2—2—2—2—2—
nonnal snGFR : Essential hypertension nh 3—3—3—3—3—3 33
Figure 5.31. Comparative dynamics of
actual and required snGFR
The exact match between Required and Resistance Adjusted snGFR after week 520
demonstrates that normal nephrons achieve the required snGFR necessary to keep FV at its
target level (Figure 5.31). This means that FV-RAS control mechanism of normal Arterioles
has become intact after the distribution of remodeled nephrons among all nephrons has fallen
below a critical point. When BP (FV) eventually starts rising after week 900, the rise is due to
the fact that Actual snGFR of normal nephrons approach their max snGFR capacity. The
progression of BP happens through insufficient capacity of remaining normal nephrons in a
way similar to the progression of BP in normal and potential-hypertensive subjects.
20
1.369M 300,871
1
Nomnal Arterioles Remodeled Arterioles
6M 400,000
1.138 M 201,742
908,423 102,613
87,897 a 0 130 260 =—390 ifn ae 910 1040 «1170
0 130 260 390 «520 650-780 «910«:1040-«1170 rea Wek
‘Time (Week) Mme (Woe
‘Nommal Arterioles: Essential hypertension ah Remodeled Arterioles : Essential hypertension nh +#——1——1——1-—-3—4-—>
Figure 5.32. Dynamics of normal arterioles Figure 5.33. Dynamics of remodeled arteriole
- high remodeled nephron loss fraction - high remodeled nephron loss fraction
The dynamics of BP in high remodeled nephron loss fraction scenario may initially
seem counterintuitive given the theories relating nephron number and hypertension (Brenner
et al., 1988). Nephrons die faster, yet the survival time of the patient increases. The critical
point is that the patient is able to quickly get rid of deleterious, high renin secreting remodeled
nephrons. Consequently, renin per nephron levels fall below the remodeling threshold renin
per nephron and remodeling loop ceases to be effective. Once remodeled nephrons fall below
levels where they would not interfere with the control of FV-RAS mechanism of normal
nephrons, proper BP control can be maintained over a long period of time.
5.4.3 Drug Intervention in Essential Hypertension
This scenario represents a drug intervention to an essential hypertension patient. A new
formulation that decreases plasma renin proportional to the dose of an R-type drug is
introduced. “R-type” denotes a cluster of drugs that affect renin-angiotensin system (Laragh,
2002).
Adjusted Renin = Plasma Renin + Effect of R-Type Drug*Plasma Renin.
Drug therapy with different doses of R-type drugs is initiated at week 30. Effect of R-
type drugs were set to -0.1, -0.2, -0.3, respectively. Comparisons with the reference essential
hypertension case demonstrate that any R-type drug affects progression of remodeling and BP
favorably (Figure 5.34 and Figure 5.35).
BP Output Remodeled Arterioles
244.60 968,742
210.75, 803,984
176.90 639,225
143.05 474,467
109.19 2 ty 309,708 ‘ "
0 20 40 60 80 100 120 140 160 180 200 0 20 40 60 8 100 120 140 160 180 200
Time (Week) ‘Time (Week)
[BP Output : Essential hypertension case. 33s 3 3 Remodeled A rterioles : Essential hypertension case. -——3—————-_+—
BP Output : Essential hypertension Rdug 1. 2-33» > >. > Remodeled A rterioles : Essential hypertension Rdg 1 ——-2——2——2—2 2
BP Output : Essential hypertension Rdrug 2 —3——3—3 33 3 — Remodeled A rterioles : Essential hypertension Rdrug 2. 333 3, 3
‘BP Output : Essential hypertension Rdrug 3 —s——4¢——4——¢ 1 4 — Remodeled A rterioles : Essential hypertension Rdrug 3. —+ 4 + +
Figure 5.34. Comparative dynamics of BP Figure 5.35. Dynamics of Remodeled Arterioles
6 CONCLUSION
The purpose of this research is to study the progression of blood pressure that results
from structural changes in the kidneys. For this purpose a system dynamics model is built that
focuses on systemic interactions among remodeling, nephron loss and renin-angiotensin
system. Blood pressure progression dynamics of normal, potential- hypertensive and essential-
hypertensive subjects are demonstrated in base runs and scenario analysis with respect to
critical parameters are conducted.
The base cases reproduce the expected dynamic behavior for key variables. The normal
subject case demonstrates how the inevitable progression of structural damage with aging is
compensated by increased excretion from remaining healthy nephrons. Normal subject does
not develop hypertension up until the later stages of his/her life. The reference cases for
potential and essential hypertensive subjects underline two different types of progression. In
the potential hypertension case the progression is driven by nephron loss due to aging, similar
to the case of normal subjects. On the other hand, in essential hypertension, progression is
reinforced with remodeling in addition to nephron loss. This reinforcing mechanism may be
responsible for fast progression of BP observed in essential-hypertensive patients who
subsequently experience malignant hypertension. Moreover, the distortion of fluid volume
and renin angiotensin control of normal nephrons by inappropriately high plasma renin
provides a realistic reproduction of renin-angiotensin system’s involvement in essential
hypertension.
The results of scenario runs hint at possible policies to control progression of BP in
essential hypertension. Whereas reducing the loss fractions of both types of nephron
subpopulations has little effect on slowing down the progression of blood pressure, increasing
the loss fraction of remodeled nephrons has significant positive impact on normalization of
BP and its maintenance at normal levels. Drug interventions could also be employed to
control hypertension. For example, in essential hypertension, drugs that affect renin-
angiotensin system would both lower the level of blood pressure and stop the progression of
remodeling.
The model can be used as a building block for more comprehensive models of long-
term structural management of kidneys. An urgent step is to validate the model with data from
longitudinal studies which focus on the number and distribution of nephrons and plasma renin
levels. A possible direction for advancing the model would be to include a structure to control
the remodeled nephrons’ renin secretion. Finally, the introduction of a drug intervention
structure could facilitate experimentation with different policies of long-term blood pressure
management. The complete model can be transformed to an interactive gaming version for
long-term hypertension management.
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