To:
From:
Date:
Subject:
Professor Andreas Linninger/Chih-Yang Hsu/Ian Gould
Félix L. Morales
December 2, 2013
Project Report
Dear Professor,
Enclosed you will find my written report for Project 1, with the title A Computational Model
for the Exploration of Vasculogenic Erectile Dysfunction. It consists of an abstract, a physiological
background, methods, and results, discussion, and conclusion, with attached appendices.
The difficulty to find complete physiological values in the literature resulted in a series of
assumptions and extra steps outlined in the attached report. Reasonable efforts were made to
ensure calculations are within acceptable parameters, but the consistency of the model may be
limited by the unavailability of reliable physiological values for comparison.
However, the model exhibits accuracy in computed values, which agree with hand-calculated
values. This enabled a detailed analysis of vasculogenic Erectile Dysfunction.
The report is attached.
Sincerely,
Félix Morales
A Computational Model for the Exploration of
Vasculogenic Erectile Dysfunction
Félix Morales
Abstract
Little attention has been given to the complete hemodynamic parameters exhibited by male
suffering from vasculogenic Erectile Dysfunction. While most of the efforts done to understand this
state include neurological mechanisms, and local genital hemodynamics, whole-body hemodynamic
characterization is lacking. To address this gap in knowledge, a computational model was used to
provide pilot values for the relevant hemodynamic properties at this particular state. A view into the
development of vasculogenic Erectile Dysfunction is obtained by describing this disease as increased
resistance to blood flow in the arteries supplying the corpora cavernosa in the penis. This increase in
resistance was done either by directly adjusting it in the model, or by adjusting the diameter of the
cavernous artery. Direct adjustment of the vascular resistance revealed that the cardiac output
follows the behavior of penile blood flow, provided other vascular resistances remain unaltered.
Upon adjustment of vessel diameter, it was found that upon a 28% decrease in maximal diameter,
penile blood flow was insufficient to attain a full erection. Furthermore, a 49% decrease resulted in
a complete inability to take the penis out of the flaccid state. It is the hope of this work to facilitate
the acquisition of insight for the diagnosis and/or treatment of Erectile Dysfunction.
Physiological background
The sexual arousal state
For a healthy adult male, the sexual arousal state is usually determined by the erection of the
penis. Following sexual stimulation, the parasympathetic division of the Autonomous Nervous
System triggers the release of vasodilators (namely, Nitric Oxide) in the arteries of the penis, which
induces the formation of cyclic Guanosine Monophosphate (cGMP) [1]. The release of cGMP results
in the relaxation of the smooth muscle cells of the deep artery of the penis or cavernous artery
(Figure 1) [2][3]. This relaxation causes the artery to dilate, facilitating the accumulation of blood in
the sponge-like structure of the corpus cavernosum penis [4]. Lastly, the expansion of this erectile
tissue causes the compression of the venous plexus draining blood out of the penis, resulting in a
net accumulation of blood at the onset of an erection [2]. This accumulation, which corresponds to
the excitement phase of the sexual response cycle proposed by Masters and Johnson, eventually
reaches steady state at the plateau phase of sexual arousal [2][4].
Figure 1. The penis is
supplied by the internal
pudendal artery, which
branches to the
bulbourethral artery, dorsal
artery, and the cavernous
artery. The cavernous
artery is responsible for the
engorgement of the penis.
Erectile Dysfunction
The inability to develop or maintain a full erection is a condition known as erectile dysfunction
(ED). The causes for ED can be organic (vasculogenic, neurogenic, drug related) or psychogenic [3].
Of particular interest for this study are the vasculogenic causes of ED. It has been demonstrated that
the insufficient perfusion of the corpus cavernosum penis by patients with vasculogenic ED is an
element of an underlying atherosclerotic process [3]. The incidence and age of patients is the same
at the onset of ED and coronary artery disease (CAD), which is in accordance with the observation
that ED and cardiovascular disease (CVD) have the same set of risk factors (hypertension, diabetes
mellitus, smoking, etc.) [3]. These findings suggest that ED can be a strong indicator of CVD, if
psychogenic causes of ED are discarded during diagnosis. In order to understand how the progress
of vasculogenic ED may influence hemodynamics in a patient, a computational representation of
the disease may yield useful patterns which may be easily monitored by medical practitioners.
Effect of Sildenafil Citrate in the treatment of vasculogenic ED
The cGMP produced in smooth muscle cells is constantly degraded by phosphodiesterase type
5 (PDE5), which is mainly found in the cells of the corpora cavernosa [1]. In order to sustain a full
erection, a constant release of NO is required to keep sufficient cGMP concentration. However, in
certain ED patients, the rate of degradation of cGMP is greater than the rate of formation, which
may result in insufficient artery dilation. Sildenafil citrate (commercially known as Viagra), treats
this problem by inhibiting PDE5, therefore allowing the patient to sustain an erection. The effect of
Sildenafil citrate is nevertheless transient [1]. This type of vasculogenic ED (in which a patient’s
artery becomes progressively less responsive to NO signaling), is difficult, of not impractical to
monitor in a patient with increased risk of developing vasculogenic ED. However, modelling this
progression through computational methods is straightforward, and may provide useful insight into
the effect of a non-responsive artery on penile blood flow.
Methods
Model
The selected model for the human circulatory system is shown in Figure 2. The organ of interest
is highlighted in green.
As seen, this model represents the circulatory system as a closed loop circuit, which is a fairly
accurate description of the system of interest. Branches represent organs within the body, and the
inlet and outlet of the network (or circuit) represent the heart. Flow through blood vessels is
represented as faces (F1, F2, and so on), whereas points where these blood vessels branch or rejoin
are assigned as points (P1, P2, etc.). In general, organs in the body are represented by three faces,
which from right to left in Figure 3, represent arteries/arterioles, capillaries, and venules/veins,
respectively. This is a simplified representation of the microvasculature on organs.
With this simplified model, the system can be described by equations similar to those used in
the analysis of circuits, but suited for fluid flow. The equations consist of conservation balances, and
constitutive equations. The general form of the conservation balance equations is,
∑ 𝐹𝑖 = 0
Eq. 1
Where F represents the algebraic sum of inflows
and outflows at a point of interest. It is
important to note that this equation is only valid
for a system in steady-state. In addition, the
relevant constitutive equation for this system is
the Hagen-Poiseuille equation:
∆𝑃𝑖 = 𝛼𝑖 𝐹𝑖
Eq. 2
Where ΔP represents the pressure drop from
one point to another, α represents the
resistance to flow, and F represents the flow
between two points. The resistance to flow (or
hydraulic resistance) is given by,
128𝜇𝑙
Eq. 3
𝛼=
𝜋𝑑4
Where µ represents the dynamic viscosity, l
represents the length of the vessel, and d
represents the diameter of the vessel. For the
case of blood, the dynamic viscosity is 0.0035
Pa*s.
The usage of these equations implies that
the following assumptions are being made
regarding the system of interest:

The fluid is incompressible and Newtonian.

The system is at steady state.

Flow is laminar.
The assumption that holds the most for a
human circulatory system is that it exhibits a
steady state behavior. Despite the underlying
assumptions, the usage of these equations has
Figure 2. The human circulatory system proven useful in making accurate predictions of
was modeled as a closed loop circuit with blood flow at specific points in the human body,
inlet/outlet representing the heart. as shown by previous work on this matter. In
Arrows represent direction of flow, and order to compute the intended solutions of
these equations in a time-efficient manner, the
are not scaled.
usage of MATLAB was required. The MATLAB
code used for this particular model is presented in Appendix 1.
By using MATLAB to solve these equations, the application of Linear Algebra concepts is
implied, by using a matrix form of these equations given by the general form,
Eq. 4
𝐴𝑥 = 𝑏
Where A represents the coefficients of the variables in the equations, x represents the unknown
variables, and b represents the target vector (which includes the boundary conditions).
Physiological parameters
For these equations to yield useful physiological values, certain parameters of the system
needed to be known. In particular, these parameters included cardiac output, mean arterial
pressure (MAP), mean venous pressure, percent of cardiac output flowing through selected organs,
diameter of major arteries and veins, diameter of cavernous artery, and blood flow through the
flaccid and erected penis.
While mean venous pressure, major vessel diameter, and blood flow through flaccid penis were
obtained directly through scientific literature search, the remaining parameters were estimated
from parameter definitions, given that these properties are mostly given in terms of common
physiologic values, such as heart rate (HR), diastolic and systolic pressures, and stroke volumes (SV).
These definitions include:
Eq. 5 [5][15]
𝐶𝑂 = 𝑆𝑉 ∗ 𝐻𝑅
1
Eq. 6 [6]
𝑀𝐴𝑃 = ((𝐷𝑖𝑎𝑠𝑡𝑜𝑙𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 ∗ 2) + 𝑆𝑦𝑠𝑡𝑜𝑙𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒)
3
In previous work with the chosen model, blood flow through the pelvic area was not specifically
included, rather included within an “other” section. Since blood flow through the penis comes from
the internal iliac artery (which supplies the pelvic area), the state of sexual arousal was modeled as
a blood flow redistribution in the “other” section, in which the blood flow through the penis was
subtracted from the blood flow through the “other” section. Finally, this decision was enforced by
the range of values found for blood flow through the fully erected penis, which did not surpass the
blood flow values of the “other” section.
Another parameter needed to solve the system are the resistances to flow in the faces of the
model. For the faces representing major blood vessels (far right and far left, vertically oriented, in
Figure 3), an estimated value of the diameter of these vessels was used through usage of Eq. 3. The
length parameter needed for this equation was obtained through the application of Euclidean
geometry on MATLAB, where the coordinates of the points in the network were read from a network
file. The blood’s dynamic viscosity was converted from Pa*s, to mm Hg*min, since the flows are
reported in L/min, and pressures in mm Hg. The remaining resistance values, which correspond to
resistances in the selected organs, were determined by application of Eq. 2, in which the flows were
already known by virtue of the percent cardiac output flowing through each organ. For the case of
pressures, usage of resistances of major blood vessels, with appropriate flows through each of them,
yielded negligible pressure drops. This allowed for proper estimation of pressure drops at each
organ. The specific pressure drops at each face were taken as percentages of the total pressure drop
at each branch (or organ). The physiologic values used to construct the model for the flaccid and
erected states are summarized in Table 1, 2, and 3.
Table 1. Physiological parameters for the baseline state obtained from the literature
Parameter
Value(s)
Source
Blood’s dynamic viscosity
0.0035 Pa*s
[7]
Mean arterial pressure
93 mm Hg
[6]
Mean venous pressure
2 mm Hg
[8]
Blood flow through penis
0.15 L/min
[9]
Diameter of main vessels
4 mm
[10]
Diameter of cavernous artery
1.83 mm
[11]
Cardiac output
5.00 L/min
Calculated from Eq. 5
Table 2. Physiological parameters for the sexually aroused state
Parameter
Value(s)
Source
Systolic pressure
140-200 mm Hg
[4]
Diastolic pressure
80-85 mm Hg
[12]
Mean arterial pressure
100-120 mm Hg Calculated from Eq. 6
Mean venous pressure
2-4 mm Hg
[8]
Diameter of cavernous artery
3.2-3.6 mm
[11][13]
Heart rate
100-175 bpm
[4]
Cardiac output
7-12.25 L/min Calculated from Eq. 5
Table 3. Percentages of Cardiac output through selected organs at baseline
Organ
Percentage
Source
Brain
14.00%
[14]
Stomach/intestines
21.00%
[14]
Hepatic artery
8.75%
[14]
Liver
29.75%
[14]
Kidneys
22.00%
[14]
Arms
9.75%
[14]
Legs
16.25%
[14]
Penis
3.00%
[14]
Other
5.25%
[14]
%pressure drop at arterioles
65.00%
[15]
%pressure drop at capillaries
20.00%
[15]
%pressure drop at venules
15.00%
[15]
It is worth noting
that
the
resistances
obtained
by
application of Eq.
2 were used not
only for the
model of the
baseline state,
but also for the
model of the
sexually aroused
state, with the
exception of the
resistance of F26 (which represents the cavernous artery), which was adjusted according to
diameters obtained through literature. This implies that the percentages shown in Table 3 were
assumed to be the essentially the same for both states. This was due to the absence of such
percentage information in the searched literature.
Modeling Vasculogenic ED as increased vascular resistance
As stated previously, some patients with vasculogenic ED (specifically, arteriogenic ED) exhibit
an atherosclerotic process, particularly in the family of arteries supplying the penis [3]. This
physiological phenomenon can be easily represented as an increased hydraulic resistance in the
face that represents the arterial supply to the penis. In the selected model, this is F26 (Figure 2).
The increase in hydraulic resistance of this face was incremented in ten-percent steps from the
baseline resistance used in the validation test for the sexually aroused state, until the baseline
resistance was doubled (100% increase). After this gradual resistance adjustment, blood flows
through the entire system were recorded as a function of the percent increase in the hydraulic
resistance. The boundary conditions (mean arterial and venous pressures) were kept constant
(116.5 mmHg and 4 mmHg, respectively). Table 4 shows the resistance values used.
Table 4. Resistances used for vasculogenic ED modeling through resistance increase
Percent increase in arterial resistance
Arterial resistance (mmHg*min/L)
0%
26.33
10%
28.96
20%
31.60
30%
34.23
40%
36.86
50%
39.50
60%
42.13
70%
44.76
80%
47.39
90%
50.03
100%
52.66
Exploring the effect of a progressively non-responsive artery in vasculogenic ED
The inability of the cavernous artery to dilate upon NO signaling can be easily modeled as a
progressive decrease in maximal arterial diameter, with proper arterial resistances computed from
Eq. 3. From the maximum dilated diameter found in the literature (3.6 mm), the arterial diameter
was decreased in 5% steps until a decrease of 85% was obtained. Decreases greater than 55% are
physiologically insignificant, considering that this decrease would result in the cavernous artery
having a diameter smaller than the baseline diameter (1.83 mm).
After computing the resistance values corresponding to these decreases (resistances at F26),
the model was tested using each of these resistances, and different sets of mean pressures within
the values found in the literature for the sexually aroused state. These pressures corresponded to
the maximum mean arterial and venous pressures attainable by a man without hypertension when
aroused; minimum mean arterial and venous pressures attainable by a man without hypotension
when aroused, and a simulation of a patient whose cardiovascular system homeostatically
compensates for the increased vascular resistance in the penis.
Penile blood flow was graphically recorded as a function of the percent decrease in the
diameter of the cavernous artery. Table 5 summarizes the diameters used. Along with these flow
values, the minimum flow required for full erection was noted, as well as the blood flow through a
flaccid penis (when arterial diameter equals baseline arterial diameter) The minimum blood flow for
full erection was determined by using a mean arterial pressure of 100 mm Hg (the minimum), a
mean venous pressure of 4 mm Hg (the maximum), and a resistance calculated from Eq 3, with the
minimum diameter of the cavernous artery at the sexually aroused state (3.20 mm), as reported in
the literature [13]. Furthermore, blood flow through a flaccid penis was determined by using a mean
arterial pressure of 120 mm Hg (the maximum), a mean venous pressure of 2 mm Hg (the minimum),
and the resistance of the cavernous artery at the baseline state. This situation represents a
completely non-responsive cavernous artery to the maximum pressure gradient attainable by a
healthy cardiovascular system.
Table 5. Diameters used for testing of the effect of diameter decrease on penile blood flow
Percent decrease in arterial diameter
Diameter (mm)
0%
3.60
5%
3.42
10%
3.24
15%
3.06
20%
2.88
25%
2.70
30%
2.52
35%
2.34
40%
2.16
45%
1.98
50%
1.80
55%
1.62
60%
1.44
65%
1.26
70%
1.08
75%
0.90
80%
0.72
85%
0.54
MATLAB code and validation
The MATLAB algorithm shown in Appendix 1 exhibits particular file reading, and data structure
assignment procedures. Given that the model selected is a modified version of a previously defined
model, the modifications were done within the MATLAB environment, and assigned to Excel files.
The need for this approach stems from an inability to perform the modifications in the network file.
To validate the predictions obtained by the model, two validation steps were implemented:
agreement between predicted values and literature values was assessed through,
𝑝𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 − 𝑙𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒
%𝑒𝑟𝑟𝑜𝑟 =
∗ 100
Eq. 7
𝑙𝑖𝑡𝑒𝑟𝑎𝑡𝑢𝑟𝑒 𝑣𝑎𝑙𝑢𝑒
The predicted values will be considered accurate if the percent error is less than ±10%. This
validation step was performed for the model describing the baseline state, and the sexually aroused
state.
The second validation step consisted of an assessment of the mathematical agreement of the
values with Eq. 1 and Eq 2. Namely, pressures returned by the model had to be between the
boundary conditions, and flows entering a point had to be equal to the flows leaving the point.
Results
Properties of the model
Equation 1 applies at every point (or node) in the network, except for the inlet and outlet, which
are the boundary conditions. Equation 2 applies at every face of the network. Hence, the amount of
equations is related with the amount of unknowns of the system, which is also known as “degrees
of freedom”. The amount of unknowns are shown in Table 6, and the complete list of equations
describing the network is given in Appendix 2.
Table 6. Degree of freedom analysis of selected model
Equation
Number
Eq 1. ΣFi=0
45
Eq 2. ΔP=αFi
56
Boundary values (P1 and P47)
2
Degrees of freedom
101
Vasculogenic Erectile Dysfunction
through resistance increase
The effect of this resistance
increase on blood flow through the
penis is displayed in Figure 3. As
expected, blood flow through this
section is decreased.
Blood flow through penis (L/min)
Blood flow through penis vs. resistance increase
0.5
0.49
0.48
y = 0.0051x2 - 0.0542x + 0.4944
R² = 1
0.47
0.46
0.45
0.44
0%
20%
40%
60%
80%
Percent increase of validation resistance
100%
Figure 3.
Upon
increasing
resistance
to flow in
the
cavernous
artery, flow
through
penis at
the erected
state is
decreased.
Nevertheless,
blood flow
variation through
1
other model0.98
0.96
represented
0.94
organs exhibited a
0.92
different behavior.
0.9
Figure 4, which
0.88
shows blood flow
0.86
variation through
0.84
the brain, shows
0.82
0.8
an example of this
0%
20%
40%
60%
80%
100%
behavior. There is
Percent increase of validation resistance
little, to none
variation in blood
Figure 4. While MATLAB returned increasing blood flow values, these flow through the
increases were insignificant, as the above plot of cerebral blood flow brain. This exact
same behavior is
demonstrates.
observed at every
other “organ” in
Cardiac output vs. resistance increase
the model, except
for the penis.
6.79
Since blood
6.78
flow
through
organs does not
6.77
increase
y = 0.0051x2 - 0.0542x + 6.7834
significantly
(or at
R²
=
1
6.76
all) as a result of
6.75
the decrease of
blood flow through
6.74
the penis, the only
possible
6.73
explanation
for
0%
20%
40%
60%
80%
100%
this overall system
Percent increase of validation resistance
blood
flow
decrease is that
Figure 5. Effect of increasing hydraulic resistance in the cavernous artery on
the cardiac output
cardiac output. Note that the regression equation is similar to Figure 4, with
decreased. Figure
different intercept.
5 evidences this.
Furthermore, it is observed that this decrease in cardiac output follows the same pattern as the
decrease in penile blood flow.
Cardiac output (L/min)
Blood flow through brain (L/min)
Blood flow through brain vs. resistance increase
Effect of a progressively non-responsive artery in vasculogenic ED
It was found that by the time the cavernous artery constricts by 28% from its maximum
attainable diameter, the blood flow through the penis is insufficient to sustain a full erection (Figure
6). This insufficient blood flow is reached at a 10% constriction when the blood pressure difference
is the least (magenta line in Figure 6). Comparing a compensating situation to the maximum blood
pressure attainable by a patient without hypertension reveals that both situations are similar when
the patient’s cardiovascular system reaches maximum pressure to account for the increasing
resistance to flow of the cavernous artery.
When the
cavernous
artery reduces
its diameter by
a 49% from
maximum
diameter, the
blood
flow
through
the
penis is already
insufficient to
even take it
out from a
complete
flaccid state.
This
event
Figure 6. Effect of diameter reduction on penile blood flow. Minimum blood
takes
place
flow for erection estimated was 0.379 L/min. Baseline flow corresponds to
sooner for the
0.195 L/min.
minimum blood
pressure situation (approximately 45% reduction).
Further observation of the values obtained through the simulation reveals that an 11%
reduction of the maximum diameter of the cavernous artery results in the minimum diameter
exhibited by this artery at the sexually aroused state, according to the literature (3.20 mm) [13].
Validation
The possibility of using the model as an accurate one comes from successful validation, which
is outlined below. Table 7 details the validation of the model by comparison to physiological values
at baseline, and Table 8 shows the same procedure for the sexually aroused state. In addition, Table
9 demonstrates mathematical agreement encountered at selected points in the model. For the
validation of baseline, the values displayed in Table 1 were used.
Similar
Table 7. Validation of model consistency at baseline
consistency
Parameter
Predicted value Literature value %error was found at
Cardiac output (F1 and F56)
5.00 L/min
5.00 L/min
0.00% every other
Flow through penis (F26,F27,F28)
0.15 L/min
0.15 L/min
0.00% point in the
Pressure drop at penis (P30 to P33) 91.00 mm Hg
91 mm Hg
0.00% model. For
further
assessment of this consistency, Appendix 3 shows the values returned by the model with the
selected boundary conditions, along with the network for easy comparison.
For the sexually aroused state, cardiac output was calculated to be between 7-12 L/min, given
a stroke volume value of 0.07 L/beat [5]. For validation purposes, the value was selected to be 7
L/min. The boundary pressures were selected to be 120 mm Hg for mean arterial pressure, and 2
mm Hg for mean venous pressure. These values are within the ranges displayed in Table 2. Due to
the lack of penile blood flow values at this state, it was assumed that the cavernous artery
minimum and maximum dilation values would yield appropriate blood flows to describe blood
flow at this state. The maximum arterial dilation (96%) was chosen to calculate the proper vascular
resistance.
A small error
Table 8. Validation of model consistency at sexually aroused state
Parameter
Predicted value Literature value %error in cardiac
Cardiac output (F1 and F56)
6.78 L/min
7.00 L/min
-3.09% output can be
Pressure drop at penis (P30 to P33)
118 mm Hg
118 mm Hg
0.00% observed, but
it is not
greater than 10%.
Table 9. Validation of model computations
Organ/face/point selected Equation/Property verified
Computation
Left Kidney, P15
Eq. 1 F14-F15=0
0.55 L/min-0.55 L/min=0
Whole system flow
Eq 1. ΣFi=0
Fout=Fin=5.00 L/min
Right leg, P26, P27, P28, P29 Pressures within boundaries 93.00, 33.85, 15.65, 2.00 mm Hg
As seen, the computational results agree with the results expected by the equations, indicating
that the MATLAB code executes properly.
Discussion
Vasculogenic Erectile Dysfunction through resistance increase
Given that the human circulatory system was modeled as a closed loop circuit, similar to an
electric circuit, the results make complete sense when using electrical circuit analysis. Since the
inlet and outlet pressures were kept constant (constant voltage), and considering that the network
connectivity resembles parallel resistances, increasing resistance in a single branch will cause a
decrease in current through that particular branch so as to maintain a constant voltage drop (a
consequence of Ohm’s Law and/or Hagen-Poiseuille equation). Furthermore, since the other
resistances do not change, the same current has to flow through them to ensure constant voltage.
From here it follows that the overall current through the circuit has to decrease in order to
account for the decreased current in just one branch (and the remaining branches unchanged).
Physiologically speaking, the previous test is not only a model of an increased occlusion of the
arteries supplying the penis, but of a homeostatic process: The circulatory system, in an attempt to
keep mean pressures constant despite an increased systemic vascular resistance, decreases the
cardiac output. This is further evidenced by the following equation:
Eq. 8 [16]
𝑀𝐴𝑃 = (𝐶𝑂 ∗ 𝑆𝑉𝑅) + 𝑀𝑉𝑃
Eq. 2
∆𝑃 = 𝛼𝐹
Where SVR stands for systemic vascular resistance, and MVP for mean venous pressure. Given that
MAP and MVP were kept constant during the test, and SVR was increased, CO had to decrease. As
can be seen, Eq. 8 is basically a rebranding of the Hagen-Poiseuille equation (or Ohm’s Law).
It is worth noting that despite this model predicts an inverse linear relationship between
hydraulic resistance and blood flow, as suggested by the whole system behavior, the decreases in
blood flow through the penis (and cardiac output) as a function of increasing resistance was best
described by a second order polynomial fit (see Figure 3 and 5). Multiple reasons for this deviation
could be stated, such as that the assumption of zero resistance through “wires” does not hold on
this model, and that despite a negligible increase in blood flow through the rest of the body, it is still
an increase. In any case, a linear relationship still holds true, as seen in Figure 6, with the caveat that
the regression coefficient for this linear fit is slightly smaller than for the second order polynomial
fit.
Figure 6. A
linear model
still exhibits
validity.
However, the
regression
coefficient is
lower than
the regression
coefficient for
a second
order
polynomial fit.
Cardiac output vs. resistance increase
Cardiac output (L/min)
6.79
6.78
6.77
y = -0.0491x + 6.7826
R² = 0.9991
6.76
6.75
6.74
6.73
0%
20%
40%
60%
80%
100%
Percent increase of validation resistance
Nevertheless, the results obtained from this test do not provide any new information about ED,
other than demonstrating that the model does behave as expected. The fact that cardiac output is
affected by changes in penile blood flow could be used as an indirect test for the onset of ED.
However, the alteration of cardiac output observed in this test could be the result of a resistance
increase in any other area of the body. Furthermore, the actual homeostatic process that occurs in
the human body, is the more or less constant cardiac output at rest. Any alteration in systemic
vascular resistance most likely will result in an altered arterial pressure to maintain proper blood
perfusion [1]. This is not only predicted by the equations previously mentioned, but is observed in
the plethora of diseases related to abnormally high blood pressures in patients. This is a result of
the limited applicability of the selected model, since it can only admit pressures as controlled
boundary conditions.
Effect of a progressively non-responsive artery in vasculogenic ED
The results of this test must be validated by the fact that the model used blood pressures within
literature ranges, which correspond to individuals with normal blood pressures. An overwhelming
majority of vasculogenic ED patients already suffer from some type of cardiovascular abnormality,
being hypertension a highly likely one [3]. In this sense, even for an ED patient with normal blood
pressures, the inability to develop a full erection may not appear until the cavernous artery has
already developed a non-responsiveness to NO close to a complete non-responsiveness. This sheds
light into why ED is not detectable until the appearance of symptoms. However, if a patient suffers
from some deficiency in heart function, the inability to develop an erection may be apparent earlier
in the development of ED, as can be seen in one of the tests shown (magenta line).
The physiological compensation test did not offer new insights, as this test was made artificially
by attempting to keep the initial blood flow by readjusting the pressure drops until the maximum
healthy drop was reached. At this point, the patient’s body cannot compensate without suffering
hypertension. Since this maximum pressure drop was the same as in other test, both curves
followed each other after they concurred.
Conclusion
The complexity of the human circulatory system can be reasonably described by simple
equations such as the Hagen-Poiseuille equation, and solutions to these equations can be easily
obtained through Linear Algebra methods. The usage of these mathematical constructs to
consistently describe physiological phenomena allows for inexpensive exploration of the
physiological effects of disease.
The present work, given the lack of reliably measured hemodynamic values for the entire
human circulatory system at the sexually aroused state, provides one of such mathematical
approaches into the physiological characterization of the healthy and diseased modes of the sexual
arousal state.
Despite some interesting results obtained by this approach, further model refinement and
testing is required to draw a complete picture into ED and its treatment. Namely, the effect of
Sildenafil citrate on human hemodynamics may be of interest by usage of the presented approach.
References
[1] G. Jackson, N. Benjamin, N. Jackson, M.J. Allen, “Effects of Sildenafil Citrate on Human
Hemodynamics,” Am. J. Cardiol., vol. 83, no. 5A, pp. 13C-20C, March 1999.
[2] H.D. Weiss, “The Physiology of Human Penile Erection,” Ann. Intern. Med., vol. 76, no. 5,
pp.793-799, May 1972.
[3] R.C. Dean, T.F. Lue, “Physiology of Penile Erection and Pathophysiology of Erectile
Dysfunction,” Urol. Clin. North Am., vol. 32, no.4, pp. 379-395, Nov. 2005.
[4] M. Zuckerman, “Physiological Measures of Sexual Arousal in the Human,” Psychol. Bull., vol. 75,
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dilation,” Circulation, vol. 86, no. 1, pp. 232-246, Jul. 1992.
[11] V.D. Okolokulak, “Vascularization of the male penis,” Rocz. Akad. Med. Bialymst., vol. 49, pp.
285-291. 2004.
[12] T. Krüger, M.S. Exton, C. Pawlak, A. von zur Mühlen, U. Hartmann, M. Schedlowski,
“Neuroendocrine and Cardiovascular Response to Sexual Arousal and Orgasm in Men,”
Psychoneuroendocrinology, vol. 23, no.4, pp. 401-411. May 1998.
[13] H. Ghanem, R. Shamloul, “An Evidence-Based Perspective to Commonly Performed Erectile
Dysfunction Investigations,” J. Sex. Med., vol. 5, no. 7, pp. 1582-1589. Jul. 2008.
[14] P.A. Iaizzo. (2013, April 29). Physiology Tutorial-Cardiovascular Function [Online]. Available:
http://www.vhlab.umn.edu/atlas/physiology-tutorial/cardiovascular-function.shtml
[15] G.A. Truskey, F. Yuan, D.F. Katz, “Introduction” in Transport Phenomena in Biological Systems,
2nd ed., Upper Saddle River, NJ: Prentice Hall, 2010, ch. 1, sec. 1.6.1, pp. 49-55.
[16] R.E. Klabunde. (2007, April 6). Cardiovascular Physiology Concepts: Mean Arterial Pressure
[Online]. Available: http://www.cvphysiology.com/Blood%20Pressure/BP006.htm
Appendix 1: MATLAB script for equation writing and solving
clear all;
close all;
drawnow;
clc;
Pin=input('Enter mean arterial pressure (mm Hg): ');
Pout=input('Enter mean venous pressure (mm Hg): ');
%Excel was used given the circumstances. Procedure for network analysis
%stayed the same.
resV=xlsread('Alphas.xlsx');
pointMx=xlsread('pointMx.xlsx');
faceMx=xlsread('faceMx.xlsx');
ptCoordMx=xlsread('ptCoordMx.xlsx');
faces=length(faceMx);
[pressures,trash]=size(ptCoordMx);
%From line 17 to 50, is all assigning the values of the A matrix
A=zeros(faces+pressures-2);
b=zeros(faces+pressures-2,1);
n1=length(find(sum(pointMx))); %To work with the useful amount of
columns in pointMx
%Assignment of coefficients of balance equations
for i=1:pressures-1
if find(pointMx(i,:)~=0)==1
else
for k=1:n1
if pointMx(i,k)~=0
A(i-1,abs(pointMx(i,k)))=sign(pointMx(i,k));
end
end
end
end
%Assignment of alphas
faceMx(:,1)=0; %Assigning zero to the elements of the first column, so my
setup in the for loop works
for i=pressures-1:faces+pressures-2 %Assignment of the coefficients of
the constitutive equations.
A(i,(i+1)-(pressures-1))=resV((i+1)-(pressures-1));
if any(faceMx((i+1)-(pressures-1),:)==1)==1
A(i,faces+(faceMx((i+1)-(pressures-1),3))-1)=sign(faceMx((i+1)(pressures-1),3));
b(i,1)=Pin;
elseif any(faceMx((i+1)-(pressures-1),:)==pressures)==1
A(i,faces+faceMx((i+1)-(pressures-1),2)-1)=-1*sign(faceMx((i+1)(pressures-1),2));
b(i,1)=-Pout;
else
A(i,faces+faceMx((i+1)-(pressures-1),2)-1)=-1*sign(faceMx((i+1)(pressures-1),2));
A(i,faces+faceMx((i+1)-(pressures-1),3)-1)=sign(faceMx((i+1)(pressures-1),3));
end
end
%Solving the system
X=A\b;
%Outputting results
fprintf('Inlet pressure in mm Hg is %d and outlet pressure in mm Hg is
%d\n',Pin,Pout);
for i=1:faces
fprintf('F%d=%s L/min\n',i,num2str(X(i)));
end
for i=faces+1:faces+pressures-2
fprintf('P%d=%s mm Hg\n',(i+1)-faces,num2str(X(i)));
end
%Printing of network equations
fid = fopen('Eq.txt','w');
%\r is added in case file is opened in Notepad (Microsoft)
fprintf(fid,'Conservation Balance Equations\r\n');
for i = 2:size(pointMx,1)-1
a = int2str(pointMx(i,1));
b = int2str(pointMx(i,2));
c = int2str(pointMx(i,3));
fprintf(fid,'F[%s]+F[%s]+F[%s]%s\r\n',a,b,c,'=0');
end
fprintf(fid,'Constitutive Equations\r\n');
%faceMx has the desired length
for i=1:size(faceMx,1)
face=int2str(i);
a1=int2str(faceMx(i,2));
b1=int2str(faceMx(i,3));
%Residual form
fprintf(fid,'(a[%s]*F[%s])-P[%s]+P[%s]%s\r\n',face,face,a1,b1,'=0');
end
fclose(fid);
Appendix 2: List of constitutive and conservation balance equations for the model
Conservation Balance Equations
F[1]+F[-41]+F[-51]=0
F[-2]+F[49]+F[0]=0
F[2]+F[-46]+F[0]=0
F[-3]+F[50]+F[0]=0
F[3]+F[-4]+F[0]=0
F[4]+F[-5]+F[0]=0
F[5]+F[-47]+F[0]=0
F[-6]+F[48]+F[55]=0
F[6]+F[38]+F[-56]=0
F[-12]+F[40]+F[44]=0
F[12]+F[-13]+F[0]=0
F[13]+F[37]+F[-38]=0
F[-7]+F[-14]+F[39]=0
F[14]+F[-15]+F[0]=0
F[15]+F[-16]+F[0]=0
F[16]+F[36]+F[-37]=0
F[7]+F[-8]+F[-17]=0
F[17]+F[-18]+F[0]=0
F[18]+F[-19]+F[0]=0
F[19]+F[35]+F[-36]=0
F[8]+F[-9]+F[-20]=0
F[20]+F[-21]+F[0]=0
F[21]+F[-22]+F[0]=0
F[22]+F[34]+F[-35]=0
F[9]+F[-10]+F[-23]=0
F[23]+F[-24]+F[0]=0
F[24]+F[-25]+F[0]=0
F[25]+F[33]+F[-34]=0
F[10]+F[-11]+F[-26]=0
F[26]+F[-27]+F[0]=0
F[27]+F[-28]+F[0]=0
F[28]+F[32]+F[-33]=0
F[11]+F[-29]+F[0]=0
F[29]+F[-30]+F[0]=0
F[30]+F[-31]+F[0]=0
F[31]+F[-32]+F[0]=0
F[-39]+F[-40]+F[45]=0
F[43]+F[-44]+F[0]=0
F[41]+F[-42]+F[-45]=0
F[42]+F[-43]+F[0]=0
F[46]+F[47]+F[-48]=0
F[-49]+F[-50]+F[52]=0
F[51]+F[-52]+F[-53]=0
F[53]+F[-54]+F[0]=0
F[54]+F[-55]+F[0]=0
Constitutive Equations
(a[1]*F[1])-P[1]+P[2]=0
(a[2]*F[2])-P[3]+P[4]=0
(a[3]*F[3])-P[5]+P[6]=0
(a[4]*F[4])-P[6]+P[7]=0
(a[5]*F[5])-P[7]+P[8]=0
(a[6]*F[6])-P[9]+P[10]=0
(a[7]*F[7])-P[14]+P[18]=0
(a[8]*F[8])-P[18]+P[22]=0
(a[9]*F[9])-P[22]+P[26]=0
(a[10]*F[10])-P[26]+P[30]=0
(a[11]*F[11])-P[30]+P[34]=0
(a[12]*F[12])-P[11]+P[12]=0
(a[13]*F[13])-P[12]+P[13]=0
(a[14]*F[14])-P[14]+P[15]=0
(a[15]*F[15])-P[15]+P[16]=0
(a[16]*F[16])-P[16]+P[17]=0
(a[17]*F[17])-P[18]+P[19]=0
(a[18]*F[18])-P[19]+P[20]=0
(a[19]*F[19])-P[20]+P[21]=0
(a[20]*F[20])-P[22]+P[23]=0
(a[21]*F[21])-P[23]+P[24]=0
(a[22]*F[22])-P[24]+P[25]=0
(a[23]*F[23])-P[26]+P[27]=0
(a[24]*F[24])-P[27]+P[28]=0
(a[25]*F[25])-P[28]+P[29]=0
(a[26]*F[26])-P[30]+P[31]=0
(a[27]*F[27])-P[31]+P[32]=0
(a[28]*F[28])-P[32]+P[33]=0
(a[29]*F[29])-P[34]+P[35]=0
(a[30]*F[30])-P[35]+P[36]=0
(a[31]*F[31])-P[36]+P[37]=0
(a[32]*F[32])-P[37]+P[33]=0
(a[33]*F[33])-P[33]+P[29]=0
(a[34]*F[34])-P[29]+P[25]=0
(a[35]*F[35])-P[25]+P[21]=0
(a[36]*F[36])-P[21]+P[17]=0
(a[37]*F[37])-P[17]+P[13]=0
(a[38]*F[38])-P[13]+P[10]=0
(a[39]*F[39])-P[38]+P[14]=0
(a[40]*F[40])-P[38]+P[11]=0
(a[41]*F[41])-P[2]+P[40]=0
(a[42]*F[42])-P[40]+P[41]=0
(a[43]*F[43])-P[41]+P[39]=0
(a[44]*F[44])-P[39]+P[11]=0
(a[45]*F[45])-P[40]+P[38]=0
(a[46]*F[46])-P[4]+P[42]=0
(a[47]*F[47])-P[8]+P[42]=0
(a[48]*F[48])-P[42]+P[9]=0
(a[49]*F[49])-P[43]+P[3]=0
(a[50]*F[50])-P[43]+P[5]=0
(a[51]*F[51])-P[2]+P[44]=0
(a[52]*F[52])-P[44]+P[43]=0
(a[53]*F[53])-P[44]+P[45]=0
(a[54]*F[54])-P[45]+P[46]=0
(a[55]*F[55])-P[46]+P[9]=0
(a[56]*F[56])-P[10]+P[47]=0
Appendix 3. Values obtained by model in validation.
F1=5 L/min
F2=0.24375 L/min
F3=0.24375 L/min
F4=0.24375 L/min
F5=0.24375 L/min
F6=1.1875 L/min
F7=1.775 L/min
F8=1.225 L/min
F9=0.81875 L/min
F10=0.4125 L/min
F11=0.2625 L/min
F12=1.4875 L/min
F13=1.4875 L/min
F14=0.55 L/min
F15=0.55 L/min
F16=0.55 L/min
F17=0.55 L/min
F18=0.55 L/min
F19=0.55 L/min
F20=0.40625 L/min
F21=0.40625 L/min
F22=0.40625 L/min
F23=0.40625 L/min
F24=0.40625 L/min
F25=0.40625 L/min
F26=0.15 L/min
F27=0.15 L/min
F28=0.15 L/min
F29=0.2625 L/min
F30=0.2625 L/min
F31=0.2625 L/min
F32=0.2625 L/min
F33=0.4125 L/min
F34=0.81875 L/min
F35=1.225 L/min
F36=1.775 L/min
F37=2.325 L/min
F38=3.8125 L/min
F39=2.325 L/min
F40=0.4375 L/min
F41=3.8125 L/min
F42=1.05 L/min
F43=1.05 L/min
F44=1.05 L/min
F45=2.7625 L/min
F46=0.24375 L/min
F47=0.24375 L/min
F48=0.4875 L/min
F49=0.24375 L/min
F50=0.24375 L/min
F51=1.1875 L/min
F52=0.4875 L/min
F53=0.7 L/min
F54=0.7 L/min
F55=0.7 L/min
F56=5 L/min
P2=93 mm Hg
P3=33.85 mm Hg
P4=15.65 mm Hg
P5=93 mm Hg
P6=33.85 mm Hg
P7=15.65 mm Hg
P8=2 mm Hg
P9=2 mm Hg
P10=2 mm Hg
P11=33.85 mm Hg
P12=15.65 mm Hg
P13=2 mm Hg
P14=93 mm Hg
P15=33.85 mm Hg
P16=15.65 mm Hg
P17=2 mm Hg
P18=93 mm Hg
P19=33.85 mm Hg
P20=15.65 mm Hg
P21=2 mm Hg
P22=93 mm Hg
P23=33.85 mm Hg
P24=15.65 mm Hg
P25=2 mm Hg
P26=93 mm Hg
P27=33.85 mm Hg
P28=15.65 mm Hg
P29=2 mm Hg
P30=93 mm Hg
P31=33.85 mm Hg
P32=15.65 mm Hg
P33=2 mm Hg
P34=93 mm Hg
P35=33.85 mm Hg
P36=15.65 mm Hg
P37=2 mm Hg
P38=93 mm Hg
P39=42.7225 mm Hg
P40=93 mm Hg
P41=54.5525 mm Hg
P42=2 mm Hg
P43=93 mm Hg
P44=93 mm Hg
P45=33.85 mm Hg
P46=15.65 mm Hg