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Vol. 168, 2711–2715, December 2002
Printed in U.S.A.
DOI: 10.1097/
From the Departments of Urology, Rabin Medical Center and Sackler School of Medicine and Department of Biomedical Engineering,
Tel Aviv University, Tel Aviv, Israel
Purpose: We examined and elucidated the physical mechanism of penile erection in normal and
pathological conditions of vascular origin.
Materials and Methods: A nonlinear lumped parameter mathematical model of penile hemo-
dynamics was developed based on physical structure and physiological function. The model was
applied to simulate the normal erectile mechanism and the pathological conditions of arterial
insufficiency and venous leakage.
Results: The model showed physiological behavior and suggested explanations for the rela-
tionships of corporeal pressure and venous flow limitation during erection. Other results related
the reduction in rigidity and tumescence to the type and severity of vascular impairment.
Conclusions: This model can be used in future studies of the diagnosis of erectile dysfunction.
KEY WORDS: penis; impotence; hemodynamics; models, theoretical
Normal penile erection is a complex integrated response
veno-occlusive function by determining the flow necessary to
mediated by vasomotor nerves controlling the flow of blood to
maintain erection. Controversies remain about what the
the penis and affecting its drainage. Hemodynamic events
common reference value should be in all studies available.6, 7
that occur during normal erection and detumescence require
Applying data obtained from normal physiological or in-
coordinated alterations in blood flow, intracavernous pres-
duced erection as well as from patients with erectile dysfunc-
sure and penile volume. Normally during erection relaxation
tion to a theoretical model of penile erection is important for
of the trabecular smooth muscle of the corpora allows high
validating the model. After it is established it can provide
intracavernous pressure with low maintenance inflow. In
insights into problems of physiological mechanisms and help
fact, trabecular smooth muscle tone is the main regulating
us understand changes in variable data during normal erec-
factor of arterial flow and the corporeal veno-occlusive mech-
tion and dysfunction. Moreover, it can serve as a tool for
analyzing the various diagnostic methods. We developed a
Erectile dysfunction of vascular origin is classified as ar-
model of penile erection and studied its relevance to the
teriogenic or veno-occlusive. The principal causes of arterio-
behavior of penile blood vessels during the flaccid and erec-
genic etiology are compromise of arterial inflow to the penis
tile phases, and dysfunction.
due to obstruction within the penis because of arteriosclerotic
changes and arterial occlusion of vessels leading to the pe-
nis.2 Venous occlusion is a mechanical hemodynamic process
depending on volume and pressure changes due to changes in
A model of penile erectile function is presented, in which
the tone of cavernous smooth muscle. Impairment may be
the corpora and tunica are regarded as a compliant chamber,
caused by reduced relaxation or increased contractility of
and the arteries and veins supplying blood and draining the
corporeal smooth muscle, or abnormal communications be-
chamber are regarded as compliant vessels subjected to ex-
tween the corpora and spongiosum.3
ternal pressure, that is corporeal pressure (fig. 1). A model of
Tests available for evaluating arterial flow and the veno-
compliant tubes subjected to external pressure has previ-
occlusive mechanism include penile duplex ultrasonography,
ously been presented.8, 9 These groups presented an analysis
which can be combined with pharmacological administration
that yielded an expression relating flow through a compliant
to the penis to obtain further dynamic information on peak
tube to the pressure in its inlet, the pressure at its outlet and
flow velocity, acceleration time and the resistance index.4, 5
to an external pressure applied to the tube. They also ana-
Penile angiography is done to investigate vessels leading to
lyzed 2 special cases, in which compliant tubes were sub-
and from the penis. Cavernosometry quantifies and maps
jected to an external pressure equal to outlet pressure, as
observed in penile arteries, or inlet pressure, as observed in
Accepted for publication June 28, 2002.
FIG. 1. Model of penis. P , arterial pressure. Q , arterial flow. Q , venous flow. P , venous pressure. P , corporeal pressure

FIG. 2. Model predicted arterial flow as function of corporeal pres-
sure at different values of arterial elasticity (E). For same corporeal
pressure lower elasticity or stiffness produced greater flow.
penile veins (fig. 1). They developed a model based on exper-
imental descriptions of the pressure-volume relationships in
collapsible blood vessel using its elasticity factor (E) and a
constant (C) related to mechanical factors.8, 9
The mathematical model is described by equations 1 to
3.8, 9 Equation 1 states, Q
IG. 4. Computer simulation of normal erection shows that ve-
1 , where
nous flow is unchanged during erection despite significant increase
A represents artery, V represents vein, P represents corpo-
in corporeal pressure. Ea, arterial elasticity.
real pressure and P represents corporeal volume. Equation
2 states, Qout
e2 PV PC /EV . Equation 3 states,
a1ln a2VC
opposing flow. Equilibrium results when flow is independent
Equation 1 indicates that inflow is a function of the difference
of pressure (fig. 3).
in arterial and corporeal pressures, and the elasticity factor.
To understand this important phenomenon intuitively we
Thus, when the pressure difference is greater, inflow is greater
can think of corporeal pressure as the driving pressure that
and decreases to zero after equilibrium is obtained, that is when
generates flow as well as the force exerted on the external
P . Since the effect of the elasticity factor is greater in the
side of the vessel, which decreases flow. Due to flow in the
exponential term, the equation also indicates that inflow is
vessel and its resistance to flow we observe a pressure drop
greater in more compliant vessels (smaller elasticity factor). It
along the vessel. Since external pressure is constant along
is reasonable since greater pressure differences in arterial and
the vessel, transmural pressure working to close the vessel
corporeal pressure, and more compliant vessels contribute to
increases along the vessel. In extreme cases if due to trans-
greater cross-sectional area, leading to smaller resistance to
mural pressure the vessel completely collapses and no flow
flow and rapid filling of the corpora.
occurs, there is no pressure drop along the vessel and pres-
Based on this equation figure 2 shows inflow as a function
sure at the collapsed point is equal to inlet pressure. This
of corporeal pressure and vessel elasticity, assuming con-
situation reduces transmural pressure to zero and the vessel
stant arterial pressure. Figure 2 also shows this relationship
opens. This mechanism prevents the vessel from full collapse
at different levels of arterial elasticity. The lower the elastic-
and, therefore, venous flow is never completely blocked. This
ity (higher compliance), the greater the inflow.
explanation is congruent with the observations regarding the
Notably figure 2 is not limited to the range of arterial
oxygenation of penile blood during erection.
pressures that are higher than corporeal pressure. In the
In the artery, the situation in which corporeal pressure is
range in which corporeal pressure is greater than arterial
greater than arterial pressure is similar to the vein. This
pressure backflow is small and limited, as in the partially
indicates a partial collapse of the artery that prevents retro-
collapsed veins where this condition occurs.
grade loss of volume during high pressure events resulting
Equation 2 indicates that outflow from the vein is never
from contraction of the ischiocavernous and bulbocavernous
zero but it is limited according to venous elasticity (fig. 3).
muscles. Since there is no evidence that venous elasticity
Increased corporeal pressure causes 2 opposing forces, in-
changes in disease states, the arteries completely control the
cluding greater driving pressure for flow in the draining vein
process. The arterial elasticity factor is expected to drop
and greater external pressure acting on the vein, that is
significantly in the transition from the flaccid to the erect
FIG. 3. Venous flow as function of corporeal pressure with fixed systemic venous pressure. E, elasticity

FIG. 7. Moderate venous leakage created significantly increased
inflow and outflow during transition phase and erection. Normal
FIG. 5. Arterial insufficiency caused by partial arterial occlusion
volume may be achieved at lower pressure and lower elasticity (Ea).
prolonged transition period and prevented buildup of sufficient pres-
sure despite increased volume. Ea, arterial elasticity.
leakage was included. This simulation resulted in graphs
(fig. 4). There was a gradual change in arterial elasticity,
state. Figure 2 shows a similar situation, while elasticity of
which dropped from 500 to 10. As arterial elasticity reached
the vein is unchanged, as indicated by the single curve (fig.
low values (trabecular relaxation or increased compliance),
inflow began to increase, while outflow increased by only a
The corporeal chamber is considered an elastic chamber
small increment, allowing corporeal filling and increased cor-
with pressure-volume relationships similar to those of most
poreal pressure. Despite increased corporeal pressure venous
biological tissues. The parameters in equation 3 are a1
draining did not increase due to the mechanism described.
0.2 and a3
10 9. To our knowledge the exact
Venous flow was limited since driving pressure was equal to
value of elasticity that represents sufficient rigidity is not
the pressure exerted externally on the vein. Thus, with vol-
known, although it is known that a buckling force of about
ume and pressure maintained (tumescence and rigidity), a
500 gm. is required to allow vaginal penetration. The rela-
low rate of flow through the penis was also maintained.
tionship of these parameters is the focus of another study. We
Arterial insufficiency caused by arterial occlusion was sim-
assumed in this particular case that rigidity is achieved at 75
ulated by increased resistance. Figure 5 shows a representa-
mm. Hg. Additional pathological conditions were added to
tive plot. Decreased arterial inflow led to a slow filling pro-
the model, including arterial occlusion in the form of a Poi-
cess, lower corporeal volume and corporeal pressure below
seuille resistance and shunts representing venous leakage, in
the elastic threshold of the tunica. This condition is mani-
which blood flows directly to the external vein with no col-
fested in reduced tumescence without rigidity.
lapsing effect due to corporeal pressure.
Arterial insufficiency may also be caused by decreased
arterial compliance (increased elasticity). This condition was
simulated by limiting the lowest elasticity that can be
Model parameters were initially set to simulate normal
achieved during erection. When simulating the normal mech-
conditions. The arteries had no occlusion and no venous
anism, the lowest value was set to 10 mm. Hg cm. 3 and in
this case the value was set to 40 mm. Hg cm. 3. Figure 6
FIG. 6. Arterial insufficiency caused by decreased arterial compli-
FIG. 8. Arterial occlusion plus venous leakage had detrimental
ance also prolonged transition period and prevented buildup of suf-
effect on corporeal pressure and volume, resulting in low elasticity,
ficient pressure despite increased volume, similar to arterial occlu-
similar to combined effect of increased arterial elasticity (Ea) and
sion. Ea, arterial elasticity.
venous leakage.

shows that the effect was quite similar to arterial occlusion.
The rate of elasticity decrease in the transition from the
In each case volume built up to almost a normal value.
flaccid to the erect state was set arbitrarily to mimic physiolog-
However, pressure did not reach the level required to pro-
ical behavior instead of using a sharp step function. It resulted
duce sufficient rigidity.
in rapidly increasing flow into the corpora and volume increase,
Figure 7 shows a case of good arterial inflow with venous
followed by a slower pressure increase. Increased pressure in
leakage. Outflow increased during erection, in contrast to the
the corpora did not generate a proportional outflow increase,
low flow rate during this phase in the normal penis. In this
but rather a limited increase in flow, as explained. Thus, the
specific case tumescence was maintained with marginal rigidity
increase in venous resistance was a function of increased cor-
achieved due to adequate arterial inflow. This condition
poreal pressure10 resulting from the partial collapse of veins.
changed dramatically due to the impaired arterial inflow
The increase in corporeal pressure affected driving and imped-
caused by partial occlusion in addition to venous leakage.
ing pressures, resulting in an equilibrium that limited outflow.
Figure 8 shows the detrimental effect of this combined dysfunc-
Since some flow always exists, pressure in the corpora could not
reach the arterial level. The study of Kursh et al supports our
Figures 9 and 10 show a 3-dimensional presentation of the
model and indicates that the flow predicted by our model during
combined effect of changes in arterial occlusion and venous
flaccid and rigid conditions is similar to experimental observa-
leakage. The graphs indicate the steady-state result of the
tions in the normal penis.11
interaction among impaired inflow, increased outflow and
When applying the model to conditions with reduced arterial
the pressure-volume relationship of the corpus cavernosum.
inflow due to reduced arterial compliance or arterial constric-
A region of tumescence could be achieved in the impaired
tion, the time constant of erection significantly increased (figs. 5
erectile mechanism, while the region of sufficient rigidity
and 6). Although tumescence may be obtained with some arte-
that could be obtained was significantly smaller, indicating
rial obstruction, pressure buildup may be severely limited, pre-
that rigidity can be reached only when inflow and outflow are
venting the achievement of functional erection. During erection
normal. Tumescence may be achieved in the presence of some
in the presence of venous leakage tumescence may be achieved
impairment of either function.
but rigidity is not (fig. 7). Figure 8 shows the detrimental effect
of both conditions occurring at the same time. The figure also
shows the changes in flow during the tumescent and erect
The erectile mechanism was evaluated with the aid of a
states. These figures are similar to those of Kursh et al, who
mathematical model developed to study the physical proper-
reported that in cases of venous leakage inflow and outflow
ties of the system and their cumulative effect on the erectile
during the erect state are greater than flow during the flaccid
process. In this series the model was used to elucidate the
state.11 These observations support our model predictions but
erectile mechanism. The study was extended to assess patho-
contradict the interpretation of experimental data reported by
logical conditions.
Saenz de Tejada et al.10
FIG. 9. Combined effect of arterial occlusion and venous leakage on corporeal pressure shown as 3-D plot. Surface indicates that
tumescence was achieved for range of combinations. Beyond tumescence range volume decreased rapidly with small increases in arterial
occlusion or venous leakage.

FIG. 10. Combined effect of arterial occlusion and venous leakage on corporeal pressure shown as 3-D plot. Surface indicates that pressure
buildup was sensitive to each variable over whole graph. Pressure decrease and sensitivity to 2 variables was also evident within range where
volume was less sensitive and tumescence was attained, indicating possibility of tumescence without pressure buildup. Combined effect of
2 variables was much greater than each separate effect.
These results are summarized in 3-dimensional (D) graphs
Physiol Rev, 75: 191, 1995
showing that the volume surface has a significant region
3. Christ, G. J.: The penis as a vascular organ. The impotence of
where it is in the normal range, while pressure is reduced at
corporeal smooth muscle tone in the control of erection. Urol
the first sign of arterial insufficiency or venous leakage (figs.
Clin North Am, 22: 727, 1995
9 and 10). The value of this 3-D presentation is that it can
4. Lue, T. F., Hricak, H., Marich, K. W. and Tanagho, E. A.:
show changes that occur simultaneously in all 3 aspects of
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accounts for all varying vascular parameters involved.
Namburi, S., Pescatori, E. S., Udelson, D. et al: In vivo assess-
This model indicates physiological behavior and is con-
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firmed by the experiments of Kursh et al in normal patients
pharmaco-cavernosometry and analysis of intracavernous
and in those with venous leakage.11 It explains the observed
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fusing the penis with oxygenated blood during erection, as
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shown by Vardi.12 This model also suggests that arterial
nile erection. IEEE Trans Biomed Eng, 47: 319, 2000
retrograde flow is possible when corporeal pressure is greater
9. Barnea, O.: A theoretical unidirectional valve based on func-
than arterial pressure but it is limited in the same fashion as
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also be used in future studies regarding the diagnosis of
10. Saenz de Tejada, I., Moroukian, P., Tessier, J., Kim, J. J.,
erectile dysfunction.
Goldstein, I. and Frohrib, D.: Trabecular smooth muscle mod-
ulates the capacitor function of the penis. Studies on a rabbit
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