Theoretical Analysis of Complex Oscillat

MICROVASCULAR RESEARCH
ARTICLE NO.

51, 229–249 (1996)

0023

Theoretical Analysis of Complex Oscillations in Multibranched
Microvascular Networks
M. URSINO,* S. CAVALCANTI,* S. BERTUGLIA,†

AND

A. COLANTUONI†

*Department of Electronics, Computer Science and Systems, University of Bologna, 40136 Bologna,
Italy; and †CNR Institute of Clinical Physiology, University of Pisa, 56100 Pisa, Italy
Received April 25, 1995
A mathematical model was used to study the origin of complex self-sustained diameter oscillations in multibranched microvascular networks. The model includes three branching levels (order
3, 2, and 1 arterioles) of a microvascular network derived from in vivo observation in the hamster
dorsal cutaneous muscle. The main biomechanical aspects covered by the model are (1) the

dependence of the elastic and active wall stress on the inner radius and (2) the static and dynamic
myogenic response. Simulations on isolated arterioles indicate that self-sustained periodic diameter oscillations may occur at constant transmural pressure. Conversely, simulations on the entire
network reveal different oscillatory patterns, including periodic, quasiperiodic, and chaotic fluctuations. Chaos in the model is revealed by the presence of a broad noise-like component in the
frequency spectrum and by the sensitivity dependence of model results on small perturbations.
Our results suggest that, owing to the intrinsic nonlinearity of the system, a contracting mechanism, such as the myogenic response, may induce different oscillatory patterns. The change from
periodic to chaotic oscillations may be a consequence of a modest variation in a parameter
(systemic pressure or arterial resistance) not necessarily related to pathophysiological conditions.
Accordingly, our in vivo observations in the skeletal muscle showed that in some instances
arteriolar vasomotion is converted from regular to highly irregular patterns in basal conditions.
Vasomotion is found to affect mean blood flow compared with the nonoscillatory steady state.
Chaotic oscillations tend to maintain a constant ratio of blood flows entering into bifurcation
vessels, whereas periodic vasomotion determines a different flow distribution at branches.
q 1996 Academic Press, Inc.

INTRODUCTION
The time pattern of arteriolar diameter changes may exhibit disparate characteristics
in in vivo microvascular preparations. Intaglietta and Breit (1991) recognized arterioles
with periodic self-sustained oscillations and with random diameter fluctuations without
any evident periodicity. The last pattern has been frequently reported in different
experimental models (Colantuoni et al., 1984, 1990; Meyer and Intaglietta, 1986;

Bertuglia et al., 1991).
The presence of periodic or aperiodic diameter fluctuations of arterioles (vasomotion) is conditioned by the interaction of many biomechanical and regulatory variables
which operate simultaneously on microvessels: the characteristics of the smooth muscle cells, the activity of local control mechanisms (myogenic, metabolic, neurogenic,
and endothelium-dependent), and the branching pattern of the microvascular network.
Owing to the nonlinear nature of arteriolar hemodynamics, arteriolar oscillations may
modify blood flow distribution to tissues and microvascular fluid exchange. Significant
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0026-2862/96 $18.00
Copyright q 1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.

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functional effects might occur when arterioles oscillate compared with those evaluated
under nonoscillatory conditions.
Factors which contribute to microvascular hemodynamics and their interactions
can be advantageously analyzed by mathematical models. The recent finding that
deterministic models can produce irregular, unpredictable fluctuations, called chaos,

can provide a deeper insight into the behavior of physiological systems. The hypothesis
that the microcirculation may present a chaotic behavior has been stressed recently,
based on the evaluation of phase–space trajectories from laser Doppler measurements
(Intaglietta and Breit, 1991) or on the calculation of the fractal dimension of perfusion
pressure oscillations (Griffith and Edwards, 1993).
In the model of Ursino et al. (Ursino and Fabbri, 1992; Ursino et al., 1992) microvascular instability and periodic oscillations in arteriolar diameter arise from the myogenic
mechanism, especially from its rate-dependent component at the level of terminal
arterioles. The model by Gonzales-Fernandez and Ermentrout (1994) incorporates
ionic transport and cell membrane potential and shows that interaction of Ca2/ and
K/ fluxes mediated by gated channels results in periodicity of these transports and
periodic changes in vessel diameter. Ackari et al. (1994) theoretically investigated the
conditions leading to low-frequency oscillations in muscular arteries. In a mathematical
model of microvascular network Kiani et al. (1994) analyzed fluctuations in blood
flow parameters caused by hemorrheological properties and by other mechanisms
(such as the myogenic response). None of these models, however, analyzed a regulated
multibranched microvascular network exhibiting transition to more complex patterns
like those observed in vivo.
The aim of our paper is to develop a mathematical model of a microvascular
branching system including first- to third-order arterioles. The model is based on the
myogenic mechanism, for which a mathematical model was studied earlier (Ursino

and Fabbri, 1992; Ursino et al., 1992) and includes the interaction of arterioles of
various branching orders. Geometrical and mechanical model parameters are obtained
from in vivo measurements in the hamster dorsal cutaneous muscle microcirculation.
The occurrence of periodic, quasiperiodic, and chaotic oscillatory patterns of vessel
diameter is then predicted with this mathematical model. Moreover, the model is used
to investigate whether oscillations may change distribution of mean blood flow at
bifurcation in the microvascular network.
MATERIALS AND METHODS
Experimental Method
Experimental measurements were made in cutaneous muscle microcirculation in
hamsters implanted with a plastic chamber, as previously described (Colantuoni et
al., 1984). Experiments were made in 10 male Syrian hamsters (Charles River, Calco
Co., Italy), weighing 80–100 g, housed individually with free access to water and
food pellets. One of the microvascular networks was chosen for the present paper
since it presented average parameters common to different animals.
Briefly, the experimental procedures consisted of the implantation of two symmetrical plastic frames which support a dorsal skinfold, while the hamsters were anesthetized (5 mg/100 g body wt, pentobarbital ip) (Nembutal; Abbott, North Chicago, IL).
A round area of the dorsum skin and underlying muscle of 15-mm diameter was
completely removed from one side of the symmetrical fold, exposing the opposite

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231

layer of skin muscle (M. cutaneus maximus) attached to the subcutaneous tissue. The
tissue was covered by a microcover glass which was fixed to one of the plastic frames,
while the other part stayed open so that the skin was in its natural environment.
Permanent catheters filled with heparinized saline were implanted in the jugular vein
and the carotid artery. They were passed under the skin to the upper posterior part of
the neck and fixed to the upper part of the window. The animals were allowed to
recover for 48 hr in an incubator at 30 { 0.57 with free access to food and water.
Unanesthetized animals were placed in a transparent tube that minimized movements without impeding respiration. Both the tube and the extending frame of the
chamber were fixed to the microscope stage. The microcirculation was studied with
fluorescent microscopy (Bertuglia et al., 1991), where fluorescein isothiocyanate
bound to dextran (MW 150,000) was injected intravenously (50 mg/100 g body wt
in 5% solution) and observed under a Leitz Orthoplan microscope fitted with a longworking-distance objective (14, na 0.12; 120, na 0.25; 132, na 0.60) and 110
eyepiece. It was then televised by a COHU 5253 SIT low-light-level camera, viewed
on a Sony PVM 122 CE monitor and recorded with a Sony U-Matic VO 5800 PS
videorecorder. Fluorescence excitation was obtained by filtering the emission of a
xenon 150-W lamp using a Leitz I2 Ploemopack filter block.
Mean arterial blood pressure (Statham PD 23 transducer connected to catheter in

carotid artery) and heart rate were monitored by a Honeywell RM 300 monitor interfaced to an IBM XT 286 personal computer. The temperature of the chamber was
controlled at 30 { 0.57 by a jet of warmed air.
The vessels in each preparation were ordered by the Strahler method (Bertuglia et
al., 1991); the scheme was applied to the terminal arteriolar networks originating from
the arterioles feeding the muscle. First, the capillaries were defined and assigned order
0. Thereafter, the terminal arterioles were assigned order 1 and vessels upstream were
assigned progressively higher orders.
Vessel diameters were measured with a computerized technique. The sequences of
interest were digitized in real time, using a custom-designed image-processing system
based on extension boards (Matrox PIP-640 B) installed in a HP Vectra RS/25C
computer. The vessel under study was aligned vertically and measured by positioning
a window across its lumen. An edge-direction algorithm detected the inside wall of
the blood vessel and computed the average diameter within the window. To individuate
the most significant frequency components of the diameter fluctuations, periods of
vasomotion lasting up to 1 hr were considered. Since during these periods vasomotion
pattern was clearly unstationary showing different aspects in the time course, several
short epochs 1-min long were extracted and, for each one, power spectral density was
calculated by using an autoregressive technique. As an advantage, autoregressive
approach, compared with the fft-based technique, permits an efficient filtering of noise
effects and a more accurate frequency resolution in the case of short-time series.

Qualitative Model Description
The characteristics of the model reproduce the network architecture, the elastic and
muscular properties of isolated microvessels located along the microvasculature, and
the static and dynamic myogenic response. A quantitative description of equations
and parameter numerical values are given in the Appendix.
Network Layout
The model includes significant simplifications. First, arterial-reactive effects—i.e.,
blood volume changes and inertia—were assumed to be negligible compared with

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FIG. 1. (a) Schematic representation of the hamster dorsal cutaneous muscle microcirculation. The portion
indicated has been reproduced in the mathematical model. The diagram is not in scale. OR 1, OR 2, OR
3: order 1, order 2, order 3 arterioles. (b) Electric analog of the network reported in a. R0 , vascular resistance
of the upstream circulation down to and including order 4 arterioles; R1 , vascular resistance of a order 3
arteriole; R2 and R3 , vascular resistances of order 2 arterioles; R4 and R5 , load resistances including order
1 arterioles. Pa , systemic arterial pressure; Pv, venular pressure.

viscous pressure losses. This simplification is commonly adopted in microvascular
hemodynamic studies. Second, we assume that venular pressure changes are negligible
throughout the simulations, in accordance with experimental findings (Borgstrom and
Gestrelius, 1987) and justified by the large venous compliance when compared to that
of the arteriolar network.
The model partly reproduces a hamster microvascular network including order 3
to order 1 arterioles. Simulation data were chosen from the hamster whose values
were closer to the average values of the examined population (Fig. 1a). There were no
significant differences in simulation results using different hamster data. The vascular
resistance of the upstream systemic circulation, including order 4 arterioles, is R0 in
the electric analog (Fig. 1b). R1 is the vascular resistance of a order 3 arteriole with
an acute branching angle from the order 4 vessel. R2 and R3 are the resistances of two
order 2 arterioles branching from the previous microvessel. Finally, R4 and R5 correspond to the resistance of the microvasculature downstream the two order 2 arterioles,
from order 1 arterioles to the venular pressure Pv . It was assumed that R4 and R5
changes are mainly affected by diameter alterations of order 1 arterioles.

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233

The resistance R0 is scarcely affected by the hemodynamic alterations within the
two downstream arteriolar branches, hence, its value was maintained constant throughout each individual simulation. However, R0 was significantly changed between simulations to study upstream influences on vasomotion pattern.
According to the Hagen–Poiseuille law, segment 1–5 resistances are inversely
proportional to the fourth power of the corresponding arteriolar radius,
Rj Å

KRj
,
r4j

j Å 1, 2, . . ., 5,

(1)

where rj is the inner radius in the segment and KRj is a constant parameter dependent
on microvessel length. The values of parameters KRj were determined from measurements of order 3, order 2, and order 1 arteriole length in hamster dorsal cutaneous
muscle microcirculation (Fig. 1a) and using the hydrostatic pressure profile in the
hamster macro- and microcirculation measured by Davis et al. (1986).
Length–Tension Characteristics of the Arterioles
The inner radius of each arteriole [Eq. (1)] was obtained from a force balance in

the vessel wall. The force per unit length acting on the wall, caused by the difference
between intravascular and extravascular pressure, dilates the vessel, and must be
equilibrated by total wall tension. According to the Laplace law,
pijrrj 0 per(rj / hj) Å (sej / smj / svj)rhj ,

j Å 1, 2, . . ., 5,

(2)

where pij is intravascular pressure in the jth segment, evaluated with reference to the
atmosphere; pe is extravascular pressure (considered equal to zero); hj is wall thickness;
and sej , smj and svj are elastic, active (muscular), and viscous circumferential wall
stresses, respectively.
The intravascular pressure, pij , was computed as the weighed average of pressures
in the inflow and outflow sections. Since smooth muscle activity is usually predominant
at the bifurcations and then propagates downstream (Meyer et al., 1987; Bertuglia et
al., 1991), scaling factors were chosen so as to emphasize the microvessel sections
close to the upstream inflow sections.
Elastic and active stresses depend on the inner radius through sharply nonlinear
relationships. These reflect different stretching of elastin and collagen fibers in the

wall and different overlapping of myosin and actin filaments in the muscle fibers. The
dependence of the elastic and active wall stress on inner radius was reproduced using
length–tension data of isolated microvessels reported by Davis and Gore (1989) in
hamster cheek pouch. According to Davis and Gore, maximal active wall stress of
2A and 3A arterioles, corresponding to order 3 and order 2 arterioles, during constriction by high K/ or norepinephrine solution is about 2.4 1 106 dyn/cm2 (1800
mm Hg).
Viscous stress in Eq. (2) is proportional to the rate of change of inner radius, hence
it falls to zero in steady-state conditions. Finally, wall thickness, as a function of inner
radius, was computed considering the wall incompressible.
We consider smooth muscle cell activity in basal condition about 13 of that during
maximal activation. As a consequence, according to Davis and Gore (1989), steadystate vessel diameter in basal conditions is 60–70% of the maximum passive diameter.

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TABLE 1a
Parameters Describing the Microvascular Network
R0 Å 5 1 107 mm Hg
sec cm03
KR3 Å 3.5 1 1006 mm Hg
sec cm
Pv Å 15 mm Hg
se01 Å 1.91 1 1002 mm Hg
sm01 Å 600 mm Hg
h2 Å 500 mm Hg sec
r02 Å 6.875 1 1004 cm
rm2 Å 11 1 1004 cm
ke4 Å 2.65 1 104 cm01
km4 Å 14 1 106 cm02

KR1 Å 1.4 1 1006 mm Hg
sec cm
KR4 Å 1.12 1 1006 mm Hg
sec cm
a Å 0.1
ke1 Å 1.5 1 104 cm01
km1 Å 6 1 106 cm02
se02 Å 1.91 1 1002 mm Hg
sm02 Å 600 mm Hg
h4 Å 500 mm Hg sec
r04 Å 5 1 1004 cm
rm4 Å 8 1 1004 cm

KR2 Å 1.06 1 1006 mm Hg
sec cm
KR5 Å 1.12 1 1006 mm Hg
sec cm
h1 Å 500 mm Hg sec
r01 Å 10 1 1004 cm
rm1 Å 16 1 1004 cm
ke2 Å 2.15 1 104 cm01
km2 Å 11 1 106 cm02
se04 Å 1.91 1 1002 mm Hg
sm04 Å 600 mm Hg

Note. Parameters in segment 3 are equal to those in segment 2. Similarly, parameters in segment 5 are
equal to parameters in segment 4. We only have KR2 x KR3 since the network is asymmetrical.

Maximal vasoconstriction in the model reduces the order 3 vessel radius by 018.53%,
while total vasodilation increases the radius by /47.2%. Analogous results were
obtained for the order 2 and order 1 arterioles.
Table 1a shows the geometrical and biomechanical parameters of the network (see
the Appendix for the meaning of symbols). Moreover, Table 1b reports the inner radii,
intravascular pressures, total wall tension, and wall thickness evaluated under steadystate conditions, assuming pa Å 100 mm Hg as mean arterial pressure under basal
conditions. The simulations in the subsequent section, however, were performed on
a network with an asymmetric layout (in particular, KR2 x KR3 in accordance with the
experimental data reported in Fig. 1a). The asymmetry introduces a small alteration
in the steady-state values of the hemodynamic quantities. More significant alterations
will be simulated by changing different hemodynamic parameters of the upstream
arterial circulation (i.e., the inflow resistance R0 and the systemic arterial pressure pa,
see Results).
The Myogenic Response
In our model feedback regulatory mechanisms and other vasoactive substances
affect the amplitude of the active length–tension curve. In this paper attention is
focused on the myogenic response, since our purpose is to demonstrate its role on
arteriolar quasiperiodic or chaotic oscillations. Indeed, other vasodilatory or vasoconstrictory variables are described as additional constant inputs acting on smooth muscle
TABLE 1b
Basal Values of the Main Hemodynamic Parameters
r1n Å 11.26 1 1004 cm
piln Å 40.5 mm Hg
T2n Å 2.97 1 1002 mm Hg cm
h4n Å 1.4 1 1004 cm

h1n Å 3 1 1004 cm
r2n Å 7.4 1 1004 cm
pi2n Å 38.8 mm Hg
T4n Å 1.38 1 1002 mm Hg cm

T1n Å 4.68 1 1002 mm Hg cm
h2n Å 2 1 1004 cm
r4n Å 4.3 1 1004 cm
pi4n Å 30.8 mm Hg

Note. These are coincident with the equilibrium values of the nonregulated symmetrical network (KR2 Å
KR3 Å 3.5 1 1006 mm Hg sec cm) with an inflow pressure pa Å 100 mm Hg. Values in segment 3 are
equal to those in segment 2, while values in segment 5 are equal to those in segment 4.

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235

TABLE 1c
Parameters Describing the Myogenic Response
Gms1 Å 40 mm Hg01 cm01
Dm1 Å 5 sec
tm2 Å 4 sec
Gmd4 Å 3200 sec mm Hg01 cm01

Gmd1 Å 1600 sec mm Hg01 cm01
Gms2 Å 40 mm Hg01 cm01
Dm2 Å 5 sec
tm4 Å 4 sec

tm1 Å 4 sec
Gmd2 Å 2400 sec mm Hg01 cm01
Gms4 Å 40 mm Hg01 cm01
Dm4 Å 5 sec

Note. Values in segment 3 are equal to those in segment 2, while values in segment 5 are equal to those
in segment 4.

cells. Additional mechanisms, such as flow-dependent or neurogenic factors, can be
included in the model with further feedback loops.
Both ‘‘static’’ and ‘‘rate-dependent’’ components of the myogenic response have
been considered, as in Ursino and Fabbri (1992). In particular, we assumed that the
amplitude of the active length–tension curve depends on the instant value and the
time derivative of total wall tension through a first-order linear differential equation.
The coefficients in this equation represent the ‘‘static’’ and the ‘‘dynamic’’ gains of
the myogenic response [Gms and Gmd in Eq. (10) of the Appendix]. A notable difference
in this model with respect to the previous one is that we incorporated the presence
of a latent period in the myogenic response. This choice is supported by several
observations on the time pattern of the arteriolar radius following rapid step pressure
changes. In particular, Grande and Mellander (1978) state that the myogenic response
starts 1–2 sec after the beginning of a transmural pressure variation at the highest
rate of change employed, but is somewhat more delayed at slower rates. Even greater
delays are evident from other experiments (Davis and Sikes, 1990). We assumed a
basal value for the latent period of the myogenic response as high as 5 sec, which
agrees with the observation by Grande et al. (1979).
We assume that the dynamic myogenic response is predominant in terminal arterioles, whereas it progressively decreases in upstream parent vessels. The gains and the
time constant of the mechanism in basal condition have been given (Table 1c) according to experimental results on myogenic response (Johnson and Intaglietta, 1976;
Grande et al., 1979; Osol and Halpern, 1985; Davis and Sikes, 1990; Davis, 1993).
Since the experimental results, however, show a wide range of values for the magnitude
of the static and dynamic responses, a sensitivity analysis of these parameters is
reported under Results.
RESULTS
Experimental Data
The hamster cutaneous muscle microcirculation consists in an arcading arteriolar
system that gives origin to terminal branchings. It was possible to differentiate two
different types of terminal networks related to the length of arteriolar orders. The first
constituted a long loop feeding the muscle fibers, the second loop was shorter and
was composed of three orders of arterioles. For the purpose of our study we used the
data of short-loop microvasculature. The arterioles were classified according to Strahler
ordering scheme, starting from capillaries assigned order 0 vessels. Order 1 were
terminal arterioles and order 4 corresponded to arcading arterioles (Fig. 1a).
Microvascular networks showed arteriolar vasomotion with different fundamental
frequencies (the frequency with the highest amplitude) and percentage amplitudes

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TABLE 2
Means of Length, Diameter of Arterioles, Frequency, and Percentage Amplitude of Vasomotion
Arteriolar
order

n

1
2
3
4

95
55
25
10

Length
(mm)
80
126
234
3147

{ 7
{ 22
{ 40
{ 629

Diameter
(mm)
7.80
11.5
20.1
35.4

{
{
{
{

1.0
2.2
3.7
9.7

Frequency
(cpm)
8.9
7.8
7.8
2.5

{
{
{
{

1.5
1.6
1.6
1.8

Amplitude
(%)
60–100
50–100
30–80
10–20

Note. Mean { SD; n, number of arterioles.

(percentage changes of mean diameter). However, order 3 arterioles originating in the
short loop dominated order 2 daughter vessel vasomotion frequency, while order 1
terminal arterioles presented a higher frequency. The diameter, length, frequency, and
amplitude of vasomotion in the different arteriolar orders of the short loop networks
are reported in Table 2. Examples of vasomotion and corresponding power spectrum
analyses are reported in Figs. 2a and 2b. During the basal observation period of
vasomotion activity different temporal and spectral patterns were observed: epochs
of regular activity alternated to highly irregular ones.
Simulations on the Isolated Arterioles
Preliminary simulations were conducted on isolated arterioles whose intravascular
and extravascular pressures were considered input quantities. The consequent values
of arteriolar diameter and resistance were computed.
We found that when a latent period is included in the myogenic response, the
arteriole may exhibit self-sustained oscillations even at constant intravascular and
extravascular pressure values (Fig. 3). Figure 4 shows how a pair of parameters
describing the myogenic mechanism should vary concurrently to maintain the isolated
microvessel in a condition of self-sustained oscillations. There are various combinations of the myogenic parameters (i.e., the static and dynamic gains and the latent
period) which can trigger self-sustained oscillations. If the latent period and/or the
dynamic gain of the myogenic response increases, self-sustained oscillations develop
even in the presence of a weak static gain. By contrast, very high values of the static
gain are required to induce self-sustained oscillations if both the dynamic response
and the latent period are low.
Simulations on the Entire Network
The entire network simulations showed a variety of different kinds of oscillatory
behavior including periodic, quasiperiodic, and chaotic patterns. These patterns may
vary significantly in response to small changes in the value of a parameter. We present
an example of the role of arterial pressure, whereas the results of other simulations
will be briefly summarized.
When arterial pressure was 90 mm Hg, periodic self-sustained oscillations, with a
fundamental frequency of about 6 cpm (Fig. 5), were observed. Therefore, the different
oscillating microvessels are phase locked and, as is evident from the three-dimensional
subspace (r1 , r2 , r4), the entire system is forced to oscillate at a single frequency.
Segments 4 and 5 oscillations are in phase opposition, i.e., segment 4 is dilated when

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237

FIG. 2. (a) Vasomotion pattern of an order 3 arteriole and the corresponding power spectrum (autoregressive modeling). (b) Vasomotion pattern of an order 1 arteriole and the corresponding power spectrum
(autoregressive modeling).

segment 5 is constricted and vice versa. The spectrum is typical of a periodic signal,
since it contains harmonic frequencies (Fig. 5).
When arterial pressure increases to 120 mm Hg, the arteriolar diameter exhibits

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URSINO ET AL.

FIG. 3. Inner-radius time changes of the simulated order 3 arteriole during experiments performed at a
constant internal pressure of 40.5 mm Hg. The values of the microvessel parameters are the same as in
Table 1.

quasiperiodic oscillations (Fig. 6). As is evident looking at the spectrum, system
dynamics are characterized by the superimposition of two independent oscillations
with incommensurate frequencies (about 6.5 and 10.5 cpm) (Fig. 6). Although the
dynamics in the three-dimensional subspace appear more complex than in the previous
case, the presence of chaos can be excluded since the results are insensitive to small
initial perturbations and the spectrum does not exhibit a noise-like component.
When arterial pressure was increased to 145 mm Hg the time pattern of diameter
changes became extremely complex, without any well-defined periodicity (Fig. 7).
The dominant oscillation frequency in the short period shows large variations. The
trajectories describe a strange attractor in the phase space (Fig. 7). The dynamic is
chaotic because of the sensitive dependence of the trajectories on small initial perturbations (Fig. 8). Moreover, the spectrum, computed over a long period, exhibits welldefined spikes superimposed on a noise-like component (Fig. 7).
Changes from periodic to quasiperiodic or chaotic patterns, consistent with those
previously shown, were also observed at constant arterial pressure, modifying R0 or
simulating microvascular networks with a different geometrical asymmetry [parameter
KR2 in Eq. (1)].

FIG. 4. Relationships between the parameters of the myogenic mechanism (static gain, Gms ; dynamic
gain, Gmd ; latent period, Dm) at the boundary from stability to instability. The graphs show how a pair of
parameters should be modified together to maintain the isolated microvessel in a condition of self-sustained
oscillation. The results concern an isolated order 3 arteriole with a constant intravascular pressure of 40.5
mm Hg. The left graph was obtained with a constant pure latency, Dm Å 5 sec, and different combinations
of the static and the dynamic gain. The right graph was obtained with a dynamic gain Gmd Å 440 sec mm
Hg01 cm01 and different combinations of the static gain and the latent period.

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239

FIG. 5. Time pattern of inner radius in segment 1 (order 3 arteriole) (top), its frequency spectrum (bottom
right), and a projection of the trajectories in a three-dimensional subspace (r1 , r2 , and r4, bottom left) of
the simulated model with R0 Å 5 1 106 mm Hg sec cm03 (upstream resistance) and Pa Å 90 mm Hg (systemic
arterial pressure). In this case the network exhibits periodic self-sustained oscillations, the spectrum is
composed of equispaced lines, and the trajectories are limit cycles. Both the spectrum and the trajectories
have been computed for a time period (400 sec) longer than the one shown at the top. In the data the initial
simulation transient is omitted.

Simulation of Blood Flow Changes at Bifurcation in the Network
Finally, we analyzed whether the presence of arteriolar self-sustained oscillations,
and their specific patterns, affects blood flow distribution at bifurcations. In these
simulations the dynamic gain of the myogenic mechanism was changed by the same
percentage amount in all the segments. We calculated the time-averaged blood flow
entering segments 2 and 3 (Fig. 9). Since alterations in the dynamic myogenic component do not affect the steady-state level of the model, the observed variations in mean
blood flow are imputable to the presence and the time pattern of the self-sustained
oscillations. When the dynamic myogenic component is weak, the model settles at a
stable equilibrium (Fig. 9) corresponding to the steady-state condition. A progressive
increase in the dynamic myogenic response determines an unstable condition, and
chaotic oscillations develop. A very high dynamic myogenic component induces more
regular oscillations with a quasiperiodic behavior.
In the simulated network vasomotion determines a significant increase in mean
blood flow (about 40%), compared with that observed without oscillations. Moreover,
when the vasomotion pattern is chaotic (points A–E in Fig. 9), the percentage changes
in mean blood flow at branching 2–3 are approximately the same in both segments.
Notably, regular oscillatory patterns (points G, H, and I in Fig. 9) cause a different
partition of blood flow at the bifurcation. In particular, the mean blood flow entering

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FIG. 6. Time pattern of inner radius in segment 1 (order 3 arteriole) (top), its frequency spectrum (bottom
right), and a projection of the trajectories in a three-dimensional subspace (r1 , r2 , and r4 , bottom left) of
the simulated model with R0 Å 5 1 106 mm Hg sec cm03 (upstream resistance) and Pa Å 120 mm Hg
(systemic arterial pressure). In this case the network exhibits quasiperiodic oscillations, the spectrum is
composed of lines at incommensurate frequencies, and the trajectories move on a closed surface. Both the
spectrum and the trajectories have been computed over a time period (400 sec) longer than the one shown
at the top. In the data the initial simulation transient is omitted.

segment 3 demonstrates a significant reduction (about 015 4 020%) compared with
the previous mean blood flow, whereas the blood flow in segment 2 does not show
significant changes.
DISCUSSION
The main findings of our study are the occurrence of self-sustained diameter oscillations in isolated arterioles and the presence of periodic, quasiperiodic, and chaotic
oscillatory patterns in a microvascular network. Furthermore, vasomotion waveforms
affect blood flow distribution at bifurcations.
Self-Sustained Oscillations in Isolated Microvessels
Contradictory results were observed in previous studies on the static and dynamic
properties of the myogenic response and their role on blood flow autoregulation
(Johnson and Intaglietta, 1976; Grande and Mellander, 1978; Grande et al., 1979;
Borgstrom et al., 1982; Osol and Halpern, 1985; Davis and Sikes, 1990). The parameters used in our model, which include the static and dynamic gain of the myogenic
mechanism and its latent period, are a reasonable compromise among the different
values previously reported.

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241

FIG. 7. Time pattern of inner radius in segment 1 (order 3 arteriole) (top), its frequency spectrum (bottom
right), and a projection of the trajectories in a three-dimensional subspace (r1 , r2 , and r4 , bottom left) of
the simulated model using a value R0 Å 5 1 106 mm Hg sec cm03 for the upstream resistance and a value
Pa Å 145 mm Hg for the systemic arterial pressure. In this case the network exhibits chaotic oscillations,
the spectrum is composed of a noisy component superimposed on spikes at incommensurate frequencies,
and the trajectories move on a strange attractor. Both the spectrum and the trajectories have been computed
over a time period (400 sec) longer than the one shown at the top. The data were obtained after a long
initial simulation, which is not shown.

By using these parameters, the model reproduces diameter oscillations occurring
when the arteriole is isolated from the circulation and its transmural pressure is at a
constant level. The bifurcation diagrams (Fig. 4) show that there are various combinations of the parameters allowing the myogenic mechanism to cause biomechanical
instability and diameter oscillations in isolated microvessels. The oscillation emerges
from a Hopf bifurcation (Glass and Mackey, 1988) and has a frequency in the range
5–10 cpm. These results are in agreement with observations on cannulated arterioles
(Duling et al., 1981; Osol and Halpern, 1988). The result that oscillations may occur
at various values of the static and dynamic gain and of the latent period indicates that
the model behavior is not critically conditioned by a particular parameter value.
A combination of model parameters which can cause oscillations in isolated microvessels is not necessarily able to induce oscillations when microvessels are joined together
to form a multibranched network. For example, in isolated vessels, the gain of the dynamic
myogenic response can be lowered to 25% of its basal value (400 sec mm Hg01 cm01
in the order 3 arteriole) before the disappearance of vasomotion, provided the latent period
is sufficiently high (Fig. 4). By contrast, in the entire network a reduction of the myogenic
dynamic gain to 60% of its basal value in all segments is sufficient to cause the stabilization
of vessel diameter and the disappearance of vascular oscillations (Fig. 9).

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FIG. 8. Effect of a small initial perturbation on model response in chaotic regimen. The continuous line
shows the inner-radius time pattern in segment 5 (order 1 arteriole) computed with the same parameters
as in Fig. 7. The dotted line represents a second simulation, performed by perturbing the segment 5 diameter
by 0.01 mm at time t Å 0 sec. The two results are identical for a certain period (about 200 sec shown)
then abruptly diverge.

FIG. 9. Normalized blood flow entering segments 2 (open circles) and 3 (asterisks) of the simulated
network (in the same conditions as those of Figs. 7 and 8) vs the normalized dynamic gain of the myogenic
response in the network. In these simulations the dynamic gain has been altered by the same percentage
amount in all the segments (100% denotes the basal value of each segment as in Table 1c). The trajectories
describing the normalized blood flow in segment 3 vs the normalized blood flow in segment 2 are also
shown in the small insets. In the low-dynamic myogenic case, the network settles at a stable nonoscillatory
condition (considered 100% of the blood flow). In the increased-dynamic myogenic response, chaotic
dynamics develop, causing a 40% increase in blood flow of both segments. Further increases in dynamic
myogenic response lead to more regular trajectories that determine a different partition of flow at the
bifurcation.

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243

Complex Patterns in Multibranched Networks: Experimental and
Theoretical Results
Arteriolar vasomotion was observed in all preparations under basal conditions and
presented higher fundamental frequencies in order 3 arterioles compared with those
previously reported (Bertuglia et al., 1991). This finding may be related to the characteristics of the terminal network used for this study since there was an arcading system
of arterioles giving rise to shorter and longer terminal branchings. The short loop was
characterized by the predominance of order 3 vessel vasomotion frequency on order
2 arterioles.
A few recent experimental studies (Yamashiro et al., 1990; Intaglietta and Breit,
1991; Griffith and Edwards, 1993) suggest that vasomotion may exhibit some aspects
of chaotic dynamics. Griffith and Edwards (1993) observed that episodes of nearly
periodic oscillations may alternate with irregular fluctuations, and the same perturbation may occasionally lead to large-amplitude oscillations or may suppress the rhythmic activity. This finding has been regarded as an example of the inherent unpredictability of vasomotion dynamics.
In this study we did not compute some index of chaos directly from the experimental
tracings, as in the previous works, but we used a different approach. We demonstrated
that vasomotion may be chaotic by formulating a deterministic mathematical model
able to predict irregular diameter fluctuations similar to those observed experimentally.
The simulation results indicate that a parameter change may induce periodic, quasiperiodic, or chaotic patterns of diameter oscillations in a multibranched network. In
particular, changes in arterial pressure, in arteriolar resistance, or in the geometrical
symmetry of the network can shift system dynamics from periodic to quasiperiodic
to chaotic. Furthermore, the waveforms and frequencies of vasomotion were similar
in the simulated and experimental situations, with both periodic and irregular patterns
of behavior (Fig. 2).
Although our data showed the presence of chaotic fluctuations when systemic arterial pressure was increased (Figs. 5–7) or upstream resistance was decreased, there
is no single, well-defined criterion to achieve chaos. Indeed, monotonic small changes
in a parameter which initially cause the appearance of chaotic patterns can subsequently lead to a stabilization of the trajectories on a new periodic attractor and
afterward may cause the reappearance of chaos. This is a typical feature of many
chaotic systems (Glass and Mackey, 1988). An example of this behavior, characterized
by an abrupt change from chaos to periodicity is evident in Fig. 9 when passing from
point F to point G.
It has been suggested that chaotic and periodic fluctuations may reflect an intrinsic
difference in the control mechanisms operating on the system (Golderberger, 1990;
Intaglietta and Breit, 1991). Whenever several control mechanisms work simultaneously, the system might exhibit a random or chaotic behavior, typical of healthy
subjects; by contrast, suppression of some mechanisms, as may occur in pathological
conditions, might result in the appearance of a periodic oscillatory behavior.
Our results suggest that the shift from a periodic to a chaotic pattern may be a
consequence of a modest change in a parameter not necessarily related to pathophysiological conditions. Accordingly, our in vivo observations in the skeletal muscle showed
that in some instances periodic vasomotion became quasiperiodic or chaotic in normal
conditions.
We conclude at present that the variability of responses from regular to chaotic

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oscillatory patterns is not necessarily representative of a pathological mechanism and
may reflect differences in the sensitivity and the initial conditions of microcirculation.
Moreover, contractile responses not only elicited by the myogenic mechanism may
induce oscillatory patterns.
In the model, chaos appears to emerge from a complex coupling among various
oscillators. The simulations of isolated microvessels demonstrate that a pressurized
arteriole may act as an autonomous oscillator. In the network, these oscillators interplay, mutually influencing their intravascular pressures. Mathematical theories of two
coupled nonlinear oscillators (Arnold, 1983) suggest that, if the amplitude and frequency of the oscillators are varied, many coupling patterns may occur. At certain
values of amplitude and frequency, two coupled oscillators may exhibit a stable
entrainment, characterized by N:M phase locking, with N and M integer numbers, i.e.,
N cycles of the first oscillator correspond to exactly M cycles of the second. In such
a condition the whole system performs periodically. However, at other values of the
amplitude and frequency, the oscillators may exhibit no stable entrainment. In this
case, the frequencies of the two oscillators are incommensurate, and the whole system
behaves quasiperiodically. Finally, if oscillation amplitudes are sufficiently high (for
instance, when arterial pressure is increased), quasiperiodic and periodic trajectories
can be converted by complex topological structures into chaos (Glass and Mackey,
1988; Arnold, 1983).
The present model is too complex for attempting a complete theoretical analysis
of its transitions toward chaos. Indeed, our aim was not to build a simple theory of
vasomotion, oriented to mathematical exploration, but rather to develop a comprehensive model based on biomechanical laws, geometrical parameters, and physiological
data. The use of a complex rather than of a simple model may permit better understanding of the mechanisms leading to vascular oscillations and, in perspective, may be
used to study the role of vasomotion in clinical and/or pathophysiological conditions.
Vasomotion Effect on Flow Partition at Bifurcation
Our results indicate that arteriolar vasomotion may significantly affect the average
rate of blood flow and may contribute to blood flow distribution in the network
(Fig. 9).
The major findings of the simulations (Fig. 9) are that vasomotion determines a
40–60% increase in the mean blood flow compared with that under steady-state
condition and that blood flow distribution at bifurcations depends on the oscillation
waveforms. Furthermore, chaotic oscillations tend to maintain a constant ratio of blood
flows in the branches, whereas more regular fluctuations cause a different partition of
flow at bifurcation thus increasing flow in one branch compared with the other. We
suggest that, during chaotic fluctuations, all arterioles exhibit similar random diameter
changes; during periodic or quasiperiodic diameter fluctuations, the time period associated with vasodilation or vasoconstriction may vary from one segment to the next.
Based on these observations it appears that the response heterogeneity and vasomotion
waveforms at bifurcations can exert a functional effective control on microvascular
perfusion.
However, although our data indicate that oscillations in arteriolar diameter may
increase blood flow, this model result strictly depends on the initial smooth muscle
cell tone. In the present simulations, the diameter in basal condition is at a position
on the ascending portion of the active stress–inner radius curve, far from the maximum
(Davis et al., 1986; Davis and Gore, 1989), suggesting that in basal condition the

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245

arterioles can increase more than reduce their diameter. Therefore, when diameter
oscillations develop, vasodilation prevails on vasoconstriction, and the effective diameter increases. Arteriolar responsiveness, however, might be different when starting
from vasodilated vessels.
In conclusion, our data support a strong correlation between myogenic mechanism
and arteriolar vasomotion. The model predicted self-sustained diameter oscillations
in isolated arterioles and more complex quasiperiodic or chaotic fluctuations in the
networks. The activity pattern of vasomotion might alter blood flow distribution at
bifurcations, thus affecting the total microcirculation hemodynamics.
APPENDIX: QUANTITATIVE MODEL DESCRIPTION
Mechanical Properties of a Single Microvessel
For the model of a single microvessel we used the Laplace law
pijrrj 0 per(rj / rj) Å sjrhj Å Tj ,

j Å 1, 2, . . ., 5,

(3)

where pij , rj , hj , and sj are intravascular pressure, inner radius, wall thickness, and
total wall stress (force per unit surface) in the jth segment and pe is extravascular
pressure.
Total wall stress is computed as the sum of elastic, active (muscular), and viscous
stress (se , sm , and sv , respectively). Analytical expressions (Ursino and Fabbri, 1992),
are
q

hj Å 0rj / r 2j / 2rjnhjn / h2jn

(4)

sej Å se0j[ekej(rj0r0j) 0 1]

(5)

smj Å sm0je
svj Å

0kmjr(rj0rmj)2

hj drj
r
.
rjn dt

(6)
(7)

A value to the parameters in Eqs. (5)–(7) has been given to reproduce the patterns
of active and elastic wall stress (Davis and Gore, 1989) in the hamster cheek pouch
and considering that the equilibrium point is located in the ascending portion of the
active stress–inner radius curve. The basal values are denoted by the subscript n in
Eqs. (3–7) and are shown in Table 1b of the text.
We assumed that feedback physiological mechanisms and other vasoactive factors
affect smooth muscle activity by modifying the amplitude of the active stress–inner
radius relationship [Eq. (6)]; i.e., we have
sm0j Å sm0jnr

3exj
,
exj / 2e00.5xj

(8)

where xj reproduces the effect of all vasoactive factors on smooth muscle tension at
the jth segment. The right-hand member of Eq. (8) has sigmoidal shape, reflecting
the existence of upper and lower autoregulation limits.
In this work we assume that smooth muscle tension is determined by: (i) the

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myogenic response (xmj) and (ii) a term (xej) which includes all the remaining vasoactive factors. We assume that interaction between these two terms is additive:
xj(t) Å xmj (t 0 Dmj) / xej(t),

j Å 1, 2, . . ., 5,

(9)

where t denotes the present instant of the simulation and Dmj is the latent period of
the myogenic response.
We assume that basal smooth muscle activity (i.e., when xj Å 0) is 13 of the maximum
activity [Eq. (8)]. Different preexisting levels of tone can be simulated by assigning
a nonzero value to the additional term xej (xej õ 0 means vasodilation, hence a lower
preexisting tone, and vice versa).
While the xej in Eq. (9) ( j Å 1, 2, rrr 5) are input quantities for the present model,
i.e., their value can be established during the simulation, the terms xmj depend on wall
tension through dynamical relationships, which reproduce the main features of the myogenic response. The following equations include both static and rate-dependent components of the myogenic mechanism, according to the data by Grande et al. (1979):
1
xh mj Å 0 r[xmj 0 Gmsjr(Tj 0 Tjn) 0 GmdjrTg j],
tmj

j Å 1, 2, . . ., 5.

(10)

The upper point in Eq. (10) represents the time derivative of the corresponding quantity. Tj in Eq. (10) is total wall tension, computed from Eq. (3). Its time derivative is
obtainable from Eqs. (3) and (4):
Tg j Å ph ijrrj / pijrrh j 0 ph er(rj / hj) 0 per(rh j / hh j)

(11)

rjrrh j
hh j Å 0rh j /
.
hj / rj

(12)

Eq. (10) presumes that the state variable xmj is equal to 0 when wall tension is at
the basal value shown in Table 2. An increase in wall tension causes an increase in
xmj , with consequent smooth muscle activation and reduction in vessel caliber. The
static and dynamic gains of the mechanism (Gmsj , Gmdj) and its time constant (tmj)
were chosen to reproduce the myogenic response (Johnson and Intaglietta, 1976;
Grande et al., 1979; Davis, 1993).
The inner radius is a state variable for the present model. In fact, from Eqs. (3)
and (7) one can write

F

G

drj rjn pijrrj 0 per(rj / hj)
Å r
0 sej 0 smj .
hj
dt
hj

(13)

The previous equations allow one to compute the time pattern of inner radius, rj ,
and its time derivative knowing intravascular and extravascular pressure in the vessel
and their time derivative. Simulating experiments on isolated arterioles, the latter
quantities are considered as external inputs. Simulating the entire network, the vessel
intravascular pressures and their time derivatives, at specific representative sections,
are obtained by the model shown in Fig. 1b.

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247

Pressure and Flow Profile in the Network
To obtain the pressure and flow profile at the different sections of Fig. 1, we can
consider the following steps:
(i) Since inner radii are state variables for the model [Eq. (13)], their values are
known at a given instant. We can compute the value of the hydraulic resistances in
the different segments as a function of inner radius, of arteriolar length, lj , and of
blood viscosity, m, by the Hagen–Poiseuille law
Rj Å

KRj 8rmrlj
Å
,
r4j
prr4j

j Å 1, 2, . . ., 5.

(14)

The constant parameters, KRj , in the order 3 and 2 arterioles (i.e., in segments 1–3 of
Fig. 1a) have been calculated with a blood viscosity Å 0.04 poise and using the length
of microvessels measured in the hamster dorsal cutaneous muscle. Parameters KR4 and
KR5 , concerning the downstream resistance from terminal arterioles down to postcapillary
venules, are given to set pressure in terminal arterioles at about 30 mm Hg (Table 1b).
Moreover, by denoting with R1S the series arrangement of resistances R0 and R1 ,
with R2S the series arrangement of resistances R2 and R4 , and finally with R3S the
series arrangement of R3 and R5 , i.e.,
R1S Å R0 / R1 , R2S Å R2 / R4 , R3S Å R3 / R5 ,

(15)

one can easily compute the input resistances at points A and B of Fig. 1b (namely,
RA and RB). We have
RB Å R2SrR3S/(R2S / R3S)

(16)

RA Å R1S / RB .
(ii) From the values of hydraulic resistances, computed by Eqs. (14)–(16), blood
flow, qj , in the jth segment, and intravascular pressure, pk , at point k of the network,
are obtained by Ki