Abstract
Vasomotion has been studied on segments of rat mesenteric and femoral arteries perfused in vitro. We have investigated1) the effect of perfusion flow on the characteristics of vasomotion and2) the nature and patterns of vasomotion. We have found that perfusion flow is not a control parameter that contributes to the genesis of vasomotion but that it affects, in most cases only slightly, the frequency and amplitude of vasomotion. We have found evidence that vasomotion is lowdimensional chaotic. The correlation dimension ranged between 2 and 4, and the average Lyapunov’s coefficient was ∼0.1. A great variety of vasomotion patterns was observed with features that are typical of nonlinear deterministic systems: regular and irregular vasomotion, quasiperiodicity, period doubling and higherorder periods, intermittency, mixed modes, and bursting activity. Vasomotion patterns appeared occasionally to be highly sensitive to perturbations in perfusion flow, which also supported the existence of nonlinear dynamics. Finally, entrainment (phase locking) was observed when arteries were perfused with oscillatory flow with frequency in the neighborhood of the frequency of vasomotion.
 chaos
 vasomotion patterns
 fractal dimension
 rat arteries
arteries often exhibit spontaneous, rhythmic activity, which is manifested by lowfrequency oscillations in vessel caliber, a phenomenon termed vasomotion. Vasomotion is predominant in microcirculation (6, 16, 20), but recently it has been shown to exist in large muscular arteries (14, 21, 23). In small arteries and arterioles, vasomotion influences perfusion (24), enhances filtration through the wall and lymphatic drainage (15), and may represent some form of homeostasis. In large arteries, the physiological significance of vasomotion remains obscure, although it has been demonstrated that vasomotion affects considerably the elastic properties of conduit vessels (26), and vasomotion in the coronary arteries has been suggested as an initiating factor of coronary spasm (23).
The mechanisms contributing to vasomotion have yet to be defined. Vasomotion has been closely linked to the myogenic mechanism (2, 5,27). A number of studies (7, 9, 18) claim that vasomotion results from instabilities in the system that controls intracellular calcium. Mechanical factors such as wall stress and intimal shear appear to modulate vasomotion (3, 4, 11, 19, 20), but it has been suggested that these mechanical factors are not control variables of the system that generates vasomotion (13). The plurality of experimental patterns observed, sensibility to initial conditions, and nonlinear (fractal) analysis of vasomotion signals suggest that vasomotion is a chaotic process (9).
The objectives of the present study are1) to examine the effect of flow on the characteristics of spontaneous vasomotion and determine whether flow is one of the control variables of the system that generates vasomotion, and 2) to assess the nature of vasomotion, examine its patterns, and determine whether lowdimensional deterministic chaos is involved.
METHODS
Experimental Procedures
The effects of pressure and flow on the characteristics of arterial vasomotion were investigated in vitro on isolated rat mesenteric and femoral arteries. Mesenteric segments, typically 6 mm in length, were taken from the midportion of the fourthorder mesenteric branch (external diameter ≈0.4 mm and length ≈12 mm, in situ). Femoral segments, typically 8 mm in length, were taken from the midportion of the femoral artery (external diameter ≈0.7 mm and length ≈30 mm, in situ). The excised arterial segments were mounted on microcannulas and perfused with Tyrode solution (Living Systems Instrumentation, Burlington, VT) using techniques similar to the work of Achakri et al. (3). Vasomotion was induced and maintained by adding a constrictor agonist in the superfusion solution (1 μM norepinephrine). Endothelial function was checked in arteries by means of acetylcholineinduced vasodilation. The experiments were terminated in cases with no response to acetylcholine. The external diameter was measured continuously using videomicroscopy, and pressure was measured upstream and downstream of the arterial segment by means of a perfusion pressure monitor. Flow was delivered by a syringe infusion pump controlled by a personal computer. The temperature in the vessel chamber (37°C) was controlled and monitored continuously. To study the effect of flow, perfusion pressure was kept constant at 90 mmHg and flow was varied between 100 and 800 μl/min in a stepwise manner. The pressure drop across the test section at the highest flow (800 μl/min) was typically in the order of 10 mmHg. Changes in pressure of 10 mmHg or less do not induce significant myogenic responses, as we have shown in earlier studies on the same arteries under similar experimental conditions (3).
Theoretical Analysis
The correlation dimension of the vasomotion time series was estimated using the method of Grassberger and Procaccia (8). In brief, this method was implemented as follows. On the basis of the diameter timeseries data, we constructed a set ofmdimensional vectorsX
_{i} = {d_{i}
(t),d_{i}
(t+ τ), ...,d_{i}
[t+ (m − 1)τ]}, where d is diameter, t is time,m is the embedding dimension, and τ is a time delay. We then calculated the correlation statistic
The choice of embedding dimension and time delay is critical for correct estimation of the correlation dimension (9, 10). In our analysis we used a time delay equal to 0.3 of the basic period of vasomotion and an embedding dimension of 8, which satisfies the Takens criterion (25). The Grassberger and Procaccia method (8) was tested first against time series from known chaotic systems (i.e., Lorenz and Henon attractors). The analysis was performed on 800s segments of continuous recording sampled at 2.6 Hz. Similar analysis has been applied to vasomotion signals by Yamashiro et al. (28) and Griffith and Edwards (13).
The largest Lyapunov’s exponent was estimated using the method of Rosenstein et al. (22).
RESULTS
Effect of Flow
A typical result of an experiment on a femoral artery is presented in Fig. 1 A, which shows the diameter at different perfusion flows and constant intraluminal pressure (90 mmHg). Flow ranged between 100 and 800 μl/min. At 800 μl/min the shear stress was ∼35 and 15 dyn/cm^{2}for mesenteric and femoral arteries, respectively. Figure1 A shows that vasomotion amplitude and frequency depend on perfusion flow. Vasomotion frequency is plotted as a function of perfusion flow for mesenteric and femoral arteries in Fig.1 B. The frequency (f) decreased monotonically in both artery groups with an increase in perfusion flow (Q˙), although the absolute value of frequency differed between the two groups. Linear regression yieldedf = 0.17 + 7.4 × 10^{−5}Q˙ and f = 0.37 + 1.07 × 10^{−4}Q˙ for femoral and mesenteric arteries, respectively. The amplitude (A) of vasomotion decreased and was severely attenuated at high flows in femoral arteries (Fig.1 C). Linear regression yielded A = 5.8 + 0.0054Q˙ and A = 8.1 + 0.0031Q˙ for femoral and mesenteric arteries, respectively. The slopes of the linear regression curves were significant.
Vasomotion Patterns
Regular and irregular vasomotion.
Vasomotion in the rat mesenteric artery exhibited a large variety of patterns. Two extreme cases are presented in Fig.2. Figure2 A shows a typical case of regular vasomotion. The frequency spectrum exhibits a single peak atf = 0.32 Hz (Fig.2 C). Figure2 B shows a case of extremely irregular vasomotion, including a multitude of frequencies and largescale variations in vessel diameter. The power spectrum is widely distributed over the range of 0–0.4 Hz (Fig.2 D). Such erratic vasomotion signals have been reported earlier in vivo and in vitro in both small and large vessels (9, 21).
Specific patterns: routes to chaos.
There is a certain parallelism between the vasomotion patterns observed in our vitro experiments and the behavior of nonlinear deterministic systems. Griffith (9) drew similar analogies in his review article. Nonlinear systems exhibit certain welldefined patterns when going from regular oscillatory behavior to chaos. These patterns signify transitional stages in the evolution toward a more complex nonlinear behavior and sometimes are called “routes to chaos.” Typical cases of such specific patterns are shown in Fig.3 and are classified as follows.
QUASIPERIODICITY.
Figure 3, A andB, shows vasomotion signals that seem to be the outcome of two distinct oscillatory subsystems, a slow and a fast subsystem. In Fig. 3 A, the fast component is at 0.34 Hz and the slow component at 0.02 Hz.
PERIOD DOUBLING AND HIGHERORDER PERIODS.
Period doubling is a typical manifestation of instability in a nonlinear system, which can be led to chaos through successive period doubling. A classic example is the logistic equation. Figure3 C shows a diameter tracing exhibiting period 2 dynamics, marked by an alternating sequence of high and lowamplitude oscillations. A typical higherorder (period 6) pattern is shown in Fig. 3 D.
INTERMITTENCY.
Regular vasomotion is sometimes interrupted by periods of irregular vasomotion, after which the oscillator returns to the same regular pattern. A tracing of such an intermittent behavior is shown in Fig.3 E. Intermittency is often present in chaotic dynamics (i.e., RayleighBénard model, Lorenz equations, etc.).
MIXED MODES.
Figure 3 F shows vasomotion with a marked mixedmode response. A highamplitude oscillation is followed by a variable number (2 or 3 in Fig.3 F) of loweramplitude oscillations. Similar phenomena were reported by Griffith and Edwards (13).
BURSTING ACTIVITY.
Another interesting mode of vasomotion is the bursting behavior, as shown in Fig. 3 G. The artery seems to rest in a relatively quiescent vasodilatatory state for a period of time (of irregular duration), after which it contracts and engages in an oscillatory activity, giving rise to irregular downward spikes, as shown in Fig. 3 G. At the end of the spiking activity, the artery returns to the quiescent mode and a completely relaxed state. Griffith and Edwards (13) observed bursting activity in histamineinduced vasomotion in rabbit ear arteries. This bursting activity was fairly unstable and led to chaos with an increase in perfusion flow.
Fractal Analysis and Chaos
We have calculated the correlation dimension (ν) of the vasomotion time series to determine 1) the minimum number of independent control variables involved and2) whether mechanical factors such as the intimal shear stress may influence the apparent complexity of the system. Figure4 A shows the estimated correlation dimension as a function of perfusion flow. The values lie in the range of 2 < ν < 4, and there was no statistically significant dependence on perfusion flow. We therefore conclude that there is a minimum number of four control variables in the vasomotiongenerating system and that the level of flow does not alter the fractal dimension of the system. Lyapunov’s exponent was estimated to be positive and typically around 0.1 (Fig.4 B).
Sensitivity to External Perturbations
Although flow seems to be an external factor influencing the amplitude and frequency of vasomotion (Fig. 1), in certain cases changes in flow resulted in largescale changes in vasomotion characteristics. Figure5 shows an example in which step changes in flow induced spectacular and reversible changes in oscillatory modes. At a baseline value of 100 μl/min, highfrequency (f = 0.36 Hz), lowamplitude (Δd = 10 μm) vasomotion prevails, with characteristics similar to the highfrequency component of vasomotion in Fig. 3 A. When flow increased to higher levels, vasomotion changed to a lowfrequency mode (f = 0.025 Hz) with a 10fold increase in amplitude (Δd = 120 μm) and characteristics comparable to the lowfrequency component of vasomotion shown in Fig. 3 A.
Entrainment
A periodic perturbation, such as oscillatory flow, was able to lock or entrain vasomotion. An example of entrainment is shown in Fig.6, in which perfusion flow was changed from constant flow (400 μl/min) to oscillatory flow for a certain period of time (∼100 s). The frequency of oscillatory flow was set at different levels ranging from 0.07 to 0.15 Hz. At constant flow, the vasomotion frequency was 0.16 Hz. We observe that in the presence of oscillatory flow, vasomotion phase locks onto the frequency of the oscillatory flow. The change in frequency is accompanied in general by an augmentation in vasomotion amplitude (see Fig. 6,top) that, however, becomes insignificant when the frequency of oscillatory flow approaches the frequency of spontaneous vasomotion (Fig. 6,bottom). No phase locking was observed for oscillatory flow with frequencies <0.07 Hz, which is approximately onehalf the frequency of spontaneous vasomotion.
DISCUSSION
We have studied vasomotion in rat mesenteric and femoral arteries in vitro, and we have found good evidence that vasomotion is lowdimensional chaotic. Mechanical forces, such as intimal shear, are not control variables of the system that generates vasomotion, and they influence slightly the characteristics of regular highfrequency vasomotion. Vasomotion patterns exhibit characteristics of nonlinear dynamic systems and show largescale sensitivity to external perturbations.
Regular vasomotion patterns are characterized by a highfrequency (a period of 3–10 s) and a lowfrequency component (a period of 40–60 s). These two regular oscillatory modes can in certain cases coexist. Griffith and Edwards (12) provided evidence that the slow oscillatory component is due to calcium release and uptake from the sarcoplasmic reticulum in smooth muscle cells, whereas the fast component is due to membrane oscillators. The interaction of the two subsystems can turn unstable, leading to chaos (9).
In all cases studied, the correlation dimension varied between 2 and 4. These results are consistent with the findings of Griffith and Edwards (13), who have showed that the fractal dimension of histamineinduced vasomotion in the rabbit ear artery was typically in the range of 2–3 and occasionally between 3 and 4. The fractal dimension was found not to depend on the level of flow and was not altered when the endothelium was functionally blocked by administration ofN ^{G}nitrolarginine methyl ester. In earlier studies we calculated the correlation dimension of vasomotion time series measured in human radial artery in vivo (1). The correlation dimension ranged between 2 and 3, which suggests that the complexity, and probably nature, of the vasomotion observed in vitro is similar to that in vivo.
The largest Lyapunov’s exponent was positive, supporting the hypothesis that vasomotion results from a chaotic system. Typical features of a chaotic system are high sensibility to initial conditions and the possibility of largescale changes in the trajectory (i.e., oscillatory patterns) following rather small perturbations by external factors. Therefore, the large spectrum of oscillatory modes and patterns as well as the extreme differences in vasomotion patterns observed for different levels of flow may be attributed to the chaotic nature of vasomotion (Fig. 5).
The results shown in Fig. 5 are not meant to contradict the dependence of vasomotion frequency and amplitude on flow shown in Fig. 1, because the results of Fig. 1 were primarily drawn from the analysis of regular, highfrequency vasomotion signals. Figure 5 shows nicely the rich and, to a large extent, unpredictable response of a nonlinear system to small external perturbations. When the weak dependence of mean diameter on mean flow is taken into account, a linear system is not capable of producing a response to changes in flow such as those shown in Fig. 5. The unpredictability of responses to external mechanical and pharmacological stimuli has been reported earlier for in vivo and in vitro preparations. For example, Griffith and Edwards (13) studied vasomotion induced by histamine in isolated rabbit ear resistance arteries and have shown that increases in flow provoked oscillations in some preparations, whereas in others they attenuated and ultimately abolished periodic oscillations. The possibility of “butterfly effects” or the application of nonlinear techniques for controlling chaos in vasomotion has been discussed extensively by Griffith (9).
We have computed the Lyapunov’s exponent and found consistently positive values in the neighborhood of 0.1. Lyapunov’s exponent characterizes the divergence over time of the phasespace trajectories. A positive Lyapunov’s exponent means that divergence will grow in time, which points toward high sensitivity in initial conditions and chaos. Similar values for Lyapunov’s exponents were estimated for vasomotion in rabbit ear arteries (10). Therefore, one of the contributions of the present study is the reconfirmation and generalization of earlier estimates of the correlation dimension and Lyapunov’s exponent (913, 28) in different arteries and species.
Phase locking or entrainment may be another indication for the existence of nonlinear dynamic mechanisms controlling vasomotion. Phase locking between weakly coupled nonlinear oscillators is a wellknown phenomenon occurring in biology (e.g., synchronized flashing of large populations of fireflies and chirping of crickets). In vivo studies have also revealed a high degree of synchronization in vasomotion between adjacent vessels in microcirculatory networks (17) as well as in large muscular arteries of the human arm (21).
The nonlinear dynamics involved in the development of vascular tone may have considerable implications in the physiology and pathology of the arterial system. Vasomotion is the manifestation of a series of complex events at the cellular (microscopic) and tissue (macroscopic) levels along which important control mechanisms (myogenic, nitric oxide synthase) are implicated. An understanding of vasomotion will enhance our basic knowledge of the control mechanisms and functional properties of arteries, and therefore we believe that vasomotion merits further investigation.
Footnotes

Address for reprint requests: N. Stergiopulos, Biomedical Engineering Laboratory, Swiss Federal Institute of Technology, PSEEcublens, 1015 Lausanne, Switzerland.
 Copyright © 1998 the American Physiological Society