comparison of chaotic and sinusoidal vasomotion in the regulation

Gndiovascular Research

ELSEVIER Cardiovascular Research 3 1 ( 1996) 388-399

Comparison of chaotic and sinusoidal vasomotion in the regulation of microvascular flow

D. Parthimos a, D.H. Edwards b, T.M. Griffith a** a Deparhnent of Diagnostic Radiology, Cardiovascular Sciences Research Group, University of Wales College of Medicine, Heath Park,

Cardiff CF4 4xN, UK b Department of Cardiology, University of Wales College of Medicine. Heath Park, CardtyCF4 4XN, UK

Received 3 1 March 199.5; accepted 4 July 1995

Abstract

Objective: In order to elucidate the physiological consequences of irregular vasomotion on microvascular flow we have compared the theoretical hydrodynamic consequences of sinusoidal and chaotic fluctuations in the diameter of a single resistance,vessel. Methods: In

initial experimental studies vasomotion was induced by histamine in isolated rabbit ear resistance arteries (m 150 pm diameter) perfused with physiological buffer under both controlled-flow and controlled-pressure conditions. The phase relationships between the observed oscillations in flow and pressure were used to validate a theoretical electrical circuit in which vasomotion was simulated as sinusoidal or as chaotic fluctuations in distal resistance, with compliance incorporated as a parallel capacitance. Results: In both the experimental and theoretical situation, oscillations in flow led those in pressure by * 90” in controlled-flow mode, whereas they were - 180” out of phase in controlled-pressure mode. In the theoretical model an increase in the amplitude of sinusoidal or chaotic diameter fluctuations enhanced flow, but “paradoxically” increased both time-averaged resistance and conductance. The model showed that with sinusoidal fluctuations the “efficiency” of perfusion (i.e., flow/viscous work expended in perfusing the vessel undergoing vasomotion) exhibited a peak whose magnitude was a function of vasomotion amplitude and the proximal capacitance in the circuit, and was attributable to transient release of charge from this capacitance. This phenomenon was not observed in simulations with chaotic vasomotion. Hydrodynamic effects specific to the presence of chaotic dynamics (e.g., abrupt increases or decreases in flow under the variation of a single parameter) were also evident when the intrinsic complexity of the vasomotion, rather than its amplitude, was varied. Conclusions: The model suggests (i) that vasomotion may serve to increase flow, (ii) that conductance provides a more accurate physiological measure of the functional consequences of active vasomotion than resistance, (iii) that chaotic vasomotion dissipates transients more readily than sinusoidal vasomotion, thereby conferring greater stability to microcirculatory perfusion and (iv) that specific modes of chaotic vasomotion may influence flow independently of their amplitude.

Keywords: Chaos; Microvascular flow; Rabbit, artery

1. Introduction

Small arteries and arterioles serve to transport nutrients and metabolites and contribute to the regulation of sys- temic and interstitial tissue pressures. It has been suggested that oscillatory perfusion may confer specific advantages over steady-state flow, but the physiological significance of vasomotion (i.e., rhythmic fluctuations in vascular cali- bre) and the contribution of vasomotion to temporal fluctn-

ations in microcirculatory flow remains the subject of investigation and considerable debate [l-6]. The potential benefits of vasomotion include functional separation of mutually incompatible detector and effector mechanisms [4], promotion of lymphatic drainage [5,6] and enhance- ment of microcirculatory mass transport [1,21. Theoretical considerations also indicate that fluctuations around a given mean diameter increase the effective time-averaged cross- sectional area of a single vessel and may thus increase

* Corresponding author. Tel.: ( + 4l- 1222) 744726; Fax: (+ 44 1222) 744726. Tie for primary review 45 days.

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D. Parthimos et al. / Cardiovascular Research 31 f 1996) 388-399 389

flow [1,7]. Vasomotion may also permit a shift in the operating point of an active arteriole to a less steep part of its diameter-resistance curve, so that resistance can be regulated without the need for precise diameter control [ 11.

There is now evidence that the highly irregular and therefore seemingly random vasomotion that can be observed both in vitro and in vivo is “chaotic”, implying a deterministic origin [8-111. In support of this, oscillatory responses induced by histamine in isolated rabbit ear arteries exhibit routes highly specific for the transition from periodic to irregular dynamics known as “period-dou- bling”, “ quasiperiodicity” and “intermittency” that char- acterize chaotic behaviour and have been observed in a wide variety of non-biological systems [8]. The functional effects of specific patterns of periodic diameter fluctuation on vascular conductance differ when averaged over a sufficiently large number of cycles [7]. In the present study we have therefore investigated whether chaotic vasomotor activity could contribute to the regulation of distal flow in ways not possible with simpler types of oscillatory behaviour. Phase relationships between oscillation in flow and pressure in isolated rabbit ear resistance arteries were determined experimentally to construct and to validate a theoretical model of flow in a single microvessel undergoing time-dependent changes in diameter. Mathematical simulations were then performed in which diameter fluctuations were either sinusoidal in nature or generated by integration of the 3-dimensional Lorenz system [ 12,131.

2. Methods

The preparation and technique for studying single rabbit ear resistance arteries in situ have been described in detail elsewhere [g-lo]. First-generation vessels (u l-l 5 cm in length and 150 pm in diameter) arising from the central ear artery were perfused by a peristaltic pump. (Watson- Marlow Type lOlU/R) with oxygenated (95% 0,/5% CO,) Holman’s buffer (composition in mM: 120 NaCl, 5 KCl, 2.5 CaCl,, 1.3 NaH2P04, 25 NaHCO,, 11 glucose and 10 sucrose, pH 7.2-7.4) at 35°C via a cannula secured in the central ear artery which was itself ligated distally so as to divert all flow into the branch vessel under study. The distal end of this artery was cut in order to allow free outflow of perfusate. Input pressure was monitored contin- uously via a side arm connected proximal to the inflow cannula, and flow by a transonic flow probe (Transonic Systems, Type 2N) included in series with the perfusion circuit (Fig. la). Data were collected digitally by a MacLab System (Type 2E). Rhythmic activity was induced by 1 or 2.5 PM histamine in the presence of NG-nitro+arginine methyl ester (L-NAME) to inhibit endothelial synthesis of nitric oxide (NO) [S-lo]. Fluctuations in flow and pressure due to the peristaltic nature of the perfusion were damped by an air-filled compliance chamber connected to the

a) Air-filled

Compliance Chamber

OXY9 B

q(t) 1

’ R, 0)

Fig. 1. (a) Schematic representation of perfusion system for studying isolated rabbit ear resistance arteries in which vasomotion was induced by histamine. The negative feedback loop regulating pump speed was acti- vated only in controlled-pressure mode. P,c, is a reference pressure close

to the operating point in controlled-flow mode. (b) Equivalent circuit used to model the consequences of vasomotion in a single artery in the presence of compliance. The circuit could be driven either by a constant voltage (PO) or current (Q,> input, thus corresponding to controlled-pres-

sure and controlled-flow techniques in tbe experimental context,

circuit via a side-arm and were not discernible on the experimental pressure and flow traces. Drugs were obtained from Sigma Limited (Poole, Dorset, UK) and stock solutions freshly prepared on the day of the experiment in saline.

Two methods of perfusion were employed: (1) controlled-flow mode in which flow could be set at a desired level by manually selecting pump speed, and (b) controlled-pressure mode in which an electronic servoloop regulated pump speed by directing input pressure to a preset level that was determined by an offset voltage. In each experiment, flow was initially set at 0.5 ml/min in controlled-flow perfusion mode. In order to disturb the system only minimally during the transition between the two methods of perfusion, the reference pressure that regulated the speed of the peristaltic pump in controlled- pressure perfusion was adjusted to give the same mean pressure as obtained in controlled-flow mode.

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390 D. Parthimos et al. / Cardiovascular Research 31 (1996) 388-399

3. Electrical analogue

The experimental perfusion system was modelled as an equivalent electrical circuit supplied either by a constant current ( = flow) or voltage (= pressure) source, thus corresponding to controlled-flow and controlled-pressure perfusion modes, respectively (Fig. lb). The resistance of the perfusion circuit, including that of the input cannula, was represented as a fixed resistance, and consequently resulted in pressure losses proximal to the arterial segment undergoing vasomotion. The air-filled compliance chamber present in the perfusion circuit was simulated by a capacitance in parallel with this time-varying resistance. The model is therefore essentially a “lumped” analogue, but linear electrical circuit theory cannot be applied in such a representation because of time-dependent variations in distal resistance. In terms of the in vivo situation the compliance in the model can be assumed to represent the total compliance of the vasculature proximal to the vessel undergoing vasomotion, rather than the compliance of the vessel itself, which will be relatively small. The behaviour of the electrical analogue can be described by the following sets of equations in the controlled-pressure and controlled-flow cases, with symbols defined as in Fig. lb:

3.1. Controlled pressure

Overall pressure PO is constant. The currents are:

q(t) =i(t) +x(t) (1)

where, substituting for their voltage-dependence, we have:

q(t) =i(t) +x(t)

_ PO-P,(t) dp,(t) p,(t) cc- - Ro dt + R,(f)

~ CdP,(t) At) - + PO-P,(t)

dt Rx(t) R,

dp,(t) *~=;[-($+&)PI(t)+$]

(2)

3.2. Controlledflow

The overall flow q(t) = Q, = constant, so that FXl. (1) gives:

dpd t) P*(t) Q,=i(t) +x(t) -Q,,=CT+-

Rx(t)

The absolute value of the external series resistance (R,) was set at 5 arbitrary units (A.U.) in all simulations as it does not contribute dynamically to the behaviour of the circuit. The mean value of the time-varying resistance (R,( t>> was chosen as 1 A.U. The value of the capacitance (C) was varied between 0.1 and 3.0 A.U. to permit evaluation of a range of dynamical behaviours. The absolute values used for the pressure or flow inputs were selected as: PO w 100 and Q, N 30 A.U. In order to corre- spond with the experimental protocols, PO and Q, were matched during theoretical simulation of the controlled- flow and controlled-pressure techniques. The diameter of the resistance artery undergoing vasomotion was assumed to vary as:

d= 1 +Au(t) (4) where u(t) is a time-dependent perturbation at time t, and A a variable amplitude. All results were normalized to the steady-state value obtained with A = 0. In the sinusoidal simulation, v(r) was modelled as a simple periodic perturbation (sin wt) whose amplitude factor (A) ranged from 0 to 0.8. Chaotic diameter fluctuations were represented as the x-variable of the continuous-time Lorenz system:

dx/dt= lO(y-x)

dy/dt=y-xz+rx (5) dz/dt=xy- 8z/3

where the “constant” coefficient r, which can be regarded as a bifurcation parameter and is a major determinant of the dynamic behaviour of this chaotic system, was varied between 30 and 210. Sustained oscillatory behaviour is observed for r > 5 24. The average amplitude of the Lorenz oscillation increases progressively with r, being proportional to 6-i [ 12,131. This rturbation was therefore normalised as u(t) = x(t)/ r - 1 to yield an F average signal of “constant” amplitude but changing dynamic complexity, and then multiplied by an amplitude factor (A) that was varied between 0 and 0.25. The maximum absolute amplitude of the normal&d Lorenz oscillations was between - - 3.4 and N 3.4, so that the overall magnitude of the simulated vasomotion (and resulting changes in resistance and conductance; see below) was comparable to that of the sinusoidal oscillation. In order to ensure that the chaotic dynamics had actually settled on its governing attractor, a large number of initial oscillations were discarded before carrying out the final computation of the relevant hydrodynamic parameters, except when studying the transient behaviour of the system.

Over sufficiently long intervals the time-averaged value of both types of perturbation is zero, which permits direct comparison of their functional consequences. Instanta- neous resistance was calculated as p,( t)/x( t) and conductance as x( t>/p,(t) (see Fig. lb). Normal&d resistance and condyctance were also calculated directly as the time- average R, of R,(r) = [l/(1 + hu(r))14 and 6x of G,(t) = [ 1 + Au(t)14 over a large number of cycles. The time-

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D. Parthimos et al./ Cardiovascular Research 31 (1996) 388-399 391

averaged “efficiency” of the circuit, defined as volume flow in the vessel undergoing vasomotion divided by viscous hydraulic work (i.e., x(t)/[P * q(f)]) was also derived and normalized to the value 1 under steady-flow conditions (i.e., A = 0).

In the sinusoidal case, oscillations with periods of 15, 10 and 5 s (denoted as angular frequencies of o = 1, 2, 3) were studied, integration of the equations of motion of the model being performed with a standard Runge-Kutta-Mer- son algorithm that employed a time step of 0.001 s. To provide varying frequencies in the chaotic case the time- step used to integrate the Lorenz equations was changed while keeping the total number of integration steps the same in each simulation. As the signals were consequently not exactly equivalent in length, effects attributable to varying residence times on different parts of the Lorenz attractor may exert a secondary influence. In the case of the Lorenz simulation the frequency of the oscillatory responses corresponded approximately to o = 1, 2 and 3 in the sinusoidal case. The chaotic data presented were generated by averaging over approximately 300, 600 and 900 oscillations, respectively.

4. Results

4.1. Experimental responses

Rhythmic fluctuations in pressure and flow were observed in 8 out of 11 arteries following activation by histamine. As previously reported [g-lo], the resulting patterns of behaviour could be either highly irregular or nearly-periodic (Figs. 2-4). The phase relationships between measured pressure and flow are illustrated for a preparation exhibiting simple periodic dynamics in Fig. 3, and for chaotic dynamics in Fig. 4. In controlled-flow perfusion mode, oscillations in flow led those in pressure by approximately 90” (e.g., - 91” and - 83” in Figs. 3b and 4b, respectively). In controlled-pressure mode they were approximately 180” out of phase (e.g., with flow leading pressure by - 188” in Fig. 3a and m 192” in Fig. 4a).

4.2. Model-generated oscillations and phase loops

Behaviour which closely resembled that observed experimentally was obtained with the electrical analogue when either sinusoidal or chaotic diameter perturbations were used to model vasomotion (Fig. 2). This theoretical model also reproduced the phase relationships between flow and pressure found under experimental conditions. In controlled-flow mode, flow led pressure by - 93” in the sinusoidal case (Fig. 3d), and by N 85” in the Lorenz case (Fig. 4d). In c on o e tr 11 d- pressure mode these phase shifts were - 203” (Fig. 3c) and - 199” (Fig. 4c), respectively.

1 min -

Hist 1j~M + L-NAME 50j~M

b) I 1

01 0 40 80 120 160 200 240 280

Time

Fig. 2. (a) Experimental trace showing pressure as a function of time in an artery constricted by 1 PM histamine in the presence of L-NAME to inhibit endothelial NO synthesis. (b) Simulated pressure trace for Lorenz dynamics with r = 48, showing the variation of p,(t) with time (see Fig. lb)

The phase shift between voltage and current in a linear circuit supplied by an oscillatory voltage or current source depends on resistance, capacitance and frequency. A simple expression for this shift cannot be derived, however, when there is a time-varying resistance as in the present case. Nevertheless, the equivalence of the phase loops in the experimental and theoretical situations under the two different perfusion conditions suggests that dynamic simi- larity was achieved in the model simulations.

4.3. Theoretical eflects of vasomotion on resistance and conductance

In model simulations time-averaged resistance and conductance both increased monotonically from the control value obtained under non-oscillatory conditions (i.e., 1; see Eq. 4) as the amplitude of sinusoidal diameter fluctuation (A) was made larger, with resistance increasing more steeply than conductance (Fig. 5a,b). The frequency (0) of the simulated vasomotion and the value of capacitance used in the calculations exerted no effect on the resulting plots of resistance and conductance against A. Consis- tently, analytical expressions derived for time-averaged resistance and conductance for a sinusoidally varying diameter fluctuation (see Appendix) were independent of w, and corresponded exactly with the results of the model simulation.

Time-averaged resistance and conductance also both increased with A in simulations with the Lorenz diameter

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392 D. Parthinws et al./ Cardiovascular Research 31 (1996) 388-399

a) t 7 E 0 lizi!

OT7 I o.77 I g- 0.70 -

E 3 0.63 -

E

- 0.56 -

3 ,o 0.49 -

LL

- 0.70 -

.; 0.63 -

g 0.56-

3 0.40 - 0

ii 0.42 - Q 1 c h &!!I

time

0.42

0.35 -IPI 45

Preisurt (mrlkg)

53 -I 63 47 49 51

Pressure (mmHg)

g 4.0 -

G 3.0 -

d,.,. \

sf 5 2.0 - 1

3 4.0 -

0 ii 3.0 -

?

dl.0. (-J

*= 2.0 - s

0.0 II 0.0 1.0 2.0 3.0 4.0

Relative Pressure

0.0 I

0.0 1.0 2.0 3.0 4.0

Relative Pressure

Fig. 3. Short segments of experimental pressure and flow oscillations and their corresponding phase relationships under conditions of (a) controlled-pressure and (b) controlled-flow perfusion in a preparation exhibiting almost sinusoidal dynamics. Note differences in phase between the two models of

perfusion. The electrical analogue exhibited similar behaviour when a sinusoidal change in diameter was used to model vasomotion (c. d). All phase loops were traversed in a counterclockwise direction.

0.65

= 0.56 .- E

= 0.61

.s

g OA4 E: 0.37

: 45

0.30

b) E 1

g-O.66 -

E

go.51 -

s 0.44 -

0 ii 0.37 -

0.30 I 45

Prksuye (;mHi)

4.0

g 3.0

z

3 2.0

d 1.0

0.0

r

g 3.0

ii

Q) 20 .L ii

2 1.0

t lmm 0.0 1.0 2.0 3.0 4.0

Relative Pressure

0.0 1.0 2.0 3.0 4.0

Relative Pressure

Fig. 4. Experimental controlled-pressure (a) and controtled-flow (b) traces and phase plots for an artery exhibiting chaotic dynamics, and equivalent chaotic vasomotion simulated by the Lorenz equations with r = 48 (c, d). The model data again closely followed by the experimental findings.

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D. Parthimos et al. / Cardiovascular Research 31 (1996) 388-399 393

perturbation. Multiple integrations were carried out with r-, the bifurcation parameter that was varied to generate dynamics of changing complexity, ranging from 40 to 80, at two effective “frequencies” o = 2 and o = 3, with two capacitances C = 1 and C = 2 (Fig. 5c,d). Non-systematic variations were apparent at different values of r, o and C, although similar trends were obtained for all sets of parameter values.

4.4. Theoretical effects of vasomotion on flow and the “ eficiency’ ’ of perfusion

With sinusoidal control, flow increased smoothly with vasomotion amplitude (A) in a sigmoidal fashion (Fig. 6a). Except at low values of capacitance, C, and frequency, w, (C=O.l and o=l)theplotsforC=O.l-3and A=l-3 were identical. The efficiency of sinusoidal perfusion, plotted against vasomotion amplitude, was a bell-shaped curve with a peak at A N 0.3 whose magnitude increased with the capacitance (Cl in the circuit and occurred at

a)

Sine

60-

40 -

20 -

O-

progressively larger values of A as C was increased (Fig. 6b,c). This “resonance” was a consequence of transients after starting the integration from the steady-state equilibrium obtained with A = 0, as its magnitude diminished progressively when a larger number of initial oscillations were excluded from the computation (Fig. 6b,c). Indeed, when a sufficiently long period of transient behaviour was rejected, at low values of capacitance (C = 0.1) and frequency (o = l), resonance was completely absent.

The consequences of the Lorenz perturbation on flow and efficiency over the range of bifurcation parameter r = 40-80 are shown in Fig. 6d and e. As in the sinusoidal case, the general trend was for flow to increase progressively with vasomotion amplitude (A). The Lorenz perturbation could either smoothly increase or decrease efficiency as A increased, but the total length of the integration time employed ultimately determined the nature of the directional change (Fig. 6e). Neither flow nor efficiency exhibited a systematic variation with C and w. In marked contrast to the sinusoidal situation, there was no bell-shaped

W

0.5 I 0.0 0.2 0.4 0.6 0.6 0.0 0.2 0.4 0.6 0.6

h h

Lorenz t-=40 to 80

cl 60 C=l,Z

d) ‘.’ 60- w2. 3

3.0

3 8 c 2.5

5 40- 3 ua g 2.0

.- i!

-0 1.5 p! 20- g

o 1.0

O- i- 0.5

0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

h h

Fig. 5. Theoretical effects of sinusoidal (a,b) and Lorenziau (c,d) diameter fluctuations on time-averaged resistance and conductance, which both increased monotically with the amplitude of the perturbation (A). In the sinusoidal case there was no dependence on the effective “frequency” of the oscillation (0) or the capacitance in the circuit (C). In the Lorenz case multiple plots were constructed by superimposing data with r varying from 40 to 80.

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394 D. Parthimos et al./ Cardiovascular Research 31 (19961388-399

efficiency “resonance” with the chaotic diameter fluctuation. The explanation for this is evident in Fig. 7 in which oscillatory activity of equivalent absolute amplitude and frequency to the sinusoidal case was initiated from a relatively high steady-state starting pressure (which consequently reflects initially high charge on the capacitance because Q = CV 1. In the sinusoidal case the system spiralled slowly to its final oscillatory state, with mean pressure (and hence the charge in Cl ultimately becoming lower than the equilibrium value for A = 0. In contrast, with the chaotic perturbation the system rapidly settled on its attractor, with pressure oscillating in an irregular fashion above and below the theoretical equilibrium at A = 0.

Transient behaviour was therefore evident with sinusoidal, but not chaotic patterns of simulated vasomotion.

The effects of altered dynamic complexity per se, rather than vasomotion amplitude, are shown in Fig. 8. Multiple simulations were carried out with the Lorenz perturbation with the bifurcation parameter (r) ranging from30to210,for C=2or3, o=2andatthreeseparate values of vasomotion amplitude (A). Over some ranges of r the calculated parameters were rather stable, with only minimal dependence on C. Highly complex effects were, however, seen in the range r = 3040, which represents the most irregular regime of the Lorenz system, and also at windows of exact integral periodicity. At the Period 3

a) ‘.I’ 112 -

3 P 1.10 -

LI. 1.06-

g .= t.o6-

3 p) 1.04-

PC 1.02 -

1.00 -

r I-

0.0 0.2 0.4 0.6 0.6

h

Sine

b) 1.016 ,-

. c=3 1.012 o=l, 2. 3

5 1.010 T~lClOOS

f u

1.006 “O”: 1.000 f-\ c=2 tc

6c

?j+g+

0.0 0.2 0.4 0.6 0.6

a

Lorenz r=40 to 80

1.012 OH, 2.3

T’3ooo5ec

r-48

d) e) 1.012

C=1. WI (a)

0.00 0.05 0.10 0.15 0.20 0.25 0.00 0.05 0.10 0.15 0.20 0.25

h J. (I): 300-r (m)r 5OOord#bns

(I): 900 Lladwom

Fig. 6. Time-averaged flow and the efficiency of the simulated perfusion. (a) In the sinusoidal case there was a sigmoidal relationship between flow and vasomotion amplitude (A) with minimal dependence on vasomotion frequency ( w) and capacitance (C). (b, c) Perfusion efficiency was highly dependent on C, exhibiting a peak near A - 0.3. This “resonance” is attributable to transients as it diminished in height on excluding a longer segment of the initial integration (7) from the final computations (3OOtJ cf. 1800 s). For a given value of T, the shape of the resonance was independent of o. (d) For different choices of C and o, flow similarly increased with A for the Lorenz perturbation at all values of r between 40 and 80. (e) At any given value of r

(illustrated for r = 48) efficiency either increased or decreased with A, according to the choice of C and o but also, importantly, according to the length of tbe signal used for the integration (i.e., 300, 500 or 900 oscillations). This apparent unpredictability of response largely reflects the influence of varying residence times on different parts of the Lotenz attractor. In marked contrast to the sinusoidal case, there was no obvious “resonance” dependent on A and C.

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a) 2.0

Q)

5 v) 1.5

3

n 1.0

g P

1 0.5

2

0.0

Sine

C=l

co=1

0.0 1.0 2.0 3.0 4.0 5.0 6.0

Relative Flow

b) Lorenz (r=80)

2.0

Ql

3 * 1.5-

i!

n l.O-

z 'i3‘

2 0.5-

2 C=l

0=l

0.0 I 0.0 2.0 4.0 6.0 6.0 10.0

Relative Flow Fig. 7. Relationship between flow and pressure on initiating the integration from a higher initial pressure (p,(t); see Fig. lb) than the equilibrium value obtained in the non-oscillatory situation with A = 0 (denoted as 0 ). (a) In the sinusoidal case the system spiralled slowly to a “steady” periodic state of low mean pressure. (b) In tbe chaotic case (r = 80) the system rapidly settled on its attractor, with pressure fluctuating both above and below the equilibrium point, thus explaining the absence of a resonant peak in efficiency.

window found at r = 100, for example, two asymmetric initial conditions chosen to start the integration (Fig. 9): if “mirror image” attractors co-exist and which one ulti- one attractor is selected by the initial condition (a, b, c>, mately governs the behaviour of the system depends on the the other can be selected by starting the integration from

a) 5 P3

P6

g4 x=0.2

s

--' F-'

33 .-

32 kxl P2

PL A=o.l _

1

W

x-o.05

0 so 50 00 120 160 160 210

r

1.10

6 G

1.W

$ 1.05 '3

= 1.04

d 1.02

I.00

I p3 I

bO.2

c=2 , I k=Y C=l "

A ArO.1

,&qFj 30 50 00 120 Is0 150 210

$ 2.0 P3

?i 150.2

g 1.5

-E

w

bO.1

00 1.0 1-0.05

0.5 4-I 30 a0 90 120 160 150 210

r

r r

Fig. 8. Multiple simulations with the Lorenz perturbation with r varying from 30 to 210 at three different fixed amplitudes (A). Only one frequency (o = 2) is illustrated. Over certain ranges or r, the intrinsic nature of the dynamics, rather than changes in A per se, influenced (a) resistance, (b) conductance, (c) flow and (d) efficiency. The most pronounced effect was apparent at r = 100, where two Period 3 attractors (~3) coexist (see Fig. 9). Which of these dominates the dynamics is sensitively dependent on the initial conditions from which the integration is started. At this periodic window, flow and conductance could thus exhibit either “beneficial” increases (continuous lines) or “detrimental” decreases (dashed lines).

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3% D. Parthimos et al./ Cardiovascular Research 31 (1996) 388-399

Fig. 9. Bifurcation map of the Lorenz system generated by Poincan? section through the plane z = r-l, which intersects all dynamically interesting trajectories of the system (adapted from Ref. 13). Over certain ranges of r the dynamics are chaotic and the intersections with 2 = r-1 then densely fill the 2-dimensional plane of the representation. Within regions of chaos there are windows of exact-integral periodicity: e.g., Period 2. 3, 4 and 5 dynamics (~2 + ~5). Two Period 3 attractors coexist at r = 100. Only one of these (right panels) is illustrated on tbe bifurcation map.

(-a, -b, -cl. As each attractor is asymmetric, the onset of Period 3 dynamics may consequently result in either a parallel increase or decrease in time-averaged conductance and flow. The effects of Period 3 dynamics on time-averaged resistance were more complex and were also influenced by vasomotion amplitude, changes in resistance and conductance occurring in opposite directions for small values of h, but in the same direction for large values of A.

5. Discussion

Previous theoretical studies have shown that the onset of periodic vasomotion results in an increase in the “effective” operating diameter of a single vessel. Sinusoidal, triangular and square wave patterns of rhythmic activity all behave similarly in respect, although the increase in time- averaged cross-sectional area is most pronounced with a square wave perturbation, and least with a triangular waveform [ 1,7]. It is therefore possible that the highly irregular patterns of behaviour that are the hallmark of chaotic behaviour confer important functional differences from simple periodic control. To evaluate this hypothesis, in the present study we have compared the theoretical effects of a

chaotic diameter perturbation generated by integration of the Lorenz equations, with those of a sinusoidal variation. The phase space trajectories of this chaotic system lie on a 3-dimensional “strange” attractor whose complexity is higher than the 2-dimensional trajectory followed by a periodic cycle. Direct comparison with a sinusoidal perturbation is nevertheless possible because the time-averaged mean of both is zero over sufficiently long periods of time. While each region of a chaotic attractor is occupied only temporarily, the ergodicity principle nevertheless ultimately guarantees full coverage of all patterns of dynamics over sufficiently long times of integration.

The vessel undergoing vasomotion was modelled as a time-varying resistance in a simplified equivalent circuit possessing a finite capacitance which may be regarded as representing the intrinsic compliance of the vasculature proximal to a vessel exhibiting rhythmic activity in vivo. This model was validated by initial experimental studies with a single, isolated rabbit ear resistance artery, perfused under both controlled-flow and controlled-pressure conditions in which histamine was used to induce rhythmic activity. Non-linear mathematical analysis of the fluctuations in pressure and flow observed experimentally in this preparation have indicated that chaotic patterns of vasomotion are generated by the interaction between at least 4 key

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control variables [8-lo]. Although this theoretically represents a higher degree of intrinsic complexity than generated by the Lorenz system, which consists of 3 coupled non-linear differential equations, the irregular appearance of the simulated responses closely matched those obtained experimentally. Furthermore, the phase differences between oscillations in pressure and flow under the two methods of perfusion were similar in the experimental and modelled situations, with both sinusoidal and chaotic patterns of behaviour. In subsequent simulations with the model, however, the circuit was driven by pressure rather than flow in order to reflect more accurately the situation in vivo, with the simplification that the supplying pressure head was steady and did not oscillate at the frequency of the heart [g-lo].

As in previous theoretical studies [ 1,7], sinusoidal fluctuations in diameter were confirmed to increase time-averaged conductance. A similar conclusion was also reached in the chaotic case. “Paradoxically”, however, both patterns of vasomotor activity also increased time-averaged resistance. By definition, instantaneous resistance is an inverse function of instantaneous conductance, so that during periods of constriction resistance is high and conductance low, whereas the converse applies when the vessel becomes dilated. Different parts of the vasomotion cycle therefore dominate the numerical computation of the time-average of these variables, allowing both to increase simultaneously. As both types of perturbation progressively enhanced time-averaged flow in an amplitude-dependent fashion, an important conclusion of the study is that changes in conductance, rather than resistance, should be regarded as providing a more accurate “physiological” measure of the functional consequences of active vasomotion. While the Lorenz equations cannot be solved exactly to determine resistance and conductance directly, analytical expressions confirmed the numerical findings in the sinusoidal case, and thus the accuracy of the method of numerical integration.

Because of inherently greater complexity, chaotic dynamics potentially confer greater flexibility than obtainable with a purely sinusoidal process. Indeed, it was found possible to effect amplitude-independent increases or decreases in flow and conductance under the variation of the single bifurcation parameter (r), which contributes to the specific dynamic form of the Lorenz oscillation. Complex effects were particularly evident over the highly irregular range where r varied between 30 and 60, and also following the onset of Period 3 dynamics at r = 100. Within this periodic window, simulations showed that there could be abrupt changes in flow when the two alternative “mirror- image” 3-100~ attractors that co-exist at the same value of r [ 131 determine the behaviour of the system. Which attractor actually dominates the overall dynamics depends critically on the initial conditions used to start the integration, and may consequently appear to be unpredictable, while nevertheless being specified deterministically. Al-

though the time-average of each of the three variables (x, y, z) contributing to the dynamics is zero within the Period 3 window, there is nevertheless considerable asym- metry when parameters such as conductance, resistance and flow are calculated: these may increase or decrease according to which attractor is ultimately selected.

A pronounced maximum in the “efficiency” of perfusion (i.e., flow/viscous work) was apparent with sinusoidal diameter control, whose location and magnitude depended strongly on C and the amplitude A of the vasomotion. The phenomenon can be considered as a “resonance” in the sense that the natural frequency of an electrical circuit is a function of both resistance and capacitance, and analytical expressions showed that time-averaged resistance/conductance were functions only of A. In the present model, however, input pressure was fixed, rather than oscillatory, so that the peak in efficiency should not be considered a resonance in the conventional sense of being a function of a forcing frequency. The height of the resonant peak diminished when larger numbers of initial “non-equilibrium” oscillations were excluded from the calculations, so that the phenomenon could be attributed to transient release of “charge” stored in the capacitance of the circuit, and thus persisted over longer integration times at high values of C. Resonance was not observed with chaotic fluctuations; efficiency then exhibiting monotonic increases or decreases as the vasomotion amplitude (h) was increased, which depended in a non-systematic fashion on the exact values of C and o chosen as well as the total length of the integration path. Analysis of the relationship between flow and pressure in the chaotic case indicated that the dynamics rapidly settled onto its attractor, thus explaining the absence of the resonance seen with a sinusoidal diameter perturbation. This observation is consistent with the hypothesis that an important functional role of chaos in biological systems may be to dissipate perturbations [ 141. In a system in which there is asymptotic stability as a periodic oscillation, the system will be modi- fied relatively slowly, although in time the effect of a given disturbance will be absorbed completely, as observed in the sinusoidal simulation. When a system is governed by a chaotic attractor, however, many more trajectories may be acceptable from a functional point of view and the system may therefore be able to dissipate transients much more rapidly, thus effectively conferring greater overall stability.

Vasomotion may become particularly evident under pathophysiological conditions when theoretical increases in conductance may serve to re-establish flow and mediate an adaptive homeodynamic response to impaired perfusion [2]. For example, vasomotion is enhanced in the distal cutaneous circulation of patients with occlusive peripheral arterial disease and during hypoxia in experimental models [ 15-171. Reductions in perfusion pressure induced by haemorrhagic hypotension also promote the appearance of large-amplitude oscillations in arteriolar diameter and mi-

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398 D. Parthimos et al./ Cardiovascular Research 31 (1996) 388-399

crocirculatory blood flow in skeletal muscle which appear to be regulated by local mechanisms as the patterns of vasomotion observed in adjacent areas of muscle do not correlate with each other [5,16-201. Vasomotion in such “pathological” situations may appear nearly sinusoidal or exhibit more complex waveforms. The present findings suggest, however, that both patterns of oscillatory behaviour will help to preserve tissue perfusion when distal perfusion pressures are low. In the sinusoidal case, however, transient local “steal” effects may then become apparent as a consequence of redistribution of flow between different tissue elements through “resonance” effects. Chaotic patterns of vasomotion might thus, at least in theory, preserve a more homogeneous pattern of perfusion.

Rhythmic activity is more prevalent in arteries from hypertensive rats than normotensive controls, a phenomenon that may protect against the potentially unstable positive feedback that is inherent to the myogenic response [21-231. Indeed, while recognising that issues of cause and effect remain controversial in the pathogenesis of hypertension, spontaneous rhythmic activity in rat aorta appears to be secondary to chronic pressure loading following experimental coarctation, consistent with the view that vasomotion may represent an adaptive response in this situation [22]. Hypertension also increases the complexity of integrated non-linear control mechanisms. In rats, fluctuations in pressure and flow in renal proximal tubules, which originate in the interaction between tubuloglomeru- lar feedback and active vasomotion in afferent arterioles, are normally periodic, but become chaotic following the onset of hypertension [24].

The high sensitivity of non-linear systems to perturbation can be exploited to bring about large changes in state with minimum expenditure of energy (25-29). It is thus possible to select the stabilize specific unstable periodic orbits, even in the biological context with carefully-com- puted perturbations - a feature unique to non-linear mechanisms of dynamic control. Indeed, we have recently shown that relatively simple techniques based on negative feedback also permit stabilisation of chaotic vasomotion in isolated rabbit ear arteries as periodic or steady-state behaviour [lo]. Perturbations may also be used to destabilize periodic or steady-state behaviour and thereby induce rather than suppress chaotic dynamics, so that regulated “control” and “anti-control” can both thus be achieved [27]. While chaotic behaviour may be inevitable in a system regulated by non-linear interactions, it is therefore possible that it confers a high degree of flexibility to the control of microcirculatory blood flow. For example, the stabilization of a particular type of asymmetric attractor could lead to a significant increase in flow, as shown in the present theoretical investigation at the Period 3 window of the Lorenz system. Sophisticated experimental ‘techniques will be re- quired to test whether “endogenous” mechanisms of chaos control actually operate in vivo, and to determine whether

the non-linear nature of microcirculatory control can be exploited therapeutically.

Acknowledgements

The work was supported by the British Medical Re- search Council. The authors wish to thank Professor A.H. Henderson for helpful comments during preparation of the manuscript, Professor G. Roberts for support and encour- agement, and W. Simons for secretarial assistance.

Appendix

For sinusoidal diameter fluctuatitns analytic expressions for time-averaged resistance CR,) and conductance (C?x> can be derived. These are:

dt

(1 +Asin(~t))~

[

5 3

I

1

= 2(1-h*) -2 (l-*2)5’*

3 (1 + A sin( wt))4 dt = 1 + 3A2 + sA4.

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