The Physics of Coronory Blood Flow - M. Zamir

130 4 Forced Dynamics of the RLC System

the corresponding time is t/tL ≈ 5. To determine these times in seconds requires an estimate of the inertial time constant tL for the coronary circulation which, of course, is not known. Indeed, one of the main aims of lumped model studies is to determine such properties of the system.

Nevertheless, we may attempt to estimate tL by recalling from sections 2.4,5 (Eqs.2.4.5,2.5.5) that for flow in a single tube the resistance to flow and the inertial constant are given by

where μ, ρ are respectively the viscosity and density of the fluid, and a, d are respectively the radius and diameter of the tube. From this, and from the definition of tL we then have

Taking ρ = 1.0 g/cm3 and μ = 0.04 g/(cm s), this becomes

where a is in centimeters.

Now, in a lumped model of the coronary circulation the characteristic parameters of the numerous tubes in the system are “lumped” together, which is equivalent to considering the system as a single tube that has these lumped properties. The problem, of course, is that these lumped properties are not known. Indeed, whether by theory or by experiment, the ultimate goal of lumped model studies is to determine not only the values of such properties but the legitimacy of the lumped parameter concept on which they are based.

Thus, from Eq. 4.7.6 we can only conclude that for a tube of 1cm radius, the equation gives a value of tL of approximately 3 seconds, while for a tube of 1mm radius, the corresponding value is 0.03 seconds. The transient state of an RLC system with these values of tL would then be approximately 15 tL in free dynamics and 5 tL in forced dynamics. The highest of these values is 45 seconds, the lowest is 0.15 seconds. Whether the true transient time of the coronary circulation lies within or indeed outside these estimates is not known. In the forced oscillatory dynamics of the coronary circulation tL would represent the time it takes the system to recover from a disturbance and return to steady state, thus knowing the actual value of this time constant would be of considerable clinical importance.

4.8 The Concept of Reactance

The concepts of reactance and impedance arise in the dynamics of the RLC system in steady state. It is important to emphasize this point because these concepts are so widely used that it is usually only implied, but rarely stated, that their use is limited to steady state dynamics only, not to the transient state. Thus, in introducing these concepts here, and using them in subsequent sections, it must be clear from the outset that we are now dealing with only the particular solution of the governing equation (Eq. 3.2.5)

and for this reason we no longer use the subscripts for the homogeneous and particular parts of the solution discussed in the previous section. It is implied here, and whenever these concepts are used, that q(t) now represents only the particular part of the solution, which, as discussed in the previous section, corresponds to the steady state dynamics of the RLC system.

Furthermore, a meaningful definition of reactance and impedance is only possible when Δp above, and consequently q(t), are simple oscillatory functions such as the trigonometric sine and cosine functions. We thus begin with the problem solved in Section 4.2, where the driving pressure drop is given by

Steady state solution of Eq. 4.8.1 with this form of the pressure drop was obtained in Section 4.2 (Eq. 4.2.13), which we now write in the form

Using standard trigonometric identities, Eq. 4.8.3 can be simplified further to

To discuss the nature and e ect of the quantity S now, we begin by noting that when S = 0, Eqs.4.8.5,6 give

132 4 Forced Dynamics of the RLC System

which we recognize as the simple expression for the flow rate through a resistance R when the driving pressure drop across it is Δp. Thus, in the dynamics of the RLC system the quantity S as defined in Eq. 4.8.4 embodies the combined e ects of inductance L and capacitance C such that when S = 0 the system behaves as a simple resistance R. When S = 0, it is clear from a comparison of Eq. 4.8.5 with Eq. 4.8.7 that S acts as an added form of resistance to flow, resulting from the presence of inductance and capacitance. It is also noted from the presence of ω in the expression for S that this form of resistance occurs only in oscillatory flow. By analogy with the same phenomenon in the flow of alternating current in an electric circuit, S is generally referred to as the “reactance”. It is a form of resistance to flow, but it di ers from R in that it only occurs in oscillatory flow. Also, unlike the viscous resistance R, the reactance S does not actually dissipate flow energy, it merely stores it and releases it within each oscillatory cycle [221].

From Eq. 4.8.4 we note that there are two ways in which the reactance S can be zero. First, when the capacitance and inductance e ects are simply absent, that is

which together lead to S = 0. Second, when the values of ω, L, C are such that

which again leads to S = 0. While the first of these circumstances is trivial, the second has clear physiological significance because it deals with the critical balance between the e ects of capacitance and inductance. To pursue this further, consider, in Eq. 4.8.4, that the frequency ω and inductance L are fixed so that the value of S now depends on the capacitance C only. It is convenient in this discussion to use the term “compliance” for the capacitance C. It is a more expressive term than “capacitance” because at higher values of the capacitance C a balloon is more elastic, more compliant, while at lower values of the capacitance C it is less elastic, less compliant.

Starting at the extreme where a balloon is rigid, compliance (C) is zero, and its reciprocal is infinite. The value of S from Eq. 4.8.4 is then infinite and negative, and the corresponding value of the phase angle θ from Eq. 4.8.6 is −π/2. Eq. 4.8.5 then indicates that the flow is leading the pressure drop by π/2. As the compliance gradually increases from this extreme value, the

values of S and θ remain negative at first but continue to increase, until at some point they both become zero. The value of C at which this point is reached is given in Eq. 4.8.11, and we shall denote this value by C0, that is

At this critical value of C the reactance is zero, the phase angle between the flow and pressure drop is zero, and the amplitude of the flow rate, which from Eq. 4.8.5 is given by

As compliance continues to increase beyond this point, the value of S becomes positive, the phase angle θ becomes positive, which means that flow is now lagging the pressure drop, and the amplitude of the flow rate begins to decrease again from its maximum value.

To illustrate these results graphically, it is convenient to use a normalized

Also, instead of using the actual capacitance C, it is more convenient to use the capacitive time constant

Since R is assumed to be constant, tC is a direct measure of C. Also, using the inertial time constant

the reactance S can be expressed in terms of these time constants as

136 4 Forced Dynamics of the RLC System

Fig. 4.8.3. Normalized flow rate (solid curve) compared with pressure drop (dashed curve) within the oscillatory cycle, and with tC = 0.03 seconds, which is just above critical value of tC = 0.0253 at which the two curves would be identical. Flow rate lags behind pressure drop and flow amplitude is below maximum.

Fig. 4.8.4. Normalized flow rate (solid curve) compared with pressure drop (dashed curve) within the oscillatory cycle, and with tC = 0.025 seconds, which is very close to critical value (at the critical value the two curves would be indistinguishable). Flow rate is in phase with pressure drop and flow amplitude is maximum.

Fig. 4.8.5. Normalized flow rate (solid curve) compared with pressure drop (dashed curve) within the oscillatory cycle, and with tC = 0.02 seconds, which is just below critical value of tC = 0.0253 at which the two curves would be identical. Flow rate leads pressure drop and flow amplitude is below maximum.

operate most optimally, in the sense of minimizing the e ects of reactance. The mere existence of these conditions is clearly of significant clinical interest because it indicates that if the coronary circulation normally operates at or near a critical value of tC , then any change in the elasticity or other properties of the system, resulting from disease or clinical intervention, may move the system away from its optimal dynamics.

Furthermore, in the present system if tL is set equal to 1.0 seconds, which is not an unreasonable estimate following the discussion in Section 4.7, these favourable conditions occur at tC = 0.0253 seconds. While these two values may be inaccurate in absolute terms, they indicate that in relative terms the capacitive time constant may be two orders of magnitude smaller than the inertial time constant.

4.9 The Concepts of Impedance, Complex Impedance

The concept of “impedance”, like that of reactance, arises in the dynamics of the RLC system in steady state under a simple oscillatory driving pressure drop and, as emphasized in the previous section, it is only valid, indeed only meaningful, in that context. Again, the concept is borrowed from the flow of alternating current in electric circuits, but it has wide applications in the dynamics of the coronary circulation, specifically in the context of lumped models of the system.