The Physics of Coronory Blood Flow - M. Zamir

240 7 Lumped Models

	25
	20					LM2
(mmHg)
	15
				p, tot
	10
flow		cap
	5
R−scaled
	0
	−5		ind
pressure,
	−10
	−15
	−200	0.2	0.4	0.6	0.8	1
			time t (normalized)

Fig. 7.4.4. Flow waves under LM 2 and a composite driving pressure wave (p), with λ = 1.0 and tL = tC = 0.2 s. This unique combination of parameter values reduces the total impedance of the system to a simple resistance R1. Inductive (ind) and capacitive (cap) flows precisely balance each other, thereby reducing the reactance of the system to zero. The R-scaled total flow wave (tot) becomes identical in form to the pressure waveform (p). The two waves are here represented by the heavy solid curve and the open circles to make them visually distinguishable.

and capacitor do not exist. Also, the expression for the harmonic impedances Zn for the system as a whole in Eq. 7.4.7 can be put in the form

	Zn	=	(1 + iωntL)(1	+ iωnλtC )
			1 + iωntC (λ + 1 + iωntL))
	R1

from which we see that when λ = 1.0 and tL = tC expression reduces to

Zn = R1

(7.4.21)

(as in Eq. 7.4.18), this

(7.4.22)

which indicates that total impedance has been reduced to simple resistance R1. Therefore, the R-scaled flow wave becomes identical in form to that of the driving pressure, as in Eq. 7.4.19. This unique situation is illustrated in Fig. 7.4.4 where the results are based on λ = 1.0 and tC = tL = 0.2 s. With these parameter values inertial and capacitive e ects exactly balance each other so that R-scaled total flow into the system follows precisely the form of the driving pressure wave.

7.5 LM3: {{R1+(pb)},{R2+C}} 241

When in an RLC system the e ects of capacitance and inductance precisely “cancel” each other, which means that the system reactance is zero, as discussed in Section 4.8, the system behaves as if the reactive elements L, C do not exist. As we saw in Section 4.8, when the driving pressure is a single harmonic (not a composite wave), the conditions under which this unique situation occurs depend not only on the values of L and C but also on the angular frequency ω (Eq. 4.8.11). The phenomenon is well known in the study of electric circuits, being referred to as “series resonance” or “parallel resonance” depending on the configuration of the RLC system [43]. In that context, interest is particularly in the value of the frequency at which the unique conditions occur, hence the reference to resonance. In the context of the coronary circulation, by contrast, the phenomenon is of interest in relation to the optimum operation of the system rather than to resonance. This is so particularly because the driving force in electric circuits is of a single harmonic form, while in the coronary circulation the driving force is of a composite waveform. Since a composite wave consists of many single harmonics which operate at di erent frequencies, the combination of values of R, L, ω at which the unique conditions occur will generally be di erent for each harmonic. What the results of this section demonstrate is that, depending on the configuration of the RLC system, these conditions may become independent of the frequency and thus apply equally to all harmonics as in the present case.

7.5 LM3: {{R1+(pb)},{R2+C}}

A feature of coronary blood flow that distinguishes it from flow in other parts of the cardiovascular system is an apparent discrepancy in the relation between pressure and flow within the oscillatory cycle. More specifically, during the systolic phase of the oscillatory cycle, when driving pressure is rising rapidly to a peak, coronary blood flow is highly diminished or even reversed, while during the diastolic phase of the oscillatory cycle, when driving pressure is coming down from its peak, coronary blood flow is at its highest [101]. In other words, coronary blood flow occurs mostly in diastole when driving pressure is diminishing rather than in systole when driving pressure is rising. The reasons for this discrepancy are not fully understood, although some possible mechanisms have been suggested.

It is important to note in this discussion that by “coronary blood flow” is meant input flow, that is flow entering the system. The reason for this is that from a practical standpoint this is the only flow to which there is reasonable access for measurements. From a clinical standpoint, of course, of more interest is flow at exit from the system, that is flow at the capillary end of the coronary circulation. The relation between this output flow and flow at entry into the system is not known because it depends on what goes on inside the system, which is what lumped models attempt to uncover. Of course, conservation of

242 7 Lumped Models

qtot

LM3

p

p _ p	b	R1	R2
pb			qcap
		qres
			C

Fig. 7.5.1. A modified windkessel lumped model with a provision for back pressure pb to simulate the e ects of surrounding tissue pressure on coronary vessels imbedded within the cardiac muscle tissue. Back pressure pb is assumed to a ect flow qres in the resistive branch of the system only. Both the primary driving pressure p and the back pressure pb are oscillatory, thus resistive flow is driven by the oscillatory di erence between them, namely p − pb. Remaining features of the model are the same as those of LM 1 in Fig. 7.3.1.

mass requires that on average, inflow and outflow must be the same. That is, under normal circumstances, the amount of fluid entering the system must equal that leaving the system in the course of one oscillatory cycle. It is at any particular moment within the oscillatory cycle that inflow and outflow are usually di erent. We return to this question later in this chapter. The point of raising this issue here is only to emphasize that the term “coronary blood flow” in the present context refers to flow at entry into the system.

In this section we consider one possible mechanism that may be responsible for the apparent discrepancy in the relation between (input) pressure and (input) coronary blood flow within the oscillatory cycle, namely that of so called “tissue pressure”, or “intramyocardial pressure”. Under this mechanism, it is postulated that during the systolic phase of the oscillatory cycle, although input driving pressure is relatively high, cardiac muscle tissue is contracting and exerting high pressure on coronary blood vessels imbedded within this tissue [84, 20, 10, 189, 68, 187, 151, 6, 41, 87, 113, 165, 202, 101]. The e ect of this is an increase in pressure within the lumen of these vessels, which leads to the notion of a “back pressure” created within the system during systole [97, 98]. The input driving pressure must overcome this back pressure in addition to pre-existing resistance within the system, hence the reduction

7.5 LM3: {{R1+(pb)},{R2+C}} 243

	R
p1	q	p2

p1 _ p2

p= p1 _ p2 q= p/R

	R
p1	q	p2 + pb

p1 _ (p2 + pb )

p= p1 _ p2			q= (p _ pb )/R

Fig. 7.5.2. Back pressure pb a ects the flow by simply changing the pressure di erence driving the flow. In the notation of the present chapter, p actually represents a pressure di erence, which in the absence of back pressure is simply the di erence (p1 − p2) between pressures at the input and output ends of the system as shown at the top. The addition of a back pressure at the output end then simply means that the pressure di erence driving the flow is now p − pb as shown at the bottom, where p continues to represent the pre-existing pressure di erence, namely p = p1 − p2.

in flow. The situation is similar to that of obstructing the output end of a trumpet, or of the exhaust system of a car, thus creating higher pressure at that end with familiar consequences. In the coronary circulation this may occur momentarily within the oscillatory cycle as the cardiac muscle contracts in the course of its pumping action, leading to reduced flow, possibly even reverse flow. The challenge for lumped model analysis is to use a configuration of RCL components in such a way as to reproduce this scenario, in particular to reproduce the possibility of drastically reduced or reverse flow during the systolic phase of the oscillatory cycle.

In a model that addresses this challenge, which is shown in Fig. 7.5.1 and which we shall refer to as LM 3, it is postulated that tissue pressure will a ect the pressure drop driving the flow along the resistive branch of the parallel system as illustrated schematically in Fig. 7.5.2. The source of the back pressure pb is taken to be a combination of pressure within the left ventricle and a constant residual pressure within the cardiac tissue which exists independently of ventricular pressure [97, 98]. The resulting combination is the pressure waveform shown in Fig. 7.5.3 along with the primary driving pressure waveform.

244 7 Lumped Models

	80
	70					LM3
	60
(mmHg)	50	p
	40
pressure
	30	pb
	20
	10
	00	0.2	0.4	0.6	0.8	1
			time t (normalized)

Fig. 7.5.3. Primary pressure wave p and back pressure wave pb used in LM 3. The source of the back pressure pb is assumed to be a combination of pressure within the left ventricle and a constant residual pressure within the cardiac tissue which exists independently of ventricular pressure.

	100
(mmHg)	80					LM3
	60
flow
		p
R−scaled	40	pb
	20		q
pressure,
	0
	−20
	0	0.2	0.4	0.6	0.8	1
			time t (normalized)

Fig. 7.5.4. Total flow wave under LM 3, with a primary driving pressure wave p(t) and a back pressure wave pb(t). Parameter values of the system are λ = 1.0 and tC = 0.1 s. The results demonstrate that back pressure can produce considerably reduced and some reverse flow within the oscillatory cycle.

7.5 LM3: {{R1+(pb)},{R2+C}} 245

The complex impedances along the two branches of this model are the same as those of LM 1 discussed in Section 7.3, namely

Zres = R1		(7.5.1)
1		(7.5.2)
Zcap = R2 +
	iωC

where the subscripts res and cap refer to the resistive and capacitive branches, respectively. The complex impedance for the entire system is given by

	1	=	1	+	1		(7.5.3)
	Z		Zres		Zcap
	Z =		R1	+ iωR1R2C			(7.5.4)
			1 + iωC(R1 + R2)

where ω is the angular frequency of oscillation of the driving pressure. For the individual harmonics of a composite pressure wave consisting of N harmonics we have, as before,

Zn,res = R1				(7.5.5)
1				(7.5.6)
Zn,cap = R2 +
		iωnC
Zn =	R1 + iωnR1R2C			(7.5.7)
	1 + iωnC(R1 + R2)

where n denotes a particular harmonic and ωn is the angular frequency of that harmonic.

The essence of the present model is that along the capacitive branch of the parallel system the oscillatory pressure driving flow is p(t), but along the resistive branch the driving pressure is p(t) − pb(t). Both pressures are oscillatory and, therefore, to follow the same analysis as before, they must be presented in their complex exponential form

pn(t) = Mnei(ωn t−φn )	n = 1, 2, . . . , N	(7.5.8)
pbn(t) = Mbnei(ωbn t−φbn )	n = 1, 2, . . . , N	(7.5.9)

where subscript b refers to properties of the back pressure wave pb.

The harmonics of the R-scaled flow rates along the two branches of the parallel system are then given by, following the same steps as for previous models and omitting some of the details,

	q	nr,res	(t) =		pn(t) − pbn(t)		(7.5.10)
					Zn,res
	R1 × qnr,res		(t) = Mn cos (ωnt − φn) − Mbn cos (ωbnt − φbn)				(7.5.11)

246 7 Lumped Models

			qnr,cap(t) =				pn(t)	(7.5.12)
							Zn,cap
R	1	×	q	nr,cap	(t) =	ωn2 tC2 λMn cos (ωnt − φn) − ωntC Mn sin (ωnt − φn)		(7.5.13)
							1 + (ωnλtC )2
The harmonics of total flow into the system are then simply given by
					R1 × qnr (t) = R1 × qnr,res(t) + R1 × qnr,cap(t)			(7.5.14)

Finally, the actual flow waves are obtained by adding their harmonics to obtain the oscillatory parts of the waves, plus the steady part of each flow if any,

R1 × qr,res(t) = R1 × q + R1 × q1r,res + R1 × q2r,res + . . . + R1 × qN r,res

	(7.5.15)
R1 × qr,cap(t) = R1 × q1r,cap + R1 × q2r,cap + . . . + R1 × qN r,cap	(7.5.16)
R1 × qr (t) = R1 × qr,res(t) + R1 × qr,cap(t)	(7.5.17)

Steady flow through the system is included this time because it is relevant to present discussion. The steady flow rate q(t) is given by

				(t) −		b(t)
		(t) =	p		p		(7.5.18)
	q
				R1

where the overline bar indicates average over one oscillatory cycle. The steady part of the flow, of course, is part of the flow in the resistive branch of the parallel system because the capacitive branch of the system supports oscillatory flow only.

Total flow under LM 3, with primary driving pressure p(t) and back pressure pb(t) is shown in Fig. 7.5.4. It is seen that the model meets the objective of demonstrating considerably reduced and some reverse flow through the system. Results using the same model parameter values but in the absence of any back pressure are shown in Fig. 7.5.5, which indicate clearly that the reduced and reverse flow observed in Fig. 7.5.4 are due entirely to the back pressure in this model. Furthermore, the individual flow waves along the two parallel branches with and without back pressure are shown in Figs. 7.5.6, 7, which confirm that the e ect of back pressure is confined to the resistive part of the flow only.

The e ects of cardiac muscle contraction on the dynamics of the coronary circulation have also given rise to a number of other concepts regarding the possible mechanisms involved. In the “waterfall” concept, the vessels imbedded within the cardiac muscle tissue are believed to collapse under muscle contraction, leading to an interval (in systole) when pressure is rising inside the vessels but there is no flow, as in a waterfall before the water level in the reservoir has reached the mouth of the fall [161, 51, 111, 205, 60]. Flow

				7.5 LM3: {{R1+(pb)},{R2+C}}			247
	100
(mmHg)	80					LM3
			q
	60
flow
		p
R−scaled	40
	20
pressure,	0
	−20
	0	0.2	0.4	0.6	0.8	1
			time t (normalized)

Fig. 7.5.5. Total flow wave under LM 3 with a primary driving pressure wave p(t) and the same parameter values as in Fig. 7.5.4 but in the absence of any back pressure, demonstrating that the reduced and reverse flow observed in that figure are due entirely to the e ects of back pressure.

	100
	80					LM3
(mmHg)	60
flow	40
R−scaled	20	cap
	0
		res
	−20
	0	0.2	0.4	0.6	0.8	1
			time t (normalized)

Fig. 7.5.6. The resistive (res) and capacitive (cap) flow waves under LM 3 with back pressure, as in Fig. 7.5.4.

248 7 Lumped Models

	100
	80					LM3
(mmHg)	60		res
flow	40
R−scaled	20
			cap
	0
	−20
	0	0.2	0.4	0.6	0.8	1
			time t (normalized)

Fig. 7.5.7. The resistive (res) and capacitive (cap) flow waves under LM 3 but in the absence of back pressure, as in Fig. 7.5.5.

DIASTOLE

dam level falling

more quickly

water level falling

SYSTOLE

dam level rising

more quickly

water level rising

Fig. 7.5.8. A schematic illustration of the waterfall concept in which the water level in the reservoir is analogous to the pressure inside the coronary vessels while the level of the dam is analogous to pressure outside the vessels caused by cardiac muscle contraction. In systole, water level in the reservoir is rising but the level of the dam is rising more quickly and no flow is possible. In diastole, water level in the reservoir is falling but the level of the dam is falling more quickly, and a point is reached where flow becomes possible.

7.6 Inflow-Outflow

249

resumes only when pressure inside the vessels exceeds the pressure outside, which strangely occurs in diastole when inside pressure is actually decreasing but at the same time the cardiac muscles are relaxing and the external pressure is decreasing more steeply. Thus, to state the waterfall analogy correctly it has to be said that flow resumes when the water level in the reservoir is actually falling but the height of the dam is falling more quickly (Fig. 7.5.8).

In other concepts the e ects of muscle contraction on the dynamics of the coronary circulation have been described as an “intramyocardial pump” e ect [29] or “variable elastance” e ect [189, 113, 114, 202].

7.6 Inflow-Outflow

One of the most important aspects of the dynamics of the coronary circulation is the fact that at almost any point in time within the oscillatory cycle, total inflow into the system is not equal to total outflow. This di erence between inflow and outflow is particularly important because, as mentioned earlier, flow measurements in the coronary circulation are extremely di cult and at best would provide only inflow data. We recall that inflow into the coronary circulation occurs via the two main coronary ostia at the root of the ascending aorta (Figs. 1.3.1, 2). Outflow from the system of course occurs at the capillary bed, hence the di culty of obtaining an actual measure of it. Yet, this outflow is what is most relevant clinically.

It is therefore important to understand the nature and source of the di erence between inflow and outflow in the dynamics of the coronary circulation, at di erent times within the oscillatory cycle and under di erent circumstances, which we examine in this section. It must be emphasized, of course, that this di erence is an oscillatory function of time that has a zero mean. In other words, under normal circumstances inflow and outflow are equal on average, the average being taken over one or at most a few cycles. Di erences occur largely within the oscillatory cycle. These di erences are important, nevertheless, because they are indicators of the interplay between the dynamics of di erent elements of the coronary circulation. More specifically, the di erence between inflow and outflow, as we shall see, is a measure of the interplay between and the relative e ects of capacitance, resistance, and inductance, thus, a change in any of these properties of the coronary circulation will lead to a change in the delicate balance between inflow and outflow within the oscillatory cycle.

In the windkessel system (LM 0) considered in Section 7.2, for example, when compliance is low, with a value of the capacitive time constant tC = 0.01 s, capacitive flow is very low and total flow through the system is dominated by resistive flow as seen in Fig. 7.2.2. Under these circumstances inflow into the system is very nearly the same as outflow, as shown in Fig. 7.6.1 where inflow is plotted against outflow at di erent points in time within the oscillatory cycle. As compliance increases, however, with tC = 0.2 s, capacitive flow