Analysis of pulsatile coronary pressure and flow velocity : looking beyond means - Appendix

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Analysis of pulsatile coronary pressure and flow velocity : looking beyond means

Kolyva, C.

Publication date

2008

Link to publication

Citation for published version (APA):

Kolyva, C. (2008). Analysis of pulsatile coronary pressure and flow velocity : looking beyond

means.

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Appendix

Theoretical background of Wave Intensity

Analysis and

in vivo applications

132

Wave Intensity Analysis

Bernhard Riemann’s method of characteristics was first presented in 1859 and has been a valuable tool for wave propagation studies ever since [10, 12]. The method may be applied to liquids, gases and solids for studying linear or non-linear wave propagation in the time domain. The method of characteristics forms the foundation of Wave Intensity Analysis (WIA).

Parker et al. [11] and Parker and Jones [10] were the first to apply the method of characteristics for solving the one-dimensional problem of flow in elastic arteries. The derivation of Wave Intensity (WI) with the method of characteristics involves advanced mathematics and only the main points will be discussed here.

The differential equations describing mass and momentum conservation in uniform, elastic arteries, assuming no viscous losses and negligible flow in/out of the arterial wall, can be transformed into a pair of hyperbolic, partial differential equations with the use of the appropriate ‘tube law’. By using a tube law that states that the cross-sectional area (A) of the artery depends only on the instantaneous, local pressure (A = A(P(x, t)) we get the following system:

0 U P U 1/ȡ 1/D U U P x t ¸¸ ¹ · ¨¨ © § ¸¸ ¹ · ¨¨ © §  ¸¸ ¹ · ¨¨ © § (9.1)

P and U are average pressure and velocity across the cross-sectional area of the artery, D is wall distensibility, ǒ is blood density and the subscripts denote partial differentiation over time (t) and along the axis of the artery (x). The characteristic directions for these equations are defined from the eigenvalues of the above system:

c U

dx/dt r , where c 1/ ȡD is wave speed. The physical interpretation of the characteristic directions is that every perturbation introduced in the flow is going to generate pressure and velocity waves that will propagate with speed U ± c along the characteristic directions (‘+’ refers to the forward and ‘-’ to the backward characteristic). Along the characteristic directions the above equations are reduced to a pair of ordinary differential equations:

0 P ȡc

1

Ut r t (9.2)

Assuming that wave speed is a function of pressure only, the above equations can be formulated in terms of the Riemann functions _r r

³

P P0 ȡc dP U R (P0 is an arbitrary

reference pressure) as:

0 ) ȡc P (U ) (R_{r t} r _t (9.3)

9

We note that along the characteristic directions the Riemann functions are constant with respect to time and are called the Riemann invariants. It can be deduced from the previous equations that:

ȡc dP dU

dR_r r (9.4)

where the operator ‘d’ is defined as the difference between two characteristics. By solving these two equations for dP and dU we get:

2 ) dR (dR ȡc dP    (9.5) 2 ) dR (dR dU   (9.6) The product 4 ) dR (dR ȡc dU dP 2 2    ˜ (9.7)

is Wave Intensity, has dimensions of power per unit area and refers to the power carried by the waves traveling in the blood stream. Forward-traveling waves make a positive contribution to the product and backward-traveling waves a negative contribution. As a result, WI is positive for dominant forward-running waves and negative for dominant backward-running waves. When local wave speed is known, it is possible to separate net WI into its forward and backward components.

Another way of deriving Wave Intensity in the arterial system, placing more emphasis on its physical meaning rather than on the mathematics is given by Jones et al. [4].

In vivo applications

A brief review of applications of WIA in the circulatory system is given in this section, in order to demonstrate the kind of mechanistic explanations WIA can provide to cardiovascular mechanics problems. Unless stated otherwise, the results mentioned in this section refer to separated WI.

The first application of WIA in aortic pressure and velocity measurements indicated, due to the timing of the waves, that aortic flow deceleration is mediated by a net expansion wave that travels from the aorta towards the periphery [11]. A decade later it was proposed that the timing and magnitude of the compression wave arriving in the aorta during mid-systole could be used to study wave reflections, since the WIA results were in good agreement with the traditionally used augmentation index in the aortic pressure waveform [7]. However, the influence of reflected waves from the periphery on the aortic pressure contour is challenged by Wang et al. who propose that when the aortic Windkessel is taken into account, the backward-traveling reflected waves are minimal [14]. In order to explain the deterioration of left ventricular (LV) function during clamping of the abdominal aorta in patients, Khir et al. studied the effect of clamping on the reflected waves arriving in the aorta from downstream [6]. They found that during clamping the reflected waves arrived earlier, had longer duration and carried more energy, increasing in that way the afterload to

134

the heart and the work of the LV and possibly triggering myocardial ischemia [6]. The effect of changes in cardiac contraction and systemic vasomotor tone on the pattern of the waves generated by the LV during aortic flow acceleration and deceleration was also studied with aortic WIA [5]. It was found that inotropic changes affect only the net forward compression wave, while changes in vascular tone affect both the net forward compression and expansion wave, but in a different way: compressive WI is reduced by vasoconstriction whereas expansive WI is reduced during vasodilation [5].

When applied in left ventricular pressure and velocity measurements, WIA showed that left ventricular diastolic suction (the property of the ventricle to tend to refill itself during diastole) can be attributed to the presence of an expansion wave that travels from the apex of the ventricle towards the left atrium after the opening of the mitral valve [15]. The energy of this wave is inversely related to end-systolic LV volume and to the rate of decrease of LV pressure [15]. The same conclusions were drawn for the right ventricle for the expansion wave that travels towards the right atrium after the opening of the tricuspid valve [13]. A modification of ‘classical’ WIA was recently proposed in order to compensate for the time-varying elastic properties of the LV during diastole. According to this modification it is a compression wave that travels from the left atrium towards the apex of the LV rather than an expansion wave traveling in the opposite direction that appears after the opening of the mitral valve [8]. When the left-heart reservoir (Windkessel) function is taken into account before WIA application, the wave energy associated with left ventricular diastolic filling is more than twice the energy calculated omitting the reservoir effect [1].

Wave Intensity Analysis has also been applied in the pulmonary circulation. Due to the considerable increase in the total cross-sectional area of the vasculature distal to the pulmonary trunk, negative wave reflections were recorded in the proximal pulmonary artery during systole [3]. These reflected waves, that are of the expansion type, augment right ventricular ejection [3]. Pulmonary artery WIA also helped reveal that waves from the left atrium are not transmitted to the proximal pulmonary artery only via the vasculature by retrograde propagation, but also directly via the heart wall [2]. Wave Intensity Analysis assessed non-invasively in the carotid arteries showed that the net compression and the expansion wave recorded to travel from the carotid arteries towards the brain in early systole and near end-ejection respectively, may be used for the non-invasive assessment of systolic and early-diastolic LV function [9].

Finally, WIA applied in various peripheral arteries illustrated how the wave pattern changes from one peripheral location to another [16].

References

[1] FLEWITT JA, HOBSON TN, WANG J, JR., JOHNSTON CR, SHRIVE NG, BELENKIE I, PARKER KH and TYBERG

JV, Wave intensity analysis of left ventricular filling: Application of windkessel theory. Am J

Physiol Heart Circ Physiol 292:H2817-2823, 2007

[2] HOLLANDER EH, DOBSON GM, WANG J, JR., PARKER KH and TYBERG JV, Direct and series transmission of left atrial pressure perturbations to the pulmonary artery: A study using wave-intensity analysis. Am J Physiol Heart Circ Physiol 286:H267-275, 2004

[3] HOLLANDER EH, WANG J, JR., DOBSON GM, PARKER KH and TYBERG JV, Negative wave reflections in pulmonary arteries. Am J Physiol Heart Circ Physiol 281:H895-902, 2001

9

[4] JONES CJH, SUGAWARA M, DAVIES RH, KONDOH Y, UCHIDA K and PARKER KH, Arterial wave intensity: Physical meaning and physiological significance, In Recent progress in

cardiovascular mechanics, pp.129-148. HOSODA H, YAGINUMA T, SUGAWARA M, TAYLOR MG and CARO CG, Editors. Harwood Academic, Chur, 1994

[5] JONESCJH, SUGAWARA M, KONDOHY, UCHIDA K and PARKER KH, Compression and expansion wavefront travel in canine ascending aortic flow: Wave intensity analysis. Heart Vessels

16:2002

[6] KHIR AW, HENEIN MY, KOH T, DAS SK, PARKER KH and GIBSON DG, Arterial waves in humans during peripheral vascular surgery. Clin Sci 101:749-757, 2001

[7] KOHTW, PEPPER JR, DESOUZA AC and PARKER KH, Analysis of wave reflections in the arterial system using wave intensity: A novel method for predicting the timing and amplitude of reflected waves. Heart Vessels 13:103-113, 1998

[8] LANOYE LL, VIERENDEELS JA, SEGERS P and VERDONCK PR, Wave intensity analysis of left ventricular filling. J Biomech Eng 127:862-867, 2005

[9] OHTE N, NARITA H, SUGAWARA M, NIKI K, OKADA T, HARADA A, HAYANO J and KIMURA G, Clinical usefulness of carotid atrerial wave intensity in assessing left ventricular systolic and early diastolic performance. Heart Vessels 18:107-111, 2003

[10] PARKER KH and JONESCJH, Forward and backward running waves in the arteries: Analysis using the method of characteristics. J Biomech Eng 112:322-326, 1990

[11] PARKER KH, JONES CJH, DAWSON JR and GIBSON DG, What stops the flow of blood from the heart? Heart Vessels 4:241-245, 1988

[12] PAYNTER HM and J. B-VI, Remarks on Riemann's method of characteristics. J Acoust Soc Am

84:813-822, 1988

[13] SUN Y, BELENKIE I, WANG J, JR. and TYBERG JV, Assessment of right ventricular diastolic suction in dogs using wave-intensity analysis. Am J Physiol Heart Circ Physiol 00853.02005, 2006

[14] WANG J, JR., O'BRIEN AB, SHRIVE NG, PARKER KH and TYBERG JV, Time-domain representation of ventricular-arterial coupling as a windkessel and wave system. Am J Physiol Heart Circ

Physiol 284:H1358-1368, 2003

[15] WANG Z, JALALI F, SUNY-H, WANG J, JR., PARKER KH and TYBERG JV, Assessment of left ventricular diastolic suction in dogs using wave-intensity analysis. Am J Physiol Heart Circ Physiol

288:H1641-1651, 2005

[16] ZAMBANINI A, CUNNINGHAM SL, PARKER KH, KHIR AW, MCG. THOM SA and HUGHES AD, Wave-energy patterns in carotid, brachial, and radial arteries: A noninvasive approach using wave-intensity analysis. Am J Physiol Heart Circ Physiol 289:H270-276, 2005