Physical Determinants of Diastolic Flow




INTRODUCTION


Comprehensive assessment of ventricular diastolic function is a complex process. Full elucidation generally requires invasive measurements, such as left ventricular (LV) end diastolic pressure, the time constant of isovolumic LV relaxation (τ), the pressure-volume (P-V) relationship of the ventricle at end diastole, and mean left atrial (LA) pressure. Such invasive measurements are inappropriate for routine clinical purposes, and thus diastolic function is generally assessed using Doppler echocardiography, largely through the observation of transmitral and pulmonary venous flow, supplemented by myocardial velocity and color M-mode Doppler information. In order to intelligently use these noninvasive indices to infer actual diastolic function of the heart, however, it is critical that a conceptual framework be in place that reflects the physical and physiological determinants of intracardiac blood flow. In this chapter, we will outline in both basic physical principles and computer simulations the relationship between basic parameters of diastolic function and the intracardiac flow patterns that can be obtained clinically.




PATHOPHYSIOLOGY


Mechanical Properties of Left Ventricular Chamber Diastole


The major function of the heart in diastole is to let the blood column flow from the antechambers (left atrium and pulmonary veins) into the left ventricle, while keeping filling pressures to a minimum. During exercise, this task is compounded by the shortening of time allowed for filling and the increase of volume needed to get into the left ventricle. Of the various chamber-wall properties that affect this process, we will briefly cover three: chamber stiffness, relaxation, and early diastolic suction.


Chamber Stiffness


Chamber stiffness can be defined as the instantaneous change of pressure for a given volume increment, mathematically the first derivative of LV pressure by volume ( dP / dV ). LV pressure is a complex nonlinear function of LV volume, meaning that stiffness changes with diastolic volume. Although the LV P-V relationship was initially considered to be exponential in shape (concave upward), more recent work suggests that it is sigmoidal in shape ( Fig. 5-1 ), with a concave downward portion to the left of the usual rising exponential. This sigmoidal curve can be expressed as:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-1-Frame class=MathJax style="POSITION: relative" data-mathml='P=A*e(V-Vd0)Kp++Pb+V≥Vd0P=-B*e(Vd0-V)Kp-+Pb-V&lt;Vd0,’>P=A*e(V−Vd0)Kp++P+bV≥Vd0P=−B*e(Vd0−V)Kp−+Pb−V<Vd0,P=A*e(V-Vd0)Kp++Pb+V≥Vd0P=-B*e(Vd0-V)Kp-+Pb-V<Vd0,
P = A * e ( V – V d 0 ) K p + + P b + V ≥ V d 0 P = – B * e ( V d 0 – V ) K p – + P b – V < V d 0 ,
where V d 0 is the inflection point, A and B are the exponential curve multipliers of the upper and lower part, P b + and P b − are pressure offsets of the upper and lower part, while K p + and K p − are the diastolic chamber stiffness indices, determining the overall steepness of the end diastolic P-V relationship (EDPVR) above and below the inflection volume. Frequently, EDPVR is conceptually simplified by assuming that V d 0 equals 0, and even further simplified by neglecting P b + , leaving us with the equation P = Ae K×EDV , or in other words ln P = a + K × EDV (see also Chapter 2 ). By differentiating this equation, we see that end diastolic stiffness dP / dV is K × Ae K×EDV or K × P EDV . Another parameter of note is the average stiffness, that is, the stroke volume divided by the LV pressure increment during diastole, a simple overall measure of how stiff the patient’s LV is during diastole over a given (working) set of conditions. The fact that myocardial relaxation is a continuous process throughout diastole results in changing relationships between pressure and volume from very stiff end systole P-V relationships to much more compliant EDPVR.


Figure 5-1


Passive pressure-volume (P-V) loop relationships. The thick continuous and thick dotted lines represent the upper and lower part of the sigmoid end diastolic pressure-volume relationship, respectively. The curvature of the upper part is determined by the chamber stiffness index, which can be interpreted as the inverse of the volume needed to increase the steepness of the P-V curve by a factor of 2.72.


Relaxation


Myocardial relaxation occurs because of calcium reuptake at the end of systole, producing a shift downward and rightward, leading to a fall in LV pressure (at a given volume). The rate of pressure decrease during relaxation depends on the velocity of calcium reuptake and on the LV volume: The smaller the volume, the lower the potential for pressure to fall, limited by the end diastolic pressure curve shown in Figure 5-1 . Rate of calcium reuptake is further modified by LV lengthening during relaxation, so true relaxation can be measured only during the isovolumic period. Several equations have been proposed to describe the rate of isovolumic pressure decay, the most general being:


<SPAN role=presentation tabIndex=0 id=MathJax-Element-2-Frame class=MathJax style="POSITION: relative" data-mathml='P(t)=(Po-Pb)×e-t/τ+Pb,’>P(t)=(Po−Pb)×e−t/τ+Pb,P(t)=(Po-Pb)×e-t/τ+Pb,
P ( t ) = ( P o – P b ) × e – t / τ + P b ,
which can be, with some caveats, simplified to:
<SPAN role=presentation tabIndex=0 id=MathJax-Element-3-Frame class=MathJax style="POSITION: relative" data-mathml='P(t)=Po×e-t/τ,’>P(t)=Po×e−t/τ,P(t)=Po×e-t/τ,
P ( t ) = P o × e – t / τ ,
and further to :
<SPAN role=presentation tabIndex=0 id=MathJax-Element-4-Frame class=MathJax style="POSITION: relative" data-mathml='τ≈IVRT/(lnPAVC-lnPMVO),’>τ≈IVRT/(lnPAVC−lnPMVO),τ≈IVRT/(lnPAVC-lnPMVO),
τ ≈ IVRT / ( ln P A V C – ln P MVO ) ,
where IVRT is the isovolumic relaxation time (the time between aortic valve closure and mitral valve opening), and P AVC and P MVO are LV pressure at aortic valve closure (AVC) and mitral valve opening (MVO), respectively, which can be approximated noninvasively by systolic blood pressure and an estimate of LA pressure.


Importantly, LV relaxation is a never-ending process that critically affects the end diastolic pressure that the patient achieves, particularly during exercise. Consider Figure 5-2A , showing a series of P-V curves representing successive intervals of τ, and Figure 5-2B , the same curves magnified on diastolic pressures. Note that after 6τ, the filling curve is virtually indistinguishable from the end diastolic P-V curve, as relaxation is 99.8% complete. When relaxation rate is normal, the end diastolic filling curve is completely relaxed at normal heart rates (12.5τ, 2C) and almost complete during exercise (5τ, 2D); with delayed relaxation, however, while the EDPV curve is fully relaxed at rest (6.25τ, 2E), it becomes stiff with exercise (3τ, 2F).




Figure 5-2


Diastolic pressure-volume (P-V) relationships shown as a function of the duration of diastole. Duration of diastole is measured by the multiples of a time constant of relaxation (τ). Panel A shows a series of P-V relationships that occur after successive intervals of τ, while panel B shows the same curves zoomed on diastolic pressures. The combination of relaxation and the end diastolic P-V (EDPV) relationship results in a dynamic diastolic part of the P-V loop represented by the round dots in A. Note that in early diastole, left ventricular pressure continues to fall even though volume is increasing, meaning that the instantaneous stiffness is actually negative, which some consider diastolic suction. When relaxation rate is normal, the end diastolic filling curve is completely relaxed at normal heart rates ( arrow 12.5τ, C) and almost complete during exercise ( arrow 5τ, D); with delayed relaxation, however, while the EDPV curve is fully relaxed at rest ( arrow 6.25τ, E), it becomes stiff with exercise ( arrow 3τ, F).


The combination of relaxation and the EDPVR results in a dynamic P-V relationship, represented by the round dots in Figure 5-2A . Note that in early diastole, LV pressure continues to fall even though volume is increasing, meaning that the instantaneous stiffness is actually negative, which some consider diastolic suction. Two other definitions of “diastolic suction” are also relevant. Consider ventricular filling from a very low end systolic volume (where the EDPVR is concave downward). If relaxation is very rapid or filling is delayed (by mitral stenosis or experimentally with a mitral occluder ), then LV pressure can fall below atmospheric pressure. Alternatively, suction has been used to refer to the small (1 to 3 mmHg) differences in pressure between the base and the apex, which assist in the low-pressure filling of the ventricle, particularly with exercise, and which will be discussed later.


Determinants of Intracardiac Blood Flow


In the most general sense, the motion of blood inside the heart, as the motion of any fluid, is determined by the Navier-Stokes equations, a complex set of four multidimensional partial differential equations, which must be solved simultaneously at every point in space and moment in time:


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∇ ⋅ v = 0
and
<SPAN role=presentation tabIndex=0 id=MathJax-Element-6-Frame class=MathJax style="POSITION: relative" data-mathml='ρDvDt=-∇P+B+μ∇2v’>ρDvDt=−∇P+B+μ∇2vρDvDt=-∇P+B+μ∇2v
ρ Dv D t = – ∇ P + B + μ ∇ 2 v
These equations appear deceptively simple but contain such complex mathematical concepts that except in the simplest of geometries, they can never be solved either analytically or with powerful supercomputers. Fortunately, considerable simplifications can be made to these equations that will facilitate a conceptual and computational approach. The most important simplification is to take the distribution of blood throughout the heart and replace it with just a few measurements at specific points inside the heart. For instance, instead of describing pressure in every cubic millimeter inside the left ventricle, we assume that LV pressure can be approximated by an average of these and give a single number for LV pressure, which is precisely how we measure and report LV pressure in practice. Similarly, instead of describing the direction and speed of blood flow at every point within the heart, we focus on points where blood velocity is maximum, such as the tips of the mitral leaflets and the pulmonary vein orifices. Finally we assume fluid incompressibility and zero viscosity and heat conduction losses. In this way, the thousands of partial differential equations that would have to be solved simultaneously throughout the heart are replaced by a few ordinary differential equations that can be solved quite easily on a personal computer.


To understand how these equations can model the flow within the heart, we first start by replacing the Navier-Stokes equations with the well-known Bernoulli equation, which applies to flow across discrete points such as valves. Here we present it with its inertial and convective terms (we still omit the viscous term, since it is negligible in almost every intracardiac situation):


<SPAN role=presentation tabIndex=0 id=MathJax-Element-7-Frame class=MathJax style="POSITION: relative" data-mathml='Δp=Mdvdt+12ρΔ(v2),’>Δp=Mdvdt+12ρΔ(v2),Δp=Mdvdt+12ρΔ(v2),
Δ p = M d v d t + 1 2 ρ Δ ( v 2 ) ,

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Mar 23, 2019 | Posted by in CARDIOLOGY | Comments Off on Physical Determinants of Diastolic Flow

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