Physical Determinants of Diastolic Flow

Comprehensive assessment of ventricular diastolic function is a complex process. Full elucidation generally requires invasive measurements, such as left ventricular end-diastolic pressure (EDP), the time constant of isovolumic left ventricular (LV) relaxation (τ), the pressure-volume (P-V) relationship of the ventricle at end diastole, and mean left atrial 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. 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 physiologic determinants of intracardiac blood flow. In this chapter we will attempt to 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 During Diastole

The major function of the heart in diastole is to let the blood column flow from the antechambers (left atrium [LA] and pulmonary veins [PV]) into LV, while keeping filling pressures to a minimum. During exercise, this task is compounded by shortening of time allowed for filling and increase of volume needed to get into the LV. Out of 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:

P = A * e^{(V - V_{d 0}) K p +} + P_{b}^{+} V \geq V_{d 0}

$P = A * e^{(V - V_{d 0}) K p +} + P_{b}^{+} V \geq V_{d 0}$

P = - B * e^{(V_{d 0} - V) 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 LnP = 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 (i.e., the stroke volume divided by the LV pressure increment during diastole), a simple measure that gives the clinician an 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.

Relaxation

Myocardial relaxation occurs due to 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 EDP curve shown in Fig. 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:

P (t) = (P_{o} - P_{b}) \times e^{- t / τ} + P_{b}

$P (t) = (P_{o} - P_{b}) \times e^{- t / τ} + P_{b}$

which can be, with some caveats, simplified to:

P (t) = P_{o} \times e^{- t / τ}

$P (t) = P_{o} \times e^{- t / τ}$

and further to :

τ \approx IVRT / (ln P_{AVC} - ln P_{MVO}),

$τ \approx IVRT / (ln P_{AVC} - ln P_{MVO}),$

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 and mitral valve (MV) opening, respectively, which can be approximated noninvasively by systolic blood pressure and an estimate of left atrial pressure.

Fig. 5.1, The diagram of passive P-V loop relationships.

Importantly, LV relaxation is a never-ending process, which critically impacts the end-diastolic pressure that the patient achieves, particularly during exercise. Consider Fig. 5.2 A, showing a series of PV curves, representing successive intervals of τ, and Fig. 5.2 B, the same curves zoomed on diastolic pressures. Note that after 6τ, the filling curve is virtually indistinguishable from the end-diastolic PV 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) (see Fig. 5.2 C) and almost complete during exercise (5τ) (see Fig. 5.2 D); with delayed relaxation, however, while the EDPV curve is fully relaxed at rest (6.25τ) (see Fig. 5.2 E), it becomes stiff with exercise (3τ) (see Fig. 5.2 F).

Fig. 5.2, Diastolic P-V relationships shown as a function of the duration of diastole.

The combination of relaxation and the EDPVR results in a dynamic PV relationship, represented by the round dots in Fig. 5.2 A. 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 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 (1–3 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 equation, a complex set of four multidimensional partial differential equations, which must be solved simultaneously at every point in space and moment in time:

\nabla \cdot v = 0

$\nabla \cdot v = 0$

and:

ρ \frac{D v}{D t} = - \nabla P + B + μ \nabla^{2} v

$ρ \frac{D v}{D t} = - \nabla P + B + μ \nabla^{2} v$

These equations appear deceptively simple but contain such complex mathematical concepts that, except in the simplest of geometries, can never be solved either analytically or with powerful supercomputers. Fortunately, considerable simplifications can be made to these equations that will allow a much simpler 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 left ventricular 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 left ventricular 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 that 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):

Δ p = M \frac{d v}{d t} + \frac{1}{2} ρ Δ (v^{2})

$Δ p = M \frac{d v}{d t} + \frac{1}{2} ρ Δ (v^{2})$

where Δ p is the pressure difference between two points, M is the inertial constant and represents the so-called effective mass being accelerated, dv / dt is the instantaneous temporal acceleration of flow through the region, ρ is blood density, and Δ( v ) is the change in the square of velocity from one point to another (with pressure in mmHg and velocity in m/sec, ½ρ v ² reduces to 4 v ² in the simplified Bernoulli equation). The first product is the inertial term of the Bernoulli equation, and corresponds to the energy used accelerating and decelerating flow; the second term is the convective term and represents kinetic energy of flow. The effective mass of blood in a narrow orifice is small and thus M is a negligible quantity in, for example, valve stenosis, but the dominant term when flow is not obstructed, such as the normal mitral valve. For constant flow through an orifice of minimal diameter D , the inertial constant M varies approximately in proportion to D , while the kinetic term ½ρ v ² varies inversely to D ⁴ .

For nonobstructive flow through the body of heart chambers, where pressure changes gradually over a distance, not at a discrete point, we must use the Euler equation, a differential version of the Bernoulli equation for pressure change along a streamline of flow:

\frac{\partial p}{\partial s} = - ρ \cdot (\frac{\partial v}{\partial t} + v \cdot \frac{\partial v}{\partial s})

$\frac{\partial p}{\partial s} = - ρ \cdot (\frac{\partial v}{\partial t} + v \cdot \frac{\partial v}{\partial s})$

where the inertial and convective terms are in the same order as the Bernoulli equation, but the discrete terms [Δ p and Δ( v )] have been replaced by their spatial derivatives, yielding the rate of pressure change per centimeter of distance along the streamline. To return the total pressure drop between points A and B along the streamline, this equation must be integrated numerically:

Δ p = - ρ \cdot \int_{A}^{B} (\frac{\partial v}{\partial t} + v \cdot \frac{\partial v}{\partial s}) d s .

$Δ p = - ρ \cdot \int_{A}^{B} (\frac{\partial v}{\partial t} + v \cdot \frac{\partial v}{\partial s}) d s .$

Fig. 5.3 shows schematically how these partial derivatives are summed together to produce the pressure gradient map. Another, practical, fact is that we can obtain the same map by applying the Euler equation to a color Doppler M mode tracing.

Fig. 5.3, Illustration of the way convective (right) and inertial (left) terms combine to produce intraventricular pressure gradient.

Clinical Relevance

In this part of the chapter we will demonstrate how interaction of solid and fluid mechanics determines intracardiac flow within three specific regions of the heart: mitral valve, pulmonary vein, and the left ventricle.

Physical Factors Governing LV Filling Velocities

To describe flow mathematically through the mitral valve, we first consider a very simplified construct, consisting of the left ventricle, which has pressure as a function of time p _V ( t ), the left atrium, which has a pressure as a function of time [ p _A ( t )], and the mitral valve, which has an area A _MV and contains a mass of blood M that is accelerating in passing from the left atrium to the left ventricle, which, to a first approximation, is the blood within a cylinder that can fit inside the narrowest portion of the mitral leaflet tips and whose length is approximately the diameter of the valve. In reality though, since pressure is applied across the mitral valve, the most relevant hydrodynamic concept is the length of this cylinder, termed the mitral inertance, related linearly to the diameter of that structure. We will use this construct to discuss three parameters of transmitral flow: mitral acceleration, peak mitral flow, and mitral deceleration.

Acceleration of Mitral Flow

To understand the acceleration of blood across the mitral valve, we apply Newton’s second law of motion that the rate at which an object accelerates is given by the force exerted on that object divided by its mass. In this case, the acceleration would be recorded by Doppler echocardiography as the velocity acceleration detected for mitral inflow, while the force is the difference in pressure on either side of the mitral valve multiplied by the area of the valve :

a = \frac{F}{m} = \frac{Δ p A}{ρ M} \approx \frac{Δ p}{ρ L}

$a = \frac{F}{m} = \frac{Δ p A}{ρ M} \approx \frac{Δ p}{ρ L}$

Here we have taken advantage of the fact that valve area appears in both the numerator and the denominator (through the definition of inertial mass) to simplify the relationship. Clearly the higher the pressure difference across the valve and the smaller the length of blood column within the mitral valve, the more rapidly the velocity will accelerate. This is why mitral velocity increases almost instantaneously in mitral stenosis, which has a very high-pressure gradient across the valve and a very small blood mass due to the small diameter of the mitral orifice. If the blood within the mitral valve was subject to an instantaneously applied pressure difference (as if the relaxation of the LV occurred suddenly), mitral velocity would start to rise linearly. In the physiologic situation, the pressure gradient is not abruptly applied but rather increases gradually (roughly linearly) with time as the ventricle relaxes, resulting in a roughly parabolic mitral velocity acceleration curve.

Peak E Wave Velocity and Conservation of Energy

Mitral valve acceleration eventually decreases with time because the maximal velocity generated by a pressure difference is limited by conservation of energy (expressed in the Bernoulli equation). Within the heart, energy in the blood takes on three principal forms: pressure (a form of potential energy), kinetic energy, and heat; the total energy in the system must remain constant. In the left atrium, where blood velocity is low, most of the energy is in the form of pressure, but as it moves toward the mitral valve, its velocity rises and it acquires a kinetic energy (½ρ v ² ), which causes the local blood pressure to fall. If no energy losses appear, pressure difference and velocity are related by the simplified Bernoulli relation, which becomes roughly Δ p = 4 v ² when pressure is measured in mmHg and velocity in meters per second. While the velocity reaches asymptotically the target gradient, reaching of this asymptote is delayed by the amount of inertance present. Reaching the asymptote can be even further delayed by gradient not being imposed abruptly, but slowly, as seen in a normally relaxing LV.

Deceleration of Mitral Flow

Returning to our general model of the mitral valve, once flow has been accelerated to maximal velocity by the pressure difference across the mitral valve, it begins falling as the pressure difference between the left atrium and left ventricle equilibrates. This is analogous to flow between two tanks stopping when the level of water in the two tanks becomes the same. When we speak of the change in pressure with a change in volume, we are dealing with the concept of compliance or its inverse, stiffness, the change in pressure for a given change in volume, which is a major determinant of the deceleration of flow across the mitral valve. Since the stiffer the ventricle, the more rapidly the pressure equilibrates across the mitral valve to decelerate the flow, we can understand why short deceleration time across the mitral valve is associated with increased ventricular stiffness. Note that in this situation, we must place atrial and ventricular stiffness together, since both chambers are involved in determining how quickly pressure equilibrates across the mitral valve. For ventricular stiffness S _V and atrial stiffness S _A , the net atrioventricular stiffness S _n is simply S _A + S _V . To obtain a mathematical expression for mitral velocity deceleration rate, we first note that for a restrictive mitral valve with area A (i.e., mitral stenosis, though even for a normal mitral valve, the overall principle holds):

Δ p = ½ ρ v^{2}

$Δ p = ½ ρ v^{2}$

which simplifies to 4 v ² when Δ p is in mmHg and v in m/sec. Differentiating this yields

d Δ p / d t = ρ v d v / d t

$d Δ p / d t = ρ v d v / d t$

But d Δ p / dt can also be expressed in terms of flow across the mitral valve ( A v) and net A-V stiffness:

d Δ p / d t = - A v S_{n}

$d Δ p / d t = - A v S_{n}$

Substituting for d Δ p / dt yields

ρ v d v / d t = - A v S_{n}

$ρ v d v / d t = - A v S_{n}$

or, rearranging,

d v / d t = A S_{n} / ρ

$d v / d t = A S_{n} / ρ$

Thus the stiffer the ventricle (or atrium) and the larger the valve area, the faster the deceleration.

Alternatively, one can represent this as a purely inertial system (with a nonrestrictive orifice), which yields a simple harmonic motion in which case the velocity across the valve is described roughly by a sine wave:

v (t) = v_{0} \sin (\frac{t}{\sqrt{m / k}})

$v (t) = v_{0} \sin (\frac{t}{\sqrt{m / k}})$

where v ₀ is the peak E wave velocity, m is mitral inertance, and k is the stiffness constant of the ventricle. The deceleration time (time for the E wave to fall from its peak to zero velocity) can be solved for as:

t_{d e c} = \frac{π}{2} \sqrt{\frac{ρ \cdot L}{A} \cdot \frac{1}{k}}

$t_{d e c} = \frac{π}{2} \sqrt{\frac{ρ \cdot L}{A} \cdot \frac{1}{k}}$

where ρ, L , and A are blood viscosity, mitral valve length, and mitral valve area, respectively. To sum it up, the stiffer the ventricle, the shorter the deceleration time. This is intuitively very understandable: Tighter springs oscillate fast. Note that in this simplification, one assumes that active relaxation is complete at the time of interest.

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