The fluid, the flow

Simplifying the situation a bit, the flow of blood (and of any fluid really) can be considered a function of the ratio of the pressure gradient to the systems resistance to that flow. This can be written quasi-mathematically by saying F is proportional to ∆P/R, where F is the flow, ∆P is the pressure gradient, and R is the resistance. Defining it in such a way allows us to grasp the fundamental basis of bulk flow: there are forces attempting to drive a fluid in one direction and there are forces attempting to prevent the fluid from flowing. Even this basic conceptual framework is enough to understand many significant portions of the circulatory system.

The entire cardiovascular pressure gradient starts and ends at the heart. Here pressures reach their peak and their valley (averaged over a full cardiac cycle). Just exiting the heart, the aorta and the elastic arteries experience very large pressures and very large pressure changes (see the difference in systolic and diastolic curves in Figure 1.3). Pressure in the arteries, often referred to simply as blood pressure, varies tremendously (see Figure 1.6), and as such must be taken from a consistent location to make sense of any time related trends. Passing through the arterioles into the capillaries, the pressure known as the capillary hydrostatic pressure continues to drop in magnitude on its way to the venous system where the cardiovascular reaches its minimum values venous pressure and, ultimately, right atrial pressure. Though in the course of normal cardiovascular function pressures will also decrease to push blood ever onward from (and back to) the heart, we must remember that it is the pressure gradients that affect flows, and not pressures themselves. 

A pressure gradient is merely the potential for fluid to flow, the resistance to that potential will determine in what manner and to what degree that fluid flows. In the human body, many factors affect the resistance to flow. Chief among these factors are those affecting vessel geometry (length, diameter, curvature, tortuosity, etc.) and blood viscosity (itself affected by temperature, hematocrit level, etc.). To see in what ways each of these parameters affects fluid flows, let us turn our attention to basic fluid modeling.

To model a fluid fully one may begin with the Navier-Stokes equations. Their derivation and proof will not be considered here and their role here is merely as a tool to explain the motion of blood in a few very specific contexts. Given this environment, we will begin by expressing the equations in cylindrical coordinates in full.

Here ρ is the density of the fluid, r represents those components in the radial direction, θ represents those components in the angular orientation, z represents those components in the longitudinal direction, r is the radius of the cylinder, v is the velocity component, g is the gravitational component, p is the pressure component, and μ is the viscosity of fluid. Though the equations may look unwieldy, they merely equate the stress in the fluid as the sum of viscous terms and pressure gradients. Meaning that a pressure difference will attempt to move a fluid and the fluid will attempt to resist this change. This is precisely how we have understood fluid flow to this point and we would do well to remember it going forward. Our use of equations will help us to systemize our reasoning.

To begin with the simplest case, let us imagine an arbitrary blood vessel in the body as a long, mostly straight, mostly cylindrical channel filled with blood. In such an environment, we can assume negligible velocities in the radial and angular directions, that the velocity in the longitudinal direction is a function of the radius, and that the effects of gravity are negligible. Furthermore let us assume that the bloods viscosity is constant and the flow is steady and incompressible. Substituting vr = vθ = 0, we can rewrite equation 1.2 as

Rearranging, we can integrate and integrate again to obtain a unbounded solution for pressure driven cylindrical flow.

For an exact solution we must set boundary conditions. We can apply a no-slip condition at the vessel wall (vz(R) = 0) and no-shear condition at the center line (true enough for estimation) to solve for the two unknown constants. Upon doing so we obtain a general velocity profile for this particular kind of pressure driven flow within a cylinder:

From this we can easily derive the famous Hagen-Poiseulle equation by integrating the velocity profile described in 1.6 over the cross-sectional area of the vessel to obtain volumetric flow rate, Q.

As the partial derivative in space approaches the length of the vessel to be measure (that is, as ∂z → L), the pressure gradient can be replaced by the ratio of the total pressure difference to the length of the vessel of interest, leaving us with a final formulation of

From this cursory examination we can see that in this sort of context, we can identify three main components of resistance: the viscosity of the fluid, the length of the vessel, and the radius of the vessel. Resistance is directly proportional to the viscosity of blood and the length of the path it travels. Though there are many ways in which these two factors can change, on a day-to-day basis, the length of vessels and the bloods viscosity can be treated as constants. The third component, the radius of the vessel, has profound ramifications for the fluid flowing within it magnifying their effects by the power of four. Intuitively, this makes sense. The majority of frictional losses in a fluid occur along the boundary layer where the effects of viscosity are at their greatest. Decreasing the radius increases the ratio of this boundary layer to the rest of the cross sectional area, subjecting the blood to more viscous forces at the expense of inertial forces.

The ratio of inertial (or momentum) forces to viscous forces was long ago popularized by Osborne Reynolds [8]. Though Reynolds was not the first to formulate this ratio or stress its importance (that honor should lie with George Stokes [9]), to this day the ratio bears his name. The Reynolds number is defined as

where ρ is again is the density of the fluid, v is the velocity of the fluid, D is the characteristic linear dimension (within a pipe, the hydraulic diameter is used), and μ is the viscosity of the fluid. For fluids of relatively constant densities and viscosities, only the velocity and the characteristic dimension affect the Reynolds number, indicating that either parameters increase aids in overcoming viscous losses. Throughout the body, the value of Re varies tremendously, from about 1 in small arterioles (where velocities are low and diameters are small) to about 4000 in the aorta (the largest vessel with the fastest velocities), indicating a wide variety of environments in which blood flows [10].

Another key characteristic affecting blood flow throughout the body occurs mere moments, mere centimeters, after blood is ejected from the left ventricle into the system circuit: the curvature of vessels. Though the approximation of vessels as long straight tubes serves a useful purpose for many applications, twists and turns both large and small exist in plenitude, and we must account for them.

When evaluating flow within curved vessels, we can turn to the work of William Reginald Dean who developed a parameter relating centrifugal forces to viscous forces, by combining the Reynolds number, Re, with a ratio of the inner tube radius, r, to the radius of curvature along the centerline of the vessel, R:

Fully developed flow through curved tubes tends to exhibit a net velocity that skews toward the outer wall of the bend. This is true for most arterial flow. From a few basic conservation laws (mass, energy, momentum), secondary flow will develop as a pair of counter-rotating vortices with flow in the middle of the vessel moving toward the outer wall. Flows with higher Dean number can separate along the inner wall of the curve. The presence of curvature in blood vessels ensures that forces from the flow (from the shear stress of blood along the wall or inertial forces from moment in the radial component of the vessel) will not be equal. The unequal distribution of forces along vessel walls requires the vasculature to compensate, often by thickening the lumen or increasing the inner diameter of the vessel. These effects and their role in shaping vessels (especially in the context of maturing physician created arteriovenous fistulas) can be discussed at length. You are spared here.

Further variables such as the temperature of the environment, the non-Newtonian viscosity of blood, elastic wall boundaries capable of passive and active contraction and dilation, and secondary body forces, coupled with the already complex nature of continuum mechanics, all conspire to make biological fluid flow an incredibly hard problem to fully solve. However, each of these additional variables effects on flow for our present study can be considered negligible compared to that of time-variance in fluid flow.

Unsteady, pulsatile flow through nearly all of the cardiovasculature dominates many aspects of the system. From the varying pressure gradients directing blood flow through the heart and the constantly loaded and unloaded stresses applied to the walls of the arteries to the frequency component of cardiac output and the time given to fill the ventricles, the pulse of the system affects it profoundly. These effects can be mechanically transduced as by the endothelial cells of the vessel wall or physiologically manifest as with the filling time. The time variance of blood flows cannot be ignored.

To model such flows fully with the Navier-Stokes equations would be difficult and nearly impossible in all but a few cases (with still fewer interesting us). Long ago, John R. Womersley developed a nondimensional parameter analyzing unsteady Navier-Stokes equations with specific reference to biological flows to determine the ratio of unsteady forces to viscous forces [11], simplifying our work and work of many others. This Womersley number, α, is found by dividing the viscous forces into the transient inertial forces to arrive at

where r is the inner radius of the tube, ω is the angular frequency of the oscillating flow, ρ is the density of the fluid, and μ is the dynamic viscosity. Since the density and viscosity of blood remain relatively constant over time, we can see that the remaining two factor influencing the Womersley number around the body as the size of the vessels and the rate of the heartbeat. When the Womersley number is low, viscous forces dominate, making velocity profiles parabolic and with the centerline velocity oscillating in phase with the driving pressure gradient. Larger Womersley numbers (>10), unsteady inertial forces dominate. The amplitude of motion due to the oscillations decreases with increasing Womersley number and the phase difference between the pressure gradient and flow grows. In this way, oscillating flows act much like low pass filters (consult [11] for a clear example of this).

A small table of typical flows, Reynolds numbers, and Womersley numbers throughout various major vessels of the body can be seen in Table 1.1.

 

Table 1.1. Diameter of blood vessels, the velocity of blood flowing within them, and their corresponding Reynolds and Womersley numbers. From this the wide range of diameters, velocities, Reynolds and Womersley numbers can be appreciated. All values were calculated assuming a heart beat of 75 beats per minute, a blood density of 1050 kg/m3 (at body temperature), and a blood viscosity of 4 centipoise. Within the body, each of these parameters exist within a range, but for an order of magnitude assessment of the situation, this table should suffice.

 

One final significant factor influencing blood flow is the Windkessel effect, which describes the interaction of the propagating stroke volume and the compliant vessels through which it travels. The introduction of the bolus of fluid into the vasculature causes the pressure to rise during systole (recall Figure 1.3), forcing blood through the vessel (a longitudinal pressure gradient), but also applying an outward pressure to the walls (a radial pressure gradient). The vasculature being composed of compliant vessels allows for a volume expansion in response to this increased radial pressure, helping to compensate for the introduction of the stroke volume.3 The elastic vessels (mainly the aorta and large arteries) act as capacitors, storing blood during systole and discharging during diastole. In this way, compliant arteries dampen fluctuations in blood pressure and assist with consis- tent tissue and organ perfusion throughout the cardiac cycle. As one ages, elastic vessels can become less compliant (hardening or becoming laden with atherosclerotic plaques), increasing pulse pressure, leading to hypertension, itself a precursor to and symptom of many cardiovascular disease (such as heart attack and stroke). For the work presented here, it is important to understand the role of the Windkessel effect as a compensatory mechanism (for the data presented on maturing fistula) while also acknowledging the complexity it adds to fluid modeling (fluid-structure interactions are difficult to simulate and replicate).

This cursory examination of fluid flow within the body excludes many deeply important aspects. Some will be discussed in other chapters (such as how fluid flow influences vascular shaping), other will not (as must unfortunately be true of any but the most fully fleshed out reviews). For the rest of this chapter, we will turn our attention to measuring fluid volumes and their movement within the human body. Specifically, we will consider the effective intravascular blood volume.

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