# Cardiovascular Hemodynamics for the Clinician

#### John Wiley & Sons

All right reserved.

ISBN: 978-1-4051-6917-2

### Chapter One

Introduction to basic hemodynamic principles

James E. Faber, George A. Stouffer

Hemodynamics is concerned with the mechanical and physiologic properties controlling blood pressure and flow through the body. A full discussion of hemodynamic principles is beyond the scope of this book. In this chapter, we present an overview of basic principles that are helpful in understanding hemodynamics.

1. Energy in the blood stream exists in three interchangeable forms: pressure arising from cardiac output and vascular resistance, hydrostatic pressure from gravitational forces, and kinetic energy of blood flow

Daniel Bernoulli was a physician and mathematician who lived in the eighteenth century. He had wide-ranging scientific interests and won the Grand Prize of the Paris Academy 10 times for advances in areas ranging from astronomy to physics. One of his insights was that the energy of an ideal fluid (a hypothetical concept referring to a fluid that is not subject to viscous or frictional energy losses) in a straight tube can exist in three interchangeable forms: perpendicular pressure (force exerted on the walls of the tube perpendicular to flow; a form of potential energy), kinetic energy of the flowing fluid, and pressure due to gravitational forces. Perpendicular pressure is transferred to the blood by cardiac pump function and vascular elasticity and is a function of cardiac output and vascular resistance.

Total energy (TE) = potential energy + kinetic energy TE = (perpendicular pressure + gravitational pressure) + kinetic energy TE = ([P.sub.Per] + [P.sub.grav]) + 1/2 [rho][V.sup.2]

where V is velocity and [rho] is blood density (approximately 1060 kg/[m.sup.3])

TE = [Psub.Per] + ([rho] x h x g) + 1/2 [rho] [V.sup.2]

where g is gravitational constant and h is height of fluid.

Although blood is not an "ideal fluid" (in the engineering sense), Bernoulli's insight is helpful. Blood pressure is the summation of three components-lateral pressure, kinetic energy (also known as the impact pressure or the pressure required to cause flow to stop), and gravitational forces. Kinetic energy is greatest in the ascending aorta where velocity is highest but even there it contributes less than 5 mm Hg of equivalent pressure.

Gravitational forces are important in a standing person. Arterial pressure in the foot will greatly exceed thoracic aortic pressure due to gravitational pull on a column of blood. Likewise, arterial pressure in the head will be less than thoracic aortic pressure. Similarly, gravitational forces are important in the venous system as blood will pool in the legs when an individual is standing. This lowers cardiac output and explains why a person will feel lightheaded if rising abruptly from a sitting or supine position. In contrast, gravity has minimal impact when a person is lying flat. Gravitational pressure equals the height of a column of blood x the gravitational constant x the fluid density. To calculate hydrostatic pressure at the bedside (in mm Hg), measure the distance in millimeters between the points of interest and divide by 13 (mercury is 13 times denser than water).

Pressure is the force applied per unit area of a surface. In blood vessels or in the heart, the transmural pressure (i.e. pressure across the vessel wall or chamber wall) is equal to the intravascular pressure minus the pressure outside the vessel. The intravascular pressure is responsible for transmural pressure (i.e. vessel distention) and for longitudinal transport of blood.

2. Blood flow is a function of pressure gradient and resistance

One of the properties of a fluid is that it will flow from a region of higher pressure (e.g. the left ventricle) toward a region of lower pressure (e.g. the right atrium) (Figure 1.1). In clinical practice, the patient is assumed to be supine (negating the gravitational component of pressure) and at rest (kinetic energy is negligible compared to blood pressure at normal cardiac output) and thus blood flow is estimated using pressure gradient and resistance.

The primary parameter used in clinical medicine to describe blood flow is cardiac output, which is the total volume of blood pumped by the ventricle per minute (generally expressed in L/min). Cardiac output is equal to the total volume of blood ejected into the aorta from the left ventricle (LV) per cardiac cycle (i.e. stroke volume) multiplied by heart rate. This formula is important experimentally but of limited used clinically because stroke volume is difficult to measure. Cardiac output in the catheterization laboratory is generally measured using the Fick equation or via thermodilution techniques, which are discussed in Chapter 6.

To compare cardiac output among individuals of different size, cardiac index (cardiac output divided by body surface area) is used. Normalization of cardiac output for body surface area is important as it enables proper interpretation of data independent of the patient's size (e.g. cardiac output will obviously differ widely between a 260-pound man and a 100-pound woman). Indexing to body surface area is also used for other measurements such as aortic valve area.

The relationship between blood flow, resistance, and pressure can be determined using a modification of Ohm's law for the flow of electrons in an electrical circuit:

where [DELTA]P is the difference in pressure between proximal and distal points in the system and R is the hydraulic resistance to blood flow.

A useful clinical equation based on Ohm's law is

Mean arterial pressure (MAP) - central venous pressure (CVP) = cardiac output (CO) x systemic vascular resistance (SVR)

Using this equation, we can calculate systemic vascular resistance knowing cardiac output, CVP, and arterial pressure. MAP is the average arterial pressure over time and is generally estimated using MAP = (1/3 x aortic systolic pressure) + (2/3 x aortic diastolic pressure). This formula was developed for heart rate = 60 bpm (at heart rate = 60 bpm, diastole is twice as long as systole) but becomes progressively more inaccurate as heart rate increases. In a patient in shock (i.e. low blood pressure and impaired tissue perfusion), measurement of CO and calculation of SVR can help identify the etiology (e.g. septic shock with high CO + low SVR or cardiogenic shock with low CO + high SVR).

3. Resistance to flow can be estimated using Poiseuille's law

Blood is not an "ideal fluid" and energy (and pressure) is lost as flowing blood overcomes resistance. Resistance to blood flow is a function of viscosity, vessel radius, and vessel length. The relationship is known as Poiseuille's law (sometimes referred to as the Poiseuille-Hagen law) and is described by the following equation:

Resistance = 8 x viscosity x length/[pi] x [radius.sup.4]

or since flow = difference in pressure/resistance:

Blood flow = [pi] x [radius.sup.4] x difference in pressure/8 x viscosity x length

Since radius is raised to the fourth power, its importance in determining resistance is paramount. A 20% increase in radius leads to a doubling in flow if all other variables are constant. Or as another example, resistance is 16 times greater in a coronary artery with a diameter of 2 mm (e.g. a distal obtuse marginal) than in a coronary artery with a diameter of 4 mm (e.g. the proximal left anterior descending).

Viscosity is also important in determining resistance (commonly abbreviated as [eta] and has units of poise = dyne s/[cm.sup.2]). It is difficult to measure directly and thus is commonly reported as relative to water. The viscosity of plasma is 1.7 x viscosity of water and viscosity of blood is 3-4 x viscosity of water, the difference being due to blood cells and particularly hematocrit.

It is important to note that Poiseuille's law only provides an approximation of resistance when used in blood vessels. The four important assumptions underlying the derivation of this equation are: (1) the viscosity of the fluid is unchanging over time or space; (2) the tube is rigid and cylindrical; (3) length of the tube greatly exceeds diameter; and (4) flow is steady, nonpulsatile, and nonturbulent. Many of these assumptions are violated when this equation is applied to blood flow in the body. Poiseuille's law is important, however, as it indicates the variables that are determinants of resistance to flow.

In the mammalian circulation, resistance is greatest at the level of the arterioles. While radius of a typical capillary is smaller than the radius of an arteriole, the number of capillaries greatly exceeds the number of arterioles, and thus the effective area is much larger. Also of importance is that arteriolar resistance can be regulated (capillaries have no smooth muscle and thus resistance cannot be regulated at that level). This enables rapid changes in vascular resistance to maintain blood pressure (e.g. in hypovolemic shock) and also enables regulation of blood flow to various organs (i.e. autoregulation). A general principle to remember is that reduction of arteriolar resistance decreases SVR resulting in an increased cardiac output while decreasing pressure proximal to the arterioles and increasing pressure distal to the arterioles.

4. Reynold's number can be used to determine whether flow is laminar or turbulent

Flow in blood vessels, as in any hydraulic system, is usually smooth and orderly because the fluid separates into an infinite number of concentric layers with different velocities. When a fluid (such as blood) flows past a solid surface (such as the vascular wall), a thin layer develops adjacent to the surface where frictional forces retard the motion of the fluid (Figure 1.2). There is a gradient of frictional resistance (and thus velocity) between fluid in contact with the solid surface and fluid in the center of the stream. If the fluid particles travel along well-ordered nonintersecting layers this is termed laminar flow. The flow resistance in laminar flow is due entirely to viscous resistance. In laminar flow, the average velocity of a fluid is one half of the maximum velocity observed in the center of the stream.

In contrast, turbulent flow is where fluid particles from adjacent layers become mixed. Turbulent flow is chaotic and less efficient because of energy losses (these losses are termed inertial resistance). In turbulent flow, the relationship between pressure difference and flow is no longer linear since resistance increases with flow. Thus, larger pressure differences are required to maintain flow. Turbulence, and associated loss of energy, is one of the causes of a drop in pressure distal to a severe stenosis.

Turbulence is important for several reasons, one of which is that it creates noise, which is the cause of some cardiac murmurs and the Korotkoff sounds (used when measuring blood pressure; Figure 1.3). Another is that turbulence alters the relationship between flow and perfusion pressure. Because of increased energy losses associated with turbulence, the relationship between perfusion pressure and blood flow is no longer linear (as described by the Poiseuille relationship) but rather greater pressure is required to maintain adequate flow (Figure 1.4).

The transition from laminar to turbulent flow can be predicted by calculating the Reynold's number, which is the ratio of inertial forces ([V.sub.[eta]]) to viscous forces ([rho]/L):

R = diameter x velocity x density/viscosity

where viscosity ([eta]) of blood is 0.004 Pa s, density ([rho]) of blood is approximately 1050 kg/[m.sup.3], velocity (V) of blood is in m/s, and the diameter of the tube is in meters. Reynold's number is dimensionless.

In a given hydraulic system there exists a critical Reynold's number below which flow is laminar. At Reynold's numbers near this critical number, a transitional zone exists where flow is neither completely laminar nor turbulent. Higher Reynold's numbers are associated with turbulent flow. In a long, straight, nonbranching pipe with nonpulsatile flow, flow is generally laminar if R < 2000 and turbulent if R > 2000. It is important to note that the Reynold's number depends on the exact flow configuration and must be determined experimentally.

In the aorta, transition from laminar to turbulent flow generally occurs at a Reynold's number between 2000 and 2500. In atherosclerotic arteries and/or at branch points, the critical Reynold's number is much lower and there can be turbulence even at normal physiological flow velocities. In severe stenoses, turbulence can be initiated at Reynold's numbers an order of magnitude less than in the theoretical, straight pipe.

The Reynold's equation is important for demonstrating variables important in determining whether flow is laminar or turbulent. As a simple approximation from this equation we see that laminar flow is difficult to maintain in conditions of high velocity (e.g. stenotic artery) and large diameter. Vessel diameter is doubly important for not only is it a direct variable in the equation, it also influences velocity. Because of the continuity equation (see item 9 below) we know that velocity increases as diameter decreases. Thus, effects of blood vessel diameter on Reynold's number are magnified. Because both velocity and diameter decrease in the microcirculation, the flow there tends to be laminar.

5. Force developed by the ventricles is a function of preload or stretch-the Frank - Starling law

The three most important factors in the regulation of ventricular function (and thus cardiac output) are preload, afterload, and contractility. Preload for the ventricles is defined as amount of passive tension or stretch exerted on the ventricular walls (i.e. intraventricular pressure) just prior to the initiation of systole. This load determines end-diastolic sarcomere length and thus the force of contraction. The Frank-Starling law states that the passive length to which the myocardial cells are stretched at the end of diastole determines the active tension they develop when stimulated to contract. The Frank-Starling law is an intrinsic property of myocytes and is not dependent upon extrinsic nerves or hormones. The general principle is that increased preload causes increased force of contraction, which increases stroke volume and thus cardiac output (Figure 1.5). The Frank-Starling law (or mechanism) helps the heart match cardiac output to venous return.

The Frank-Starling law was derived from independent work by these two investigators. In the 1890s, Otto Frank measured pressure developed by isolated beating frog ventricles against an occluded aorta under varying preloading conditions. He found that as end-diastolic volume increased, ventricular systolic pressure and the maximum rate of pressure development ([dP/dT.sub.max]) increased. Approximately 20 years later, Ernst Starling found similar results using a heart-lung preparation in an anesthetized dog in which he controlled heart rate, venous pressure, venous return, and arterial resistance.

For any given heart, there is not just a single Frank-Starling curve. Rather there is a family of curves, each of which is determined by the afterload and inotropic state (i.e. contractility) of the heart. While changes in venous return cause a ventricle to move along a single Frank-Starling curve, changes in contractility and afterload cause the heart to shift to a different Frank-Starling curve.

6. Wall tension is a function of pressure and radius divided by wall thickness-Laplace's relationship

Laplace's law describes the relationship between the transmural pressure difference and the tension, radius, and thickness of the vessel wall or ventricular chamber (Figure 1.6). Simply stated, Laplace's law is

Wall tension = pressure x radius/wall thickness

The pressure inside a blood vessel or ventricle exerts a distending force (tension) on the walls proportional to the magnitude of the pressure and radius. Thus, wall tension on the aorta is high. In chronic hypertension, aortic wall thickness increases as an adaptation to normalize wall tension. Similarly, LV hypertrophy develops in response to chronic pressure elevations and/or dilation, again as an adaptation.

(Continues...)

Excerpted from Cardiovascular Hemodynamics for the Clinician Copyright © 2008 by George Stouffer. Excerpted by permission.
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