Flow and Pressure Measurement

8.1 Introduction

From the measurement system for measuring hypertension in humans with an indirect method, to the system for measurement of flow and pressure waveforms in the uterine artery of cows, flow and pressure measurement systems have wide ranging applications in biology and medicine. A few of the varied methods for making these measurements will be discussed later.

Measurement of flow and pressure are two of the most important measurements in biological and medical applications. Because there is a complex control system in the human body, controlling pressure and flow, the interactions of the measuring system with the physiological control system increases the complexity of the measurement. The invasive nature of flow and pressure measurement also makes it difficult to obtain accurate measurements in some cases. Sometimes, a noninvasive or indirect measurement is possible.

According to the American Heart Association Council on High Blood Pressure Research, accurate measurement of blood pressure is critical to the diagnosis and management of individuals with hypertension. Pickering et al. (2005) wrote:

The auscultatory (indirect measurement) technique with a trained observer and mercury sphygmomanometer continues to be the method of choice for measurement in the office.

8.2 Indirect Pressure Measurements

An important example of a noninvasive, indirect measurement is the measurement of blood pressure using a sphygmomanometer. This type measurement is routinely used in clinical practice because of its relative ease and low cost.

A sphygmomanometer has a cuff that can be wrapped around a patient's arm and inflated with air. The device includes a means for pressurizing the cuff, and a gauge for monitoring the pressure inside the cuff. When the pressure inside the cuff is increased above the patient's systolic pressure, the cuff squeezes the arm and blood flow through the vessels directly under the cuff is completely blocked. The pressure inside the cuff is released slowly, and when the cuff pressure falls below the patient's systolic pressure, blood can spurt through arteries in the arm beneath the cuff. The operator measuring blood pressure typically places a stethoscope distal to the cuff and just over the brachial artery. Turbulent flow of blood through the compressed artery makes audible sounds, known as Korotkoff sounds. At the time these Korotkoff sounds are initially detected, the blood pressure is noted, and this value becomes the measurement of the systolic pressure.

The nature of the sound heard through the stethoscope changes as the sphygmomanometer pressure continues to drop. The sounds change from a repetitive tapping sound to a muffled rumble and then the sound finally disappears. The pressure is once again noted at the instant that the sounds completely disappear. This value is the diastolic blood pressure.

Indirect measurement of blood pressure using a sphygmomanometer has the advantage of being noninvasive, inexpensive, and reliable. One disadvantage of the system is the relative inaccuracy of the system associated with operator error, the relatively low resolution, and the difficulty of measuring the diastolic pressure. Since the measurement of diastolic pressure depends on the detection of the absence of sound, the method is inherently dependent on the ability of the operator to clearly hear very quiet sounds and to judge the point in time at which those sounds disappear.

8.3 Direct Pressure Measurement

When more accurate pressure measurements are required, it becomes necessary to make a direct measurement of blood pressure. Two methods that will be described next, involve using either an intravascular catheter with a strain gauge measuring transducer on its tip, or the use of an extravascular transducer connected to the patient by a saline-filled tube.

8.3.1 Intravascular: strain gauge-tipped pressure transducer

Transducers are devices that convert energy from one form to another. It is often desirable to have electrical output represent a parameter like blood pressure or blood flow. Typically, these devices convert some type of mechanical energy to electrical energy. For example, most pressure measuring transducers convert stored energy in the form of pressure into electrical energy, perhaps measured as a voltage.

A pressure transducer very often uses a strain gauge to measure pressure. We can begin the topic of strain gauge tipped pressure transducers by considering strain. Figure 8.1 shows a member under a simple uniaxial load in order to demonstrate the concept of strain. As the load causes the member to stretch some amount AL, strain can be measured as AL/L. A strain gauge could be bonded onto a diaphragm on a catheter tip.

Two advantages of a strain gauge pressure transducer, compared to an extravascular pressure transducer described in Sec. 8.3.2, are good frequency response, and fewer problems associated with blood clots. Some disadvantages might include the difficulty in sterilizing the transducers, the relatively higher expense, and transducer fragility.

As the strain gauge lengthens due to the load, the diameter of the wire in the strain gauge also decreases and therefore the resistance of the wire changes. Figure 8.2 shows a drawing of a strain gauge with a load F applied. It is possible to calculate the resistance of a wire based on its length, its cross-sectional area, and a material property known as resistivity. Equation 8.1 relates strain gauge resistance R to those properties.

where R = resistance, in ohms, O

p = resistivity, in ohm X meters, Om L = length, in meters, m

A = wire cross-sectional area, in square meters, m2

Figure 8.1 Demonstrating strain in a tensile specimen, AL/L.

Figure 8.1 Demonstrating strain in a tensile specimen, AL/L.

F

1

1

Figure 8.2 A strain gauge with force F applied.

Figure 8.2 A strain gauge with force F applied.

We can now examine the small changes in resistance due to the changes in length and cross-sectional area by using the chain rule to take the derivative of both sides of Eq. (8.1). The result is shown in Eq. (8.2).

We can now divide Eq. (8.2) by pL/A to get the more convenient form of the equation shown below in Eq. (8.3).

dR dL dA dp

If you stretch a wire using an axial load, the diameter of the wire will also change. As the wire becomes longer, the diameter of the wire becomes smaller. Poisson's ratio is a material property which relates the change in diameter to the change in length of wire for a given material. Poisson's ratio is typically written as the character n, which students should not confuse with kinematic viscosity, which is also typically written as n. Poisson's ratio is defined in Eq. (8.4).

dL/L

In Eq. (8.4), D is the strain gauge wire diameter, and L is the strain gauge wire length.

It is also useful here to show that since A = p/4 D, dA/A is related to dD/D so that we can write Eq. (8.3) in the more convenient terms of wire diameter instead of cross-sectional area.

In Eq. (8.5), dD is equal to D2 — D1 and D2 + D1 = 2D, where D is the average diameter between D1 and D2. Therefore, the relationship between dA and dD is shown by Eq. (8.6).

Now it is possible to write an equation for the change in resistance divided by the nominal resistance as shown in Eq. (8.7), dR dL 2dD dp

R L D p or by combining with the definition of Poisson's ratio we get Eq. (8.8).

dL/L

dR dL dL dp dL dp

The term (1 + 2v) dL/L represents a change in resistance associated with dimensional change of the strain gauge wire. The second term dp/p is the term that represents a change in resistance associated with piezoresistive effects, or change in crystal lattice structure within the material of the wire.

The gauge factor for a specific strain gauge is defined by Eq. (8.9).

Gauge fa ctor G — (jRRR b — (1 1 2v) 1 (¿M <8.«

For metals like nichrome, constantan, platinum-iridium, and nickel-copper the term (1 + 2v) dominates the gauge factor. For semiconductor materials like silicon and germanium, the second term dominates the gauge factor.

A typical gauge factor for nichrome wire is approximately 2, and for platinum-iridium approximately 5.1. Strain gauges using semiconductor materials have a gauge factor that is two orders of magnitude greater than that of metal strain gauges. Semiconductor strain gauges are therefore more sensitive than metallic strain gauges.

If we know the gauge factor of a strain gauge and if we can measure AR/R then it is possible to calculate the strain, AL/L. However, the change in resistance, AR is a very small change. Even though it is an electrical measurement, we need a strategy to detect such a small change in resistance.

A Wheatstone bridge is a device that is designed to measure very tiny changes in resistance. See the circuit diagram in Fig. 8.3

In Fig. 8.3, I1 is the current flowing through the two resistors, R1. The current flowing through the two resistors can be shown to be the excitation voltage driving the circuit divided by 2R1 as shown in Eq. (8.10).

The voltage at point 1, between the two resistors is

The current I2, flowing through the right side of the circuit, through the strain gauge and the potentiometer is

Wheatstone bridge

Wheatstone bridge

The voltage at point two, on the right side of the circuit, between the strain gauge and potentiometer is

Vout for the bridge is now given by the difference between the voltage at points 1 and 2. See Eqs. (8.14) through (8.16).

and since AR/R is much, much greater than AR/AR,

Therefore, to measure (AR/R) we can use a bridge and measure (Vout/Vexcitation). It is possible to obtain AR/R by multiplication of (Vout/Vexcitation) by four. Now, if we divide AR/R by the gauge factor, G, we will obtain the strain.

If you question the necessity of the bridge in our measurement, compare the circuit in Fig. 8.4. If you use a known resistance in series with the strain gauge and measure resistance across the strain gauge, by measuring voltage, you will obtain the following:

The current I flowing through the circuit is

The output voltage of the circuit that is used to measure the change in resistance is

Since R/2R >> R/AR, this circuit yields a fairly constant output of 1/2 V.

^excite

Figure 8.4 A voltage measuring circuit that does not use a bridge. This circuit can not accurately measure small changes in resistance.

Finally, pressure is proportional to the strain measured by the transducer and the strain gauge can be calibrated to output a voltage proportional to pressure. One can imagine a pressure transducer with a diaphragm that moves depending on the pressure of the fluid inside the transducer. A strain gauge mounted on the diaphragm of the transducer measures the displacement, which is proportional to pressure.

In a strain gauge-tipped pressure transducer, a very small strain gauge is mounted on the tip of a transducer, and the strain gauge tip deflects proportionally to the pressure measured by the strain gauge. The resultant strain may also be converted to a voltage output which may be calibrated to the pressure being measured.

8.3.2 Extravascular: catheter-transducer measuring system

Figure 8.5 shows a schematic of an extravascular pressure transducer. The transducer is connected to a long thin tube called a catheter. The catheter can transmit pressure from the blood vessel of interest to the extravascular pressure transducer. The pressure is transmitted through a column of heparinized saline. Heparin is used to prevent the blood from clotting and clogging the end of the catheter.

Although the extravascular pressure measurement system is a nice compromise between relatively low cost and accuracy, there are several potential problems associated with the system that I will mention here.

■ Air bubbles in the system, when present, have an important effect on the system's ability to measure high frequency components of the pressure waveform. When you have air bubbles in the brake lines of your automobile, the brakes feel spongy and are less responsive. In

Transducer

Catheter

Figure 8.5 An extravascular pressure transducer connected to a long thin catheter which can be inserted into a blood vessel to transmit the pressure in that vessel.

the same fashion, an extravascular transducer with air bubbles in the catheter will be spongy and may not respond accurately to the pressure.

■ Blood clots can form in the catheter. The blood clots will restrict flow and either plug the catheter or cause a significant pressure drop between the vessel and the extravascular transducer.

■ The use of the extravascular pressure transducers require a surgical cut down. This disadvantage is also present in other direct pressure measurements but is not a disadvantage for indirect pressure measurements.

8.3.3 Electrical analog of the catheter measuring system

In Chap. 7, Sec. 7.7, a solution was developed that was published by Greenfield and Fry in 1965 that shows the relationship between flow and pressure for axisymmetric, uniform, fully developed, horizontal, Newtonian, pulsatile flow. The Fry solution is particularly useful when considering the characteristics of a transducer and catheter measuring system. By developing an electrical analog to a typical pressure measuring catheter, it will be possible for us to use some typical, well-known solutions to RLC circuits to characterize things like the natural frequency and dimensionless damping ratio of the system. From our circuit analog, we will be able to understand better the limitations of our pressure measuring system and to predict important characteristics.

It was possible to simplify the Fry solution to the following first order ordinary differential equation with terms that represent fluid inertance and fluid resistance as was shown next and in Chap. 7, Eq. (7.78).

In Eq. (7.78), P1 represents the pressure in the artery being measured, P2 represents the pressure at the transducer, / represents the length of the catheter, Q is the flow rate of the saline in the catheter, and dQ/dt is the time rate of change of the flow rate. Hydraulic inertance and hydraulic resistance are represented by L and Rv, respectively, and are defined in Eqs. (8.19) and (8.20).

Rv pR2

Cv 8m p R4

In Eq. (8.19), p represents the fluid (saline) density in the catheter and R represents the radius of the catheter. Three empirical proportionality constants are represented by cu, c1,and cv. In Eq. (8.20), m represents fluid viscosity (saline viscosity).

The volume compliance of the transducer, C, represents the stiffness of the transducer or the change in volume inside the transducer corresponding to a given pressure change. The volume compliance is written in Eq. (8.21).

By separating variables and integrating with respect to time, it is possible to solve for flow rate Q as a function of the change in the pressure in the transducer as shown in Eqs. (8.22) and (8.23).

dt dt

The time rate of change of Q, as a function of compliance and P2 can now be written:

Now it becomes possible to substitute Eqs. (8.23) and (8.24) into Eq. (7.88).

L and Rv are defined by Eqs. (8.19) and (8.20), respectively. C is the volume compliance of the transducer. The length of the catheter is represented by /, and the pressure in the blood vessel and the transducer are P1 and P2, respectively.

Next rewrite the derivative terms in the somewhat simplified form where dp/dt is written as P and d2pldt2 as P and we arrive at Eq. (8.26).

Now we can define a term, E, that is equal to 1/C or the inverse of the volume compliance of the transducer. The term E is known as the volume modulus of elasticity of the transducer. Then, by multiplying both sides of the equation by the catheter length,/, we arrive at Eq. (8.27). Notice that Eq. (8.27) is a second order, linear, ordinary differential equation with driving function EP1. This type of equation is typical in many types of electrical and mechanical applications and one that we will use several times.

For the mechanical analog system shown in Fig. 8.6, the equation defines a typical spring, mass, damper system where the spring constant is k = E, the damping coefficient for the damper is b = /Rv, and the mass term is m = /L. The canonical form of the second-order differential equation describing the spring, mass, damper system, which practically all mechanical engineers have seen, is shown in Eq. (8.28).

8.3.4 Characteristics for an extravascular pressure measuring system

For all second order systems, there are several system characteristics that may be of interest. In this section, we will discuss those characteristics, including static sensitivity, undamped natural frequency, and damping ratio.

For the mechanical system from Eq. (8.28), the characteristics are well known and are written in Eqs. (8.29) to (8.31).

In Eq. (8.29), y represents displacement of the mass shown in the mechanical spring mass and damper system in Fig. 8.6. The figure shows a time varying driving force aF(t) driving the mass up and down while it is attached to a spring with spring rate k, and a damper with associated constant b.

Figure 8.6 A spring, mass, damper analog to the extravascular pressure measuring system. The driving force for the system in the picture is aF(t).

(Analogous to £P1) (Analogous to P2)

(Analogous to £P1) (Analogous to P2)

Figure 8.6 A spring, mass, damper analog to the extravascular pressure measuring system. The driving force for the system in the picture is aF(t).

Undamped natural frequency k rad m s

Damping ratio

By making the appropriate substitutions from Eq. (8.27), we can now solve for the specific characteristics associated with our extravascular pressure measurement system.

For the static system, displacement y does not change. In the pressure measuring system, the pressure does not change. All of the higher order terms like y, y, P2, and P2 are now zero. For the pressure measuring system, it is desirable that for every pressure input an equivalent pressure output occurs, so let us design the static gain to be unity. Equation 8.28 for the static system is my + by + ky= aF(t) or ky = aF(t)

For the analogous pressure measuring system k = E and a = E so the static sensitivity is 1, or E/E.

Again, this means simply that the measurement system is designed to measure the input value and repeat it as the output value.

The undamped natural frequency and dimensionless damping ratio of the system are given by Eqs. (8.32) and (8.33).

Undamped natural frequency

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