# P2 82J

The relationship between the natural frequency, the damped natural frequency, and the damping ratio is shown below in Eq. (8.55).

By measuring the amplitude ratio of successive positive peaks, it is possible to determine the logarithmic decrement of the system. From the logarithmic decrement, it then becomes straight forward to measure the damped natural frequency, the damping ratio, and the undamped natural frequency.

If the natural frequency of the catheter measuring system is very low, some relatively low-frequency harmonics of the pressure waveform can be amplified, distorting the pressure waveform significantly. If possible, choose a catheter-transducer measuring system with a natural frequency that is 10 times greater than the highest harmonic frequency of interest in the pressure waveform.

### 8.4 Flow Measurement

Measurement of blood flow is an important indicator of the function of the heart and the cardiovascular system in general. Cardiac output is the measurement of the flow output of the heart and is a measure of the ability of the heart and lungs to provide oxygenated blood to tissue throughout the body. In this chapter we will have a look at several techniques for measuring cardiac output as well as techniques for measuring blood flow in specific vessels as a function of time.

Adolph Fick was born in 1829 in Kassel, Germany. Fick studied medicine, later became interested in physiology, and took a position in Zurich where he was already making contributions to the scientific literature in physics at the age of 26. Fick's first contribution as a physicist was a statement that diffusion is proportional to concentration gradient. Fick is most famous because of a brief, obscure publication in 1870, in which he described how mass balance might be used to measure cardiac output—the Fick principle.

### 8.4.1 Indicator dilution method

The general formulation of the indicator dilution method of measuring blood flow is one that is based on the Fick principle. The substance is injected into the blood stream in a known quantity. The flow rate of blood through a given artery, which is also the time rate of change of volume of blood moving through the same artery, can be predicted if the rate of injection of some mass of the indicator is known, along with the change in concentration of the indicator in the blood. For example, imagine that a bolus of indicator dye is injected into the blood stream. The concentration of the dye at a single point in some vessel could be measured and will be high immediately upon injection and will reduce with time as the dye is first diluted and then cleared from the system. The concentration will be a function of time. For this case change in concentration, AC, would be the change in concentration over time at a specific point.

The general formulation for the flow rate is given by Eq. (8.56), where m is the given quantity of indicator substance injected, dm/dt is the time rate of injection of m, Vis the blood volume, and C is the concentration of m in blood. Depending on the specifics of the method used, AC may be a change in concentration at a single point in a blood vessel at two different times, or it may be the instantaneous difference in concentration between two different locations. Examples demonstrating the indicator-dilution method are described in more detail in the following sections.

8.4.2 Fick technique for measuring cardiac output

The Fick principle relates the cardiac output, Q, to oxygen consumption and to the arterial and venous concentration of oxygen as shown in Eq. (8.57).

For Eq. (8.57), Q is the cardiac output in L/min, dm/dt is the consumption of O2 in L/min, Ca is the arterial concentration of O2 in L/L, and Cv is the venous concentration of O2 in L/L. Using a device called a spirometer, it is possible to measure a patient's oxygen consumption. It is also possible to measure both arterial and venous oxygen concentration from a blood sample collected through a catheter.

Oxygen is the indicator used in this method which enters the blood stream through the pulmonary capillaries. Cardiac output is calculated based on the oxygen consumption rate and the oxygen concentration in arterial and venous blood.

### 8.4.3 Fick technique example

A patient's spirometer oxygen consumption is 250 mL/min while her arterial oxygen concentration is 0.2 mL/mL and her venous oxygen concentration 0.15 mL/mL. Cardiac output is calculated by dividing the oxygen consumption rate by the change in concentration, (0.22 — 0.15) L/L. The patient's cardiac output is 5 L/min.

8.4.4 Rapid injection indicator-dilution method—dye dilution technique

Indocyanine green is used clinically as an indicator to measure blood flow. This method could be used to measure cardiac output but is also applicable to local flow measurements like cerebral blood flow or femoral arterial flow. A curve is generated using a constant flow pump and blood is continuously drawn from the vessel of interest into a colorimeter cuvette to measure color (dye concentration).

Figure 8.9 shows a dye concentration curve that is generated by this method. A bolus of dye is rapidly injected into the vessel of interest. The solid line in the figure represents the fluctuation in concentration of the o O

Recirculation of dye

Curve if recirculation were not possible

Time

Figure 8.9 Recirculation curve used for measuring blood flow with indocyanine green dye dilution method.

Recirculation of dye

Curve if recirculation were not possible

Time

Figure 8.9 Recirculation curve used for measuring blood flow with indocyanine green dye dilution method.

dye, in the blood, after injection. The variable m represents the amount of dye injected. Q represents the flow rate and C represents the dye concentration. The area under the curve in Fig. 8.9 represents the amount of dye injected. The flow rate can be calculated as shown in Eq. (8.58).

In Eq. (8.58) Q is approximately constant and C is a function of time.

### 8.4.5 Thermodilution

The technique of thermodilution uses heat as an indicator for measuring blood flow, thereby avoiding the use of dyes like indocyanine green. Cooled saline can be injected into the right atrium while a thermistor placed in the pulmonary artery measures temperature. Temperature is used as a measure of the concentration of the indicator substance, in this case heat. Flow can then be calculated using Eq. (8.59).

Pbcb

where Q = flow rate, m/s q = heat content of injectate, J pb = density of blood, kg/m3 cb = specific heat of blood, J/(kg °K) AT = temperature change, 0K.

8.4.6 Electromagnetic flowmeters

A conductor moving through a magnetic field generates an electromotive force (EMF) in that conductor. The EMF results in the flow of current which is proportional to the speed of the conductor. Based on this principle, an electromagnetic flowmeter can measure flow of a conducting fluid when that conducting fluid flows through a steady magnetic field. Figure 8.10 shows a schematic of a blood vessel between permanent magnets that generate a magnetic field within the blood vessel. A current is generated, between the two electrodes shown, that is proportional to the average blood velocity across the cross section of the vessel. The EMF generated between the electrodes placed on either side of the vessel is given in Eq. (8.60).

where u = the instantaneous blood velocity, m/s L = the length between electrodes, m B = the magnetic flux density, T

T has the units Wb/m2

The variable e represents the EMF or voltage generated in response to blood velocity. For a uniform magnetic flux density B and a uniform velocity profile, the expression may be simplified to the scalar Eq. (8.61).

Figure 8.10 Aschematic of an elec-tromagnetiu flowmeter with magnetic field B and blood velocity u .

Some potential sources of error in electromagnetic flowmeters include:

1. Current flows from high to low velocity areas.

2. Current can be shunted through the vessel wall. This effect varies with hematocrit.

3. Fluid outside the vessel wall can shunt current.

4. Nonuniform magnetic flux density will cause a variation in the flowmeter output, even at a constant flow rate.

One typical type of electromagnetic probe is the toroidal type cuff perivascular probe. The toroidal cuff uses two oppositely wound coils and a permalloy core. The probe should fit snug around the vessel and this requires some constriction during systole.

### 8.4.7 Continuous wave ultrasonic flowmeters

Another type of flowmeter used in medical applications is the continuous wave Doppler flowmeter. This type of flowmeter is based on the Doppler principle, and when a target recedes from a fixed sound transmitter, the frequency is lowered because of the Doppler effect. For relatively small frequency changes, the relationship in Eq. (8.62) is true.

In Eq. (8.62), Fd is the Doppler shift frequency, Fo is the source frequency, u represents the target velocity in this case an erythrocyte, and c represents the velocity of sound. For two shifts, one from source and one to target:

Consider now a probe at an angle a with respect to the patients skin, as shown in Fig. 8.11. The signal can now be calculated as:

Some problems that occur with continuous wave Doppler ultrasound are:

1. The Doppler shifted frequency is actually not a single frequency, but a mixture of many frequencies.

Figure 8.11 A Doppler continuous wave probe elevated at an angle a with respect to the patient's skin.

2. The velocity profile of the flow field is not constant across the vessel cross section.

4. Turbulence & tumbling of cells also causes a Doppler shift.

5. Simple meters do not measure reverse flow.

### 8.4.8 Continuous wave Doppler ultrasound example

Using a continuous wave Doppler with a carrier frequency of 7 MHz a = 45°, blood velocity of 150 cm/s, and speed of sound = 1500 m/s, find the Doppler shifted frequency. Is it in the audible range?

_ 2Foucosa

10 kHz

Yes. The frequency, 10 kHz is in the audible range of a normal, healthy human.

8.5 Summary and Clinical Applications

Chapter 8 introduces flow and pressure measurement schemes. A very quick review to remind the reader of the important clinical applications of those measurements is given below.

Indirect pressure measurements. According to the American Heart Association, high blood pressure is listed as a primary or contributing cause of death in more than a quarter of a million Americans each year.

Nearly one in three U.S. adults has high blood pressure and 30 percent of those people do not know it. Indirect blood pressure measurement using a sphygmomanometer is the method of choice for measuring blood pressure in the doctor's office and is critical to the management regimen of this disease.

Intravascular pressure measurement. Cardiologists sometimes insert a strain gauge tipped pressure catheter into an artery to determine whether a specific stenosis is the cause of decreased blood flow. If the pressure downstream from the blockage is much lower than the pressure immediately upstream, this indicates that the lesion is the cause. This procedure can also be used to evaluate the effectiveness of catheterization and stenting. This use of intravascular pressure measurement is an important aspect of the diagnosis of coronary arterial disease.

Extravascular pressure measurement. Pulmonary capillary wedge pressure is the pressure measured by inserting a balloon-tipped catheter from a peripheral vein into the right atrium, through the right ventricle and then positioning the catheter within a branch of the pulmonary artery. Measurement of pulmonary capillary wedge pressure (PCWP) gives an indirect measure of left atrial pressure and is particularly useful in the diagnosis of left ventricular failure and mitral valve disease. The catheter used for this procedure has one opening (port) at the tip of the catheter and a second port several centimeters proximal to the balloon. These ports are connected to extravascular pressure transducers allowing the pressure measurement.

Cardiac output measurement. Cardiac output is one of the main determinants of organ perfusion (Kothari et al., 2003). During a coronary artery bypass, dislocation of the heart changes the cardiac output and so it is important for the anesthesiologist to have a reliable tool for assessing hemodynamic status to avoid situations with disastrously low cardiac output. There are a number of methods used to assess cardiac output, including the Fick technique, rapid injection dye dilution and thermodilution.

Electromagnetic and continuous wave Doppler flowmeters. These two devices are two tools in the physicians' arsenal that can be used to measure blood flow in a variety of arteries. For example, physicians use Doppler flowmeters to diagnosis peripheral arterial disease. The physician measures flow waveforms and could compare, for example, the waveforms from arteries in the left arm to those from the right arm. A significant difference in the waveforms between the two vessels could indicate a stenosis in one of the vessels.

While electromagnetic flowmeters are more invasive, they are commonly used as the gold standard for measuring blood flow in animal experiments in laboratory settings. In Chap. 10, a model of flow through the mitral valve in the porcine model is presented (Szabo et al., 2004). That model was developed from data that was collected at the University of Heidelberg and the flow measurements were made using the electromagnetic flowmeter.

Bibliography

John WC Jr., Michael RN, Walter HO, et al. Medical Instrumentation Application and

Design. John G. Webster, ed. 3rd ed. NY. Houghton Mifflin Company. Cromwell L, Weibell FJ, Pfeiffer EA. Biomedical Instrumentation and Measurements.

2nd ed. Prentice-Hall. Englewood Cliffs, NJ; 1980. Togawa T, Tamura T, Oberg P. Biomedical Transducers and Instruments. Boca Raton, FL. CRC Press; 1997.

Khandpur RS. Biomedical Instrumentation, Technology and Applications. NY. McGraw-Hill; 2005.

Pickering TG, Hall JE, Appel LJ, Falkner BE, Graves J, Hill MN, Jones DW, Kurtz T, Sheps SG, Roccella EJ. Recommendations for Blood Pressure Measurement in Humans and Experimental Animals, Hypertension. 2005; 45:142. Kothari N, Amaria T, Hegde A, Mandke A, Mandke NV. Measurement of cardiac output:

Comparison of four different methods, IJTCVS. 2003; 19:163-168. Szabo G, Soans D, Graf A, Beller C, Waite L, Hagl S. A new computer model of mitral valve hemodynamics during ventricular filling, Euro. J. Cardiothorac Surg. 2004; 26:239-247.