Applied BioFluid Mechanics  Lee Waite and Jerry Fine
.pdf232 Chapter Eight
L 
L 
∆L
P
Figure 8.1 Demonstration of strain in a tensile specimen, L/L.
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 crosssectional area, and a material property known as resistivity. Equation (8.1) relates strain gauge resistance R to those properties.
R 5 rL/A 
(8.1) 
where R resistance, in Ohms,
r resistivity, in Ohm meters, m
F
Figure 8.2 Strain gauge with force F applied.
F
Flow and Pressure Measurement 
233 
L length in meters, m
A wire crosssectional area in square meters, m2
We can now examine the small changes in resistance due to the changes in length and crosssectional area by using the chainrule to take the derivative of both sides of Eq. (8.1). The result is shown in Eq. (8.2).

r 

rL 

L 

(8.2) 

dR 5 A dL 2 
A2 dA 1 A dr 


We can now divide Eq. (8.2) by rL/A to get the more convenient form of the equation shown in Eq. (8.3).
dR 5 dL 2 dA 1 dr 
(8.3) 

R L A 
r 



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).
Poisson’s ratio 5 y 5 
2dD/D 
(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 5 p4 D2, 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 crosssectional area.


p 

p 





p 






sD22 2 D12d 


sD2 
2 D1dsD2 
1 D1d 


sD2 
2 D1ds2Dd 

dA 
5 
4 
5 
4 
5 
4 

















A 


p 
D2 



p 
D2 



p 
D2 



4 



4 




4 













(8.5)
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).
dA 
5 
2dD 
(8.6) 
A 
D 

234 Chapter Eight
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 5 dL 
2 2dD 1 
dr 
(8.7) 
r 

R L 
D 


or by combining with the definition of Poisson’s ratio y 5 2dD/D dL/L
dR 5 dL 
1 2y dL 1 
dr 
5 s1 1 2yd dL 1 
dr 
(8.8) 
r 
r 

R L 
L 

L 


The term (1 2v)dL/L represents a change in resistance associated with dimensional change of the strain gauge wire. The second term dr/r 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).


R 


L 


r 

L 

(8.9) 

Gauge factor ; G 5 a R 



L b 5 s1 1 2yd 1 a 
r 

L b 







For metals like nichrome, 
constantan, platinumiridium, and nickel 





> 

> L 

> 




copper the term (1 2v) dominates the gauge factor. For semiconductor 

materials like silicon and germanium, the second term 








a 

r 

Lb 










r 









dominates the gauge factor.
A typical gauge factor for nichrome wire is approximately 2, and for platinumiridium 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 R/R, then it is possible to calculate the strain L/L. However, the change in resistance R 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
Flow and Pressure Measurement 
235 
Figure 8.3 A bridge for measuring small changes in resistance.
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).
I1 5 Ve/s2 # R1d 
(8.10) 
The voltage at point 1, between the two resistors is: 

E1 5 I1 # R1 5 Ve/2 
(8.11) 
The current I2, flowing through the right side of the circuit, through the strain gauge and the potentiometer is:
I2 5 Ve/s2R 1 Rd 
(8.12) 
The voltage at point two, on the right side of the circuit, between the strain gauge and potentiometer is:
E2 
5 I2 
# R 5 Ve 
R 
(8.13) 

2R 1 




R 
Vout for the bridge is now given by the difference between the voltages at points 1 and 2. See Eqs. (8.14) through (8.16).
E1 2 E2 
5 sVe/2ds1 2 s2R/s2R 1 Rdd 
(8.14) 

E1 2 E2 
5 sVe/2ds R/s2R 1 Rdd 
(8.15) 

and since 2R is much, much greater than 
R 


E1 2 E2 5 sVe/4ds 
R/Rd 
(8.16) 
236 Chapter Eight
Figure 8.4 A voltage measuring circuit that does not use a bridge. This circuit cannot accurately measure small changes in resistance.
Therefore, to measure ( R/R) we can use a bridge and measure
(Vout/Vexcitation). It is possible to obtain R/R by multiplication of (Vout/Vexcitation) by 4. Now, if we divide R/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: 

I 5 Ve/s2R 1 Rd 
(8.17) 
The output voltage of the circuit that is used to measure the change in resistance is:
Vout 5 I R 5 sVe # Rd/s2R 1 Rd 
(8.18) 
Since R/2R is R/ R, this circuit yields a fairly constant output of 1/2V. Finally, strain 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 gaugetipped pressure transducer, a very small strain gauge is mounted on the tip of a transducer, and the strain gauge tip
238 Chapter Eight
8.3.3 Electrical analog of the catheter measuring system
In Chap. 7 Sec. 7.8, 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, wellknown 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 better able to understand the limitations of our pressure measuring system and to predict important characteristics.
It was possible to simplify the Fry solution to the following firstorder ordinary differential equation with terms that represent fluid inertance and fluid resistance as was shown below and in Chap. 7, Eq. (7.89).
P1 
2 P2 
5 L 
dQ 
(7.89) 

/ 
1 RvQ 



dt 

In Eq. (7.89), 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).
L 5 
s1 1 c1dr 
5 
cur 
(8.19) 

pR2 
pR2 







RV 5 
cv8m 

(8.20) 


pR4 




In Eq. (8.19) r 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).
C 5 dV 
(8.21) 
dP2 

Flow and Pressure Measurement 
239 
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).
dV 5 C dP2 
(8.22) 

dV 

dP2 
(8.23) 
Q 5 dt 
5 C 
dt 
The time rate of change of Q, as a function of compliance and P2, can now be written:
dQ 
d2P2 
(8.24) 
5 C 
dt2 

dt 

Now it becomes possible to substitute Eqs. (8.23) and (8.24) into Eq. (7.89) above.
LC 
d2P2 
1 RvC 
dP2 
5 
P1 
2 P2 
(8.25) 
dt2 
dt 

/ 






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 pressures in the blood vessel and the transducer are P1 and P2, respectively.
Next rewrite the derivative. terms$in the somewhat simplified form where dPdt is written as P and ddt2P2 5 P and we arrive at Eq. (8.26).
$ 
. 

/LCP2 1 
/RvCP2 1 P2 5 P1 
(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 secondorder, linear, ordinary, differential equation with driving function EP1. This type of equation is typical in many, many types of electrical and mechanical applications and one that we will use several times.
$ 

. 

/{LP2 1 
/{RvP2 1 {E P2 5 EP1 
(8.27) 

m 
b 
k 

For the mechanical analog system shown in Fig. 8.6, the equation defines a typical spring, mass, damper system where the spring constant is
240 Chapter Eight
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).
k E, the damping coefficient for the damper is b 5 /Rv, and the mass term is m /L . The canonical form of the secondorder differential equation describing the spring, mass, damper system, which practically all engineers have seen, is shown in Eq. (8.28).
$ . 
(8.28) 
my 1 by 1 ky 5 aFstd 
8.3.4 Characteristics for an extravascular pressure measuring system
For all secondorder systems, there are several system characteristics that may be of interest. In this section, we 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), (8.30), and (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 timevarying 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.
static sensitivity ; k 5 a/y 
(8.29) 







undamped natural frequency ; vn 5 Å 
k rad 
(8.30) 

m 

s 

Flow and Pressure Measurement 
241 

damping ratio ; z 5 

b 
5 
b 

(8.31) 




bcr 

2 
2km 


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

. 


/{L P2 1 
/{Rv P2 1 {E P2 5 {E P1 
(8.27) 

m 
b 
k 
a 

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:
$ 
. 
(8.28) 
my 1 b y 1 ky 5 aFstd or ky 5 aFstd 

{ 
{ 

00
For the analogous pressure measuring system k E and a E so the static sensitivity is one, 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 ; vn 5 Å 

E rad 
5 Å 

EpR2 rad 
(8.32) 

/L 


s 


/cur 

s 













/ 
cv8m 




/Rv 




pR4 






Damping ratio ; z 5 






5 


Å 








(8.33) 

2E/L 






cur 


2 


2 
















E/pR2 















Sections 8.3.6 and 8.3.7 describe two examples of secondorder, pressure measuring systems. Case 1 is an undamped catheter measuring system, and Case 2 is the undriven, damped system.
8.3.5 Example problem—characteristics of an extravascular measuring system
A liquidfilled catheter, 100 cm long, with an internal radius of 0.92 mm, is connected to a strain gauge pressure sensor. The curve in Fig. 8.7 shows a