MODELLING OF TRANSIENT RESPONSE FOR CATHETER – PRESSURE SENSOR SYSTEM

 

Vasile Manoliu

 

Depart. of Electrical Machines and Drive Systems, Faculty of Electrotechnics, University “POLITEHNICA” of Bucharest, Splaiul Independentei 313 - 77206, Bucharest, Romania

 

 

Abstract Blood pressure measurements are essential for control of circulatory systems with artificial hearts or circulatory assist devices and are necessary in the back-up monitoring of these systems. Extracorporeal pressure transducers connected to a fluid-filled catheter have been most commonly used for short term measurements of blood pressures in animal experiments of artificial hearts/circulatory assist devices. With appropriate approximations on can reduces a catheter-sensor lumped-parameter model to a second order system. In this paper a transient response and the frequency response of the catheter-sensor system by means of the analogous electric circuit is studied. The effects of changes in the hydraulic system owing to air bubbles are, also, considered.

 

1. INTRODUCTION

 

The heart is an extremely active organ, beating approximately 30 million times, and ejecting some 2,1 million litres of blood, per year. The cardiac cycle represents the events associated with the flow of blood through the heart during one heart beat. Since alternate myocardial contraction and relaxation mainly achieve the movement of blood, the cycle employs the terms systole (i.e. contraction) and diastole (i.e. relaxation). Blood circulates because the heart “pump” establishes pressure gradients.

An understanding of the dynamic properties of a pressure-measurement system is important for preserve the dynamic accuracy of the measured pressure. In the clinical situation errors in measurement of dynamic pressure can have serious consequences (e.g. an underdamped system can lead to overestimation of pressure gradients across stenotic heart valves).

Pressure sensors can monitor blood pressure in postsurgical patients as part of a closed-loop feedback system. Such a system injects controlled amounts of the drug nitroprusside to stabilize the blood pressure.

The liquid-filled catheter sensor is a hydraulic system that can be represented by either distributed- or lumped-parameters models; for the clinical situation, the single-degree-of-freedom (lumped-parameter) model is easier to work, and the accuracy of the results obtained by using these models is acceptable.

 

2. HYDRAULIC – ELECTRICAL ANALOGY

 

In order to understand how the supply of blood to a tissue is regulated in order to maintain cellular, tissue and organ system homeostatic processes, on need to consider three interrelated physical aspects of circulation: blood flow, blood pressure and peripheral resistance. The latter two influence the rate of blood flow.

Blood circulates in the systemic and pulmonary circuits, and the rate of flow is dependent upon two factors: arterial blood pressure and the peripheral resistance provided by blood vessels and blood viscosity. The rate of flow is inversely proportional to the resistance since for a given pressure, the higher the resistance, the lower the flow rate.

Since the left ventricle pumps blood in a pulsating manner and tissue flow generally varies accordingly, the average pressure is very important. The difference between the blood pressure at the base of the aortic arch and the right atrium represents the circulatory pressure. Due to relatively small pressures in the venous system, arterial pressure approximates to the circulatory pressure.

Peripheral resistance, R_s, is the generally used term since most friction is encountered in the peripheral circulation. Resistance is related to blood viscosity n, the length and the diameter of blood vessels:

 

                                                                           (1)

 

The average pressure, , is determined only of cardiac output CO and peripheral resistance:

 

                                                                              (2)

 

Cardiac output is the volume of blood ejected into the aorta each minute.

Arterial compliance determine only how quickly is obtained a new level of average pressure  (i.e. time constant) when, for example, CO is sudden modified.

The proportionality relation between cardiac output, CO, and venous pressure, p_v, is nonlinear:

 

                                                                              (3)

 

the proportionality factor, G, is named heart contractility.

A constant compliance model  (or the linear three-element Windkessel model) of arterial system is useful when pressure variation within the cardiac cycle is small and at about normal mean blood pressure. When the mean blood pressure is high, or when the pressure variation within a single cardiac cycle is large, the nonlinear pressure dependent compliance model will more accurately reflect the behavior of the arterial system.

Another parameter, proportional with blood density r, characterize uniform blood flow, is named inertance:

 

                                                                             (4)

 

with l and A length and cross-sectional area of blood vessel, respectively.

Based on previous considerations, when the model is represented with lumped parameters (circuit elements), can be used the correspondence system between physiologic (hydraulic) and electrical elements, presented in Table 1.

 

Table 1. Hydraulic-electrical analogy

 

Hydraulic	Electrical
pressure                       p [mm Hg ]	voltage                          U [V]
flow                              Q [ml / min]	current                           I  [A]
flow resistance R [mm Hg´min / ml]	ohmic resistance           R [W]
Compliance                  C [ml /mm Hg]	capacitance                   C [F]
Inertance                      M [g / cm4]	inductance                    L [H]
Contractility                  G [ml /(min´mmHg)]	conductance                 G [W-1]
Volume                                    V [ml]	charge                           q  [C]

 

3. ANALOGOUS ELECTRIC SYSTEMS

 

The modeling approach taken here develops a lumped-parameter model for the catheter and sensor separately.

Figure 1 shows the physical model of a catheter-sensor system.

 

 

Fig. 1. a) Physical model of a catheter–sensor system. b) Analogous electric system for this catheter–sensor system.

 

An increase in pressure at the input of the catheter causes a flow of liquid to the right from the catheter tip, through the catheter, and into the sensor, this liquid shift causing a deflection of the sensor diaphragm, which is sensed by an electromechanical system. The subsequent electric signal is then amplified.

A liquid catheter has inertial, frictional and elastic properties, represented by inertance, resistance and compliance, respectively. Similarly, the sensor has the same properties, for the diaphragm using in addition the compliance C_d. In fig. 1.b is shown an electric analog of the pressure-measuring system.

The analogous circuit in Figure 1.b) can be simplified to that shown in Figure 2.

Compliance of the sensor diaphragm, C_d, is much larger than those of catheter or sensor cavity, provided that the saline solution is bubble-free and the catheter material is  relatively noncompliant. The resistance and inertance of the liquid in the sensor can be neglected compared to those of the liquid in the catheter.

The liquid resistance R_c of the catheter is due to friction between shearing molecules flowing through the catheter; Poiseuille’s equation can be used for calculate R_c knowing the values of catheter length, its radius and the liquid viscosity.

The liquid inertance L_c of the catheter is due primarily to the mass of the liquid.

Thus, on can neglect the resistive and inertial components of the sensor with respect to those of the liquid catheter (the liquid-filled catheter is longer than the cavity of the sensor and of smaller diameter).

Using the analogy between input voltage and applied pressure and applying the Kirchhoff’s voltage law in Figure 2, gives:

 

                                 (5)

 

where the compliance C_d of the sensor diaphragm is the inverse of the volume modulus of elasticity of the sensor diaphragm.

 

 

Fig. 2. a) Simplified analogous circuit. b) Analogous circuit for catheter-sensor system with a bubble in the catheter. c) Simplified analogous circuit for catheter-sensor system with a bubble in the catheter, assuming that L_cd and R_cd are negligible with respect to R_c and L_c.

 

The natural undamped frequency is:

 

                                                                           (6)

 

and the damping ratio:

 

                                                                                                         (7)

 

By substituting in these relations the definition expressions for R_c and L_c, can be shown that:

 

                                                       (8)

 

and

                                                           

                                                                    (9)

 

 

where:                                                         

The transient response and the frequency response of the catheter-sensor system by means of the analogous electric circuit can be studied. In addition, on can study the effects of changes in the hydraulic system by adding appropriate elements to the circuit. For example, an air bubble in the liquid makes the system more compliant (catheter properties proximal to the bubble are resistance R_c and inertance L_c and distal to the bubble: resistance R_cd and inertance L_cd). Thus its effect on the system is the same as that caused by connecting an additional capacitor in parallel to that representing the diaphragm compliance.

 

4. APPLICATION

 

A 5-mm-long air bubble has formed in the rigid-walled catheter connected to a sensor. The catheter is 1m long, internal radius is 0.46 mm and filled with water at 20°C; volume modulus of elasticity of the diaphragm is 0,49×10¹⁵  N/m². The isothermal compression of air DV/Dp is 1ml per cm of water pressure per liter of volume. On can obtain the frequency-response curve of the system with and without the bubble.

The analogous circuit for the hydraulic system with and without the bubbles is shown in Figure 2b) and c).

The values of natural frequency and the damping ratio without the bubble is:

f_n = 91 Hz and .

To calculate the new values of natural frequency and the damping ratio for the case in which a bubble is present (5mm long), on suppose the total compliance: C_t = C_d + Cb  or:

 

                                                                                                                       (10)

                                                            

 

The volume of the bubble is:

 

                                                                

 

One centimeter of water pressure is 98,5 N/m². Thus: DV/Dp = 3,38×10^-14 m⁵/N = C_t.                                                  

 

On can determine the new values for natural frequency and damping ratio, assuming that the only parameter that changes is the value of DV/Dp. Thus:

                                                        

                                                    

 

The frequency response for the system without (continuous line) and with the bubble (circles line) present is shown in Figure 3.

 

 

Fig. 3. Frequency-response curves for catheter-sensor system with and without bubbles.

 

Note that the bubble lowers f_n and increases z. This lowering of natural frequency may cause distortion problems with the higher harmonics of the blood-pressure waveform.

 

5. CONCLUSION

 

The measurement of blood pressures is essential for control of circulatory systems with artificial hearts or circulatory assist devices. In this paper was investigate a second-order system for a catheter-sensor lumped parameter model. The presence of air bubbles may cause distortion problems affecting information on arterial properties.

 

REFERENCES

 

[1] V. Manoliu – “Design and modelisation elements in bioengineering” (in romanian), U.P. Bucharest, 1999.

[2] J. G. Webster (ed.) – “Medical instrumentation – Application and design”, Ed. J. Wiley&Sons, 1998.