Frequency characteristics of pressure transducer kits with inserted pressure-resistant extension tubes

Abstract

The accurate monitoring of arterial blood pressure is important for cardiovascular management. However, the frequency characteristics of pressure transducer kits are influenced by the length of the pressure-resistant tube. To date, there have been few studies addressing the frequency characteristics of pressure transducer kits with inserted pressure-resistant extension tubes (pressure-resistant extension tube (ET) circuits). In this study, we examine ET circuits from the viewpoint of the frequency characteristics of pressure transducer kits. DT4812J transducer kits (length 150 cm; Argon Medical Devices, TX, USA) were used. Three original ET circuits were prepared, with the pressure-resistant tube of the DT4812J being extended with a 30-cm length of pressure-resistant tube (180ET circuit), a 60-cm length of pressure-resistant tube (210ET circuit), and a 90-cm length of pressure-resistant tube (240ET circuit). Each of these circuits was evaluated as part of this study. The natural frequency of the original DT4812J circuit was 45.90 Hz while the damping coefficient was 0.160. For the 180 ET circuit, the natural frequency and damping coefficient were 36.4 Hz and 0.162, respectively. For the ET210 circuit, the natural frequency and damping coefficient were 30.3 Hz and 0.175, respectively. For the ET210 circuit, the natural frequency and damping coefficient were 25.3 Hz and 0.180, respectively. As a result of extending the circuit, it was found that the natural frequency decreased drastically, while the damping coefficient increased slightly. When the extension of a pressure transducer kit is required, we should pay careful attention to the major decrease in the natural frequency, which may influence the pressure monitoring.

Appendices

Appendix 1: Basic parameters of sine wave

Generally, the sine wave shown in Fig. 7 can be expressed mathematically by the following expression [17].

$${\text{y}} = {\text{Y}}_{\text{m}} \sin \, \left( {\omega {\text{t}} - \theta } \right)$$

(6)

where y is the amplitude at any time. Y_m is maximum amplitude. T is the wavelength period. In addition, the sine wave function has a relationship defined by Eq. (7).

$$\sin \,{\text{x}} = \sin \, \left( {{\text{x}} + 2\pi } \right)$$

(7)

Fig. 7

Sinusoidal curve

Therefore, Eq. (8) must be because the plane angle has a period of 2π (rad).

$$\omega {\text{T}} = 2\pi$$

(8)

$${\text{i}} . {\text{e}} . \quad \omega = 2\pi /{\text{T}}$$

(9)

ω is the angular velocity (or angular frequency). The frequency (f) is expressed as follows;

$$f = 1/{\text{T }}\left[ {\text{Hz}} \right]$$

(10)

The relationship between the frequency (f) and wavelength period are expressed as follows;

$$\omega = 2\pi f$$

(11)

When a sine wave progresses as an undulation, the relationship between the traveling wave (v; m/s) and wavelength (λ; m) is expressed as follows;

$$\lambda = v/f$$

(12)

Appendix 2: Propagation of a pressure wave in the catheter

It is convenient to apply electric circuit theory to the transmission of pressure in a catheter [13, 18]. The electrical equivalent circuit of a catheter filled with a saline solution becomes the distributed constant line shown in Fig. 8.

Fig. 8

Electrical equivalent circuit. Lc [kg/m⁴] and Rc [kg/m⁴ s] are the liquid inertia and viscous resistance of the catheter per unit length, respectively. Cc [m⁵/Newton] is the compliance of the catheter per unit length. e _i and e ₀ are the entrance to and exit from the electric equivalent circuit

Lc and Rc can be approximated as follows.

$$Lc \,\fallingdotseq \,\rho_{c} /\pi r_{c}^{2} \left[ {{\text{kg}}/{\text{m}}^{4} } \right]$$

(13)

$$Rc\, \fallingdotseq \,8\mu_{c} /\pi r_{c}^{4} \left[ {{\text{kg}}/{\text{m}}^{4} \,{\text{S}}} \right]$$

(14)

The distributed constant line shown in Fig. 8 indicates that the blood pressure pushes the saline for 1 min, and reaches the length corresponding to the next minute.

Therefore, the blood pressure at the catheter tip section end (entry) is transmitted towards the exit while gradually being temporally delayed. At the same time, the blood pressure wave is attenuated by the viscous resistance of the catheter. These phenomena could be indicated by introducing a propagation coefficient (γ) into the following Eq. (15), (16), or (17) [17, 19].

$$\gamma = \alpha + j\beta$$

(15)

$$\alpha \left( {{\text{Nep}}/{\text{m}}} \right) = \left[\sqrt {({\text{Rc}}^{2} + \omega^{2} Lc^{2} )\omega^{2} \,Cc^{2} } {-}\omega^{2} {\text{Lc}}^{2} {\text{Cc}}/2\right]^{1/2}$$

(16)

$$\beta \left( {{\text{rad}}/{\text{m}}} \right) = \left[ {\left( {{\text{Rc}}^{2} + \omega^{2} \,{\text{Lc}}^{2} } \right)\omega^{2} \,{\text{Cc}}^{2} + \omega^{2}\, {\text{LcCc}}/2} \right]^{1/2}$$

(17)

In Eq. (16), α is the decay constant for which the pressure wave attenuates only e^−α. β is the phase constant.

Regarding the voltage phase of the circuit shown in Fig. 8 at an optional point in time, the voltage vector (V) could indicate the following;

$$V = Ae^{{{-}\gamma x}}$$

(18)

$${\text{i}} . {\text{e}} .\quad {\text{V}} = {\text{Ae}}^{{{-} \, (\alpha + j\beta )x}}$$

(19)

The instantaneous values of the above-mentioned Eq. (19) are expressed as follows;

$$v = \sqrt 2 \left| {\text{A}} \right|{\text{e}}^{{{-}\alpha x}} { \sin }\left( {\omega {\text{t}}{-}\beta x{-}\theta_{0} } \right)$$

(20)

A is an integral constant. Figure 9 shows the instantaneous values of the voltage at an optional point in time. Therefore, the wavelength (λ) is expressed as follows;

$$\beta \lambda = 2\pi$$

(21)

$$\therefore \quad \lambda = 2\pi /\beta$$

(22)

Fig. 9

Instantaneous values of voltage at an optional point in time. λ is the wavelength

Because the density of the saline and the viscosity coefficient are 10³ and 10⁻³ kg/m s, respectively, it becomes Rc ² ≪ ω² Lc ². Thus, for a practical catheter, α, β and v are approximated as follows;

$$\alpha \,\fallingdotseq \,Rc\sqrt {Cc/ Lc } /2$$

(23)

$$\beta \,\fallingdotseq\, \omega \sqrt {LcCc }$$

(24)

$$v \, \fallingdotseq \, 1/\sqrt {Cc/ Lc }$$

(25)