Ind Eng Chem Fundam 1986, 25, 462-470
462
Dynamics of Fluid-Filled Catheter Systems by Pulse Testing Joel 0. Hougen‘ The University of Texas, Austin, Texas 78746
Stephen T. Houged Victoria Vascular Associates, Victoria, Texas 77904
Thomas J. Hougen5 Boston Floating Hospital, New England Medical Center Hospitals, Tuffs University School of Medicine, Boston, Massachusetts 02 I 1 I
The principal purpose of this work was to evaluate the applicability of pulse testing as a technique for the experimental determination of the dynamic properties of fluid-filled catheters. This procedure requires that the system be excited with only a single appropriate pressure pulse at the inlet end of the catheter. Time histories of measures of inlet and outlet pressures may be converted into frequency response form, via a Fourier transformation computation routine. Dynamic parameters may then be readily obtained from graphical presentations of such information or by other methods. An additional objective was to determine the effects of leads of various lengths between the end of the catheter and the pressure sensor, since this is the arrangement commonly used in clinical practice. Lead lengths from 0 to 83 cm were used. I n all cases the dynamic behavior could be related to the response of second-order linear differential equations with parameters dependent upon catheter type and lead length. For the systems studied, damping factors varied from 0.07 to 0.7 and undamped natural frequencies varied from 18 to 42 rad/s, with no significant dependency upon temperature in the region from 27 to 38 O C . As lead length was increased beyond about 65 cm, behavior became increasingly independent of the specific catheter type and more dependent on length of iead. Simplicity and ease of execution, relative ease of data reduction, and extensive documentation of successful applications make the pulse technique an attractive method for determining the dynamic performance of liquid-filled catheter systems.
Limitations in the dynamic response of conventional catheter-transducer systems used for measuring blood pressure have been recognized for many years. Numerous publications have appeared reporting on these shortcomings, presenting confirmatory test data and, in some instances, suggesting remedial methods to procure more acceptable performance. Several comparisons of the performance of the conventional catheter systems to those having a pressure transducer at or near its tip have also appeared. The common pitfalls into which the unwary user may wander in the interpretation of data obtained with conventional systems have also been frequently emphasized. Four comprehensive reviews of the literature as well as definitive studies of such systems are those of Hansen (1949), Fry (1960), Yanoff et al. (1963), and Hellige (1976). T o date, the experimental descriptions of the dynamics of catheter systems have been obtained either from the transient response to an abrupt change in pressure a t the tip of the catheter (a step or “pop” test) or as the response to a steady sinusoidal signal imposed at the catheter’s tip. No consideration appears to have been given to pulse testing, although two papers have casually mentioned the method (Bottaccini et al., 1967; Cronvich and Burch, 1969).
Professor emeritus. Brother of t h e late Olaf A. Hougen and father of S.T. Hougen and T. J . Hougen Vascular surgeon. §Associate Professor of Pediatrics (Cardiology)
*
0196-43 1318611025-0462$0 1.50/0
The present study had three objectives: to demonstrate the feasibility of the pulse method as a technique by which to obtain dynamic response information about fluid-filled catheter systems, to present such information for some typical catheters, with and without leads of different lengths between the exterior end of the catheter and the sensing transducer, and to gain an understanding of the problems and requirements associated with the measurement of blood pressure to assist in the development of an apparatus by which this can be accomplished in a precise and convenient manner. In conducting a pulse test, the input is forced with one, and only one, appropriate pulse and the time histories of input and output are recorded. Data from these records are used to compute the trapezoidal approximation of the Fourier transforms, from which the frequency response is obtained provided the system tested is essentially linear (Dreifke, 1961). Most of this study was devoted to testing catheter systems simulating clinical conditions where a narrow flexible tube is used to connect the exterior end to the sensing transducer, but a number of tests of catheters only were also made. Dynamic error in the response of conventional systems results from the physical arrangement wherein the pressure sensor is remote from the point a t which the pressure measurement is desired, the sensor or transducer being mounted externally, either connected directly to the end of the catheter or separated from it by a length of small-bore flexible tubing. c3 1986
American Chemical Society
Ind. Eng. Chern. Fundarn., Vol. 25, No. 4, 1986
When the fluid within the catheter (and interconnecting tubing or lead) has a low viscosity and the compliance of the transducer force-summing member is finite, a lightly damped system may be created so that the terminal pressure will oscillate in response to an abrupt change in pressure a t the catheter tip. The pressure signal developed by the exterior sensor can then easily exceed in amplitude, and indeed the entire measured time history can depart drastically from that of the input. In fact, several studies imply that some catheter-transducer systems can be described by a lightly damped second-order ordinary differential equation with constant coefficients of the form shown in eq 1, (Hansen, 1949; Wood et al., 1954; Fry et al., 1957; Yanhoff et al., 1963; Akers et al., 1966; Melbin and Spohr, 1969; Shapiro and Krovetz, 1970; Falsetti et al., 1974; Barquest and Schmaltzel, 1975; Barry et al., 1975; Stanton et al., 1979; Glanz and Tyberg, 1979). In this relation w, is the undamped natural angular frequency (radians per unit time) and l is the damping factor, which if less than unity ensures an oscillatory system. If the coefficients wn and {are known, the behavior of pressure p in response to any realistic forcing function can be computed. Published data present some evidence of the need for a third-order model, and other studies indicate nonlinear behavior caused by compliance of both catheter and sensor and distributed impedance (Fry, 1960). When the step test is used, the pressure at the tip of the catheter is abruptly decreased from some initial value to zero. This method may be called the response to initial condition forcing from which the desired system parameters w, and { are theoretically recoverable from the solution to the differential equation, shown as eq 1, subject to the stated boundary conditions and provided second-order behavior is valid.
p ( t ) = p , t c 0; p ( t ) = 0, t > 0 The direct sinusoidal test is performed by imposing a sinusoidally varying pressure a t the tip of the catheter. Assuming an appropriate selection of forcing frequencies, the frequency response, which consists of both the ratio of the magnitude of the steady-state output to that of the input sine wave and the phase angle difference, is obtained a t the selected frequencies. These are sometimes denoted as the dynamic amplitude ratio (DAR) and the dynamic response angle (DRA). Any interpretations from the data derived from either test are subject to the assumptions implied by linearity. For some systems either method of testing has disadvantages. Step testing demands that the change imposed be indeed a step, Le., that the change occur a t time O+. While testing time can be brief, the results recovered may be limited in reliability if the time history of the input does not conform to the mathematical concept. The more responsive the system, the more error will be introduced by deviations of the input from a true step. Moreover, the information which can be retrieved from step tests, under the best of conditions, is always quite minimal compared to that obtained by using other forcing functions. The frequency response method also has disadvantages, especially for slowly responding systems and where testing time carries a penalty. Where other than electrical signals are used for inputs, difficulties are frequently encountered in producing a purely sinusoidal input form. However, data reduction can be automated even for obtaining phase angle relationships, and this is sometimes an advantage.
-
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Figure 1. Arrangement of catheter and transducers: (a) testing a t controlled temperature; (b) testing a t ambient conditions.
In addition, deviations from linearity will be readily evident as manifested in a nonsinusoidal output time history, provided, of course, the input is satisfactory, although little insight is obtained about the nature and sources of the nonlinearities. In only two of the numerous papers scanned (Bottaccini et al., 1967; Cronvich and Burch, 1969) was any mention made of pulse inputs as test signals for system excitation and the use of the Fourier transformation of input-output pulses as a method of extracting frequency response information. This is surprising because the method was used for testing aircraft as early as 1954 (Muzzey and Kidd, 1954; Smith and Triplett, 1954) and was described in detail by Draper et al. (1953). A very detailed study of the pulse method has been conducted by Dreifke (19611, and several applications for determining the dynamics of industrial processes have been presented by Lees and Hougen (19561, Hougen et al. (1963), Dreifke et al. (1962), and Hougen (1979). In any event, the use of the pulse testing technique for testing catheter-transducer systems appeared to be a “natural” in view of the pulsatile nature of the signals to be measured. That is, the exciting pulse can be made to have a t least some of the characteristics of naturally occurring pulses and thus be free from the criticism of being too severe, as the step test may be. And since, in theory, only one appropriate input pulse is needed to obtain data from which to compute the frequency response information, testing time will be minimal. One purpose of the work herein described was to determine the effects of lead length (between outer end of catheter and transducer) on the performance of several widely used cardiovascular catheters rather than being restricted only to the catheter itself. This does not appear to have been the principal objective in previous studies. Since a number of catheters were tested with no lead, results of this study may be compared to those obtained by others who have investigated such systems. Methods The arrangement for supporting catheters for the pre-
464
Ind. Eng. Chem. Fundam.. VOI. 25. NO. 4, 1986 01 w.,er (Ira", "SS"8d
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Unfiltered outputs from the transducer-bridge circuits formed the inputs to Hewlett-Packard high-impedance, high-gain, dc amplifiers (Model 8803A) which, when used with the Hewlett-Packard Model 7418A oscillograph, can detect changes as small as 1 wV. The frequency response of the amplifiers were essentially flat to 200 Hz; the oscillograph pen motors were flat to 30 Hz. Prior to testing, the catheter, lead (if any), and transducers were filled completely with deaerated distilled water, great care being taken to ensure the absence of air. After the calibrated transducers were balanced and the performance of the entire system was verified, an abrupt pulse in pressure was produced at the input by manually depressing and releasing the water surface appearing at the top of the 'j8-in. pipe coupling located at the inlet end of the catheter assembly (see Figure 2a). The desired input excitation was a smooth pulse with a duration of sufficient length to produce an acceptable output. For the systems studied, input pulse durations varied from ahout 0.1 to 0.2 S.
Considerable experience and skill were required to produce a satisfactory input for the following reasons: (1) The input pressure must be consistent with the allowable I u 1 maximum limits of the transducers (1 psi) and the disEI.0"IC.I . . @ r J placement of the recording pens (50-mm full scale). (2) canactor The time duration of the input pulse must he comparable b to the duration of pressure pulses occurring in the vascular Figure 2. Catheter terminals and sensing elements: (a) input end: systems of humans or animals and yet be fast enough to (b) output end. excite the dynamics of the system. liminary tests under constant-temperature conditions is Not all tests produced satisfactory records, excessive pen shown in Figure la. The catheter was suspended inside deflection being the most common fault, but too small or a concentric pair of glass tubes about 45-50 mm in diamoddly shaped inputs also occurred on occasion. Tests were eter, with the catheter passing through the center hole of continued until at least four satisfactory sets of records rubber stoppers. The annular space between the glass had been obtained. tubes was sealed by O-rings so that the enclosure could be To determine whether the test results were sensitive to adapted to accommodate catheters of different lengths. the magnitude of the input pulse height, two levels of input At the inlet end the catheter passed through an O-ring pressure excursions were applied in a given series of tests. compression fitting in a 'j8-in. pipe tee. The transducer If the response characteristics of transducers, amplifiers, (Statham PMGTC, f l psi, 350) was connected to the opand pen motors are such that the ratio of the amplitude posite end of the tee. A short nipple and coupling in the of an output sine wave to that of the input wave (norstem position faced vertically upward, as shown in Figure malized) is unity for all frequencies needed to identify the 2a. Water from a temperature-regulated source was circomponent being tested, then in Laplace notation, assumculated through the glass tubes so that most of the catheter ing each component is indeed a discrete linear element, was immersed in water at the selected temperature. the performance function of that component may be deIf the output transducer were close-connected (no lead), fined as the catheter terminated in a male-to-male Luer-Lok couG(s) = (output/input)(s) 1 (2) pling. A short section of rigid tubing 50 mm long with an i.d. of 2 mm was connected to the opposite side, the end As will be shown later, the highest frequency of interest of which terminated near the midpoint of the compression is about 100 radjs (16 Hz), while the measuring system fitting to which the transducer was attached. An O-ring components having the lowest frequency response are the formed the seal, as seen in Figure 2b. galvanometer pen motors which exhibit constant ampliThe narrow plastic lead (i.d. 1.12 mm) was fitted at each tude ratios a t frequencies as high as about 30 Hz (188 end with 18-gauge, blunt-end needle cones, the i.d. of the radjs). The normalized performance functions of all metal inserts also being 1.12 mm. One end was connected measuring system components may thus be assumed to be to the Luer-Lok; the other was sealed with an O-ring in unity. the compression fitting attached to the transducer. The Of course, there is some degree of coupling between arrangement is shown in Figure lb. The lead and catheter transducer and catheter, since it is obvious that the more were anchored to a rigid support to prevent spurious compliant the transducer force-summing diaphragm, the motion. more liquid will be displaced in the catheter conduit during a test. Thus, experimental results will be unique to each When it was discovered that bath temperatures in the range from 27 to 38 "C did not produce discernible difcathetertransducer combination, which seems to be subferences in the dynamic response of several catheters, the stantiated by others. water bath was discarded and the catheters merely supThe extent of coupling as well as the nonlinearities can, ported on a rigid member to which the transducers were in part, be observed by obtaining data using input forcing also secured, as shown in Figure lh. The arrangement of functions with different magnitudes, either forcing sinuthe data system components is also shown here. soids or driving pulses. This was the reason for using The transducers used were capable of following oscilpulses of two distinctly different peak heights in this study. lations a t a frequency of 200 Hz and, when excited with No detectable differences in the results obtained were noted, except for the most responsive catheter with no 10 V, gave a full-scale output of about 50 mV.
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
l5 10
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extension lead. These differences were small indeed.
Data Reduction While certain qualitative inferences may be drawn by inspecting the time histories of the output, more quantitative information is obtained from frequency responses. The ratio of the Fourier transform of the output to that of the input pulses gives the performance function of a particular catheter system, computed as
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-2.8
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Radians/Second
Figure 4. Bode plot obtained by processing data from Figure 3 and comparison to theoretical frequency responses corresponding to second-order systems with damping factors of 0.3 and 0.4.
where PFGw) is the performance function in the frequency domain. Since both f(t);, and f(t),ut are continuous functions of a bounded variation that return to their initial values after a finite time and remain so, as time progresses, the Fourier integrals exist and may be evaluated for any value of w. The results are thus identical with those obtained through direct sinusoidal forcing. The integrals may be numerically evaluated by approximating the oscillograph records as the tops of a number of trapezoids whose bases subdivide the abscissa into n equals increments. Typically
where At is a sectionally fixed increment width. The performance function of the system tested then becomes, in the frequency domain k=n-1
C
PFGw) =
e-jwkAtfout(kAt)
k=l
(5)
k=n-1
e-JwkAtfi,(kAt) k=l
For convenience, the computing program was written to permit time history records to be divided into two sections of different increment widths. This makes it possible to resolve slowly decaying records with fewer data. The initial point of the output time history was assumed to occur a t the same time as the beginning of the forcing pulse.
Determination of Dynamic Parameters A set of oscillograph records is shown in Figure 3. The catheter was a 7F end-hole wedge balloon type (Critikon,
Inc., Tampa, FL) with a 54-cm lead, representative of a moderately responsive system. The illustration shows the subdivision of each time history. The input was divided into 19 increments of 0.01 s each (20 data points); the output was split into two sections, the first section divided into 25 increments of width 0.01 s (26 points) and the second into 25 increments of width 0.02 s (a total of 51 points for the output time history). Pen displacements were read with a precision of about 1/5 division (0.2 mm) with the aid of a hand lens. The printed computer output listed items needed to identify the records, the data, increment widths, scaling factors, the real and imaginary parts of both input and output and their transforms (amplitude and phase angle), and the ratio and log of the ratio of amplitudes and the phase angle differences for each designated pair of records, all corresponding to preselected frequencies from zero to as high as found necessary, usually less than 100 rad/s. In addition, the area under each time history was computed, i.e., the values of the transform at zero frequency. The normalized amplitude ratios and phase angles for the performance function, PFGo), were then plotted on standard graph paper by using log of the magnitude, or dynamic amplitude ratio (DAR), and the phase angle difference, or dynamic response angle (DRA) (in deg), vs. log of the frequency (in rad/s). Such a representation is commonly referred to as a Bode plot. A typical result is shown in Figure 4, which was obtained by processing data taken from Figure 3. This Bode plot indicates excellent conformity to a second-order linear system with a damping factor lying between 0.3 and 0.4 and an undamped natural angular frequency of about 20 rad/s. The dynamics of second-order systems are completely specified by the damping factor and the undamped natural angular frequency, commonly denoted as { and w,, re-
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Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986
Table I. Summary of Dynamic Parameters of Catheter Systems with Different Lengths of Lead Connecting End of Catheter and Transducer lead damping length, freq, rad/s log factor, test catheter description cm W, wP (DAR) iI 7F wedge pressure type, 11/*cm3 max capacity, 120 cm, i.d. 1.80 mm 0 25 23 0.29 0.27 21 0.36 54 17 0,171 0.42 64 19 15 0.140 0.43 83 18.9 15 0.11 I1 Gensini, 8-100; 100 cm, i.d. 1.73 mm 0.07 41.5 40 0 0.84 54 0.25 20 19 0.32 64 22 18.5 0.145 0.388 21 18 0.17 0.36 83 111 8F AEEEI, 1308-35-08 UMI USA, 100 cm, i.d. 1.80 mm 64 22.8 19 0.21 0 325 64 21.8 19 0.20 0.335 64 22.8 20 0.203 0.332 IV 24 21 0.19 7F AIFHC. NIHG 5440 USCI USA, 80 cm, i.d. 1.47 mm 64 0 34 24 21 0.17 64 0.36 V 7F CAFAE, 1308-35-07 UMI USA, 123 cm, i.d. 1.65 mm 21 24 0.19 64 0.343 24 21 0.18 0.35 64 VI 6F CAIFD, 1308-33-85 UMI USA, 82 cm, i.d. 1.45 mm 24 21 0.12 64 0 417 24.7 20 0.12 64 0.417 VI1 6F CBBJH, 1308-35-86 UMI USA, 76 cm, i.d. 1.45 mm 19.2 16.5 0.17 54 0.36 21 19 0.20 64 0.34 18.9 15 0.11 83 0.43 VI11 5F CJHCF, 1308-35-85 UMI USA, 78 cm. i.d. 1.14 mm 20.6 17 0.135 64 0.40 20.8 18 0.181 64 0.35 IX 5 Swan-Ganz, flow directed, balloon type, ‘ 1 5 cm3 capacity, 72 cm. i.d. 1.25 mm 23 12.5 54 0.60 18.5 64 0.70 18.5 64 0 70
spectively. For such systems, the maximum dynamic amplitude ratio is
which occurs a t the forcing frequency
in radians per unit time. Estimations of [, w,, and wp were facilitated by comparing experimental Bode plots to plastic templates of the frequency responses of second-order systems having specific damping ratios. While exact fits to specific profiles could not be expected, except fortuitously, the theoretical forms were very useful for enhancing the consistent evaluations of these parameters. Use of the templates is illustrated in Figure 4 where the amplitude ratios and phase angles for second-order systems with damping factors of 0.3 and 0.4 are shown and may be compared to the computed frequency response for which data appear. The experimental data show that this catheter system behaves very much like a second-order lumped parameter system with a damping factor lying between 0.3 and 0.4, a peak in log (DAR) of 0.19, and an undamped natural frequency (at which the DRA is -90 deg) of about 20 rad/s. From a plot of log (DAR),, vs. {, derived from eq 6, { is found to be 0.34. With this value of {, the ratio of up to u, is obtained from eq 7 . In this case wp/w, = [ l - 2(0.34)2]’/2 = 0.88
(8)
w, is the frequency a t which the DRA is -90 deg, and appears to be close to 20, so that upshould be about 17.5 rad/s, which is consistent with the amplitude data, although it is difficult to ascertain the precise frequency at which this maximum DAR occurs by this method. More sophisticated techniques for retrieving parameters from dynamic data are available but were not warranted.
Scope of Study Nine catheters were tested: two with lead lengths of 0, 54,64, and 83 cm, one with lead lengths of 54 and 83 cm, and the remainder with leads 64 cm long. Table I identifies the catheters and lists other pertinent information. As mentioned previously, all catheters were tested at two levels of input excitation to note any dependency upon the magnitude of the input pulse. Several satisfactory test records were always obtained in each case, from which computed frequency responses could be compared to one another for consistency and reproducibility.
Experimental Results Examination of Time Histories. Typical oscillograph records for the most responsive catheter (Gensini) with leads of 0-84 cm (see test 11, Table I) are shown in Figure sa-d. Because of the high sensitivity of this catheter, especially when no lead was used, the input amplifier attenuation was set a t 200 (0.2 in. of water/mm of pen deflection) with the output a t 500 (0.5 in. of water/mm of pen deflection), except when a lead length of 83 cm was used and both attenuators were set at 500. The deflection of the input pen for the latter test should therefore be multiplied by 2.5 to be comparable to the other input records, as indicated by the broken line in Figure 5d. This series of time histories clearly shows the damping effect introduced by an interconnecting lead between catheter and transducer. Since it was known that such catheter arrangements behave approximately as lightly damped second-order systems, the damped angular frequency can be estimated from the oscillograph time histories following the first cycle of oscillation. For the Gensini catheter mentioned above, for which the time histories appear in Figure 5a-d, the damped frequency changes from approximately 40 rad/s (7 Hz) to 21 rad/s (3.3 Hz) for 0 and 84 cm of lead. respectively. The appearance of an artifact on the input signal is also clearly seen in these records. This is associated with the physics of the input system and is not affected by the catheter arrangement. The frequency of this oscillation,
Ind. Eng. Chem. Fundam., Vol. 25,
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No. 4, 1986 467
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Figure 5. Time histories obtained from pulse tests of Gemini catheter with leads of different lengths: (a) no lead, (b) 54 em, (e) 64 em, and
(d) 83 em.
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F i g u r e 6. Time histories obtained from Dulse tests of a Swan-Ganz catheter with leads of different lengths: (a) 54 em. (b) 64 em. (e) 83 cm (larger input), (d) 83 em (smaller input).
especially evident in Figure 5b, is about 520 rad/s (83 Hz) and is far removed from the frequencies pertinent to the system being tested. For this reason, this input artifact has no discernible effect on the output time histories obtained in this study and was largely ignored when selecting data for computation of input Fourier transforms. The artifact could be attenuated by inserting a loose cotton plug
in the 'Ia-in. pipe coupling a t which the input signal was introduced; in almost all tests such a plug was in place. As the lead length was changed from 54 cm (short) to the longest used in this study (83 cm), the differences in response were difficult to detect by inspection of the time histories. This will be noted from Figure 5a-d and Figure 6a-d, where the results of testing a Swan-Ganz catheter
488
Ind. Eng. Chem. Fundam., Vol. 25, No. 4 , 1986 15 t-
8 Lead Lenglh 8 3 c m
c
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b
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Figure 7. Time histories obtained from pulse test of a Gensini catheter with inputs of two different magnitudes: (a) input pulse peak 12 in. of water (22 mmHg); (b) input pulse peak 2.6 in. of water (4.9 in. of water).
with three different lead lengths are shown (test IX, Table I). The sensitivity of the measuring system was identical for the records shown in Figure 6a-c, the overall sensitivity of each channel being 0.5 in. of water/mm of pen deflection ( A = 500). Figure 6d shows the time histories obtained when testing the Swan-Ganz catheter with a long lead (83 cm) when the input pulse peak was 3.5 in. of water. These records may be compared to Figure 6c, where the input
pulse peak was 13 in. of water. The oscillograph records in Figure 7a,b show the responses of a Gensini catheter with a long lead (83 cm) and suggest that the damped frequency changes from 21 to about 22 rad/s as the pulse peak of the input decreases from 12 to 2.6 in. of water (22 to 4.9 mmHg). For a Swan-Ganz catheter with the same lead length (Figure 6c,d), the effects are not discernible as the pulse peak is decreased from 13 to 3.5 in. of water (24 to 6.7 mmHg). From these results, it is concluded that, when lead length exceeds 50 cm, there is virtually no dependency of frequency response upon the magnitude of the input pulse within the range encompassed in this study. This conclusion is supported by results obtained from the information to be presented below. It is also apparent that, as lead length increases, behavior becomes less dependent on catheter type and more upon lead length, albeit each catheter-lead combination may have unique characteristics. Frequency Response. The frequency response information obtained from the Fourier transform calculations was plotted on standard graph paper showing log of the normalized magnitude or dynamic amplitude ratio, DAR (scale 21/2 in./decade), and the phase angle difference (in deg) between output and input, the dynamic response angle, DRA (scale 20 deg/in.), vs. log of the frequency (in rad/s). These scales were chosen to correspond to those for which plastic profiles or templates of frequency response characteristics for first- and second-order linear systems are available. Table I summarizes the results of an analysis of the Bode diagrams. In several instances the data represent averages of two or more tests; in other cases results from a single test are shown, chosen as the best of a series. The remarkable adherence to purely second-order performance will be noted as especially evident from Figure 4, as well as from Figure 8a,b where the computed fre-
b
a
\
Angular Frequency
I
- Radians/Second
Figure 8. Frequency response of Swan-Ganz catheter with two different lead lengths: (a) 54 cm and (b) 83 cm.
Ind. Eng. Chem. Fundam., Vol. 25, No. 4, 1986 469
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Figure 9. Frequency response of two catheters without leads: (a) Gensini and (b) 7F wedge.
quency response for a Swan-Ganz catheter with a 54-cm lead (test IX) fits almost exactly the theoretical profiles corresponding to the DAR and DRA of a second-order system with a damping factor of 0.6 (Figure 8a). When lead length with this catheter is increased to 83 cm, the damping factor approaches 0.7 (Figure 8b), a condition for which such systems will exhibit no peak in the frequency response. The time histories corresponding to the Bode plots of Figure 8, a and b, are shown in Figure 6, a and c, respectively. Catheters used with no leads are very responsive, having the characteristics of very lightly damped second-order systems. For the Gensini, the damping factor is about 0.07, for a 7F wedge, about 0.27, and evidently somewhat higher for a catheter of the Swan-Ganz type. The frequency response for a Genoini catheter with no lead, derived from time histories almost identical with those shown in Figure 5a, appears in Figure 9a. Computed amplitude and phase angles leave much to be desired, indicating difficulties of extracting frequency response data from pulse-driven lightly damped systems. For such systems, Dreifke (1961) has shown that the slowly decaying oscillatory response can be severely truncated, yet the dynamic parameters can be retrieved. (After a cycle or two following the termination of the input, the output becomes decreasingly independent of the forcing pulse.) The output time history from which Figure 9a was computed was truncated a t about the point indicated in Figure 5a. Refinements in data reduction techniques for such systems can be developed but were considered outside the immediate scope of this study. The undamped natural angular frequency for the Gensini with no lead is about 41 rad/s, which is consistent with the damped frequency derived from the time histories of Figure 5a. The frequency response of a 7F wedge catheter with no lead is shown in Figure 9b. The damping factor is about 0.27, and the undamped natural angular frequency is about 26 radls.
For a given catheter, the dynamic characteristics change as lead length increases, as evidenced by the increase in damping factor. As lead length increases to 83 cm, performance becomes less dependent on the catheter itself and several types exhibit almost identical behavior with lead lengths of 64 cm.
Comparison to the Results of Some Previous Studies As can be seen from the list of references, many studies of the dynamics of catheter-transducer systems have been conducted. Strict comparisons of results from different laboratories are difficult to make in view of the differences in experimental apparatus, especially transducers. The small dimensions of the records of responses, as they appear in publications, also contribute to the difficulties of comparing critical parameters. The greatest disparity between the results presented here and those of others is in the frequency a t which the resonant frequencies occurs. This is obviously dependent upon the dimensions of the catheter, the transducers, the lead, if any, and the dynamic characteristics of the measuring system. Air bubbles are to be assiduously avoided (Bottaccini et al., 1967; Cronvich and Burch, 1969). Assuming their absence, in general, the undamped natural frequencies reported here are considerably less than those reported elsewhere. Cronvich et al. (1969) show frequency responses of a number of needles of different lengths (7-2.6 cm) and gauges (18-30) where the characteristics change from lightly damped with peak frequencies between 9 and 15 Hz (18 gauge, 7 cm) to apparent first-order characteristics (30 gauge, 2.6 cm). Wood et al. (1954), in testing several transducer systems, report peaks in frequency response from about 3 to 60 Hz using a 5 French 100-cm catheter. Fry et al. (1957) report resonant frequencies from 37 to 74 Hz for catheters of several lumen dimensions and
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lengths (4F, 105 cm to QF,150 cm). Falsetti et al. (1974) show natural frequencies for 75 catheters encompassing a range from 10 to 90 Hz, and Rarquest (1975) found peak frequencies from 10 to 80 Hz for lengths of 30-60 cm and i.d.’s from 0.86 t o 1.19 nm. These are in contrast to the values reported herein where undamped natural frequencies for catheters with no lead varied from 3 to 6.7 Hz (19 to 42 radlsj. The most likely explanation for these apparent discrepancies lies in the particular transducers used, which in these studies had a volume of ahout 4 . j mL, somewhat greater than those usually employed in cardiac laboratories. It may. in fact, be possible to draw from studies of pneumatic transmission lines to estimate the critical dynamic parameters from the properties of the system and the fluid contained in the catheter and lead as reported by Hougen et al. (1963) and by Hougen (1979). Here it was shown that the relation given as eq 9, in the Laplace domain, serves to describe the dvnamics of meumatic transmission lines well past their resonant frequencies
P(L,S) ___ -
P(s)
s
+
1 2.0
+1
where
r=
RL(Y2+ Q/uL)”’ 2pc
(10)
is the equivalent second-order damping factor and Un
=
C
L(y2+ Q / U L ) ~ ’ ~
Acknowledgment We thank Ellen McCarty for excellent assistance in the preparation of this manuscript. Nomenclature (I = cross-sectional area of tube ( ’ = velocity of sound in fluid d = internal diameter of tube I, = tube length P, = pressure at input end of tube I)
dn2
pendent only upon the damping factor and undamped iiatural frequency. These dynamic parameters can be derived from frequency response information which, in turn, may be computed from the Fourier transformation of time histories obtained from pulse tests.
Akers. W. W.; Barnard, A. C. L.; Bourland, H. M.; Hunt, W. A,; Timlake, W. P.; Varley, E. Biophys. J . 1966, 6 , 725. Barquest, J. M.; Schmaltzel, J. L. Presented at the American Society of Engineering Education Conference, Colorado State University, Fort Collins, June 1975. Barry, W. H.;Marlon, A. M.; Adams. M.; Harrison, D. C. Cardiovas, Res 1975, 9 , 433. Bottaccini. M. R . ; Burton, D. L.; Lim, T. P. K. J . Appl. Physiol. 1967, 22.
832.
(11)
is the equivalent second-order undamped natural angular frequency. Since the conduit diameter is not necessarily constant for catheter-lead combinations commonly used in practice, these relations cannot be used to predict the dynamic characteristics of all fluid-filled catheter systems. Further experimental work will be required before useful relations can be developed. Conclusions Pulse testing is an acceptable technique for procuring dynamic characteristics of fluid-filled catheters. Testing is easily and rapidly executed and requires no special signal-generating apparatus. This makes the method a logical choice for determining the dynamic parameters of any system. The transient time histories of pressure from water-filled catheters, of the type considered here, closely resemble responses from systems described by second-order ordinary linear differential equations. Consequently, the relationship between input and output for such systems is de-
Cronvich, J. A.; Burch, G. E. Am. Heart J . 1969, 77, 792. Draper. C. S.; McKay, W.; Lees, S. Instrument Engineering; McGraw-Hill: New York, 1953; Vo!. 11. Dreifke, G. E. Ph.D. Dissertation, Washington University, St. Louis, MO, 1961. Dreifke, G. E.; Mesmer, G.; Hougen, J. 0. Trans. Instrum. Soc. Am. 1962, I . 353. Falsei.-H. L.; Mates, R. E.; Carroll, R. J.; Gupta, R. L.; Bell, A. C. Circulation 1974. 4 9 . 165. Fry, D. L.; Noble, F. W.; Mallos, A. J. Circ. Res. 1957, 5 , 40. Fry, D. L . Physiol. Rev. 1960, 4 0 , 753. Glanz, S. A.; Tyberg, J. V. A m . Physiol. Soc. 1979, H376. Hansen, A. T. Acta Physiol. Scand. 1949, 19 (Suppl. No. 68), 7. Hellige. G Basic Res. Cardiol. 1976, 7 1 , 319 (part l), 389 [part 2). Hougen, J. 0 . : Walsh, R. A.; Martin, 0 . R. Control Eng. 1963, 5 , 114. Hougen, J. 0. Measurements and Control Applications, 2nd ed.; Instrument Society of America: Research Triangle Park, NC, 1979 Lees. S.;Hougen, J. 0. Ind. Eng. Chem. 1956, 4 8 . Li, K.-J.; van Brummelen, A. G. W.; Noordergraaf, A. J , Appi. Physiol. 1976. 4 0 , 839. Melbin, J.; Spohr, M. J . Appl. Physiol. 1969, 2 7 , 749. Muzzey, C. L.; Kidd, E. A. Report No. CAL-60. 1954; Cornell Aeronautical Laboratories. Inc., Buffalo, NY. Patel, D J ; Mason, D. T.; Ross, J., Jr.; Braunwald, E. A m . Heart J . 1965, 69, 785. Shapiro, G. G.; Krovetz, L. J. Am. Heart J . 1970, 8 0 , 226. Smith. G A . ; Triplett. W. C. Trans. AMSE 1954, 7 6 , 1383. Wood, E . H ; Leusen, I . R.; Warner, H. R.; Wright, J. L. Circ. Res. 1954, 2 ,
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Yanoff. H. M.; Rosen, A. L.: McDonald, Bioi 1963, 8 , 407
N. M.; McDonald, D A. Phys. Med
Recpu’ed for reLieu June 18,1986 Ac c e pt e d J u l y 15, 1986
