INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

EARLY PRESSURE AND FLOW MEASUREMENTS and subsequent calculations of human adult aortic input impedance were made by Gabe et al. (3) and Patel et al. (11). Subsequent measurements were performed by Mills et al. (5), Nichols et al. (8), Murgo et al. (7), and O'Rourke and Avolio (9). Although considerable variability was found, in many individuals the modulus fell to a shallow minimum of 8-10% of the zero frequency (Z₀) resistance (~1,300 dyn · s · cm⁵) at 4-5 Hz, and a peak occurred at ~7 Hz. Other individuals exhibited modulus curves with dual minima. The phase of the impedance dropped from zero at Z₀ to about 40° at ~1 Hz and then rose to a zero crossing at ~3-4 Hz and rose gradually to ~25° at 5 Hz. Considerable individual variation also existed in the phase.

Murgo et al. (7) found different impedance spectra in groups of young (24 ± 2 yr), older (33 ± 3 yr), and oldest (40 ± 4 yr) adults. In the young group, the modulus displayed a strong minimum at low frequency, followed by relatively constant modulus at higher frequency. The oldest group showed a minimum at higher frequency followed by a maximum at twice that frequency, suggestive of a dominant reflection site within the vasculature. Increased impedance phase was also found in the oldest group. Both of these phenomena were explained by the degeneration (stiffening) of the aorta with age, which would increase wave speed and reduce compliance.

Several differences between infant and child vs. adult impedance were expected. First, because of somewhat smaller aortic pulse pressure but much smaller cardiac stroke volume, it was expected that infant/child vascular systems would exhibit lower compliance. [Compliance may be roughly estimated as stroke volume divided by pulse pressure. Lower compliance is possible in spite of the increased tissue elasticity of younger vessels (13), because compliance is a volumetric strain response, whereas elasticity is a lineal strain response.] Lower compliance would decrease the impedance phase, particularly at low frequency. Second, because average aortic pressures are somewhat smaller in infants and children, but average flow rates (cardiac outputs) are much lower, total vascular resistance must be increased. Because peripheral resistance (R_p) normally provides most of the total vascular resistance, this factor would raise the modulus not only at Z₀ but across all frequencies. Third, because of the shorter vessel lengths in infants and children, shorter wave reflection times and, therefore, modulus minima at higher frequencies were expected.

Several lumped-parameter models [with resistance (R), compliance (C), and inertance (L) elements] have been applied to the adult systemic vasculature (Fig. 1). The two-element model (2) (termed RC) provides the important characteristic of a drop in impedance modulus along with negative impedance phase with increasing frequency, although these characteristics are accompanied by an asymptotic modulus at high frequency of zero and an asymptotic phase at high frequency of 90°, both of which are nonphysiological. The three-element model (19) (termed RCR) provides a significant improvement over the two-element model in that its modulus has a nonzero high-frequency asymptote and its phase has a high-frequency asymptote of zero. However, Burkhoff et al. (1) found that, whereas the RCR model provided a reasonable representation of afterload for predicting integrated measures of cardiac performance, such as stroke volume, stroke work, and average systolic and diastolic aortic pressures, it did not provide realistic aortic pressure and flow waveforms, and it significantly underestimated peak aortic flow. The four-element model (14) (termed RLRC) provides potential for improvement with a minimum modulus at intermediate frequency and increasing modulus for higher frequency (whereas the RCR model has a monotonically decreasing modulus) and positive phase at high frequency. An alternative four-element model (17) (termed RLRC2) also provides a minimum modulus at intermediate frequency but with a constant high-frequency asymptote compared with the increasing asymptote of the RLRC model. The RLRC2 model provides a transition from negative to positive phase at intermediate frequency but with a high-frequency asymptote of zero compared with the 90° asymptote of the RLRC model. Five-element models have also been proposed and compared with more simple models (14, 18).

View larger version (22K): [in this window] [in a new window]   Fig. 1.   Lumped-parameter models of the systemic circulation. R, resistance; C, compliance; L, inertance; Rc, characteristic aortic resistance; Rp, peripheral resistance. RC, 2-element model; RR, 3-element model; RLRC, 4-element model; RLRC2, alternative 4-element model (see text).

This study provides what is thought to be the first report of simultaneous high-fidelity measurements of aortic pressure and flow and calculations of impedance in infants and children. The fits of the RCR, RLRC, and RLRC2 models to the results were evaluated.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Six patients at the Primary Children's Medical Center (Salt Lake City, UT) were enrolled in this study between June 1996 and August 1997. Approval to conduct the investigation was obtained from the local Institutional Review Board, and written informed consent was obtained from the parents of the patients before enrollment in the study. All patients were admitted for repair of simple cardiac defects and had normal aortic anatomy, stable cardiac function, and no previous history of cardiac or thoracic surgery. The patients ranged in age from 0.8 to 4.0 yr (2.2 ± 1.4 yr, mean ± SD) and in weight from 7.2 to 16.4 kg (11.5 ± 3.9 kg). Five of the patients underwent surgery for closure of atrial septal defects, and one underwent surgery for pulmonary valvotomy and patch angioplasty of the right pulmonary artery. The ages and weights of the patients, along with average aortic pressures and flows, are presented in Table 1.

The patients, measurement procedures, and flow and pressure data are the same as those used by Pantalos et al. (10) in the investigation of intra-aortic balloon pump timing errors. Briefly, a transit time ultrasonic perivascular flow probe (10 or 14 mm, 100-Hz frequency response, Transonic Systems, Ithaca, NY) was positioned around the aorta ~1 cm downstream of the aortic valve after surgical exposure of the heart but before cannulation for cardiopulmonary bypass. A high-fidelity catheter tip pressure transducer (5-Fr MPC-500, 5-kHz frequency response, Millar Instruments, Houston, TX) was placed in the aortic root through a hole, and purse-string suture, that was later used for the cardioplegia infusion line at the time cardiopulmonary bypass was initiated. The catheter was advanced down the ascending aortic lumen so that the sensor was at the level of the flow probe. Consequently, the pressure and flow measurements were made simultaneously at the same location in the ascending aorta. Flow and pressure waveforms were recorded on FM tape (MR-30 data tape recorder, 313-Hz frequency response; TEAC America, Montebello, CA).

Segments of the analog recordings were subsequently digitized (GW Instruments MacADIOS analog-to-digital board in a Macintosh 8100 computer running Superscope 2.17 data- acquisition software) over 5 s with 0.005-s sampling interval (10,000 samples). The beginnings of the waveforms were selected from digitized data by identifying the sharp rise in pressure between diastole and systole. The beginning and end of each pressure cycle were similarly identified. Ten cycles were used, except for patient 6, whose heartbeat frequency was low enough that only eight cycles of digitized data were available in the 5-s sampling window. A Fourier transform was calculated for each cycle of pressure and flow. Input impedance was then calculated for each cycle as the ratio of pressure and flow. The real and imaginary parts of the impedance values were averaged, and standard deviations were calculated. A threshold of quality for the resulting average impedance values was established. Harmonics that had standard deviations of either real or imaginary parts >20% of the zeroth harmonic modulus were excluded, along with all higher harmonics for the same patient data. The maximum number of harmonics was arbitrarily limited to 10. These impedance values were compared with the average impedance of five adults (8).

The RCR model was fit to the impedance of each patient and to the adult average. Two fitting procedures were evaluated. First, with the sum of characteristic aortic resistance (R_c) and R_p set equal to the measured modulus at Z₀, R_c and C were found iteratively such that the normalized root mean square difference (NRMS) in measured and modeled impedance was minimized

(1)

where Re and Im are the real and imaginary parts of the measured impedance (Z) and the modeled impedance (A), and m is the number of harmonics satisfying the accuracy criterion. Second, with the sum of R_c and R_p again set equal to Z₀, R_c and C were found iteratively such that the first harmonic modulus and phase were fit exactly. The NRMS was also calculated for this second fitting method.

The four-element models were fit to the measured patient impedance values and to the adult average by procedures similar to the first method above. For the RLRC model, with the sum of R_c and R_p set equal to Z₀, R_c, C, and L were found iteratively such that NRMS was minimized. For the RLRC2 model, with R_p set equal to Z₀, R_c, C, and L were found iteratively such that NRMS was minimized. For each fit, the sensitivity of NRMS to parameter values was tested, including, for the four-element models, variations in L and C with constant LC product (constant crossover frequency; see DISCUSSION).

In addition, arterial compliance was also calculated for the patient data by the area (4) and pulse pressure methods for comparison to the results of the curve fits described above. These calculations were based on the first complete cycle only. Mean systolic and mean diastolic pressures (Table 1) were also calculated from the digitized data for the first cycle only.

General correlation coefficients (r) and 95% confidence intervals (CI₉₅) were calculated for the correlations of results with both patient weight and age. ANOVA comparisons were performed for the results from pediatric patient vs. adults and for results from infants (patients 1 and 3) vs. the group of children (patients 2, 4, 5, and 6). (Although patient 1 was not technically an infant, being slightly older than 1 yr, the similarity of the response of patient 1 and true infant patient 3 promoted this grouping and description of the groups for comparison purposes.)

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Example pressure and flow waveforms used for impedance calculations are shown in Fig. 2. The cycle periods were constant for each patient within ±1 sampling interval except for patient 5, for whom the variation was ±1.5 sampling intervals, and patient 6, for whom the variation was ±2.5 sampling intervals. Mean pressures and flows for each 10-cycle (8-cycle for patient 6) data set are given in Table 1. Although mean flow increased with patient weight and was reasonably correlated (r = 0.886, CI₉₅ = 0.265 < r < 0.988), mean pressure was not well correlated (r = 0.214, CI₉₅ = 0.723 < r < 0.874). Mean resistance calculated by the ratio of mean pressure and mean flow decreased with patient weight and was highly correlated as shown in Fig. 3.

View larger version (30K): [in this window] [in a new window]   Fig. 2.   Example of aortic pressure (AOP; top) and aortic flow (AOQ; bottom) waveforms (patient 2). Ticks on abscissas indicate 1-s intervals.

View larger version (15K): [in this window] [in a new window]   Fig. 3.   Mean R calculated as the ratio of mean pressure and mean flow. Correlation with patient weight is as follows: r = 0.988, 95% confidence interval (CI95) = 0.999 < r < 0.888.

Mean harmonics of pressure, flow, and impedance, along with standard deviations, are provided in Table 2. The impedance values listed are limited to the harmonics satisfying the accuracy criterion (that standard deviation of either real or imaginary parts of the impedance be no greater than 20% of the zeroth harmonic modulus). The accuracy criterion corresponded in all but one case with a modulus of flow <0.02 l/min or a modulus of pressure <0.15 mmHg. (The exception was patient 6, for whom the normalized standard deviations of the real and imaginary parts were 20.46 and 26.05, respectively, for a combination of a relatively low-pressure modulus of 0.21 mmHg and a relatively low-flow modulus of 0.028 l/min.) For patient 1, the number of harmonics was arbitrarily limited to 10, although the accuracy criterion was not exceeded. Sampling frequency may also limit the resolution of the Fourier transforms; however, in these measurements, the sampling frequency (200 Hz) greatly exceeded the frequency of the highest harmonic considered (25.4 Hz for the 10th harmonic for patient 1).

Normalized impedance modulus and phase for all patients are shown in Fig. 4. Two types of responses were found. Type A had low inertance with constant high-frequency asymptote of impedance modulus and a positive, but relatively small, impedance phase at high frequency. Type B had high inertance with impedance modulus that increased at high frequency and high-impedance phase at high frequency. The type A response, which was also facilitated by high R_c, occurred in the two youngest patients (patients 1 and 3), and the type B response, with lower R_c, occurred for the older patients (patients 2, 4, 5, and 6). A comparison of the fits of the lumped-parameter models, using an example of each type of response, is shown in Fig. 5.

View larger version (27K): [in this window] [in a new window]   Fig. 4.   Normalized aortic input impedance of all six patients. Symbols represent individual patients.

View larger version (2222K): [in this window] [in a new window]   Fig. 5.   Comparison of fits of lumped-parameter models to low-L, high-Rc patient 1 (A) and high-L, low-Rc patient 2 (B). , Patient data.

The total vascular resistances (Z₀) of the models were specified by the fitting procedures to be equal to the average impedance moduli at Z₀ (Table 2) for each patient. The Z₀ values, which resulted from the Fourier analysis, thus were nearly identical to the average resistances calculated from the average pressures and flows (Fig. 3). Z₀, therefore, also decreased with patient weight and was highly correlated (r = 0.988, CI₉₅ = 0.890 < r < 0.999). Although R_c in the models showed a general decrease with increasing patient weight, the correlations were not strong (Fig. 6). R_p, however, decreased with patient weight and was well correlated (Fig. 7). C increased with patient weight for all models (Fig. 8). L decreased with patient weight for the RLRC2 model, but no clear trend was evident for the RLRC model (Fig. 9).

View larger version (18K): [in this window] [in a new window]   Fig. 6.   Characteristic aortic resistance for each model. Correlations with patient weight: RCR, r = 0.824 and CI95 = 0.980 < r < 0.037; RCR2, r = 0.842 and CI95 = 0.982 < r < 0.097; RLRC, r = 0.819 and CI95 = 0.980 < r < 0.023; RLRC2, r = 0.494 and CI95 = 0.932 < r < 0.530.

View larger version (17K): [in this window] [in a new window]   Fig. 7.   Rp for each model. Correlations with patient weight are as follows: RCR, r = 0.954 and CI95 = 0.995 < r < 0.630; RCR2, r = 0.975 and CI95 = 0.997 < r < 0.785; RLRC, r = 0.953 and CI95 = 0.995 < r < 0.627; RLRC2, r = 0.988 and CI95 = 0.999 < r < 0.890.

View larger version (23K): [in this window] [in a new window]   Fig. 8.   Arterial C for each model. Correlations with patient weight are as follows: RCR2, r = 0.907 and CI95 = 0.363 < r < 0.990; RLRC, r = 0.852 and CI95 = 0.131 < r < 0.984; RLRC2, r = 0.952 and CI95 = 0.618 < r < 0.995. Statistics for RCR model are not given because of inaccuracy of C values. For reference, arterial C calculated by area (Ref. 4) and pulse pressure methods are also shown: RC, area, incisura to end diastole; RC2, area, peak after incisura to end diastole; RC3, pulse pressure. Correlations with patient weight are as follows: RC, r = 0.748 and CI95 = 0.161 < r < 0.971; RC2, r = 0.910 and CI95 = 0.377 < r < 0.990; RC3, r = 0.961 and CI95 = 0.682 < r < 0.996.

View larger version (19K): [in this window] [in a new window]   Fig. 9.   Arterial L for the two 4-element models (RLRC and RLRC2). Correlations with patient weight are as follows: RLRC, r = 0.222 and CI95 = 0.876 < r < 0.719; RLRC2, r = 0.691 and CI95 = 0.963 < r < 0.275.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Considerable variation among patients in pressure and flow was evident in the waveforms (example in Fig. 2) and in their averages (Table 1). Because the patients were selected on the basis of absence of left ventricular and aortic abnormalities, the variability is not likely due to abnormal cardiac or vascular performance. That all patients had normal left ventricular performance appears to be supported by the pressures and flows being in an acceptable range and by the reasonable correlation of aortic flow with patient weight. The variability in pressure may be due in part to subject variability, including response to anesthesia. The anesthetic induction and maintenance agents used with these patients (fentanyl, pancuronium bromide, isoflurane, and nitrous oxide) are known to cause varying degrees of myocardial depression and arterial vasodilation, which affect the stroke volume and vascular resistance. The patient-to-patient variability in the amount of agent needed to achieve the desired anesthetic plane and the individual cardiovascular response to the agents, coupled with the normal range of cardiovascular regulation, may have contributed to the subject-to-subject variability found in this patient population.

The correspondence of high standard deviation (>20%) of impedance to low pressure and flow moduli approaching the accuracy of the respective measurements suggests that the quality of the impedance averages for increasing frequency was limited at this level by the accuracy of the pressure and flow measurements rather than by inherent variations in the input impedance. Note that considerable fluctuations in impedance (exceeding in some cases twice the value of the individual modulus) nonetheless exist in the harmonics that satisfy the accuracy criterion. These fluctuations may be representative of primary changes in vascular response (due, for instance, to varying vasoconstriction) or of changes in vascular response due to interacting secondary functions (for instance, respiration) but may also derive in part from inherent nonlinearities in vascular response.

For compliant vessels, greater pressure increases cross-sectional area for flow, resulting in lower impedance to flow. The influence of this effect was investigated by testing the correlation of Z₀ with zeroth harmonic pressure for each of the cycles. Decreases in Z₀ with increasing pressure were found in all patients, although correlations, other than for patient 1, were not strong (patient 2: r = 0.153, CI₉₅ = 0.714 < r < 0.528; patient 3: r = 0.392, CI₉₅ = 0.819 < r < 0.315; patient 4: r = 0.561, CI₉₅ = 0.880 < r < 0.107; patient 5: r = 0.312, CI₉₅ = 0.787 < r < 0.395; patient 6: r = 0.365, CI₉₅ = 0.809 < r < 0.343). In the most highly correlated subject (patient 1, Fig. 10), a maximum zeroth harmonic pressure variation among cycles of ~10% corresponded to a maximum Z₀ variation of ~20%. Correlations between higher harmonics of impedance and pressure were not evident. These small differences do not explain the much larger fluctuations observed among cycles in the higher harmonics of pressure, flow, and impedance but suggest that linearity of response should not be assumed in cases in which pressure varies more than in the present measurements (~10%).

View larger version (18K): [in this window] [in a new window]   Fig. 10.   Correlation of average zeroth harmonic impedance with average zeroth harmonic pressure for patient 1: r = 0.900 and CI95 = 0.976 < r < 0.624.

The excellent correlation of mean resistance with patient weight (Fig. 3) appears to indicate that this measure of systemic vascular response was normal despite the effects of anesthesia and other influences and, furthermore, that patient weight was a good scaling parameter (correlation with patient age was slightly lower: r = 0.928, CI₉₅ = 0.993 < r < 0.471, as it was for most results). The average resistance of the heaviest patient was only slightly higher than that of adults, despite the child's weight being only about one-quarter that of adults. On the other hand, the resistance of the lightest patient was approximately three times that of adults. One might expect that conductance (inverse of resistance) per unit weight may be constant, and, indeed, this parameter varied only from 3.2 × 10⁵ to 4.0 × 10⁵ cm⁵ · dyn¹ · s¹ · kg¹ in the patients but was ~1.1 × 10⁵ cm⁵ · dyn¹ · s¹ · kg¹ in adults. Conductance per unit weight was only weakly correlated with weight among the patients [r = 0.222, CI₉₅ = 0.719 < r < 0.876, and P = 0.493 between infants and children (ANOVA)] but was significantly different in infants and children compared with adults (P < 0.001).

Decreases in R_c (Fig. 6) and R_p (Fig. 7) with patient weight were also seen. Because of the wide range of R_c in the patients, the difference in R_c between infants and children vs. adults was not significant (RCR: P = 0.379, RCR2: P = 0.315, RLRC: P = 0.378, RLRC2: P = 0.196), but the difference between infants and children was significant except for the RLRC2 model (RCR: P = 0.015, RCR2: P = 0.059, RLRC: P = 0.018, RLRC2: P = 0.676). In the heaviest patient, R_p was only ~10% higher than that of adults, whereas R_c was still approximately three times higher.

The ratio of R_c to total vascular resistance also dropped with increasing weight (except for the RLRC2 model, which exhibited an R_c value for best fit that was influenced by its parallel inertance). This trend is consistent with the obvious growth of the arteries from childhood to adulthood; however, because R_c represented a small fraction of the total vascular resistance, its development had little impact on Z₀. The normalized R_c values (0.058-0.212) were higher than those for adults [~0.028-0.056 from data by Nichols et al. (8)], but, because of the wide range, the difference was not statistically significant among these groups except for the RLRC2 model (RCR: P = 0.323, RCR2: P = 0.222, RLRC: P = 0.321, RLRC2: P = 0.083). The difference between infants and children, however, was significant (RCR: P = 0.006, RCR2: P = 0.072, RLRC: P = 0.006, RLRC2: P < 0.090). The difference in the R_p-to-R_c ratio (Fig. 11) was significant for comparisons of both infants and children vs. adults (RCR: P = 0.049; RCR2: P = 0.028; RLRC: P = 0.046; RLRC2: P < 0.001) and for infants vs. children (RCR: P = 0.018; RCR2: P = 0.093; RLRC: P = 0.017; RLRC2: P = 0.022).

View larger version (17K): [in this window] [in a new window]   Fig. 11.   Ratio of Rp to Rc (). Correlations with patient weight are as follows: RCR, r = 0.665 and CI95 = 0.319 < r < 0.959; RCR2, r = 0.582 and CI95 = 0.436 < r < 0.946; RLRC, r = 0.642 and CI95 = 0.355 < r < 0.956; RLRC2, r = 0.425 and CI95 = 0.919 < r < 0.590.

The reduction of R_p to a level near that of adults suggests that the peripheral circulation develops rapidly early in life, whereas the remaining difference between child and adult R_c suggests that development of the proximal arteries may follow the slower geometric growth of the body itself. The difference in the development rates is supported by the above comparisons of infant, child, and adult R_p-to-R_c ratio. The reasons for this difference in development rates remain to be investigated. One attractive hypothesis is that R_p, as well as other parameters, may decrease rapidly to reduce hydraulic power requirements. However, in this data set, power increased with patient weight (r = 0.771, CI₉₅ = 0.109 < r < 0.528) and even increased slightly in terms of power per unit weight (r = 0.173, CI₉₅ = 0.743 < r < 0.863).

The decrease in the R_c-to-Z₀ ratio with increasing weight is also evident in Fig. 4; however, other distinguishing features in the impedance plots also stand out. With the exception of patients 1 and 3, the patient data exhibited increasing modulus and phase with increasing frequency at high frequency. The phase increased to 60° or more for patients 2, 4, 5, and 6. This behavior, which is characteristic of inertial flow, is much stronger than for adults. Higher inertance, which scales with l/A, where  is blood density, l is vessel length, and A is cross-sectional area, might be expected in infants and/or children if A scaled with l². Indeed, the modeled inertance values were approximately an order of magnitude larger than those for adults for the RLRC model (Table 3), although the wide range of the infant/child values made this difference statistically insignificant (P = 0.340). Although no trend with increasing weight was evident for L for the RLRC model, L in the RLRC2 model decreased with weight (Fig. 9).

On the other hand, patients 1 and 3 exhibited no significant trend in modulus for high frequency, and phase remained below 35° for patient 1 and lower than 15° for patient 3. Patient 3 was the youngest and lightest subject, whereas patient 1 was the second youngest and third lightest; however, the connection between age or weight and the less inertial impedance behavior of these patients is unclear.

The RLRC model produced the best fits of the impedance data for all patients. Furthermore, this model also produced the lowest values of the Akaike information criterion (AIC) and the Schwarz criterion (SC; see Ref. 17), two indexes that account in different ways for the expectation that models with more parameters should provide better fits (Table 3). A survey of the curve fits for each patient (examples in Fig. 5) reveals the following characteristics.

First, the RCR model was severely limited in its ability to match infant/child impedance because its phase cannot be positive. The curve fits in this work were based on errors between modeled and actual impedance at all harmonics satisfying the accuracy criterion, which included high-frequency harmonics. These curve fits produced unrealistic values of C because increasing compliance produced better (though still poor) fits of the higher harmonics of impedance as the modeled impedance phase approached zero. The best fits in several cases would actually be obtained with infinite C; however, the tabulated C values (Table 3) are those for which further increases in C produced insignificant improvements in NRMS. Better estimates of C were obtained with the RCR model by using fits based on the zeroth and first harmonics only (designated RCR2 in Fig. 14). Yet another compliance estimation method for the RCR model is based on a fit of the zeroth harmonic, an estimate of R_p from an average of the high-frequency harmonics of the impedance modulus and then a calculation of C based on an approximate fit of the first harmonic impedance modulus (18a). This method, however, depends on the first harmonic modulus being larger than the estimated R_p, which was not the case for the infants and children. The method resulted in imaginary values for C.

The area method of Liu et al. (4), another compliance estimation method based on the decay of pressure during diastole according to an RC model of cardiac afterload, produced greatly different C values depending on the interval of diastole chosen. C estimated by the area method is given by

(2)

where is the integrated pressure from time 1 to time 2, and P₁ and P₂ are the beginning and ending pressures. If the interval from the incisura to the end of diastole was used, the small pressure difference in the denominator of Eq. 2 produced large C values. If the interval from the peak in pressure after the incisura to end diastole was used, the much larger pressure difference produced much smaller C values. These two compliance values (for models RC and RC2 in Fig. 8) essentially bracketed the compliances resulting from the curve fits of the lumped-parameter models. C estimated by the pulse pressure method (for model RC3 in Fig. 8) was comparable to the higher area method results. C correlated strongly with patient weight (Fig. 8) and was significantly different in infants and children relative to adults (RCR2: P = 0.024, RLRC: P < 0.001, RLRC2: P < 0.001, pressure wave forms were not available to calculate RC, RC2, and RC3 compliances, and statistics for the RCR model are not given because of the inaccuracy of the C values for this model). C in infants and children was approximately 10 times lower than that in adults. C was not significantly different between infants and children except for the RC3 model (RC: P = 0.383, RC2: P = 0.203, RC3: P = 0.053, RCR2: P = 0.218, RLRC: P = 0.506, RLRC2: P = 0.188).

Second, for the patients with strong inertial character, the RLRC2 model produced impedance modulus curves with sharp minima, which adversely influenced the fits. In addition, its transition from negative phase to positive phase was abrupt and was followed by decreasing phase toward an asymptote of 0° (see Fig. 5B). This model did not appear to fit the character of these patients. For patients 1 and 3, the mismatch of the shape of the modeled impedance to the patient data was less dramatic but still noticeable.

The poor performance of the RLRC2 model can be explained by examining its impedance equation

(3)

which can be nondimensionalized as

(4)

where Z* = Z/Z₀, n = /₁, where  is angular frequency and ₁ is first harmonic angular frequency, L* = ₁L/Z₀,  = R_c/Z₀, C* = ₁R_pC, and Z₀ = R_p for this circuit. The imaginary part of the impedance disappears (the phase crosses through zero) for small nL*/ (L*/ is the ratio of aortic flow dissipation time to cycle period, indicating the relative impedance of the parallel L and R_c flow pathways) and large nC* (C* is the ratio of pressure dissipation time to cycle period, indicating the relative impedance of the parallel R_p and C flow pathways) when the inertance and compliance resonate for a normalized angular frequency of approximately n_X = (L*C*)^1/2. The crossover corresponds to the first imaginary term X1 in Eq. 4, canceling the second imaginary term X2. In the patients with inertial character, the normalized crossover frequency was ~1.4-1.8, which was close to the resonant frequency. At the crossover frequency, nL*/ was small, allowing the first real term R1 to be small. In addition, nC* was large, allowing the second real term R2 also to be small. The impedance, therefore, exhibited a precipitous drop near the crossover frequency (Fig. 12).

View larger version (24K): [in this window] [in a new window]   Fig. 12.   Normalized modulus terms vs. normalized frequency (Z*; see Eq. 4) for patient 2 for RLRC model (A) and RLRC2 model (B). Thick line is resultant modulus. X1, X2, R1, R2: 1st and 2nd imaginary and 1st and 2nd real terms of Eq. 4.

The physical interpretation of this modeled behavior is that the flow substantially bypassed the resistance of the proximal arteries and found a sufficiently large compliance downstream such that it offered little back pressure. For the patients with inertial behavior studied here, this behavior appears unrealistic. For patients 1 and 3, nC* was large enough, but nL*/R_c* was not small enough at the crossover frequency to result in a sharp minimum in modulus.

For the adults represented by Nichols et al. (8), nC* was large, but nL*/R_c* was not small at the crossover frequency; thus a shallow minimum results. In Eq. 4, both R2 and X2 drop sharply with increasing frequency. When the falling X2 matches the magnitude of the rising X1, these two terms cancel at the crossover frequency n_X. Similarly, R1 rises to match the magnitude of the decreasing R2 at a normalized frequency of n_R =  ^1/4 × (L*C*)^1/2 for small nL*/R_c* and large nC*. When these frequencies are close to one another, which occurred for the range of patient values of of 0.2-0.5, the potential for a sharp minimum in modulus exists. (Note in Fig. 12 that the R1-R2 match occurs near the X1-X2 crossover.)

This sharp minimum is prevented if the magnitude of R1 rises above that of X1 before the crossover frequency. This matching frequency, called the bypass frequency, is given by n_b = /L* and corresponds to the frequency at which the impedance of the inertance begins to exceed that of the R_c. Thus the ratio of the bypass and crossover frequencies controls the sharpness of the modulus minimum. If n* = n_b/n_X is greater than one, then a sharp minimum will occur. For patient 2, shown in Fig. 12, n* = 6.71, the bypass frequency occurred beyond the maximum frequency on the graph, and a sharp minimum resulted. The RLRC2 model provided reasonable fits for patients 1 and 3 and for the adult, for which n* = 1.68, 1.07, and 0.65, respectively, but excessively sharp minima for the other patients with larger n* (n* = 6.7, 7.8, 5.2, and 7.8 for patients 2, 4, 5, and 6, respectively).

This sharp minimum, which depends on a delicate balance of inertial and compliant impedance, does not occur with the RLRC model, which produced the best quantitative fits and also provided qualitative shapes better matched to the patient data. Its advantageous features for the patient data included a more gradually and monotonically increasing phase with increasing frequency at high frequency and a softer minimum in modulus. The impedance of the RLRC model is given by

(5)

which may be nondimensionalized as

(6)

where the dimensionless parameters are the same as for the RLRC2 model, except  = R_p/Z₀ and Z₀ = R_c + R_p for this circuit. For the RLRC model, the minimum modulus is defined by where the imaginary terms cancel and where the second real term in Eq. 6 is small. (Note in Fig. 12 that R1 =  provides a floor for the modulus.) In both the infant/child and adult data, the second real term was only a few percent of at the crossover frequency, which was again near (LC)^1/2.

The physical interpretation is that the aortic compliance is large enough to accommodate the flow through the R_c, thereby decreasing impedance to near its minimum, before the frequency becomes high enough for inertance to influence the impedance. In this model, however, impedance is limited to a minimum of by the series resistance. Inertance serves only to increase impedance above the value of with increasing frequency. This concept is consistent with flow through relatively stiff conduits and branches, in which fluid inertia may grow to dominate viscous dissipation in the flow behavior, but viscous dissipation does not disappear and is never bypassed.

The RLRC model performed better than the RCR and RLRC2 models for all infant, child, and adult data in terms of lower NRMS, AIC, and SC (Table 3). The RLRC2 model produced lower NRMS values than the RCR model except for the adult data and produced lower AIC and SC except for the adult data and patient 3. Stergiopulos et al. (17) advocated the RLRC2 model over the RLRC and RCR models for fitting the response of dogs and adult humans; however, the present results are, to our knowledge, the first quantitative comparison of the fits of the two four-element models to human data. Yoshigi and Keller (20) compared the performance of 18 models with two to five elements (including RCR, RLRC, and RLRC2) in matching the impedance of chick embryos and found that the RLRC model and a three-element model resulting from the elimination of the proximal resistor from the RLRC model produced the best fits.

Westerhof et al. (18a) determined that, among many adult mammals, the ratio of R_c to Z₀ (R_c/Z₀) was a constant. In the six patients examined in this project, R_c/Z₀ dropped with increasing weight for the RCR, RCR2, and RLRC models but increased for the RLRC2 model, although none of the correlations were strong. The value of R_c/Z₀ of ~0.1 for the heaviest subjects compared with the Westerhof et al. constant value of 0.055 and the value of 0.042 obtained from the Nichols et al. (8) data suggests that this ratio may change during the development of humans. Such development appears to include a rapid reduction in R_p and a slow reduction in R_c, as discussed above.

Westerhof et al. (18a) also found that the ratio of pressure dissipation time (R_pC) to heart cycle period T (R_pC/T) was a constant, hypothesizing that a lower limit of this ratio was necessary from a cardiovascular function perspective to maintain diastolic pressure high enough to provide adequate coronary flow. C* = R_pC (the same as R_pC/T except for a factor of 2) increased for all models, although correlations were not strong except for the RCR2 and RLRC2 models (Fig. 13). The strong increase in C dominated the decrease in R_p and the increase in T with weight to cause the increase in the ratio. The differences in this parameter were strong between infants and children for the RLRC and RLRC2 models (RCR2: P = 0.575, RLRC: P = 0.073, RLRC2: P = 0.007) and between infants and children vs. adults only for the RCR2 model (RCR2: P = 0.071, RLRC: P = 0.443, RLRC2: P = 0.156). The value of R_pC/T of ~3 for the heaviest subjects compared with the values calculated from the Nichols et al. (8) data of 3.7 and 4.1 for the RCR2 and RLRC models, respectively, suggests that this ratio may continue to rise into adulthood. However, the Westerhof et al. (18a) value of 2.19 for adult humans suggests that this ratio may reach a maximum at some intermediate age before falling to the adult value. Such behavior would be consistent with a drop in compliance in older individuals with constant R_p and heart rate. More data are required to conclusively support either suggestion.

View larger version (23K): [in this window] [in a new window]   Fig. 13.   Ratio of pressure dissipation time to heart cycle period (C*). Correlations with patient weight are as follows: RCR2, r = 0.991 and CI95 = 0.916 < r < 0.999; RLRC, r = 0.310 and CI95 = 0.670 < r < 0.896; RLRC2, r = 0.930 and CI95 = 0.484 < r < 0.993. Statistics for RCR model are not given because of inaccuracy of C values.

The behavior of compliance per unit weight depended strongly on the model. Compliance per unit weight correlated strongly with weight only for the RLRC2 model (RC: r = 0.707, CI₉₅ = 0.246 < r < 0.965; RC2: r = 0.795, CI₉₅ = 0.046 < r < 0.977; RC3: r = 0.807, CI₉₅ = 0.013 < r < 0.978; RCR2: r = 0.860, CI₉₅ = 0.159 < r < 0.984; RLRC: r = 0.469, CI₉₅ = 0.553 < r < 0.928; RLRC2: r = 0.936, CI₉₅ = 0.517 < r < 0.993). The mean compliance per unit weight of the patients was nearly the same as in adults for the RLRC model but varied significantly between infants and children vs. adults for the RCR2 and RLRC2 models (RCR2: P < 0.001, RLRC: P = 0.931, RLRC2: P < 0.001). No significant differences were found between infants and children except in the RC3 model (RC: P = 0.428, RC2: P = 0.265, RC3: P = 0.053, RCR2: P = 0.225, RLRC: P = 0.695, RLRC2: P = 0.188).

The ratio of flow dissipation time L/Z₀ to heart cycle period T (L*) was also evaluated (Fig. 14). This flow dissipation time represents the characteristic time for the inertia in the flow to be dissipated by the total vascular resistance. No strong correlation was apparent in this ratio with patient weight. However, L* varied significantly between infants vs. children but not between infants and children vs. adults (RLRC: P = 0.165 and RLRC2: P = 0.880 for infants and children vs. adults; RLRC: P = 0.039 and RLRC2: P = 0.063 for infants vs. children). L* was low in infants, high in children, and low in adults. High L* was connected with inertial behavior in the children.

View larger version (19K): [in this window] [in a new window]   Fig. 14.   Ratio of flow dissipation time to heart cycle period (L*). Correlations with patient weight are as follows: RLRC, r = 0.270 and CI95 = 0.693 < r < 0.887; RLRC2, r = 0.662 and CI95 = 0.958 < r < 0.324.

Scaling arguments were used to compare wave reflection effects in the pediatric patients to those in adults. Wave speed (W) may be estimated by

(7)

where Va is arterial volume. If  is assumed to be constant with patient weight and Va is assumed to scale directly with patient weight, then the 2-16 times smaller compliance in infants and children might combine with the 4-10 times smaller volume to produce little difference in wave speed. Indeed, when compliance per unit weight was calculated from the pediatric patients and compared with values for the average adult, no trend or strong correlation with weight was found for the RLRC model. With the assumption that artery length scales with the cube root of weight, wave reflection times estimated from the wave speeds and artery lengths were 1.6-2.2 times shorter in the infants and children than in the average adult. However, the heart cycle periods in the patients were also 1.5-2.3 times shorter. Thus no significant differences in wave reflection effects between infants and children vs. adults were found. These estimates appear to support the speculation that wave reflection time is regulated to minimize cardiac work by avoiding reflection of high-pressure pulses during the same systolic period (6).

In summary, these results provide the first calculations of aortic input impedance in pediatric patients. For four of the six patients, the results exhibited strong inertial character compared with adults, with increasing modulus and large positive phase with increasing frequency at high frequency. The RLRC lumped-parameter model reproduced the character of the impedance data best. For this model, compliance and R_p/R_c differed greatly from those of adults. Furthermore, R_c and R_p/R_c varied considerably between infants and children, and R_p and Z₀ were well correlated with patient weight. These comparisons demonstrate that the overall character of the circulation is different in infants and children compared with adults and suggest that significant changes are in progress during development from infant to child.