## Abstract

Aortic impedance data of infants, children and adults (age range 0.8–54 yr), previously reported by others, were interpreted by means of three alternative four-element windkessel models: W4P, W4S, and IVW. The W4P and W4S are derived from the three-element windkessel (W3) by connecting an inertance (*L*) in parallel or in series, respectively, with the aortic characteristic resistance (*R*_{c}). In the IVW, *L* is connected in series with a viscoelastic windkessel (VW). The W4S and IVW (same input impedance) fit the data best. The W4S, however, suffers from the assumption that *R*_{c} is part of total peripheral resistance (*R*_{p}). The IVW model offers a new paradigm for interpretation of resistive properties in terms of viscous (*R*_{d}) properties of vessel wall motion, distinguished from *R*_{p}. Results indicated that rapid reduction of *R*_{d}/*R*_{p} during early development is functional to modulation of decay time constant (τ_{d}) of pressure in diastole, such that normalization over heart period (τ_{d}/T) is independent of body size. Estimates of total arterial compliance (*C*) vs. age were fitted by a bell-shaped curve with a maximum at 33 yr. With body weight (BW) factored out by normalization, the *C*/BW data scattered about a bell-shaped curve centered at 66 mmHg. Inertance was significantly higher in pediatric patients than in adults, in accordance with a lower cross-sectional area of the vasculature, commensurate to a lower aortic flow. Changes of arterial properties appear functional to control the ratio of pulsatile power to active power and keep arterial efficiency as high as 97% in infants and children.

- arterial compliance
- arterial inertance
- aortic input impedance
- viscoelasticity
- viscoelastic windkessel

quantitative evaluation of physical properties of arterial systems is important in understanding the dynamics of pressure-flow relationships and implications of alterations in these properties with respect to vascular-ventricular interaction, pressure monitoring, and logical approach to therapy (24, 29, 41). Because overall arterial system properties are presently impossible or impractical to measure directly, they must be estimated indirectly from measurements of pulsatile pressure and flow. Evaluation of aortic input impedance from Fourier analysis of pulsatile pressure and flow has been generally recognized as an important approach in hemodynamics to characterize the systemic arterial circulation and the load it imposes on the left ventricle (LV) (18, 25–32). To interpret impedance data in terms of resistive, reactive, and wave propagation properties, formulation of appropriate model structures is needed (4–15, 36–40, 44). The three-element windkessel model (W3) has been the most widely used and accepted lumped-parameter model of the systemic circulation, for its succinct representation of arterial properties such as total peripheral resistance (*R*_{p}), total arterial compliance (*C*), and aortic characteristic impedance (*R*_{c}) (29, 40). Recent studies by Stergiopulos et al. (37, 38), however, have demonstrated that the W3 model can produce realistic aortic pressures and flows, but only with parameter values of *C* and *R*_{c} that overestimate and underestimate, respectively, the vascular properties. A four-element windkessel model (W4P), derived from the W3 model by introducing an inertial term (*L*) in parallel with *R*_{c} (9, 14, 15, 38), yielded an improvement of data fit and vascular parameter estimation in dogs and humans. The inertial term was given the hemodynamics meaning of total arterial inertance (38). An alternative formulation of the four-element windkessel, obtained from the W3 by connecting *L* in series with *R*_{c} (W4S model), was supposed to be inadequate for a suitable approximation of aortic impedance because it is expected to produce an impedance modulus that deviates towards unacceptable high values with increasing frequency (38). In contrast with this assumption, Yoshigi and Keller (44), who systematically constructed and analyzed 18 analog circuit models to characterize embryonic arterial impedance, had previously reported that the W4S reproduced impedance modulus fluctuation and phase zero-crossing best. In a more recent study, Sharp et al. (36) performed simultaneous high fidelity measurements of aortic pressure and flow in infants and children and applied windkessel-derived models to evaluate the development of systemic arterial input impedance. Among the W3, W4S, and W4P models, the W4S reproduced the impedance data best, while model parameter estimates suggested that significant changes are in progress during vascular development (36). Especially, a significant difference was observed in the development rate of *R*_{c} and *R*_{p}. The reasons for this difference could not be found and remained a challenging issue for further investigation. We hypothesized that this issue could be addressed by involving in the analysis an alternate four-element windkessel, disregarded by Sharp et al. (36), which consists of an inertial element connected in series with a viscoelastic windkessel (VW) (8, 12, 13, 16). This model, referred to as inertance viscoelastic windkessel (IVW) (13), is characterized by a two-zeros and one-pole, flow-to-pressure transfer function identical to that of the W4S (8). Thus, these two configurations are equally suitable to fit the impedance data but are characterized by different physiological interpretations. The IVW model was successfully used previously to describe and interpret the input impedance of terminal aortic circulation in the dog and to explain oscillatory phenomena (8, 13). In the present study, this model configuration was applied to the aortic input impedance of infants, children and adults, compared with the W4S and the W4P, to improve understanding of the development of systemic arterial mechanical properties from infancy to adulthood.

## METHODS

#### Experimental data.

To gain a broader view of vascular development from infancy to adulthood, the aortic impedance data reported by Sharp et al. (36) for infants and children were linked not only with an average aortic impedance reported by Nichols et al. (Fig. 4 in ref. 28), but also with aortic impedance data of individual adults, redrawn from Murgo and Westerhof (Fig. 4 in ref. 26). Details on impedance evaluations are given in these previous works. The average impedance redrawn from Nichols et al. (28) is the same used by Sharp et al. (36) for their study. The age and body weight (BW) of all subjects, along with the mean aortic pressure (MAP) and flow (CO), and *R*_{p} are presented in Table 1. Individual pediatric patients are denoted as C1 to C6. According to Sharp et al. (36), patients C1 and C3 are to be considered infants, whereas the other four patients (C2, C4, C5, and C6) are children. All together, the pediatric patients were denoted as C-group. Individual adults from (26) are denoted as A1 to A5, whereas the average adult from (28) is denoted as AM. All together, these data from adults constitute the A-group.

#### The models.

The structures of windkessel models considered in the present study are displayed in Fig. 1. An inductor, *L*, connected in parallel with *R*_{c} in the three-element windkessel model (W3, ref. 40) yields a four-element model denoted as W4P (9, 14, 38). Connection of *L* in series with *R*_{c} yields an alternative four-element structure denoted as W4S (8). In these models, *L* is supposed to account for the inertial properties of blood motion, *C* is total systemic arterial compliance, *R*_{p} is total peripheral resistance, and *R*_{c} is aortic characteristic resistance. In the W4S model, *R*_{p} is the sum of resistance (*R*) and *R*_{c}. Connection of *L* in series with a viscoelastic version of the windkessel (8, 13) yields the inertance-viscoelastic windkessel (IVW). In this model structure the electrical analog of the Voigt cell, consisting of a resistor *R*_{d} and a capacitor *C*, is assumed to characterize viscoelasticity (12, 16).

#### Parameter estimation.

Total peripheral resistance, *R*_{p} (i.e., the zero-frequency impedance) was computed as the ratio MAP/CO. Input impedance patterns predicted by the W4P, W4S, and IVW models were fitted to individual experimental impedance data to estimate model parameters, save *R*_{p}. Parameter estimation was performed iteratively, such that the normalized root mean square difference (NRMSE) between observed impedance data (D_{i}) and modeled impedance (*Z*_{i}) was minimized (36): (1)

In Eq. 1, Re and Im are real and imaginary parts of measured and modeled impedances, and *m* is the number of harmonics taken into account (26, 28, 36).

Robustness of the parameter estimation process relates to the uniqueness of solution for the parameters and asymptotic standard errors of parameter estimates (4, 17, 19). The problem of uniqueness of solutions for parameters of the two competing models was addressed by structural identifiability analysis (1, 19) as described in the Appendix. Precision of parameter estimates was expressed as percent coefficient of variation: (2) where *p*_{i} is the *i*-th component of the model parameters vector p̄ and SD(*p*_{i}) is the standard deviation of *p*_{i}, which is calculated as the square root of the diagonal terms of the inverse of the Fisher information matrix (17).

#### Statistical analysis.

All data and results are given as means ± SE. Shapiro-Wilk test implemented in Microcal Origin Software was used to evaluate the hypothesis that each data vector or parameter vector considered for statistical analysis had a normal distribution (significance was set at 5% level). For normally distributed samples, two-tailed Student's *t*-test was applied to analyze the differences between two samples. Wilcoxon rank sum test was used to compare samples which were not normally distributed. Comparisons among more than two normally distributed samples were performed with parametric one-way ANOVA. Kruskal-Wallis nonparametric one-way ANOVA was used to compare samples which were not normally distributed. The method of excess variance, also called extra sum of squares (33), was used for statistical comparison of NRMSE from the three competing models. In all cases, a value of *P* < 0.05 was considered statistically significant.

## RESULTS

Example impedance data fits in an infant and an adult are displayed in Figs. 2 and 3, respectively. In these figures the solid line represents the impedance modulus and phase angle predicted by both the W4S and IVW models, while W4P model prediction is represented by the dashed line. The mean NRMSE of data fits (Eq. 1) produced by both the W4S and IVW models, in infants and children, was significantly lower (excess variance analysis, *P* < 0.05) than that produced by the W4P in the same subjects. No significant difference in the quality of data fit was observed in adults. This result indicates that the W4S and IVW models yield a significant improvement over the W4P in fitting impedance data of pediatric patients.

The plots of MAP, CO, and *R*_{p}, as a function of age and BW, are displayed in Fig. 4. With increasing age and BW, MAP (panels *A* and *B*) and CO (panels *C* and *D*) data showed an exponential increase described by the equation: (3) where *Y* is either MAP or CO, and *X* is either age or BW; β is an appropriate asymptotic parameter; and τ is a parameter that characterizes the development rate. Estimates of τ (denoted as τ_{a} and τ_{w} for the functional dependence on age and BW, respectively) and related percent parameter estimation error (CV%) are reported in Table 2. The *R*_{p} data (Fig 4, *E* and *F*) were fitted by an exponentially decreasing function: (4) where *Y* is *R*_{p}, *X* is either age or BW, and α and δ are appropriate parameters. Estimates of τ_{a} and τ_{w} are given in Table 2 with related CV%. The arrow pointer in Fig. 4*E* and *F* indicates that the *R*_{p} value of the 1.5-yr-old child (C2) falls between the values measured in infants C1 and C3, rather than approximating the value measured in the 2-yr-old child (C4).

CO is the product of stroke volume (SV) and heart rate (HR). Fig. 5 shows an exponential decrease of HR with increasing age (panel *A*) and BW (panel *B*). Estimates of τ_{a}, τ_{w} and CV% are given in Table 2. The concomitant increase of SV from infancy to adulthood shows an exponential trend with increasing age (panel *C*) and a linear trend with increasing BW (panel *B*, SV = 1.3·BW, r = 0.99). Mean SV/BW ratio of the C-group (1.30 ± 0.17 mL/kg) was practically the same as that of the A-group (1.30 ± 0.04 mL/kg).

Mean estimates (and related CV%) of *R*_{c} and *R*_{d} parameters provided by the three competing models are compared in Table 3. Functional dependence of *R*_{c} and *R*_{d} on age and BW, as predicted by the three models, was approximated by exponentially decreasing curves as shown in Fig. 6. Estimates of τ_{a} and τ_{w} are given in Table 2 with related CV%. Values of *R*_{c} obtained from the W4P (panels *A* and *B*) showed great variability in infants and children, and the exponential approximation of the decay with increasing age and BW from infancy to adulthood was characterized by high estimation errors on τ_{a} and τ_{w} estimates (Table 2). This is in part due to a huge value of 1,950 dyn·s·cm^{−5} estimated in the 1.5-yr-old child C2. In Fig. 7 the *R*_{c} and *R*_{d} estimates provided by the three models for each individual are plotted vs. the corresponding mean impedance moduli between 2 and 10 Hz (denoted as *Z*_{cexp}). A good linear correlation was found between the W4S-based estimates of *R*_{c} and *Z*_{cexp} data (dashed line through closed triangles in Fig. 7*A*, r = 0.98). By contrast, the dash-dot line (Fig. 7*A*) through the *R*_{c} estimates provided by the W4P (open triangles) showed a deviation towards much higher values in pediatric patients and was characterized by a low correlation coefficient of 0.68. A good linear correlation characterized the scatterplot of IVW-based estimates of *R*_{d} vs. *Z*_{cexp} (solid line through closed squares in Fig. 7*B*, r = 0.97).

Mean estimates and CV% of *C* provided by the three competing models are compared in Table 4. Functional dependence of *C* on age over the full range of 0.8–54 yr (Fig. 8*A*) was described by a bell-shaped curve of the type previously proposed by Langewouters et al. (22) for approximation of pressure dependence of arterial compliance: (5) where the variables *Y* and *X* are for *C* and age, respectively. *P*_{m} parameter is the maximum compliance, which is found when age is equal to *P*_{0}, while *P*_{1} is the half-width age, that is, at the age *P*_{0} ± *P*_{1} the compliance is reduced to *P*_{m}/2. Fitting Eq. 5 to *C* data yielded, for the three models, the estimates of *P*_{m}, *P*_{0} and *P*_{1} reported in Table 5 together with related CV%. With respect to BW, compliance showed a linear (r ≥ 0.86) increase from infancy to adulthood (Fig. 8*B*).

Mean estimates of *L* (and related CV%) over the C-group and the A-group provided by the W4P, W4S, and IVW are compared in Table 6. In accordance with the theoretical solution predicted by equations A6 and A14 (see Appendix), the W4S and IVW models produced the same values of *L*. Estimates from these two models and from the W4P indicate that infants and children are characterized by having aortic impedance data with much stronger inertial character than adults. Functional dependence of *L* estimates on age and BW, as predicted by the three models, showed exponential trends (Fig. 9), with values of τ_{a}, τ_{w}, and related CV% given in Table 2. Estimates of *L* produced by the W4S and the IVW showed great variability in infants and children, and the exponential approximation of the decay with increasing age and BW from infancy to adulthood was characterized by high estimation errors on related τ estimates (Table 2). This is in part due to a huge value of 9.9 dyn·s^{2}·cm^{−5} estimated in the 1.5-yr-old child C2, i.e., the one who presented the unexpectedly high *R*_{p} value indicated by the arrow in Fig. 4, *E* and *F*.

## DISCUSSION

The description of quantitative models of vascular development from infancy to adulthood is important to consider since arterial properties are widely used for clinical investigation in humans (24, 29, 41). The study by Sharp et al. (36) was the first report of simultaneous high-fidelity measurements of aortic pressure and flow, and calculation of aortic impedance in infants and children. Interpretation of impedance data with four-element windkessel models helped to improve understanding of the overall mechanical properties of the circulation in pediatric patients compared with adults. However, an alternative four-element model, denoted as IVW (8, 13), that is supposed to incorporate viscoelastic (rather than purely elastic) properties of circulation was missed by Sharp et al. (36). The behavior of this model in interpreting results as they relate to the issue of age-dependency and vascular growth was analyzed in the present study and compared with that of the W4P and W4S. Our analysis took advantage of additional data from 5 adult human subjects taken from Murgo and Westerhof (26). The small number of cases, however, prevented any conclusion on the role of gender.

Reliability of the parameter estimation procedure used in the present study is supported by the fact that our estimates of W4P and W4S model parameters found in infants and children were similar to those reported by Sharp et al. (36). This was true also for the parameter estimates obtained from the average adult (case AM in Table 1), except for the *L* estimate provided by the W4P, which needs to be reconsidered. According to Sharp et al. (36), this estimate of *L* was 21 dyn·s^{2}·cm^{−5} (RLRC2 model in their Table 3 and Fig. 9). By contrast, our present application of the W4P model to the same impedance data yielded a value of 2.7 dyn·s^{2}·cm^{−5}, i.e., one order of magnitude lower. We consider our estimate of *L* correct in light of the fact that it is consistent with the mean value of 2.8 ± 0.5 dyn·s^{2}·cm^{−5} obtained from our application of the W4P model to the aortic impedance of the A-group (Table 6).

Possible problems in model identifiability (4, 17, 19) need to be considered before proceeding any further with interpretation of results. It is generally accepted that three-element windkessel models admit unique solutions for the parameters, with small parameter estimation errors (4, 10, 12). When an inductance element is added, uniqueness of the solutions for the parameters and the asymptotic standard errors of parameter estimates need to be tested. We addressed the uniqueness problem for the three competing models by structural identifiability analysis (1, 19), as described in the Appendix. With respect to the precision of parameter estimates, the W4S and IVW models provided *L* values with mean CV% of 20% (Table 6). This error reduced to 12% for *L* estimates from the W4P (Table 6). Mean CV% associated with estimates of *C*, *R*_{c}, and *R*_{d} ranged from 4% to 12% (Tables 3 and 4).

A substantial difference among the models displayed in Fig. 1 is that the W4P and W4S are characterized by the *R*_{c} parameter, which is supposed to represent the aortic characteristic impedance (38, 40), while in the IVW an *R*_{d} parameter is introduced, which has a different topological location and a different physiological meaning. Interpretative problems related to the estimates of these parameters in infants, children, and adults are going to be discussed by comparing the W4P and W4S first, and then considering the IVW.

Theoretically, characteristic impedance depends on the physical properties of the vessel under study, while input impedance oscillates at low frequencies because of waves reflected from more distal points and approaches characteristic impedance at higher frequencies. Thus, a rough estimate of *R*_{c} has traditionally been determined experimentally as the arithmetic mean of the input impedance moduli over 2–4 Hz (26–30, 40). In the present study, the averaging was performed in the range between 2 and 10 Hz. This average was referred to as *Z*_{cexp}. A good correlation is shown in Fig. 7*A* (dashed line) between *Z*_{cexp} and the *R*_{c} estimates provided by the W4S model. By contrast, the straight line (dash-dot) that fits the *R*_{c} estimates provided by the W4P model diverges towards a significant overestimation of *Z*_{cexp} in infants and children. Accordingly, the mean *R*_{c} estimates provided by the W4P in pediatric patients (893 ± 216 dyn·s·cm^{−5}) were more than twice (*P* < 0.05) the mean *Z*_{cexp} (382 ± 97 dyn·s·cm^{−5}), while in the same patients *Z*_{cexp} and the mean of *R*_{c} estimates provided by the W4S (329 ± 108 dyn·s·cm^{−5}) were not significantly different. Thus the W4S fits the impedance data better than the W4P, especially in infants and children, as judged from comparing NRMSE of data fit, and is characterized by a *R*_{c} parameter that is better representative of the impedance moduli in the range of major physiological interest, as judged from regression analysis of *R*_{c} vs. *Z*_{cexp} data (Fig. 7*A*) and from *R*_{c} and *Z*_{cexp} mean values. Thus, in accordance with Sharp et al. (36), we considered the *R*_{c} estimates provided by the W4S more suitable for further analysis of development rate compared with that of *R*_{p}.

By linking the results from six infants and children with those from six adults, we were able to characterize the development rate of *R*_{c} as a function of both age and BW, by the parameters τ_{a} and τ_{w}, respectively (Fig. 6*A–D*, and Table 2). These could be compared with the corresponding values that characterize the development rate of *R*_{p} (Fig. 4*E* and *F*, and Table 2). Our results show that after rapid reduction from birth, *R*_{p} reaches the typical value of ∼1,000 dyn·s·cm^{−5} around 10–12 yr of age (Fig. 4*E*). The time constant of this reduction with increasing age (τ_{a} = 2.2 yr) is more than three times higher than the corresponding time constant of *R*_{c} reduction (τ_{a} = 0.6 yr). Indeed, the time constant of *R*_{c} is dominant and characterizes the exponential reduction (τ_{a} = 0.6 yr, CV% = 32%) of the *R*_{c}/*R*_{p} ratio shown in Fig. 10*A*. Similar considerations hold for the functional dependence on BW (Fig. 10*B*). Rapid reduction of *R*_{p} is consistent with rapid development of circulation in early life, critical hallmarks of which are vasculogenesis, the de novo formation of blood vessels, and angiogenesis, the budding of new conduits from pre-existing vessels (35). We convey with Sharp et al. (36) that a reason for the faster change in *R*_{c} cannot be found if we consider that in the W4S structure, *R*_{c} is part (up to 20% in the C-group) of total peripheral resistance (*R*_{p} = *R* + *R*_{c}). This sum also contradicts the meaning of (real) aortic characteristic impedance, which by definition should not incorporate properties of the resistance vessels (arterioles). A further limitation (4, 12) is that the continuous component of pulsatile flow (the cardiac output, CO) is supposed to cross the series resistance, *R*_{c}, thus causing a pressure drop (product of CO times *R*_{c}) quantified as 6% of MAP in infants and children, and 4% of MAP in adults.

The IVW model appears suitable to resolve these limitations. This model provides exactly the same input impedance data fit as the W4S, but is characterized by a *R*_{d} parameter (rather than *R*_{c}), which is supposed to represent viscous losses of vessel wall motion (12, 13, 16). The values of *R*_{d} estimated in infants, children and adults are well correlated with *Z*_{cexp} (Fig. 7*B*). Especially in infants and children, mean *R*_{d} (388 ± 139 dyn·s·cm^{−5}) was not significantly different from mean *Z*_{cexp} (382 ± 97 dyn·s·cm^{−5}). This correlation indicates that the viscous losses of vessel wall motion contribute importantly to determine the impedance modulus in the frequency range of major physiological interest. Indeed, just at the fundamental frequency (HR) the relatively low impedance of the *C*-*R*_{d} cell bypasses the *R*_{p} and, together with *L*, determines the input impedance pattern. An advantage of the IVW over the W4S is that the low-resistance component, *R*_{d}, is detached from *R*_{p}. Thus, the ambiguity of a resistor, *R*_{c}, that is part of *R*_{p} and plays the role of characteristic impedance is eliminated. The damping resistor *R*_{d} does not interfere in the relation between CO and MAP. Rather, it becomes important in modulating the arterial wall response to pulsatility as was shown previously for the terminal aortic circulation (13). Assumption of a viscoelastic, rather than purely elastic, windkessel opens a new paradigm for interpretation of the development of the buffering function exerted by arterial system. According to the elegant description by Safar and Boudier (35), the early phases of vascular growth involve dramatic changes of blood flow and functional coupling of the heart and the arterial system. An optimal design of the arterial tree is such that the pressure rise during systole is minimized (so that myocardial oxygen demands are minimized), and pressure is maintained as high as possible during diastole (to assure blood flow). The process of continuous adaptation of the vessel structure to changing hemodynamic load involves rapid changes in wall thickness, with elastin and collagen accumulation and changes in cross-sectional area and aortic length. Variation in the contribution of vascular smooth muscle cells to accumulation of elastin and the intermolecular collagen cross-linking (2, 3, 35) may affect the viscous, besides the elastic, properties of the viscoelastic windkessel. The rapid reduction of *R*_{d} (Fig. 6*E* and *F*) in the IVW and the concomitant increase of *C* (see Fig. 8 and following paragraph of this Discussion) during development appear to be overall markers of these phenomena.

The main purpose for which windkessel models were developed was to gain insight into the overall elastic arterial properties that permit a damping mechanism through which the cyclic blood flow coming from the heart is changed into flow at the arteriolar (resistant) level (23, 29, 37, 41). The capacitive element of windkessel models is a marker of the overall elastic arterial properties. Unfortunately, the lack of a “gold standard” has brought up the growth of a variety of model-based methods to estimate overall arterial compliance in the absence of generally accepted requisites helpful to state what is best (23, 24, 29, 37). Based on this consideration, it is not possible to definitely state which one, among the W4P, W4S, and IVW models, produces the best estimates of *C*. However, some considerations can be made as to the plausibility of changes of these estimates with age and body size in relation to what is commonly understood in physiology. All three sets of *C* vs. age estimates provided by W4P, W4S, and IVW models could be fitted with bell shaped-curves (Fig. 8*A* and Table 5) that peaked about the age of 33 yr. Thus, in accordance with physiological expectation, the development of the buffering function of thoracic aorta, associated with accumulation and reorganization of elastin and collagen fibers (2, 3, 35), is reflected in the increasing side of the bell-shaped curve that predicts a continuous increase of windkessel compliance from infancy to young adult age. In the elderly, the aorta and elastic arteries stiffen (21, 22, 29, 34, 42). This well-known phenomenon appears reflected in the descending side of the bell-shaped curve (Fig. 8*A*). The increasing trend of compliance with BW (Fig. 8*B*) appears consistent with the increase in vascular dimensions and with the need to accommodate increasing CO as confirmed by the linear relation between *C* estimates from the IVW and CO depicted in Fig. 11 (r = 0.88). The IVW is preferred for its ability to resolve limitations in *R*_{c} of the other two models, as discussed above.

Recently, more attention has been paid to the role of BW, which has been shown to be significantly associated with increased arterial stiffness independently of MAP (21, 42). The scatterplot of *C*/BW estimates vs. MAP displayed in Fig. 12 is an attempt to factor BW out in analyzing the changes of overall elasticity with pressure. In this figure the *C*/BW data are approximated by the bell-shaped curve of Eq. 5 (22), where *Y* and *X* are for *C*/BW and MAP, respectively. *P*_{m} parameter is the maximum normalized compliance, which is found when mean pressure MAP is equal to *P*_{0}, while *P*_{1} is the half width pressure; that is, at the MAP level of *P*_{0} ± *P*_{1}, the normalized compliance is reduced by a half. Estimates of *P*_{m}, *P*_{0}, and *P*_{1} were 0.42 (CV% = 15%) 10^{−4} cm^{5}·dyn^{−1}·kg^{−1}, 66.4 (CV% = 5.6%) mmHg, and 26.3 (CV% = 27%) mmHg, respectively.

In the IVW, the *R*_{p}, *R*_{d}, and *C* terms play a role in the discharge of the capacitor during diastole (diastolic decay of pressure). The related time constant, τ_{d} (Eq. A12), showed a bell-shaped trend (Fig. 13) over the age range of 0.8–54 yr, with a maximum of 2.43 (CV% = 12.7%) s at 28.3 (CV% = 13.1%) yr, and half-width age of 24.9 (CV% = 23.3%) yr. This trend is similar to that found for compliance (compare Fig. 13 with Fig. 8*A*). During early development, while both *R*_{d} and *R*_{p} are decreasing, *C* increases to adapt the behavior of the arterial system as a buffering chamber. As a consequence, the time constant τ_{d} increases, thus contributing to keep diastolic pressure higher during diastole and assuring the adequate level of oxygen supply to tissues. Within the A-group, with *R*_{d} and *R*_{p} keeping constant (Figs. 4*E* and 6*E*), the decrease of τ_{d}, visible in Fig. 13 in the range over 30 yr, reflects the arterial stiffening process with aging.

The increase of τ_{d} during development between infant and child associates with a reduction of HR with increasing age and body size (Fig. 5*A* and *B*), which is supposed to reflect the assessment of a favorable correspondence between cardiac performance and timing of wave reflection, an important factor in limiting aortic systolic pressure and aiding coronary perfusion (7, 11, 29). On average, no difference (*P* < 0.05) was found in the ratio of τ_{d} to the heart period T between pediatric patients and adults (Fig. 14). This result confirms in humans the previous finding that the normalized arterial decay time over heart period is independent of animal size (39).

Because τ_{d} (Eq. A12) can be written as follows: (6) the rapid reduction of the *R*_{d}/*R*_{p} ratio appears functional to correct the increase of the time constant (*R*_{p}·*C*) of the purely elastic windkessel (Fig. 15) in such a way that the normalized arterial decay time over heart period is maintained irrespective of body size.

One more point of discussion relates to the topological location and meaning of the *L* component. It has been demonstrated in previous reports and confirmed here that the introduction of *L* in windkessel structures plays a key role in fitting the impedance data and interpreting alterations of impedance patterns from infancy to adulthood (36). Because in the W4S and IVW models *L* is connected in series at the input of three-element windkessel configurations (Fig. 1), these models provide *L* estimates lower (*P* < 0.05) than those provided by the W4P, where *L* is connected in parallel with *R*_{c} (Table 6). The arrangement of an inertance in series rather than in parallel was questioned by Stergiopulos et al. (38), who considered it logical to have the inertial term in parallel with *R*_{c} (W4P model) because this implies that at very low frequencies, where the local properties of the aorta (*R*_{c}) play a negligible role, it is the total arterial inertance that dominates, whereas at high frequencies it is *R*_{c} that determines the arterial impedance. Stergiopulos et al. (38) also claimed that limitations of the series arrangement of *R*_{c} (W4S model) are *1*) it increases the impedance modulus at all frequencies and results in a model that behaves even less well than the three-element windkessel, and *2*) at higher frequencies, the model impedance becomes too large and differs strongly from the physiological values. In contrast with these supposed limitations, results of the comparative analysis of NRMSE of data fit performed here and those by Sharp et al. (36) show that, in pediatric patients, the W4S and IVW models yield a significantly better fit of impedance data than the W4P. In adults, our present results showed that all three competing models are characterized by producing the same quality of data fit. Although the impedance modulus of the W4S and IVW models is expected to deviate towards unacceptable high values (38), this happens in a high frequency range that is beyond the range of physiological interest (Figs. 2 and 3). Poor performance of the W4P model in the presence of a strong inertial character (infants and children) has been extensively discussed by Sharp et al. (36).

Once windkessel models with *L* in series are considered acceptable, there is a need to discuss the plausibility of *L* estimates as markers of the inertial properties of blood motion in relation to their changes from infancy to adulthood. Considering that inertance is inversely related to cross-sectional area (29, 38) and that, consistently, the *L* estimates provided by both W4S and IVW models decrease with increasing body size from infancy to adulthood (Fig. 9 and Table 6), this parameter can be assumed, to some extent, to be a marker of the inertial properties of blood motion. This conclusion holds also for the W4P model. The fact that in the W4P and W4S, the inductor, *L*, is coupled with the resistor, *R*_{c}, in parallel or in series, respectively, affects the parameter estimation and limits the interpretation of their individual roles (36). This limitation is emphasized in the W4S where *R*_{c} is also part of *R*_{p} as discussed above. These limitations do not affect the IVW model, where the relationship between pulsatile pressure and flow at the input of the arterial system are governed by well distinguished inertial, viscoelastic, and resistive properties. If the *L* estimates of this model are reliable, the high value of 9.9 dyn·s^{2}·cm^{−5} estimated in the 1.5-yr-old child, C2, might be a marker of an anomaly in the very early vascular development. Indeed, in this child (Fig. 4*E* and *F*, arrow), *R*_{p} falls between the values measured in infants rather than approximating the value measured in the two-yr-old child, C4.

If we compare the change of the mean *L* estimates between the C-group and the H-group with the corresponding mean CO measurements (Fig. 16), an inverse relationship is found that is consistent with physiological expectation. An increase of CO occurs with increasing vascular diameter (20, 43) and the expected decrease of inertance.

Eventually, assuming the IVW representation of the arterial system, the suggestive hypothesis arises that early development of *R*_{p}, *R*_{d}, *C*, and *L* between infant and child is functional to favorable settlement of aortic pressure power components (6, 28–31). The average pressure power over a cardiac cycle (active power, Ẇ_{a}), is the sum of steady power (Ẇ_{s}), given by the product of MAP and CO, and pulsatile power (Ẇ_{p}), which is half the sum of the products of amplitudes of corresponding pressure and flow harmonics times the cosine of their phase difference. For a given *R*_{p} and CO, the Ẇ_{s} component is the minimum power required to deliver the given net (mean) flow to the tissues. The Ẇ_{p} component is energy per unit time lost in pulsation and does not contribute to the net flow that is necessary for tissue perfusion. In the expression of Ẇ_{p}, the product of amplitudes of corresponding pressure and flow harmonics can be replaced by the product between impedance moduli and squared amplitude of corresponding flow harmonics. Taking into account that *R*_{d} estimates of the IVW correlate with the average of impedance moduli within 2–10 Hz (Fig. 7*B*), the *R*_{d} term representing viscous losses of vessel wall motion is an important determinant of Ẇ_{p} (*R*_{d} does not contribute to Ẇ_{s}, while *R*_{c} in the W4S does). By reducing the impedance modulus in the frequency range of major physiological interest, reactive and resistive properties contribute to optimize arterial efficiency, defined as E_{art} = 1 − Ẇ_{p}/Ẇ_{a} (6, 31). From pressure and flow components reported by Sharp et al. (36) in their Table 2, we computed Ẇ_{p}, Ẇ_{s}, and their sum, Ẇ_{a}, in the six pediatric patients. Mean Ẇ_{p} was 24.3 ± 9.2 mW, and the ratio Ẇ_{p}/Ẇ_{a} was 0.033 ± 0.007, such that vascular development occurred in these patients with an efficiency, E_{art}, as high as 0.97 ± 0.01.

#### Summary and conclusions.

In summary, we used previously reported impedance data of pediatric (36) and adult subjects (26, 28) and three different four-element windkessel models (Fig. 1) to gain a broader insight into the development of aortic input impedance than was reported previously (36). Our results confirm that the connection of an inertance in series to a three-element windkessel produces best fits of impedance data in the presence of a strong inertial character. This conclusion holds for both the W4S and IVW model structures, which are characterized by the same generalized flow-to-pressure transfer function (Eq. A1) and provide the same estimates of *L*. However, these two models provide different physiological interpretations of impedance development as it relates to resistive and viscoelastic properties. The W4S model suffers from the assumption that *R*_{c} has to conciliate the representation of aortic characteristic impedance with part of total peripheral resistance, *R*_{p}. This contradiction becomes especially evident in infants and children, where *R*_{c} can be as much as 20% of *R*_{p} and causes a pressure drop at zero-frequency that has no physiological explanation. A further challenging point of discussion is that no reason can be found for the faster change of *R*_{c} compared with *R*_{p} during early development as far as *R*_{c} is part of *R*_{p} (36). The IVW model offers a new paradigm for interpretation of the development of systemic arterial mechanical properties, under the assumption that vessel resistance to blood motion, represented by *R*_{p}, is distinguished from a resistance, *R*_{d}, representative of viscous losses of arterial wall motion. Viscoelasticity gains importance towards the periphery of the arterial system and is logically involved in a lumped model of aortic input impedance that, by definition, is supposed to account for the overall system properties and not only of the proximal arteries. Rapid reduction of *R*_{d} during early development contributes to modulate the decay time, τ_{d}, of pressure in diastole and maintain the ratio τ_{d}/T independent of body size. Estimates of compliance, *C*, were fitted by a bell-shaped curve with a maximum at the age of 33 yr. When BW was factored out by normalization, the *C*/BW data also scattered about a bell-shaped curve with a maximum at 66 mmHg. Estimates of inertance were significantly higher in infants and children than in adults. This is consistent with a lower cross-sectional area of the vasculature commensurate to a lower CO. Changes of resistive, inertial, and viscoelastic properties appear functional to control the ratio of pulsatile power to active power and keep arterial efficiency as high as 97% in infants and children.

## APPENDIX

The notion of identifiability is fundamentally a problem in uniqueness of solutions for specific attributes of certain classes of mathematical models. The usual question is whether or not it is possible to find a unique solution for each of the unknown parameters of the model, from data collected in experiments performed on the real system. It is a critical aspect of the modeling process, especially when the parameters are supposed to represent physical attributes of interest and the model is needed to quantify them. If the model is deterministic and the data are noise-free, the problem is generally a nonlinear algebraic one and, unfortunately, solution of this algebraic problem is generally both nontrivial and nonunique for all but the simplest of models. In the presence of real, noisy data, the problem is compounded. Nevertheless, structural identifiability conditions for the noise-free case are minimal, necessary conditions for achieving a successful estimation of model parameters of interest from real input-output data (19). Structural identifiability analysis was put on a formal basis by Bellman and Åström (1). This analysis is used here to address the noise-free problem for the W4S, W4P, and IVW models as one that must be resolved, ascertaining identifiability under ideal circumstances prior to attempting the difficult problem of parameter estimation with real data as discussed in the main text.

#### Uniqueness of W4S and IVW parameter estimates.

The following general expression applies to both the W4S and IVW model structures for characterization of the flow to pressure transfer function in the ascending aorta as ratio of polynomials in the power of *s* (Laplace operator): (A1) This ratio of polynomials in the power of *s* is characterized by four observational coefficients, denoted as a_{1}, a_{2}, τ_{d}, and G, which can be determined from the experimental data (1, 19).

#### Relation of equation A1 to W4S model parameters.

The following relations hold among observational coefficients a_{1}, a_{2}, τ_{d}, and G, and the physical W4S model parameters *L*, *R*, *R*_{c}, and *C*: (A2) (A3) (A4) (A5) Given that G = *R*_{p} is computed as the ratio of MAP to CO, the following unique solution exists for the physical parameters *L*, *R*_{c}, *R*, and *C*: (A6) (A7) (A8) (A9)

#### Relation of equation A1 to IVW model parameters.

The following relations hold among observational coefficients a_{1}, a_{2}, τ_{d}, and G, and the physical IVW model parameters *L*, *R*_{p}, *R*_{d}, and *C*: (A10) (A11) (A12) (A13) Given that G = *R*_{p} is computed as the ratio of MAP to CO, the following unique solution exists for the physical parameters *L*, *R*_{d}, and *C*: (A14) (A15) (A16) Equations A6 and A14 indicate that the W4S and the IVW models admit the same unique solution for the inertial parameter, *L*.

#### Uniqueness of W4P model parameter estimates.

An inertance connected in parallel with *R*_{c} yields the W4P model, which is characterized by the following generalized flow to pressure transfer function, *Z*_{W4P} (*s*): (A17) This ratio of polynomials in the power of s is characterized by five observational coefficients, denoted as α_{1}, α_{2}, β_{1}, β_{2}, and G, which can be determined from the experimental data. The following relations hold among these and the physical parameters *L*, *R*_{p}, *R*_{c}, and *C*: (A18) (A19) (A20) (A21) (A22) Given that *R*_{p} is computed as the ratio of MAP to CO, the following solution exists for the parameters *R*_{c}, *L*, and *C*: (A23) (A24) (A25)

#### Frequency response of the W4P, W4S, and IVW models.

The frequency responses *Z*_{W4P} (jω), *Z*_{W4S} (jω), and *Z*_{IVW} (jω) of the W4S, W4P, and IVW models, respectively, can be expressed in terms of phenomenological parameters like damping factor (ζ), natural frequency (ω_{n}), and time constants (τ_{i}) as described previously by Burattini et al. (8). Relations with physical parameters have also been analyzed in Ref. 8.

## Acknowledgments

This work was supported in part by the Italian Ministry of Instruction, University, and Research (PRIN-COFIN 2004 grant to R. Burattini).

## Footnotes

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