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Eur J Obstet Gynecol Reprod Biol. Author manuscript; available in PMC 2010 November 29.

Published in final edited form as:

Published online 2009 March 20. doi: 10.1016/j.ejogrb.2009.02.030

PMCID: PMC2993013

NIHMSID: NIHMS112029

Yanmei Zhu,^{1} Benjamin J. Sprague,^{1,}^{2} Terrance M. Phernetton,^{2} Ronald R. Magness, Ph.D.,^{2,}^{3,}^{4} and Naomi C. Chesler, Ph.D.^{1}

The publisher's final edited version of this article is available at Eur J Obstet Gynecol Reprod Biol

See other articles in PMC that cite the published article.

Changes in uterine vascular impedance may yield diagnostic insight into physiological and pathological changes in uterine vascular resistance and compliance during the ovarian cycle and pregnancy. Herein, our objectives were to develop a model to simulate uterine vascular impedance in order to gain insight into the vascular size and stiffness changes that occur during the ovarian cycling and pregnancy.

Two electrical analogue transmission line models were developed and evaluated based on goodness-of-fit to experimental impedance measurements, which were obtained in nonpregnant luteal and follicular phase (NP-L and NP-F) and pregnant (P) ewes (n=4-8 per group). First, an anatomically-based, multi-segment, symmetric, branching transmission line model was developed. Parameter values were calculated based on experimental measurements of size and stiffness in the first three generations of the uterine arterial tree for NP-L, NP-F and P ewes. Then, a single segment transmission line model was developed and effective parameter values were optimized to best-fit the measured impedances.

The anatomically-based multi-segment model did not yield the expected good agreement with the experimental data (R^{2}>0.5 for all groups). In contrast, the impedance spectra predicted by the single segment model agreed very well with experimental data (R^{2}=0.93, 0.82, and 0.84 for NP-L, NP-F and P, respectively; P<0.0001, all groups). Furthermore, the changes in the best-fit model parameters for NP-F and P compared to the NP-L were consistent with the prior literature on the effects of the ovarian cycle and pregnancy on vascular resistance and compliance. In particular, compared to NP-L, NP-F had decreased longitudinal and terminal resistance with a modest increase in compliance whereas pregnancy caused more dramatic drops in longitudinal and terminal resistance and a significant increase in compliance.

The single segment transmission line model is a useful tool to examine changes in vascular structure and function that occur during the ovarian cycle and pregnancy.

During the ovarian cycle, changes in sex hormone (i.e., estrogen and progesterone) levels increase uterine blood flow approximately 2-fold (1-3). Since little to no change in perfusion pressure occurs (1-3), uterine vascular resistance (UVR) decreases approximately 2-fold. In healthy pregnancies, uterine blood flow increases by 20-50-fold, again with no change in perfusion pressure (2, 4-7); thus, UVR decreases 20-50-fold. The failure of UVR to decrease adequately with pregnancy is a hallmark of preeclampsia (8-11). Decreases in uterine vascular compliance also occur during pregnancy (12, 13). A decrease in systemic arterial compliance has been shown to occur with preeclampsia, which contributes to insufficient uterine perfusion (14, 15), but the consequences of preeclampsia for uterine artery compliance changes during pregnancy are not known. Furthermore, the ways in which these resistance and compliance changes affect blood flow waveforms, which are measured clinically for diagnosis of preeclampsia with or without intrauterine growth retardation (14, 16, 17), are not well understood.

Uterine vascular impedance is a function of blood pressure and flow that, unlike UVR, is sensitive to changes in waveform that are caused by longitudinal arterial resistance and arterial compliance. Whereas UVR measures the opposition to steady flow, impedance measures the opposition to pulsatile flow and is affected by pulse wave reflections in the vascular bed. Also unlike UVR, impedance is complex and frequency-dependent. Thus, it can be difficult to gain simple insights into vascular structure and function changes from an experimentally measured impedance spectrum. A computer model of pulsatile blood flow dynamics in a vascular network that captures the dependencies of hemodynamic function (i.e., impedance, based on pulsatile pressure-flow relationships) on vascular structure (i.e., size, number and mechanical properties) can be a useful tool to explore mechanisms by which physiological and pathological changes in blood flow patterns arise during the ovarian cycle and with pregnancy.

Many features of pulsatile pressure and flow in a vascular system can be modeled by an electrical circuit model in which blood flow and pressure are represented by electrical current and voltage, respectively (18-20). Either lumped parameter or transmission line models can be used, but only transmission line models capture details of wave reflections that are critical to pulsatile pressure-flow relationships (21). Three dimensional finite element (or finite volume or finite difference) numerical models can also be used to predict blood pressure and flow patterns but only if the walls are modeled as compliant and physiologically realistic outlet boundary conditions are used. The use of compliant walls in a complex, branching vascular network requires significant computational resources; furthermore realistic pressure and flow data below the arteriolar level in the uterine circulation are not available. Thus, here we used transmission line models to predict uterine vascular pulsatile pressure-flow relationships during the ovarian cycle and with pregnancy.

A transmission line model of the uterine vasculature was first proposed by Mo *et al.* (22). This model represented each arterial segment as a resistor (*R*) in series with an inductor (*L*) to model the vessel longitudinal impedance, and one capacitor (*C*) to model the elasticity of the vessel wall. A terminal load resistor (*R _{L}*) ensured appropriate steady state behavior. Hill

In this study, two transmission line models were developed and evaluated for their ability to predict ovine uterine vasculature impedance measured at two time points in the ovarian cycle and in pregnancy. The first was an anatomically-based, multi-segment, symmetric branching model in which parameter values were selected based on experimental measurements. The second was a single-segment, effective model in which parameters values were optimized to best-fit experimentally obtained impedance spectra. The effects of the ovarian cycle and pregnancy on parameter values were investigated. In addition, we investigated the effects of the single-segment model parameter values on impedance spectra and blood flow.

Based on the work by Mo *et al*. (22), we modeled a single artery in the uterine arterial tree with a distributed resistor (*R*) in series with a distributed inductor (*L*) representing the vessel longitudinal resistance and inertance, respectively, and one distributed capacitor (*C*) representing the elasticity of the vessel wall. For any given arterial segment in the transmission line model, the propagation coefficient (*γ* ) and characteristic impedance (*Z*_{C}) for each segment are calculated as:

$$\gamma =\sqrt{(R+jwL)\left(jwC\right)}$$

(1)

$${Z}_{c}=\sqrt{\frac{R+jwL}{jwC}}$$

(2)

where $j=\sqrt{-1}$ and *w* is the radial frequency (*w* = 2 π *f* where *f* is a frequency component of the heart rate) (24). For a segment of length *l*, if the downstream load impedance is *Z _{L}*, the input impedance upstream is (24):

$${Z}_{\text{in}}={Z}_{C}\left[\frac{{Z}_{L}+{Z}_{C}\phantom{\rule{thinmathspace}{0ex}}\mathrm{tanh}\left(\gamma l\right)}{{Z}_{C}+{Z}_{L}\phantom{\rule{thinmathspace}{0ex}}\mathrm{tanh}\left(\gamma l\right)}\right]$$

(3)

A preliminary analysis of uterine arterial tree structures in pregnant and nonpregnant ewes (see Figure 1) revealed no significant differences in number of arteries in the first three generations. Therefore, for our anatomically-based transmission line model, we used the same basic structure for nonpregnant and pregnant cases. In particular, a three-level symmetric branching tree structure in which four identical daughter branches (tertiary, *k*=3) originated from each of two identical daughter branches (secondary, *k*=2), which originated from the main uterine artery (primary, *k*=1), was assumed (Figure 2A). Then, for all identical segments at a given level *k*:

$${Z}_{k,\text{in}}={Z}_{k,C}\left[\frac{{Z}_{k,L}+{Z}_{k,C}\phantom{\rule{thinmathspace}{0ex}}\mathrm{tanh}\left({\gamma}_{k}{l}_{k}\right)}{{Z}_{k,C}+{Z}_{k,L}\phantom{\rule{thinmathspace}{0ex}}\mathrm{tanh}\left({\gamma}_{k}{l}_{k}\right)}\right]$$

(4)

And because of the symmetry of the branching structure,

$${Z}_{k,L}=\frac{1}{{n}_{k+1}}{Z}_{(k+1),\text{in}}$$

(5)

where *n _{k+1}* is the number of identical elements that originate from the end of segment

A representative Mercox casting of the uterine arterial tree from a single horn of a nonpregnant ewe.

Transmission line models used to simulate the ovine uterine vasculature impedance. A. Anatomically-based, multi-segment, symmetric branching model and B. Single-segment effective model.

Parameter values for this multi-segment, symmetric branching model were based on experimental measurements of primary, secondary and tertiary arteries in the uterine vascular bed in nonpregnant (both follicular and luetal phase) and pregnant ewes. To calculate these parameter values, we made several assumptions consistent with the prior literature (18, 19, 22). First, blood was assumed to be Newtonian and incompressible such that the Navier-Stokes equations could be used. Second, blood flow in each arterial segment was assumed to be fully-developed and symmetric about the axis. Third, leakage from the small branches of the main artery that supply the cervix area were ignored. And, fourth, we assumed that artery walls were linearly elastic and nonviscous with elastic modulus *E _{k}*. Given these assumptions, the parameter values

$${R}_{k}=\frac{8\phantom{\rule{thinmathspace}{0ex}}\mu}{\pi {{r}_{k}}^{4}}\left[\frac{Ns}{{m}^{6}}\right]$$

(6)

$${L}_{k}=\frac{9\phantom{\rule{thinmathspace}{0ex}}\rho}{4\pi {{r}_{k}}^{2}}\left[\frac{N{s}^{2}}{{m}^{6}}\right]$$

(7)

$${C}_{k}=\frac{3\pi {{r}_{k}}^{3}}{2{E}_{k}{h}_{k}}\left[\frac{{m}^{4}}{N}\right]$$

(8)

where *r _{k}* is the vessel internal radius at a physiological pressure,

We also developed a non-anatomical single-segment transmission line that “effectively” represents the uterine vasculature (Figure 2B). Hence, we call it an single-segment effective transmission line model. In this case, the parameters *R*, *L*, *C* and transmission line length *l* do not reflect characteristics of individual arterial segments but the overall arterial network behavior. For a single-segment transmission line, *Z _{1,in}* is given by equation (4) with

Both the multi-segment and single-segment effective models were implemented in Matlab (The MathWorks Inc. Natick, MA). Predicted input impedance spectra were calculated based on the appropriate set of parameter values in the frequency range of interest (0-10 Hz). The maximum frequency (10 Hz) was chosen to be five times the normal heart rate (~2 Hz). The models were first used to repeat Hill’s work (23) on wave transmission model in the umbilicoplacental circulation. Once the computations were thus checked, each model was used to calculate the uterine vascular impedance magnitude spectra in the nonpregnant luteal phase (NP-L), nonpregnant follicular phase (NP-F) or pregnant (P) state. Goodness-of-fit was evaluated based on the square of the correlation coefficient between the predicted and measured impedance spectra (R^{2} value) and the significance of the correlation (p value).

Procedures for animal handling and protocols for experiments were approved by the University of Wisconsin-Madison Research and Animal Care and Use Committees of both the Medical School and the College of Agriculture and Life Sciences. Data from three groups of multiparous female sheep of mixed western breeds were used: two nonpregnant groups in different phases of the ovarian cycle and one late-gestation, pregnant group. The nonpregnant (NP) animals (n=13) were experimentally synchronized and sacrificed during two distinct controlled points of the ovarian cycle. These were the late follicular phase during the peri-ovulatory period (NP-F, n=7) and the late luteal phase 10-11 days post ovulation (NP-L, n=6) as described previously (26). The pregnant group (P, n=10) were studied at 120-130 days gestation. The natural gestation length for sheep is 145-147 days.

Uterine arterial blood pressure (*UAP*) and heart rate (*HR*) were measured via a fluid-filled catheter inserted near the uterine artery, after anesthetization. Uterine blood flow (*UBF*) through the primary uterine artery was measured using a 3 mm or 6 mm ultrasound probe (3RS and 6RS probes, Transonic Systems Inc. Ithaca, NY), depending on the size of the vessel. *UAP*, *HR* and *UBF* were sampled simultaneously at 1000 Hz using WinDaq Pro Data Acquisition software (DATAQ Instruments, Akron, OH).

To calculate experimental uterine vascular impedance spectra, *UAP* and *UBF* waveforms for ten sequential cardiac cycles were isolated and independently analyzed. Values were averaged for each animal. Coefficients of variation within each animal were less than 8%. Digital signal processing prior to frequency analysis included Hann windowing and zero-padding. These processed waveforms were then resolved into their frequency components using a discrete fast Fourier transform. The ratio of pressure to flow at each harmonic of the heart rate was computed as previously described (11, 17, 27). Here we consider only impedance magnitude because the distance between the sites of pressure and flow measurement made the impedance phase calculation unreliable and because the impedance magnitude typically contains more clinically relevant information (28).

After the hemodynamic measurements were obtained, right and left uterine artery outer diameters were measured (OD* _{in vivo}*). Following sacrifice and hysterectomy, the uterine artery branching networks from each horn of the uterus were dissected and kept intact. Within 30 minutes, Mercox acrylic casting material was perfused into the artery network until the left and right uterine artery outer diameters matched the OD

Second generation arteries from each uterine arterial network were harvested for measurement of circumferential elastic modulus *E* as described in detail elsewhere in this issue (29). Briefly, arteries were isolated, harvested and perfused *ex vivo*. Outer diameter and length were measured during pressurization from 0 to 120 mmHg in 10 mmHg increments. Circumferential stress and strain were calculated using standard formulations and the slope in a physiological pressure range was taken as *E*. For additional methodological details, we refer the reader to (29).

The measured impedance magnitude spectra for the luteal and follicular phase and pregnant ewes are shown in Figure 3. As expected, pregnancy decreased the impedance dramatically compared to the two nonpregnant cases for all frequencies. Impedance magnitude in the follicular phase was also lower than in the luteal phase at all frequencies. In both nonpregnant cases, the impedance reached a minimum at the 6^{th} harmonic and a maximum between the 8^{th} and 9^{th} harmonics. In the pregnant case, the impedance reached a minimum near the 5^{th} harmonic and a maximum at the 8^{th} harmonic, at which point it was nearly equal to the input resistance (at 0 Hz) in magnitude.

Experimental measurements of ID* _{m}*, WT

Experimental measurements of uterine artery inner diameter (ID_{m}), wall thickness (WT_{m}) and length (Length_{m}) in each of the first three generations of the uterine arterial network (*k*=1, 2, 3) for luteal and follicular phase and pregnant ewes. Circumferential **...**

Parameters values for the multi-segment transmission line model in which, for each generation of the uterine arterial tree (*k*=1, 2, 3), longitudinal resistance and inertance, and transverse compliance (*R*, *L*, and *C*, respectively) per unit length (*l*), were **...**

The predicted input impedance magnitude spectrum at the primary uterine artery (*Z _{1,in}*) was computed using equations (4-5) in an iterative fashion. As shown in Figure 4, the agreement with the experimentally obtained impedance spectra was poor. All correlation coefficients (R

Experimental measurements (symbols) and multi-segment model prediction (line) of uterine vascular impedance magnitude for (A) luteal and follicular phase (R^{2}= 0.30 and 0.27, respectively; p>0.05) (B) and pregnant ewes (R^{2} = 0.04; p>0.05). **...**

Since elastic modulus is known to increase with increasing distance from the heart (30), we also tested the effects of letting ½ *E _{1}* =

The best-fit *R*, *L*, *C* and *l* values for the single-segment transmission line model of uterine vascular impedance for the luteal and follicular phase and pregnant ewes are given in Table 3. In contrast to the anatomically-based multi-segment model, the single-segment model yielded impedance spectra that agreed well with experimental data (R^{2}=0.94, 0.84 and 0.87 for luteal, follicular and pregnant, respectively; Figure 5). All correlations were also highly significant (p<0.0001 for all groups).

Experimental measurements (symbols) and single-segment effective model prediction (line) of uterine vascular impedance magnitude for (A) luteal and follicular phase (R^{2}=0.93 and 0.82, respectively; p<0.0001) and (B) pregnant ewes (R^{2}=0.84; p<0.0001). **...**

Parameters values for the single-segment transmission line model in which longitudinal resistance and inertance, and transverse compliance (*R*, *L*, and *C*, respectively) per unit length (*l*), were modeled with distributed elements based on best-fit to the **...**

Compared to the luteal phase, best-fit length was relatively unchanged in the follicular phase but inertance *L* was nearly half, compliance *C* was approximately double and longitudinal resistance was five orders of magnitude less ($\sim 1\times {10}^{5}\phantom{\rule{thinmathspace}{0ex}}\frac{Ns}{{m}^{6}}$). With pregnancy, length approximately tripled, *L* decreased 30-fold, *C* increased 6-fold, and *R* remained small ($\sim 1\times {10}^{5}\phantom{\rule{thinmathspace}{0ex}}\frac{Ns}{{m}^{6}}$). *R _{L}* changed according to the measured changes in

To demonstrate the dependence of the shape of the impedance spectra on parameter values, we changed parameter values by plus or minus 20% of the original value (one at a time) and calculated the impedance spectra. Due to the large number of degrees of freedom in the multi-segment model, we used the single-segment model and only present data for the luteal phase and pregnant spectra (Figure 6). As expected, increasing *UVR* (which is equal to *R _{L}* in this case) increased the input resistance. Increasing

Predicted impedance spectra for (A, B, C) luteal phase and (D, E, F) pregnant ewes for variations in (A, D) *UVR*, (B, E) *C* and (D, F) *L* in the single-segment effective transmission line model. In each case, parameter values were varied by ± 20% **...**

As a consequence of these changes in spectra, for any pressure waveform, the flow waveform into the uterine vasculature will change. To demonstrate, we imposed a sinusoidal pressure waveform at 2 Hz (*HR*=120 bpm) with mean pressure of 100 mmHg and a pulse pressure of 40 mmHg. The resulting predicted mean UBF for the range of *UVR*, *C* and *L* values used above (Figure 6) are shown in Figure 7. As expected, the mean *UBF* is strongly inversely dependent on *UVR*. Since decreasing *C* tends to increase impedance near 2 Hz, the consequence of decreasing *C* is to decrease blood flow for the same driving pressure. Changing *L* by ±20% of the best-fit value has little effect on the impedance magnitude near 2 Hz, so varying *L* in this range has minimal effect on mean blood flow. Overall, the effects of these parameters in the luteal and follicular (not shown) phase and in pregnancy are similar.

Here we present two transmission line models of the uterine vasculature in order to gain insight into the vascular size and stiffness changes that occur during the ovarian cycle and with pregnancy. Our major findings are that the effects of the ovarian cycle and pregnancy on the vasculature can be represented by decreased longitudinal and terminal resistance with a modest increase in compliance in the follicular phase (compared to the luteal phase) and more dramatic drops in longitudinal and terminal resistance and a significant increase in compliance with pregnancy.

We note that the anatomically-based, multi-segment, symmetric branching transmission line model poorly predicted the measured impedance spectra for all groups (NP-L, NP-F and P). The values used to calculate the model parameters, including *r _{k}*,

Using a single-segment effective transmission line model and the method of least squares to obtain optimized parameter values enabled excellent fitting of the experimental hemodynamic data (R^{2}>0.8 for all groups). We note that our optimization algorithm may have found local minima and not the global minima of the target functions. However, the changes in length predicted by the NP-L, NP-F and P optimizations agree well with our experimental data (Table 1), which suggests that the minima represent physiologically meaningful states. Furthermore, we ran the optimizations for several different sets of initial values and consistently obtained the set of best-fit parameter values reported here (Table 3).

The predicted uterine vascular changes in pregnancy – a decrease in *R* and increase in *C* – agree with our own experimental data (Table 1 and (29)) and others’ data obtained in other species (12, 13, 32, 33). The effects of the ovarian cycle on uterine artery size or stiffness have not been as well studied. Our own experimental data have demonstrated that *UVR* decreases in the follicular phase compared to the luteal phase and secondary generation uterine artery elastic modulus tends to decrease as well (Table 1 and (29)). These findings are consistent with the single-segment effective model predictions that terminal load impedance decreases and compliance increases (somewhat) with the follicular phase. Our model results also suggest two other potential effects of the ovarian cycle on uterine arterial size and stiffness that warrant further investigation. First, the decrease in longitudinal resistance with minimal lengthening suggests that mid-size arteries -- larger than resistance arteries and smaller than the main conduit arteries -- dilate significantly in response to circulating hormone levels. Second, the two-fold drop in inertance suggests an increase in arterial size at one or several levels in the arterial tree.

It is worth noting here that the single-segment model is an “effective” model; that is, it simulates the impedance spectra with parameter values that are devoid of physical meaning. In particular, considering equations (6) and (7), the changes in *R* and *L* predicted between the luteal and follicular phase (Table 3) cannot be explained by a change in radius in a single artery with the characteristics of the single-segment transmission line. Instead, *R* and *L* and also *l* and *C* represent bulk characteristics of the system and likely reflect changes in multiple or different segments of the actual uterine arterial tree. While these bulk characteristics are devoid of precise anatomical information, they do predict the hemodynamic behavior of the uterine arterial network during the ovarian cycle and with pregnancy, and may lead to clinically useful insights in cases of abnormal uterine arterial hemodynamics.

Finally, we investigated the impact of single model parameters on the shape of whole impedance spectra and mean blood flow. In accordance with linear transmission line theory, we found that the input resistance is determined by *UVR*, the magnitude of the first minimum is determined by *L*/*UVR C*), and the frequencies of the first minimum and second peak are determined by 1/*LC* . In the frequency range of the normal sheep heart rate (~2 Hz) (34), mean *UBF* decreases with increasing *UVR*, increases with increasing *C*, and decreases slightly with increasing *L* in both the luteal phase and with pregnancy. These findings are in accordance with prior data showing that transformation of the uteroplacental bed (to decrease *UVR*) and remodeling of large arteries (to increase compliance and diameter) (12, 13, 32, 33, 35-37) are all important to the dramatic increase in blood flow that accompanies a healthy pregnancy.

In this study, electrical analog transmission line models were used to simulate the input impedance of the ovine uterine vascular circulation for nonpregnant and pregnant conditions. Two models, including an anatomically-based, multi-segment, symmetric branching model with parameter values based on experimental measurements and a single-segment effective model with parameter values based on optimization to measured impedance spectra, were developed. The impedance characteristics of the uterine circulation were better simulated by the latter method for all conditions. The variations in the parameter values over the ovarian cycle and with pregnancy were in agreement with available data on the effects of the ovarian cycle and pregnancy on uterine artery dimensions and mechanical properties. The single-segment model was also used to explore the dependencies of impedance spectra and mean blood flow at physiological conditions (heart rate and pressure) on uterine vascular structure and function. In future work, we plan to use this model to investigate parameters that may be useful for the diagnosis of pathologies of pregnancy such as preeclampsia with and without intrauterine growth retardation.

A single-segment transmission line model demonstrates that impedance changes with ovarian cycling and pregnancy are caused by changes in longitudinal and terminal resistance and compliance.

The present study was supported in part by NIH grants HL49210, HL087144, HD38843, (RRM) and HL086939 (NCC). The authors would also like to thank Professors Susan Hagness and Nico Westerhof for insightful comments on the modeling approaches.

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