Vascular Resistance

The parameters of the three-element Windkessel outflow models were calculated as described below. Given a target diastolic (P_d) and systolic (P_s) pressure, and flow rate at the inlet (Q_in(t)), the initial estimate for the net peripheral resistance (R_T) was calculated as [50 (link)]
$$R_{\mathrm{T}}=\frac{P_{\mathrm{m}}-P_{\text{out}}}{{\overline{Q}}_{\text{in}}},P_{\mathrm{m}}=P_{\mathrm{d}}+\frac{1}{3}(P_{\mathrm{s}}-P_{\mathrm{d}}),$$ where Q̄_in is the mean flow rate and P_m is the mean blood pressure, assumed uniform throughout the arterial network. We then calculated the resistance R₁ + R₂ at the outlet of each terminal vessel that yields the desired flow distribution and satisfies
$$\frac{1}{R_{\mathrm{T}}}=\sum _{j=2}^M\frac{1}{R_1^j+R_2^j},$$ where M is the number of terminal branches and j = 1 corresponds to the aortic root. For each individual outlet, the proximal resistance (R₁) is assumed to be equal to the characteristic impedance of the upstream 1-D domain; i.e.
$$R_1=\frac{{\rho }_{\mathrm{f}}{c}_{\mathrm{d}}}{A_{\mathrm{d}}},$$ where c_d and A_d are, respectively, the wave speed and area at diastolic pressure (P_d). This choice of R₁ minimizes the magnitude of the waves reflected at the outlet of the 1-D domain [38 ].
The total compliance (C_T) was calculated from either (i) the time constant τ = 1.79 s of the exponential fall-off of pressure during diastole given in [51 ] or (ii) using an approximation to
$C_{\mathrm{T}}=\frac{\mathrm{d}V}{\mathrm{d}P}$ , where V(t) is the total blood volume contained in the systemic arteries. According to [50 (link)],
$$C_{\mathrm{T}}=\frac{\tau }{R_{\mathrm{T}}},$$ which can be calculated once R_T is determined using Eq. (13). Alternatively,
$C_{\mathrm{T}}=\frac{\mathrm{d}V}{\mathrm{d}P}$ can be approximated by [50 (link)]
$$C_{\mathrm{T}}=\frac{Q_{max}-Q_{min}}{P_{\mathrm{s}}-P_{\mathrm{d}}}\mathrm{\Delta }t,$$ where Q_max and Q_min are the maximum and minimum flow rates at the inlet and Δt is the difference between the time at Q_max and the time at Q_min. We use both Eqs. ( 16) and (17) depending on the available input data.
According to [52 (link)] we have
$$C_{\mathrm{T}}=C_{\mathrm{c}}+C_{\mathrm{p}},C_{\mathrm{c}}=\sum _{i=1}^N{C}_{0\mathrm{D}}^i,C_{\mathrm{p}}=\sum _{j=2}^M\frac{R_2^j{C}^j}{R_2^j+R_1^j},$$ where C_c is the total arterial conduit compliance, C_p is the total arterial peripheral compliance, N is the total number of vessels in the 1-D domain, M < N is the number of terminal branches (j = 1 denotes the inlet and is not included in the sum), R₁, R₂, and C are parameters of the three-element Windkessel model (Fig. 1) and C_0D is the compliance of each vessel, which is calculated as
$$C_{0\mathrm{D}}=\frac{A_{\mathrm{d}}L}{{\rho }_{\mathrm{f}}{(c_{\mathrm{d}})}^2},$$ where L is the length of the vessel. We calculated C_p = C_T − C_c and distributed it following the methodology described by Alastruey et al. [52 (link)] More specifically, we have
$${\stackrel{\sim }{C}}^j=C_p\frac{R_{\mathrm{T}}}{R_2^j+R_1^j},$$ where C̃^j is the terminal compliance of each branch distributed in proportion to flow as described by Stergiopulos et al. [2 (link)]. We then introduced a correction factor to arrive at the final value of C^j:
$$C^j={\stackrel{\sim }{C}}^j\frac{R_2^j+R_1^j}{R_2^j}=C_{\mathrm{p}}\frac{R_{\mathrm{T}}}{R_2^j}.$$
This expression follows from a linear analysis of the 1-D equations in a given arterial network in which each terminal branch is coupled to a three-element Windkessel model [52 (link)].
For all of the simulations, the Windkessel compliances and resistances (C^j, j = 2, …, M), (
$R_1^j$ and
$R_2^j$ , j = 2, …, M) were iteratively calculated to achieve physiologically realistic pressure ranges. To reach a desired pulse pressure (P_pulse = P_s − P_d) and diastolic pressure (P_d) at a particular vessel, we calculated
$R_{\mathrm{T}}^0$ and
$C_{\mathrm{T}}^0$ given by Eqs. (13) and (16) or (17) using the iterative formulae
$$R_{\mathrm{T}}^{n+1}=R_{\mathrm{T}}^n+\frac{\mathrm{\Delta }{P}_{\mathrm{m}}^n}{{\overline{Q}}_{\text{in}}},\mathrm{\Delta }{P}_{\mathrm{m}}^n=P_{\mathrm{d}}-P_{\mathrm{d}}^n,$$
$$C_{\mathrm{T}}^{n+1}=C_{\mathrm{T}}^n-\frac{Q_{max}-Q_{min}}{{(P_{\text{pulse}}^n)}^2}\mathrm{\Delta }t\mathrm{\Delta }{P}_{\text{pulse}}^n,\mathrm{\Delta }{P}_{\text{pulse}}^n=P_{\text{pulse}}-P_{\text{pulse}}^n,$$ where the superscript n is the iteration number of the windkessel parameter estimation process performed using the 1-D formulation, and
$P_{\mathrm{d}}^n$ and
$P_{\text{pulse}}^n$ are the diastolic and pulse pressure, respectively, at a specific target location in the 1-D model, typically the inlet, at each iteration. Equations (22) and (23) follow from a first-order Taylor expansion of Eqs. (13) and (17) around the current mean and pulse pressures
$P_{\mathrm{m}}^n$ and
$P_{\text{pulse}}^n$ , respectively, with
$\mathrm{\Delta }{P}_{\mathrm{m}}^n$ approximated using the change in diastolic pressure. The total compliance was adjusted by altering the total peripheral compliance C_p, since the total conduit compliance C_c is a function of the vessel geometry and wall stiffness. This process was iterated using the 1-D model until
$P_{\mathrm{d}}^n$ and
$P_{\text{pulse}}^n$ were smaller than 1% of the target P_d and P_pulse, respectively. Fig. 2 shows the evolution of the systolic, mean and diastolic pressure, net peripheral resistance and total compliance calculated using the 1-D formulation to match the target systolic and diastolic pressures for the baseline aorta model. The final values of the Windkessel compliances and resistances were used in the 3-D counterparts of the 1-D models.
Other methods have been proposed in the literature to estimate the parameters of the outflow boundary conditions. A root-finding method is described by Spilker and Taylor [53 (link)] in the context of 3-D models with compliant arterial walls. Devault et al. proposed a Kalman-filter based methodology in a 1-D model of the circle of Willis [54 (link)].

Reservoir pressure (figure 2) was calculated from the ensemble averaged radial tonometric waveforms recorded by the Sphygmocor device without the application of a generalized transfer function. In brief, sphygmocor *.txt files were saved and data subsequently imported into Matlab (Mathworks, Inc, Natick, Massachusetts, USA) for analysis using customised programs based on (10 (link);21 (link)). Further details are provided in Supplementary data. Reservoir pressure is assumed to vary temporally in the same way throughout aorta and large elastic arteries, but with a time lag that depends on the location and wave propagation characteristics of the arteries (13 (link)). Mass conservation in an arterial system containing N vessels requires
$$Q_0\left(t\right)=\sum _{n\in N}{C}_n\frac{d\overline{P}(t-{\tau }_n)}{\mathit{dt}}+\sum _{n\in N}\frac{\overline{P}(t-{\tau }_n)-P_{\infty }}{R_n}$$
where Q₀(t) is the volume flow rate at the aortic root, C_n is the compliance of the vessel segment n, $\overline{P}\left(t\right)$ is the reservoir pressure at the aortic root, $\overline{P}(t-{\tau }_n)$ is the reservoir pressure in vessel n, R_n is the resistance of vessel n, τ_n is the time it takes for a wave to travel from the aortic root to vessel n and P_∞ is the pressure at zero flow.
Excess pressure in vessel n (XSP_n) is defined as the difference between the measured pressure P_n(t) and the reservoir pressure
$${\mathit{XSP}}_n\left(t\right)=P_{n\left(t\right)}-\overline{P}(t-{\tau }_n)$$
Hydraulic work done by the ventricle (W) depends on the volume flow rate Q_ot and the pressure in the aortic root
$$W=\underset{0}{\overset{\tau }{\int }}\overline{P}\left(t\right){\mathrm{Q}}_0\left(t\right)\mathit{dt}$$
where t corresponds to the cardiac period. This work can be separated into reservoir and excess work
$$W=\underset{0}{\overset{\tau }{\int }}\overline{P}\left(t\right){\mathrm{Q}}_0\left(t\right)\mathit{dt}+\underset{0}{\overset{\tau }{\int }}{\mathit{XSP}}_0\left(t\right){\mathrm{Q}}_0\left(t\right)\mathit{dt}\equiv \overline{W}+\mathit{XSW}$$
where reservoir work $\left(\overline{W}\right)$ is the hydraulic work done by the ventricle against the reservoir pressure and the excess work (XSW) is the work done against the excess pressure at the aortic root. For a given flow, Q_o(t) the integral of the excess pressure (XSPI) is therefore an index of the excess work done by the ventricle.

Pulsed-wave and color Doppler ultrasound examination of the uterine and umbilical arteries was performed in patients with preeclampsia (Acuson, Sequoia, Mountain View, CA) using a 3.5 or a 5 MHz curvilinear probe. Transducers were directed toward the iliac fossa, the external iliac artery was imaged in a longitudinal section, and the uterine artery was mapped with color Doppler as it crossed the external iliac artery. Pulsed-wave Doppler was performed of both uterine arteries and when three similar consecutive waveforms were obtained, the resistance index (RI) of the right and left uterine arteries was measured and the mean RI of the two vessels was calculated. Uterine artery Doppler velocimetry30 (link) was defined as abnormal if either the mean RI was above the 95^th percentile for gestational age31 (link)or in the presence of a bilateral early diastolic notch.32 (link) The Doppler signal of the umbilical artery was obtained from a free floating loop of the umbilical cord during the absence of fetal breathing and body movement. When three similar consecutive waveforms were obtained, the pulsatility index (PI) was measured. Umbilical artery Doppler velocimetry was defined as abnormal if either the PI was above the 95^th percentile for gestational age using the reference range proposed by Arduini and Rizzo33 (link)or in the presence of abnormal waveforms (absent or reversed end diastolic velocities) as described by Trudinger et al.34 (link) The patients were classified into the following 4 groups: 1) Normal Doppler velocimetry in the uterine and the umbilical arteries; 2) Doppler abnormalities in the uterine artery alone; 3) Doppler abnormalities in the umbilical artery alone; and 4) Doppler abnormalities in both vessels.