Pulse Pressure

Tonometry waveforms were signal-averaged using the electrocardiographic R-wave as a fiducial point. Systolic and diastolic cuff blood pressures obtained at the time of the tonometry acquisition were used to calibrate the peak and trough of the signal-averaged brachial pressure waveform. Diastolic and integrated mean brachial pressures were used to calibrate carotid pressure tracings.24 (link) Calibrated carotid pressure was used as a surrogate for central pressure.24 (link) Central pulse pressure was defined as the difference between the peak and trough of the calibrated carotid pressure waveform. Carotid-brachial pulse pressure amplification was defined as brachial pulse pressure divided by central pulse pressure. Augmentation index was computed from the carotid pressure waveform as previously described.25 (link) Carotid-femoral (aortic) and carotid-radial (muscular artery) pulse wave velocities were calculated from tonometry waveforms and body surface measurements, which were adjusted for parallel transmission in the brachiocephalic artery and aortic arch by using the suprasternal notch as a fiducial point.26 (link) The carotid-femoral transit path spans the aorta, making carotid-femoral PWV a measure of aortic stiffness. In contrast, the carotid-radial transit path spans the subclavian, brachial and radial arteries, making carotid-radial PWV a measure of muscular artery stiffness.

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)].