Estimating Central Pulse Pressure From Blood Flow by Identifying the Main Physical Determinants of Pulse Pressure Amplification

Several studies suggest that central (aortic) blood pressure (cBP) is a better marker of cardiovascular disease risk than peripheral blood pressure (pBP). The morphology of the pBP wave, usually assessed non-invasively in the arm, differs significantly from the cBP wave, whose direct measurement is highly invasive. In particular, pulse pressure, PP (the amplitude of the pressure wave), increases from central to peripheral arteries, leading to the so-called pulse pressure amplification (ΔPP). The main purpose of this study was to develop a methodology for estimating central PP (cPP) from non-invasive measurements of aortic flow and peripheral PP. Our novel approach is based on a comprehensive understanding of the main cardiovascular properties that determine ΔPP along the aortic-brachial arterial path, namely brachial flow wave morphology in late systole, and vessel radius and distance along this arterial path. This understanding was achieved by using a blood flow model which allows for workable analytical solutions in the frequency domain that can be decoupled and simplified for each arterial segment. Results show the ability of our methodology to (i) capture changes in cPP and ΔPP produced by variations in cardiovascular properties and (ii) estimate cPP with mean differences smaller than 3.3 ± 2.8 mmHg on in silico data for different age groups (25–75 years old) and 5.1 ± 6.9 mmHg on in vivo data for normotensive and hypertensive subjects. Our approach could improve cardiovascular function assessment in clinical cohorts for which aortic flow wave data is available.


Amplification term
The time-domain solution of a low frequency approximation for the amplification term T 2 px, tq is given by x η πr 4 0 Q in ptq´4 x η 3πr 4 0ˆρ r 2 0 η`8 Cηx 2 πr 4 0˙d Q in ptq dt .
(S9) Figure S1 shows each term in Equation (S9) at the outlet of the ascending aorta and brachial artery. The Asc. Aorta Brachial  Figure S1. Components of the amplification term, T 2 px, tq, with time at the outlet of the ascending aorta (left) and brachial artery (right) of the 25-year-old subject in the in silico dataset. The ρr 2 0 {η term in Equation (S9) is shown in the outer plots. The inflow (red lines) and 8Cηx 2 {πr 4 0 (dotted lines) terms are shown in the inner plots.
continuous black line corresponds to the term with the characteristic time ρr 2 0 η , which clearly dominates over the other two. As a result, T 2 can be simplified to The same behaviour is observed in all the segments along the aortic-brachial arterial path. To reduce the number of parameters in the models and given that the differences between the average (r 0 ) and diastolic (r d ) luminal radii are small, we have used r 0 « r d in all our calculations.

AMPLIFICATION TERM VERIFICATION
According to Equation (10) in Section 2.3.2 Approximate Amplification Term, pulse pressure amplification (∆PP), along an arterial segment is proportional to the segment length (l), and the maximum temporal rate of decrease in late systolic-flow (´dQ in {dt| min ); and inversely proportional to the square of the luminal diastolic radius (r d ). To verify these proportionalities, for all 729 in silico subjects aged 25 years old and all 729 aged 75 years old, we extracted ∆PP, r d , l,´dQ in {dt| min , for all the arterial segments in the aortic-brachial arterial path. Then to analyze the effect of changing one variable at a time, we used´∆PP r 2 d {pdQ in {dt| min q as the dependent variable to observe the effect of the length ( Figure S2A), ∆PP{plpdQ in {dt| min qq to observe the effect of the luminal diastolic radius ( Figure S2B) and ∆PP r 2 d {l to observe the effect of the maximum temporal rate of decrease in late systolic-flow ( Figure S2C).
Length and temporal rate of decrease in late systolic-flow exhibited linear correlations, shown with grey lines in Figures S2A and S2C, respectively. The smallest correlation coefficient was R 2 " 0.91. Radius exhibited a power-law decay with exponents of´1.73 and´2.18 for the 25-and 75-year-old in silico subjects, respectively (shown with a grey line in Figure S2B). Both exponents are close to the expected value of´2.

IN SILICO VERIFICATION OF CPP ESTIMATION WITH THE SINGLE VESSEL MODEL
This section contains the results for cPP and ∆PP estimation with variations in cardiovascular properties using the single-vessel model. Mean and standard deviation (SD) for stroke volume, heart rate, left ventricular ejection time, total vascular resistance and total vascular compliance correspond to the 25 year-old in silico subjects. The length and the radius of each vessel of the 25-year-old baseline subject were changed by 14% and 11%, respectively. These percentages were calculated from the  Figure S3. Effect of cardiovascular properties on central pulse pressure, cPP, and its amplification from the aortic root to the outlet of the brachial artery, ∆PP. Aortic root (A) and brachial artery (B) flow wave (first column), cPP (second column), and ∆PP (third column) for the 25-year-old baseline subject (black) and with a standard deviation (SD) decrease (blue) and a SD deviation increase (red) in (A) stroke volume (SV), heart rate (HR) and left ventricular ejection time (LVET), and (B) total vascular resistance (R T ) and total vascular compliance (C T ); and with a 14% decrease (blue) and 14% increase (red) in the total network length (L T ) and with a 11% decrease (blue) and 11% increase (red) in the average radius of the network (xr N y). The closed dots were calculated using the reduced model and the open dots were calculated using Equations (9) and (10) of the single-vessel model. Legends in the first column indicate the maximum temporal rate of decrease in late systolic-flow in mL/s 2 for all flow waves.