ORIGINAL RESEARCH published: 23 February 2021 doi: 10.3389/fphys.2021.608098 Estimating Central Pulse Pressure From Blood Flow by Identifying the Main Physical Determinants of Pulse Pressure Amplification Joaquín Flores Gerónimo 1*, Eugenia Corvera Poiré 2,3 , Philip Chowienczyk 4 and Jordi Alastruey 1,5 1 Department of Biomedical Engineering, School of Biomedical Engineering and Imaging Sciences, King’s College London, London, United Kingdom, 2 Departamento de Física y Química Teórica, Facultad de Química, Universidad Nacional Autónoma de México, Ciudad Universitaria, Mexico City, Mexico, 3 Universitat de Barcelona Institute of Complex Systems (UBICS), Universitat de Barcelona, Barcelona, Spain, 4 Department of Clinical Pharmacology, British Heart Foundation Centre, St Thomas’ Hospital, King’s College London, London, United Kingdom, 5 World-Class Research Center, Digital Biodesign and Personalized Healthcare, Sechenov University, Moscow, Russia 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 (1PP). The main purpose of this Edited by: Giovanni Biglino, study was to develop a methodology for estimating central PP (cPP) from non-invasive University of Bristol, United Kingdom measurements of aortic flow and peripheral PP. Our novel approach is based on a Reviewed by: comprehensive understanding of the main cardiovascular properties that determine 1PP Chloe May Park, along the aortic-brachial arterial path, namely brachial flow wave morphology in late University College London, United Kingdom systole, and vessel radius and distance along this arterial path. This understanding was Ramakrishna Mukkamala, achieved by using a blood flow model which allows for workable analytical solutions in the University of Pittsburgh, United States frequency domain that can be decoupled and simplified for each arterial segment. Results *Correspondence: Joaquín Flores Gerónimo show the ability of our methodology to (i) capture changes in cPP and 1PP produced by joaquin.flores@kcl.ac.uk 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 Specialty section: ± 6.9 mmHg on in vivo data for normotensive and hypertensive subjects. Our approach This article was submitted to Vascular Physiology, could improve cardiovascular function assessment in clinical cohorts for which aortic flow a section of the journal wave data is available. Frontiers in Physiology Keywords: pulse pressure amplification, 1-D model, peripheral pressure, central pressure, analytical solutions, Received: 18 September 2020 hemodynamics Accepted: 18 January 2021 Published: 23 February 2021 Citation: 1. INTRODUCTION Flores Gerónimo J, Corvera Poiré E, Chowienczyk P and Alastruey J (2021) Peripheral systolic blood pressure is the most commonly used measure of circulatory function and Estimating Central Pulse Pressure From Blood Flow by Identifying the cardiovascular risk. However, a significant predictive benefit has been observed when measuring Main Physical Determinants of Pulse central (aortic) blood pressure (cBP) (Agabiti-Rosei and Muiesan, 2015; Williams et al., 2018) Pressure Amplification. and it has been suggested that cBP should be a better indicator of risk (Agabiti-Rosei et al., Front. Physiol. 12:608098. 2007; Avolio et al., 2009; Sharman et al., 2013; McEniery et al., 2014; Agabiti-Rosei and Muiesan, doi: 10.3389/fphys.2021.608098 2015; Williams et al., 2017) since it is more representative of the load exerted on major organs Frontiers in Physiology | www.frontiersin.org 1 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow (Herbert et al., 2014; Agabiti-Rosei and Muiesan, 2015; Williams et al., 2004), mass index (Pichler et al., 2016), pulse transit time et al., 2017). For instance, elevation of cBP induces coronary (Gao et al., 2016; Natarajan et al., 2017), and wave reflection arteriosclerosis which in turn can lead to adverse events such coefficient (Gao et al., 2016), and lower age (Wilkinson et al., as stenosis and myocardial infarction (Agabiti-Rosei et al., 2007) 2001; Herbert et al., 2014) and pulse wave velocity (Hashimoto and causes chronic kidney disease that can advance to end-stage and Ito, 2010; Pierce et al., 2012), and is significantly influenced renal disease (Safar et al., 2002; Ohno et al., 2016). Additionally, it by cardiovascular risk factors, such as hypertension and obesity has been suggested that cBP assessment can improve therapeutic (Herbert et al., 2014). Experimental and computational models decisions since anti-hypertensive drugs have different effects on have been used to study the effect on 1PP of cardiovascular peripheral and central pressure values (Sharman et al., 2013; properties (Karamanoglu et al., 1995; Figueroa and Humphrey, McEniery et al., 2014; Agabiti-Rosei and Muiesan, 2015). 2014; Mynard and Smolich, 2015; Gaddum et al., 2017) and The morphology of the peripheral blood pressure (pBP) age (Charlton et al., 2019), showing that 1PP raises with wave, usually assessed non-invasively in the arm, differs increasing ventricular inotropy (contractile state of the ventricle), significantly from the morphology of the cBP wave, whose tapering, peripheral load and vessel length; and decreasing wall direct measurement is highly invasive. In particular, pulse thickness and age. However, there are currently no methods pressure (PP)—the difference between systolic and diastolic based on the physics of blood flow in the systemic arterial blood pressures—often increases from central to peripheral tree that enable explicit analytical identification of the main arteries leading to the so-called pulse pressure amplification cardiovascular determinants of 1PP—and hence estimation of (1PP) (Figure 1). The cBP wave can be estimated using cPP from peripheral PP (pPP)—from data that can be acquired population-based generalized transfer functions (GTFs) from non-invasively for a specific subject. a calibrated pBP wave; however the debate continues on the The main purpose of this study was to develop a methodology suitability of this approach for all patients and conditions (Avolio for estimating cPP from non-invasive measurements of blood et al., 2009; Shih et al., 2011). Several studies have proposed an flow (aortic or brachial) and pPP. Our novel approach is based adaptive transfer function technique which personalizes some of on a comprehensive understanding—using the physics of blood the parameters of a single-tube transmission line model coupled flow—of the main cardiovascular properties that determine to an impedance boundary condition that reflects incoming 1PP along the aortic-brachial arterial path. The methodology pulse waves (Swamy et al., 2009; Hahn et al., 2012; Gao et al., presented in this study was assessed using in silico data 2016; Natarajan et al., 2017). Such an approach can improve generated by blood flow modeling and in vivo data measured in the accuracy of the estimated cBP waveform compared to the normotensive and hypertensive subjects. These datasets included GTF approach. In addition, the pressure wave measured in the reference cPP values. common carotid artery has been used as a surrogate for the cBP wave; however reproducible pressure waves are more difficult 2. MATERIALS AND METHODS to obtain at the carotid artery than at the radial artery due to anatomical reasons (Avolio et al., 2009). 2.1. In vivo Data Understanding the effect of cardiovascular parameters on We used previously available measurements of cBP and pBP 1PP could improve cBP assessment from non-invasive pBP waveforms in a group of normotensive volunteers (n = 26) measurements; specially central PP (cPP) assessment which has and hypertensive subjects (n = 57) (Fok et al., 2014a; Li et al., been shown to be of greater predictive value for cardiovascular 2017). Subjects were recruited from those who were evaluated outcomes than brachial PP (Safar et al., 2002; Williams et al., for hypertension at Guy’s and St Thomas’ Hypertension Clinic. 2006). Large 1PP values have been associated with male sex Although they were referred for evaluation of hypertension, (Segers et al., 2009; Herbert et al., 2014), higher heart rate blood pressure settled in some subjects and the sample included (Wilkinson et al., 2002), height (Asmar et al., 1997; Camacho some who were normotensive (Li et al., 2017). Subjects with significant valvular disease, impaired left ventricular systolic function (ejection fraction <45%), and arrhythmia were excluded. 60% of the hypertensive subjects were on treatment with anti-hypertensive medications. Characteristics of both groups are given in Table 1. Radial and carotid pressure waveforms were obtained by applanation tonometry performed by an experienced operator using the SphygmoCor system (AtCor, Australia). Ensemble-averaged carotid pressure was used as surrogate for ascending aortic pressure (Chen et al., 1996). Approximately 10 cardiac cycles were obtained and ensemble FIGURE 1 | Central and peripheral pressure waveforms. Central pulse averaged. Brachial blood pressure was measured in triplicate pressure, cPP, is mainly determined by the blood flow ejected by the ventricle by a validated oscillometric method (Omron 705CP, Omron into the aorta, Qin , and the total arterial compliance, CT (Vennin et al., 2017). Health Care, Japan) and used to calibrate radial waveforms Peripheral PP amplification, 1PP, is mainly determined by the rate of change and, thus, to obtain a mean arterial pressure (MAP) through of Qin with time, t, in late systole and the radius, r, and length, l, of the brachial artery, as shown in this study. integration of the radial waveform. Carotid waveforms were calibrated from MAP and diastolic brachial blood pressures Frontiers in Physiology | www.frontiersin.org 2 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow TABLE 1 | Characteristics of the datasets: normotensive (second column) and hypertensive (third column) in vivo subjects and in silico subjects (fourth column). Characteristic Normotensives Hypertensives In silico (n = 26) (n = 57) (n = 4374) Age (years) 44 ± 14 40 ± 13 50 ± 17 Sex (male %) 58 65 100 Height (m) 1.72 ± 0.03 1.72 ± 0.09 1.76 ± 0.00 Weight (kg) 79.2 ± 7.7 79.4 ± 13.4 - Heart rate (bpm) 62 ± 8 62 ± 10 76 ± 9 rAo (cm) 0.96 ± 0.12 0.98 ± 0.11 1.97 ± 0.19 FIGURE 2 | Schematic representation of the three models used in the study. rBra (cm) 0.20 ± 0.03 0.20 ± 0.03 0.45 ± 0.05 (A) Full 116-artery model with cardiovascular properties taken from Charlton SV (mL) 55.4 ± 15.8 52.4 ± 13.9 60.4 ± 12.4 et al. (2019). (B) Reduced 15-artery model containing the aortic-brachial arterial path of the full model. (C) Single-vessel model of the brachial artery (1Qin /1t)Ao (mL/s2 ) 1550 ± 464 1580 ± 405 1966 ± 270 with the nomenclature used throughout the study. Tapering in the brachial dQin /dt min,Bra (mL/s2 ) 96 ± 35 98 ± 31 434 ± 83 artery of the full and reduced models was represented by five straight arterial cPP (mmHg) 39.7 ± 12.3 45.6 ± 13.5 35.0 ± 15.3 segments of different radii. The single-vessel model allowed us to identify pPP (mmHg) 51.0 ± 12.0 54.1 ± 14.7 50.2 ± 13.6 analytically the main physical determinants of 1PP. 1PP (mmHg) 11.2 ± 6.4 11.5 ± 7.5 15.5 ± 5.2 RT (mmHg s/mL) 1.96 ± 0.55 2.20 ± 0.70 1.27 ± 0.31 CT (mL/mmHg) 1.15 ± 0.48 1.04 ± 0.40 0.85 ± 0.30 2.3. Blood Flow Models Three blood flow models with decreasing level of mathematical Values are percentage or mean ± SD. rAo , aortic radius; rBra , brachial radius; SV, stroke complexity were employed in this study (Figure 2). Firstly, volume; (1Qin /1t)Ao , approximate maximum temporal rate of decrease in late-systolic flow at the ascending aorta; dQin /dt min,Bra , maximum temporal rate of decrease in late- we used the 116-artery model of the systemic circulation systolic flow at the brachial artery; cPP, central pulse pressure; pPP, peripheral pulse described above. Blood pressure and flow waves at any pressure; 1PP, pulse pressure amplification; RT , total vascular resistance; CT , total point in the arterial network were simulated using our in- vascular compliance. rBra and dQin /dt min,Bra for the in vivo datasets were estimated from the corresponding data in the aorta using Equations (16) and (17), respectively. house, linear, one-dimensional (1-D) formulation, which enables analytical solutions in the frequency domain and has previously been verified against computational 1-D and 3-D solutions (Flores Gerónimo et al., 2016). This model—hereafter referred to on the assumption of equality of these pressures at central as the “full model”—provided reference cBP and pBP waves to and peripheral sites (Pauca et al., 1992). Ultrasound imaging assess the accuracy of the other two simpler models. was performed by an experienced operator using the Vivid- The second model simulated blood flow in the 15 arterial 7 ultrasound platform (General Electric Healthcare, United segments that make up the upper thoracic aorta and left brachial Kingdom). Velocity above the aortic valve was recorded using artery of the full model. This model hereafter referred to as the pulsed wave Doppler obtained from an apical 5-chamber view. “reduced model”—has the same inflow waveform prescribed at All ultrasound measurements were averaged over at least 3 the aortic root as the full model. Such inflow waveform was cardiac cycles. Cross-sectional area of the aortic valve (obtained obtained by the AorticFlowWave script described in Charlton in the parasternal long-axis view) was used to estimate the aortic et al. (2019) for the desired inflow characteristics (heart rate, radius. These data were acquired in a previous study approved by stroke volume and left ventricular ejection time). Figure 3A the London Westminster Research Ethics Committee, for which shows the inflow waveform for the 25 year-old baseline subject. written informed consent was obtained (Li et al., 2017). The vascular network of the reduced model was obtained from the full model by lumping groups of arteries into optimized three-element windkessel (WK) models that preserved the total vascular compliance and resistance of the full model. Six 2.2. Dataset of in silico Pulse Waves sets of WK parameters were calculated, for the six terminal We used an existing dataset containing in silico cBP and pBP segments of the reduced model, from the flow and pressure waves measured, respectively, at the aortic root and outlet of waves produced by the full model using a gradient descent the brachial artery, and blood flow waves measured at the aortic algorithm previously described in Fossan et al. (2018). Table 2 root, in a group of 4,374 virtual subjects. Characteristics of the in shows the vascular parameters of the reduced model that was silico group are given in Table 1. Pulse waves were simulated for able to reproduce pressure and flow waves along the aortic- subjects of each age decade, from 25 to 75 years old, using a 116- brachial arterial path with relative errors smaller than 3 and artery, one-dimensional (1-D) model of blood flow in the larger 10%, respectively (Figure 3). All relative errors in this figure were systemic arteries of the thorax, limbs, and head (Figure 2A). calculated using the full model as a reference, as described in The model cardiovascular properties were identified through a Supplementary Section 1. comprehensive literature review and simulated pulse waves were The reduced model provided accurate and workable analytical verified by comparison against clinical data [see Charlton et al. solutions for blood pressure and flow waves along the aortic- (2019) for full details]. brachial arterial path. It was further simplified into a third model: Frontiers in Physiology | www.frontiersin.org 3 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow FIGURE 3 | Reduced vs. full model pressure and flow waves. (A) Schematic representations of both models (left) and comparison of corresponding pressure and flow waves at four locations along the aortic-brachial arterial path (right). Each plot shows relative errors (expressed as percentages) for average (RMS), maximum (Max), systolic (Sys) and characteristic pressure points P1 and P2 calculated as described in Supplementary Section 1. (B) Pressure wave along the same arterial path, for the full (left) and reduced (right) model. Characteristic pressure points P1 (yellow), systolic pressure (green), and P2 (blue) are shown in each plot. a straight single-vessel model of the brachial artery which enabled based on the formulation described by Flores Gerónimo et al. us to identify analytically the main physical determinants of 1PP, (2016) (see Supplementary Section 2); i.e., as described in the next sections. p̂ = T̂1 + T̂2 , (1) 2.3.1. Single-Vessel Model where Blood pressure in the frequency domain, p̂(x, ω), along the single- " # vessel model (Figure 2C) can be described by the summation cos(kc l)ẐM + sin(kc l) Q̂in Tˆ1 = cos(kc x) , (2) of an attenuation, T̂1 (x, ω), and an amplification, T̂2 (x, ω), term cos(kc l) − ẐM sin(kc l) M Frontiers in Physiology | www.frontiersin.org 4 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow TABLE 2 | Vascular parameters of the reduced model. Arterial Length rd Eh R1 R2 CWk segment (cm) (mm) (Pa m) (mmHg s mL−1 ) (mmHg s mL−1 ) (mL mmHg−1 ) 1. Asc. aorta 6.00 18.1 975.7 - - - 2. Aortic arch I 2.00 15.8 851.0 - - - 3. Aortic arch II 3.90 15.2 817.8 - - - 4. L. subclavian 3.40 6.4 346.4 - - - 5. Brachial I 8.44 5.4 292.3 - - - 6. Brachial II 8.44 4.8 262.9 - - - 7. Brachial III 8.44 4.3 234.4 - - - 8. Brachial IV 8.44 3.7 207.5 - - - 9. Brachial V 8.44 3.1 173.2 1.548 10.982 0.046 10. Brachiocephalic 3.40 8.6 461.1 0.123 4.523 0.252 11. L. Com. carotid 6.95 4.4 239.6 0.406 10.333 0.047 12. L. vertebral 3.70 2.2 152.2 1.582 57.410 0.008 13. Desc. Thor. aorta I 5.20 12.8 691.7 - - - 14. Intercostal 8.00 1.5 139.7 9.115 3.081 0.051 15. Desc. Thor. aorta II 10.40 9.7 520.3 0.125 0.964 0.683 They were taken from the 25-year-old baseline subject in Charlton et al. (2019). Segment numbers are displayed in Figure 2B. The remaining parameters are: blood density ρ = 1, 060 Kg m−3 , blood viscosity η = 2.5 × 10−3 Pa s and outflow pressure Pout = 33 mmHg. rd is the diastolic luminal radius, E is the Young modulus, h is the wall thickness, R1 , R2 , and CWk are the two resistances and compliance, respectively, of the outflow windkessel models. L, left; Com, common; Desc, descending; Thor, thoracic. Q̂in and C = 0.205 mm2 /mmHg. The approximation for r0 ≈ rd Tˆ2 = − sin(kc x) , (3) M was used given that the difference between r0 and rd was smaller than 2.5%, which allowed us to reduce the number of parameters with x the axial direction, ω the angular frequency, l the vessel in the model. The WK parameters CWk , R1 , R2 and Pout are length, Q̂in the Fourier transform of Qin (t) the flow wave at the equal to those of the WK model coupled to the outlet of the inlet, and segment ‘Brachial V’ in the reduced model (see Table 2). We note iωCη iωCA0 K(ω) that PP along the vessel is indeed slightly attenuated by T1 and k2c (ω) = , M 2 (ω) = , considerably amplified by T2 . Next, we derive an approximate A0 K(ω) η expression for the amplification term T2 . R1 + R2 − iωR1 R2 CWk Ẑ(ω) = . (4) 1 − iωR2 CWk 2.3.2. Approximate Amplification Term A Taylor series expansion of the amplification term T̂2 (x, ω) η is the blood viscosity, A0 is the average luminal cross- about ω = 0 yields 3π r0 r2 sectional area, C = 2Eh d is the vessel compliance – with E  2 Cηx2   the Young modulus, h the wall thickness, and r0 and rd the xη ρr0 T̂2 ≈ − 1 − iω + Q̂in , (5) average and diastolic luminal radii, respectively—CWk is the WK- A0 K 0 6η 6A0 K 0 model compliance,h and R1 and i R2 are the WK-model resistances. K(ω) = − iωρ η 1 − kr2J01J0(kr 0) (kr0 ) is the dynamic permeability where where K 0 = r02 /8 is the steady state permeability. The time- J0 and J1 are the Bessel functions of order zero and one, domain solution of Equation (5) is given by respectively, k2 = iωρη and ρ is the blood density. Inverse Fourier ρr02 8Cηx2   transforms can be applied to obtain time-domain blood pressure, 8x η 4x η dQin (t) T2 (x, t) = − 4 Qin (t) − + . (6) p(x, t), and its attenuation and amplification components T1 (x, t) πr0 3πr04 η πr04 dt and T2 (x, t), respectively. Figure 4 compares the pressure waves simulated by the For all segments of the aortic-brachial arterial path, the term reduced and single-vessel models along the brachial artery. It ρr02 containing dQin /dt dominates. Moreover, the viscous time η includes the terms T1 and T2 calculated using the single-vessel 8Cηx2 model (Equations 2 and 3), with Qin (t) the flow wave measured is at least three orders of magnitude larger than π r04 (see at the inlet of the brachial artery of the reduced model, l = 42.2 Supplementary Section 2.1). As a result, Equation (6) can be cm the total length of the brachial artery, r0 ≈ rd = 4.3 mm, further approximated as the average radii at diastolic pressure of the 5 arterial segments that make up the brachial artery in the reduced model, and 4x ρ dQin (t) T2 (x, t) = − . (7) Eh = 234 Pa m the average Eh value along the brachial artery, 3πr02 dt Frontiers in Physiology | www.frontiersin.org 5 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow FIGURE 4 | Analytical pressure wave decomposition. The total pressure wave calculated using the reduced model (first column) at proximal (top), midpoint (middle), and distal (bottom) sites along the brachial artery was decomposed into the attenuation, T1 (second column) and amplification, T2 (third column) terms calculated using the single-vessel model (Equations 2 and 3). The approximated total (sum of Equations 2 and 7) and T2 pressures (computed using Equation 7) are shown in dotted lines (first and third columns, respectively). Horizontal dashed lines highlight pulse pressure (PP) values. Vertical dashed lines (bottom plots) indicate the time of minimum dQin /dt. According to this equation the main determinants of T2 along an with PPIn the pulse pressure at the inlet. Figure 4 (third column) arterial segment are its radius, length, and the rate of change of compares the exact T2 term along the brachial artery given Qin with time. Moreover, the amplification term T2 is maximum by Equation (3) with the corresponding approximated term when dQin /dt is minimum and, hence, systolic blood pressure, calculated using Equation (7). To avoid high-frequency noise, PSys (x), along an arterial segment can be estimated as the time derivative of Qin was filtered by computing the Fourier series of dQin /dt and keeping terms up to 15 Hz to reproduce 4xρ dQin the filtered signal. The vertical dotted lines in the first and third PSys (x) = PIn Sys − , (8) columns indicate the time of the maximum temporal rate of 3πr02 dt min decrease in late-systolic flow, dQin /dt min , which approximately corresponds to the time of peak pressure for the reduced model where PInSys is the systolic pressure at the inlet and dQin /dt min and the exact T2 term. The approximated pressure given by denotes the maximum temporal rate of decrease in late-systolic the sum of Equations (2) and (7), and shown on Figure 4 flow at the inlet. Assuming equal diastolic pressure at any point (first column) with dotted lines, overestimates systolic pressures along the arterial segment, PP and 1PP can be expressed as a calculated by the reduced model, shown on Figure 4 (first function of the distance x from the inlet as column) with continuous lines, by less than 6%. 4xρ dQin PP(x) = PPIn − , (9) 3πr02 dt min 2.4. Methodology for Estimating cPP Based on the previous analytical expressions for the amplification 4xρ dQin term T2 , two methods can be developed for estimating cPP at the 1PP(x) = − , (10) 3πr02 dt min aortic root from the flow at the aortic root or the brachial artery Frontiers in Physiology | www.frontiersin.org 6 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow FIGURE 5 | The two cPP estimation methods. (A) Pulse pressure, PP(x), along the aortic-brachial arterial path with distance, x, from the aortic root. The start of the brachial artery is indicated by the dashed vertical line. PP values calculated along the path using the reduced model (dots). Methods 1 and 2 estimate cPP (solid lines) based on the approximate expression for PP(x) given by Equation (11), starting from the PP value at the outlet of the brachial artery, pPP. (B) Approximate calculation  of the maximum temporal rate of decrease in late-systolic flow, dQin /dt min,Ao , from the aortic flow waveform, Qin (t). and pPP at the outlet of the brachial artery. Both are illustrated in 2.4.2. Method 2 Figure 5. This method assumes that all the change in PP along the aortic- brachial arterial path occurs in the brachial artery. As a result, 2.4.1. Method 1 Equation (12) becomes This method assumes that Equation (9) can be applied to the entire aortic-brachial arterial path as 4 ρ lBra dQin cPP = pPP + , (15) 3π (rBra )2 dt min,Bra 4xρ dQin PP(x) = cPP − , (11) where lBra and rBra are the length and mean radius of the brachial 3π (r̄)2 dt min,Ao artery, and dQin /dt min,Bra is the maximum rate of decrease of late systolic blood flow at the inlet of the brachial artery. with x the distance from the aortic root, cPP the PP at the aortic  Equation (15) was used to estimate cPP for the in silico dataset. root, dQin /dt min,Ao the maximum temporal rate of decrease in The following correlations and assumptions were used for the in late-systolic flow at the aortic root and r̄ the mean radius of the vivo dataset. According to the in silico dataset, rBra is strongly aortic-subclavian arterial segments. Note that the suffix Ao stands correlated with the radius of the aortic root, rAo (both in for aortic. Equation (11) can be used to compute cPP from the PP cm) through at the brachial artery, pPP, as rBra = 0.246 rAo − 0.04, (16) 4 ρ lAo−Bra dQin cPP = pPP + , (12) with a correlation coefficient R = 0.98. Moreover, 3π (r̄)2 dt min,Ao dQin /dt min,Bra is correlated with the corresponding value at the aortic root (both in mL/s2 ), through where lAo−Bra is the length of the aortic-brachial arterial path. Equation (12) was used to estimate cPP for the in silico dataset. dQin dQin The following assumptions were used for the in vivo dataset. = 0.0763 + 22, (17) dt min,Bra dt min,Ao When using in vivo aortic flow data with low temporal resolution, such as the Doppler ultrasound data used in this study, with R = 0.80. We therefore estimated the brachial radius and dQin /dt min,Ao can be approximated by (Figure 5B) maximum temporal rate of decrease of brachial flow from the corresponding in vivo data at the aortic root.  dQin   1Qin  The length of the brachial artery was estimated by multiplying ≈ , (13) the aortic-brachial length computed using Equation (14) by a dt min,Ao 1t Ao factor of 0.73. This corresponds to the ratio of the brachial artery length to the total aortic-brachial length measured in the in with 1Qin the end-systolic flow minus the peak flow and 1t the silico dataset. time of end of systole minus the time of peak flow. lAo−Bra was estimated using the empirical expression (Sugawara et al., 2018) 3. RESULTS lAo−Bra (mm) = 37.9 × sex(male = 1, female = 0) 3.1. Approximate Pressure Calculations +1.4 × age(years) + 2.5 × height(cm) − 14.8. (14) Using the Single-Vessel Model Figures 6A,B shows the ability of the single-vessel model to The radius of the aorta was used as a surrogate of the mean radius describe—through Equations (8) and (9), respectively—systolic of the aortic-subclavian arterial segments, that is, r̄ ≈ rAo . pressure, PSys , and PP for each segment along the aortic-brachial Frontiers in Physiology | www.frontiersin.org 7 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow FIGURE 6 | Calculation of systolic blood pressure, PSys , and pulse pressure, PP, using the single-vessel model. PSys (A), and PP (B) (solid lines) calculated using Equations (8) and (9), respectively. The dots show the corresponding values obtained by the reduced model. Luminal diastolic radius, rd (C), and maximum temporal rate of decrease in late-systolic flow, −dQin /dt|min from the reduced model (D) at the inlet and outlet of each segment of the aortic-brachial arterial path, for the baseline 25 (left) and 75 (right) year-old subject from the in silico dataset. Dashed vertical lines indicate the start of the brachial artery. arterial path, for the 25 (left) and 75 (right) year-old baseline Relative errors were smaller than 1.3% for PSys and 13.7% for PP subjects of the in silico dataset. Results are compared with for the 25 year-old in silico subject and 1.8% for PSys and 2.5% for corresponding PSys and PP values obtained from reduced models PP for the 75 year-old in silico subject, along the aortic-brachial of each subject. The inlet systolic pressure, PInSys , in Equation arterial path. The larger luminal radii and smaller rate of blood In (8) and inlet pulse pressure, PP , in Equation (9) for the first flow decrease in late systole in the 75-year-old subject led to a arterial segment (ascending aorta) were taken from the values smaller pulse pressure amplification from aortic root to brachial provided by the corresponding reduced model. For the remaining artery, compared to the 25-year-old subject. segments, PIn In Sys and PP were assumed to be equal to the values Starting from the PP values at the outlet of the brachial artery calculated at the outlet of the adjacent upflow segment using of the reduced model and assuming that the PP at the inlet Equations (8) and (9), with the luminal diastolic radius and of an arterial segment is equal to the PP at the outlet of the maximum temporal rate of decrease in late-systolic blood flow adjacent one, Equations (9) and (10) applied to each arterial shown in Figures 6C,D computed from the reduced model. segment from the brachial artery to the aortic root enabled Frontiers in Physiology | www.frontiersin.org 8 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow us to calculate cPP and 1PP with variations in cardiovascular 12) overestimated cPP (Figure 8A, bottom) and Method 2 properties. Cardiovascular properties variations for the 25-year- underestimated cPP (Figure 8B, bottom). The mean ± SD error old in silico subjects led to relative errors smaller than 18% for was 3.3 ± 2.8 mmHg for Method 1 and -1.2 ± 2.7 mmHg for cPP and 24% for 1PP when the morphology of the aortic flow Method 2. Estimated cPP values were closer to measured cPP wave was modified to account for changes in stroke volume, values for Method 2. When only the 75-year-old subjects were heart rate and left ventricular ejection time (results are shown considered (blue dots), the agreement between cPP obtained in the Supplementary Figure 3A). On the other hand, variations from Method 1 and measured values was much closer, with R2 in total vascular resistance, total vascular compliance, total path values and mean ± SD errors of 0.99 and 0.1 ± 2.7 mmHg for length and average network radius (for the same aortic root Method 1, respectively, and 0.99 and −3.4 ± 2.7 mmHg for waveform) led to relative errors smaller than 20% for cPP and Method 2, respectively. 29% for 1PP (Supplementary Figure 3B). According to Equation (10), 1PP along an arterial segment is 3.3. In vivo Verification of cPP Estimation proportional to the segment length and the maximum temporal Methods rate of decrease in late systolic-flow; and inversely proportional We also tested the accuracy of the two new cPP estimation to the square of the vessel radius. These proportionalities methods on the in vivo data (Figure 8, right). For the whole were verified in all the arterial segments of the aortic-brachial dataset, estimated cPP was strongly correlated with measured arterial path for the 25- and 75-years-old in silico subjects. cPP, though with R2 values smaller than those obtained for the in Proportionalities for length and temporal rate of decrease in silico dataset: 0.74 for Method 1 and 0.75 for Method 2. Method late systolic-flow exhibited linear correlations with the smallest 1 led to a smaller mean error than the one obtained for the in correlation coefficient (R2 ) of 0.91. Proportionality in radius silico dataset (−2.0 vs. 3.3 mmHg). However, the SD was greater exhibited a power-decay with exponents of −1.73 and −2.18 in the in vivo dataset (7.0 vs. 2.8 mmHg). For Method 2, both for the 25- and 75-year-old in silico subjects, respectively, both the mean error and SD were larger in the in vivo dataset (5.1 ± exponents are close to the expected value of −2. Results are 6.9 vs. −1.2 ± 2.7 mmHg). For both methods, no considerable shown in the Supplementary Figure 2. differences in mean, SD or R2 values were observed between normotensive (Method 1: −2.4 ± 6.7 mmHg, R2 = 0.72; Method 2: 4.9 ± 6.4 mmHg, R2 = 0.75) and hypertensive (Method 1: 3.2. In silico Verification of cPP Estimation −1.9 ± 7.4 mmHg, R2 = 0.74; Method 2: 5.3 ± 7.4 mmHg, R2 Methods = 0.74) subjects. Figure 7 shows the ability of the cPP estimation methods to describe—through Equation (12) for Method 1 and Equation (15) for Method 2—changes in cPP and 1PP, respectively, 4. DISCUSSION with variations in cardiovascular properties. Mean and standard deviation (SD) for stroke volume, heart rate, left ventricular We have developed and tested two methods for estimating cPP ejection time, total vascular resistance and total vascular from aortic or brachial blood flow and peripheral blood pressure. compliance correspond to the 25-year-old in silico subjects. The Both methods originated from a model of blood flow in the length and the radius of each vessel of the 25-year-old baseline larger arteries of the systemic circulation based on physical subject were changed by 14% and 11%, respectively. These principles (conservation of mass and linear momentum). The percentages were calculated from the covariance (SD/Mean) of mathematical complexity of this full model was simplified based the aortic arch length (Bensalah et al., 2014) and brachial artery on the physics of blood flow, resulting in a reduced model of radius (van der Heijden-Spek et al., 2000), respectively. Both blood flow in the upper thoracic aorta and left brachial artery that methods were able to capture changes in cPP and 1PP produced allows for workable analytical solutions in the frequency domain. by the variations in cardiovascular properties. Excluding radii This reduced model enabled us to obtain an approximation variations, Method 1 led to relative errors smaller than 20% for blood pressure, in time domain, along each segment of the for cPP and 28% for 1PP. Excluding radii variations, Method aortic-brachial arterial path from which the main cardiovascular 2 led to smaller relative errors: 18% for cPP and 24% for determinants of 1PP could be identified, leading to the two 1PP. Except for radii variation where a competing effect arises, new cPP estimation methods. Both methods were able to (i) approximate PP and 1PP values followed the trends provided by capture changes in cPP and 1PP produced by variations in the reduced model. For all the cardiac variations 1PP increases cardiovascular properties and (ii) estimate cPP with mean with the maximum temporal rate of decrease in late-systolic differences smaller than 3.3 ± 2.8 mmHg on in silico data for flow and length; and is almost not affected by the total vascular different age groups (25–75 years old) and 5.1 ± 6.9 mmHg on resistance and total compliance as described by Equation (10) in vivo data for normotensive and hypertensive subjects. and consequently by Equations (12) and (15). We tested the accuracy of the two new cPP estimation 4.1. Main Cardiovascular Determinants of methods on the dataset of in silico pulse waves (Figure 8, left). Pulse Pressure Amplification For the whole dataset, estimated cPP was strongly correlated We have obtained a simple expression highlighting the main with measured cPP, with R-squared (R2 ) values of 0.97 for cardiovascular determinants of 1PP along an arterial segment. both methods (Figures 8A,B, top). Overall, Method 1 (Equation According to Equation (10), 1PP is directly proportional to the Frontiers in Physiology | www.frontiersin.org 9 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow FIGURE 7 | Effect of cardiovascular properties on central pulse pressure, cPP, and its amplification from the aortic root to the outlet of the brachial artery, 1PP. Aortic root (A) and brachial artery (B) flow wave (first column), cPP (second column), and 1PP (third column) for the 25-year-old baseline subject (black) and with a standard deviation (SD) decrease (blue) and a SD increase (red) in (A) stroke volume (SV), heart rate (HR) and left ventricular ejection time (LVET), and (B) total vascular resistance (RT ) and total vascular compliance (CT ); and with a 14% decrease (blue) and 14% increase (red) in the total network length (LT ) and with a 11% decrease (blue) and 11% increase (red) in the average radius of the network (hrN i). The closed dots were calculated using the reduced model, the open triangles were calculated using Equation (12) of Method 1 and the open dots were calculated using Equation (15) of Method 2. Legends in the first column indicate the maximum temporal rate of decrease in late systolic-flow in mL/s2 for all flow waves. Frontiers in Physiology | www.frontiersin.org 10 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow FIGURE 8 | Estimated (cPPE ) vs. measured (cPPM ) central pulse pressure. Correlation and Bland-Altman plots comparing cPPE values calculated using Methods 1 (A) and 2 (B) with (cPPM ) values for the in silico (left) and in vivo (right) datasets. In vitro 75-year-old subjects are shown in blue dots. In vivo hypertensive subjects are shown in red dots. Dotted horizontal lines show the limits of agreement (±1.96SD) between estimated and reference cPP values. Frontiers in Physiology | www.frontiersin.org 11 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow distance along the arterial segment and the maximum temporal the height which in turn is significantly different with gender rate of decrease in late-systolic flow, and inversely proportional (Max Roser and Ritchie, 2019), however, in vivo measurements to the square of the vessel radius. Vascular geometry, therefore, of radius and proximal flow at the brachial artery would be plays a very important role in changing PP from central to needed to corroborate the correlations between 1PP and peripheral arterial sites. This result is in agreement with previous its determinants. studies in which 1PP was associated with higher height and, hence, greater vessel length (Asmar et al., 1997; Camacho et al., 2004) and greater tapering and, hence, greater decrease in vessel 4.2. Aging and Pulse Pressure radius (Belardinelli and Cavalcanti, 1992; Mynard and Smolich, Amplification 2015). Mynard and Smolich (2015) found, using computational Aging is associated with increased PP (Pinto, 2007) and decreased modeling, that brachial artery tapering is a major factor leading 1PP (Herbert et al., 2014). It has been shown to increase PP more to wave intensity amplification in the arm. They suggested that rapidly in central rather than distal arteries and, consequently, this may explain why amplification of wave intensity, which is it has been suggested that 1PP attenuation is predominantly directly related to the rate of change of pressure with time (and caused by an increase in central systolic BP (cSBP) which, in hence PP), was not found in other regions where the degree of turn, is induced by a rise in arterial stiffness (Safar et al., 2002). tapering was lower. Our analytical results further support this Other studies have highlighted the importance of ventricular finding. We have shown that changes in PP mainly occur in the dynamics—in addition to arterial stiffness—in determining PP brachial artery rather than in the aorta (Figure 6B), despite the (Fok et al., 2014b; Vennin et al., 2017; Li et al., 2019). Our rate of decrease in systolic flow being greater in the aorta than in study shows that the decrease in 1PP with age is mainly caused the brachial artery (Figure 6D). This is because (i) the brachial by the increase in arterial radius and decrease in the rate of artery is longer than the upper thoracic aorta and (ii) the vessel change of aortic flow with time in late systole (see Equation radius, which has an inversely quadratic contribution to 1PP in 10) strongly correlated with left ventricular ejection dynamics. Equation (10), is much smaller in the brachial artery (Figure 6C). The mechanisms behind the age-related increase in diameter are Equation (10) highlights the predominant role that flow wave still unclear, though they may be the result of a compensatory morphology in late systole plays in 1PP. Cardiac properties adaptation to plaque formation and/or increases in wall thickness such as stroke volume (SV), heart rate (HR) and left ventricular (Thijssen et al., 2015). ejection time (LVET) introduce variations in late-systolic aortic flow resulting in changes in 1PP (Figure 7). In particular, increasing SV or HR and decreasing LVET augments the rate 4.3. Central Pulse Pressure Estimation of decrease in late-systolic aortic flow and causes a greater 1PP, Methods as predicted by Equation (10) and consequently by Equations We have developed two methods that estimate cPP from (12) and (15). This result is in agreement with previous studies measurements of the aortic flow wave and pPP that can be underlining the potential importance of ventricular dynamics in obtained by non-invasive means; e.g., Doppler ultrasound or determining 1PP (O’Rourke, 1970; Asmar et al., 1997; Gaddum magnetic resonance imaging for aortic flow and an oscillometric et al., 2017); e.g., O’Rourke (1970) found that, in human subjects, device for pPP. We have tested both methods on a dataset 1PP increases with decreasing LVET. Equation (10) provides a of in silico pulse waves and on a cohort of normotensive and mechanistic explanation of this result since a smaller LVET would hypertensive subjects, which covered a wide range of pressure lead to a greater temporal rate of decrease in late-systolic flow wave morphologies and cPP values including those seen in and, hence, greater 1PP. hypertensive subjects (Table 1). Stronger linear correlations Interestingly, 1PP is independent of the total compliance between estimated and reference cPP values were observed for of the network (Figure 7B), again in agreement with Equation the in in silico data compared to the in vivo data, the same was (10) and despite cPP decreasing with the increasing compliance. true for the smaller values for the mean and SD of the errors for Indeed, cPP is mainly determined by total arterial compliance cPP estimates (Figure 8). This is due to the larger experimental and left ventricular ejection dynamics affecting SV (Vennin et al., errors of the in vivo measurements, compared to the uncertainties 2017). Except for radii variation, where the effects of the radius of the in silico data due to model assumptions. and the maximum temporal rate of decrease in late systolic-flow Recent (2017) clinical guidelines for the validation of non- (induced by radii variation) are competing, approximate PP and invasive central blood pressure devices propose a mean absolute 1PP values followed the trends provided by the reduced model difference ≤ 5 mmHg with a SD ≤ 8 mmHg compared with (Figure 7). the reference cSBP (Sharman et al., 2017). Method 1 led to Our analytical results are in concordance with the mean absolute differences within recommended values (2.0 ± 7.0 experimental observation of a linear increase in 1PP with mmHg), though Method 2 did not (5.1 ± 6.9 mmHg). Direct the distance along an arterial segment (Gaddum et al., 2017) implementation of Method 2 requires measurements of the flow and further analysis of cardiac variables could be performed wave, vessel length and vessel radius in the brachial artery, which in experimental setups to corroborate our analytical results. were not available for our in vivo cohorts and could lead to Observed tendencies of in vivo measurements, like the difference smaller mean absolute differences within recommended values. in 1PP with gender (Segers et al., 2009), can be analyzed in terms Indeed, Method 2 led to smaller differences than Method 1 when of our results, namely, the brachial artery length correlates with tested using in silico measurements in the brachial artery. Frontiers in Physiology | www.frontiersin.org 12 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow By using transfer functions, the influence of the large Although comparison with the dataset of in silico pulse waves has arteries and peripheral load on pulse pressure amplification provided a wider range of pressure and flow wave morphologies has been studied (Karamanoglu et al., 1995). Using a single- and hemodynamic conditions, indicating that both methods tube transmission line model enables personalization of some would be equally valid in pathological conditions, such as systolic of the transfer function parameters, improving the accuracy hypertension in older subjects and in heart failure, this needs of the estimated cBP waveform (Swamy et al., 2009; Hahn testing prospectively. et al., 2012; Gao et al., 2016; Natarajan et al., 2017). Further improvement in cBP assessment could be achieved by including 4.5. Perspectives additional patient-specific information to transfer functions Assessment of cPP has been shown to be of greater based on the results of our study; i.e., the geometry of the predictive value for cardiovascular outcomes than brachial artery and the flow morphology. It is worth mentioning brachial PP. It is, therefore, essential to understand the that our model approach differs from the lossless transmission hemodynamic determinants of PP amplification. The line model approach used by Swamy et al. (2009), Hahn results that we have presented here show that blood flow et al. (2012), Gao et al. (2016), Natarajan et al. (2017). First, is a key determinant of PP amplification, confirming the starting from continuum mechanics (fluid dynamics, and a importance of measuring flow and the key role played model for vessel elasticity), our approach accounts for the spatial by ventricular ejection. Therefore, conditions and drugs and temporal variations of blood pressure and flow in the that influence cardiac function may influence pulse wave brachial artery. Second, it accounts for viscous losses along morphology independent of arterial function. Moreover, the vessel. Third, it decomposes pressure along the vessel into the two methods proposed here allow for a more regular space-varying attenuation and amplification terms, whereas the assessment of a patient’s cPP, due to their non-invasive transmission line model does a decomposition into forward and nature which removes the risk of complications due to backward waves that are independent of position (the distance cardiac catheterization. between the peripheral and central positions is introduced by adding a time delay). The main determinants of 1PP are also 5. CONCLUSION different in both models: the pulse transit time and reflection coefficient in the transmission line model, and the inflow wave We have identified the main determinants of pulse pressure morphology and arterial geometry in our model. Notably, the amplification—highlighting the important role of flow main determinants of 1PP in our model are independent of morphology—and presented two methods based on the peripheral vascular properties. physics of blood flow for estimating central pulse pressure Machine learning approaches could also be used to estimate from non-invasive measurements of aortic flow and peripheral cPP from aortic or brachial flow and pPP. However, like blood pressure. We have tested both methods on in silico data transfer function methods, they should be trained on datasets for different age groups (25–75 years old) and in vivo data for covering the large range of physiological and pathophysiological normotensive and hypertensive subjects. Our approach could conditions encountered in healthy and diseased subjects, to improve cardiovascular function assessment in clinical cohorts make them widely applicable. The ability to numerically generate for which aortic ultrasound or magnetic resonance imaging data datasets containing hemodynamic data for thousands of virtual are available. subjects (Charlton et al., 2019) could facilitate this process. DATA AVAILABILITY STATEMENT 4.4. Limitations This study is subject to several important limitations. Our The raw data supporting the conclusions of this article will be non-invasive measurements of pressure and the flow were made available by the authors, without undue reservation. not simultaneous and subject to experimental error. Carotid pressure is also an imperfect surrogate of aortic pressure and AUTHOR CONTRIBUTIONS subject to calibration errors. Both Methods 1 and 2 require a flow wave measurement in addition to the peripheral pulse All authors contributed to the article and approved the pressure. Calculation of the maximum temporal rate of flow submitted version. decrease in late systole requires differentiation of the flow wave, which can be challenging when working with noisy flow waves; FUNDING e.g., those acquired by Doppler ultrasound. To facilitate this calculation, we have approximated this maximum temporal JFG acknowledges support of a postdoctoral fellowship from rate by the slope given from peak flow to end of systolic CONACyT [CVU:288182]. ECP and JA acknowledge the UK- flow (see Figure 5B). Further validation of both methods using Mexico Visiting Chair for a mobility grant. ECP acknowledges more accurate techniques for measuring flow, such as magnetic funding from the Centre for Mexican Studies UNAM-UK, resonance, would be valuable, as well as further validation of for financing a short stay; CONACyT (Mexico), through Method 2 using measurements of blood flow and vascular project no. 219584 and agreement no. 2018-000007- 01EXTV- geometry in the brachial artery. We tested both methods only 00183; the Faculty of Chemistry UNAM, through PAIP 5000- in normotensive subjects and in otherwise hypertensive subjects. 9011; and DGAPA, UNAM, through a PASPA programme, Frontiers in Physiology | www.frontiersin.org 13 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow during a sabbatical leave. PC and JA were partially funded EPSRC, BHF, Wellcome Trust, NIHR, GSTT or Ministry by the British Heart Foundation (BHF) [PG/15/104/31913 of Science. and PG/17/50/32903]. The authors also acknowledge financial support from the Wellcome EPSRC Centre for Medical ACKNOWLEDGMENTS Engineering at King’s College London [WT 203148/Z/16/Z], Department of Health through the National Institute for Health The authors thank Drs. Ye Li (King’s College London), Jorge Research (NIHR) Cardiovascular MedTech Co-operative at Aramburu (Navarra University), and Peter Charlton (King’s Guy’s and St Thomas’ NHS Foundation Trust (GSTT) and College London) for data and programming support. the Ministry of Science and Higher Education of the Russian Federation within the framework of state support for the SUPPLEMENTARY MATERIAL creation and development of World-Class Research Centers Digital biodesign and personalized healthcare [075-15-2020- The Supplementary Material for this article can be found 926]. The views expressed are those of the authors and online at: https://www.frontiersin.org/articles/10.3389/fphys. not necessarily those of the UNAM, CONACyT, AMEXCID, 2021.608098/full#supplementary-material REFERENCES contractility/relaxation dynamics. Hypertension 63, 1050–1055. doi: 10.1161/HYPERTENSIONAHA.113.02955 Agabiti-Rosei, E., Mancia, G., O’Rourke, M. F., Roman, M. J., Fossan, F. E., Mariscal-Harana, J., Alastruey, J., and Hellevik, L. R. (2018). Safar, M. E., Smulyan, H., et al. (2007). Central blood pressure Optimization of topological complexity for one-dimensional arterial blood flow measurements and antihypertensive therapy. Hypertension 50, 154–160. models. J. R. Soc. Interface 15:20180546. doi: 10.1098/rsif.2018.0546 doi: 10.1161/HYPERTENSIONAHA.107.090068 Gaddum, N., Alastruey, J., Chowienczyk, P., Rutten, M. C. M., Segers, P., and Agabiti-Rosei, E., and Muiesan, M. L. (2015). Understanding and treating central Schaeffter, T. (2017). Relative contributions from the ventricle and arterial tree blood pressure. Dialog. Cardiovasc. Med. 20, 169–184. Available online at: to arterial pressure and its amplification: an experimental study. Am. J. Physiol. https://www.dialogues-cvm.org/ Heart Circ. Physiol. 313, H558–H567. doi: 10.1152/ajpheart.00844.2016 Asmar, R., Brisac, A., Courivaud, J., Lecor, B., London, G., and Safar, M. (1997). Gao, M., Rose, W. C., Fetics, B., Kass, D. A., Chen, C.-H., and Mukkamala, R. Influence of gender on the level of pulse pressure: the role of large conduit (2016). A simple adaptive transfer function for deriving the central blood arteries. Clin. Exp. Hypertens. 19, 793–811. doi: 10.3109/10641969709083187 pressure waveform from a radial blood pressure waveform. Sci. Rep. 6:33230. Avolio, A. P., Bortel, L. M. V., Boutouyrie, P., Cockcroft, J. R., McEniery, doi: 10.1038/srep33230 C. M., Protogerou, A. D., et al. (2009). Role of pulse pressure Hahn, J., Reisner, A. T., Jaffer, F. A., and Asada, H. H. (2012). Subject-specific amplification in arterial hypertension. Hypertension 54, 375–383. estimation of central aortic blood pressure using an individualized transfer doi: 10.1161/HYPERTENSIONAHA.109.134379 function: a preliminary feasibility study. IEEE Trans. Inf. Technol. Biomed. 16, Belardinelli, E., and Cavalcanti, S. (1992). Theoretical analysis of pressure 212–220. doi: 10.1109/TITB.2011.2177668 pulse propagation in arterial vessels. J. Biomech. 25, 1337–1349. Hashimoto, J., and Ito, S. (2010). Pulse pressure amplification, doi: 10.1016/0021-9290(92)90289-D arterial stiffness, and peripheral wave reflection determine pulsatile Bensalah, M. Z., Bollache, E., Kachenoura, N., Giron, A., De Cesare, A., flow waveform of the femoral artery. Hypertension 56, 926–933. Macron, L., et al. (2014). Geometry is a major determinant of flow reversal doi: 10.1161/HYPERTENSIONAHA.110.159368 in proximal aorta. Am. J. Physiol. Heart Circ. Physiol. 306, H1408–H1416. Herbert, A., Cruickshank, J. K., Laurent, S., and Boutouyrie, P. (2014). Establishing doi: 10.1152/ajpheart.00647.2013 reference values for central blood pressure and its amplification in a general Camacho, F., Avolio, A., and Lovell, N. H. (2004). Estimation of healthy population and according to cardiovascular risk factors. Eur. Heart J. pressure pulse amplification between aorta and brachial artery using 35, 3122–3133. doi: 10.1093/eurheartj/ehu293 stepwise multiple regression models. Physiol. Meas. 25, 879–889. Karamanoglu, M., Gallagher, D. E., Avolio, A. P., and O’Rourke, M. F. doi: 10.1088/0967-3334/25/4/008 (1995). Pressure wave propagation in a multibranched model of the Charlton, P., Mariscal Harana, J., Vennin, S., Li, Y., Chowienczyk, P., human upper limb. Am. J. Physiol. Heart Circ. Physiol. 269, H1363–H1369. and Alastruey, J. (2019). Modelling arterial pulse waves in healthy doi: 10.1152/ajpheart.1995.269.4.H1363 ageing: a database for in silico evaluation of haemodynamics and pulse Li, Y., Gu, H., Fok, H., Alastruey, J., and Chowienczyk, P. (2017). Forward and wave indices. Am. J. Physiol. Heart Circ. Physiol. 317, H1062–H1085. backward pressure waveform morphology in hypertension. Hypertension 69, doi: 10.1152/ajpheart.00218.2019 375–381. doi: 10.1161/HYPERTENSIONAHA.116.08089 Chen, C.-H., Ting, C.-T., Nussbacher, A., Nevo, E., Kass, D. A., Pak, P., et al. Li, Y., Jiang, B., Keehn, L., Gu, H., Boguslavskyi, A., Cecelja, (1996). Validation of carotid artery tonometry as a means of estimating M., et al. (2019). Hemodynamic mechanism of the age-related augmentation index of ascending aortic pressure. Hypertension 27, 168–175. increase in pulse pressure in women. Hypertension 73, 1018–1024. doi: 10.1161/01.HYP.27.2.168 doi: 10.1161/HYPERTENSIONAHA.118.12402 Figueroa, C. A., and Humphrey, J. D. (2014). Pressure wave propagation in full- Max Roser, C. A., and Ritchie, H. (2019). Human Height. Our World in Data. body arterial models: a gateway to exploring aging and hypertension. Proc. Available online at: https://ourworldindata.org/human-height IUTAM 10, 382–395. doi: 10.1016/j.piutam.2014.01.033 McEniery, C. M., Cockcroft, J. R., Roman, M. J., Franklin, S. S., and Flores Gerónimo, J., Alastruey, J., and Corvera Poiré, E. (2016). A novel analytical Wilkinson, I. B. (2014). Central blood pressure: current evidence and approach to pulsatile blood flow in the arterial network. Ann. Biomed. Eng. 44, clinical importance. Eur. Heart J. 35, 1719–1725. doi: 10.1093/eurheartj/ 3047–3047. doi: 10.1007/s10439-016-1625-3 eht565 Fok, H., Guilcher, A., Brett, S., Jiang, B., Li, Y., Epstein, S., et al. Mynard, J. P., and Smolich, J. J. (2015). One-dimensional haemodynamic modeling (2014a). Dominance of the forward compression wave in determining and wave dynamics in the entire adult circulation. Ann. Biomed. Eng. 43, pulsatile components of blood pressure. Hypertension 64, 1116–1123. 1443–1460. doi: 10.1007/s10439-015-1313-8 doi: 10.1161/HYPERTENSIONAHA.114.04050 Natarajan, K., Cheng, H.-M., Liu, J., Gao, M., Sung, S.-H., Chen, C.-H., et al. (2017). Fok, H., Guilcher, A., Li, Y., Brett, S., Shah, A., Clapp, B., et al. Central blood pressure monitoring via a standard automatic arm cuff. Sci. Rep. (2014b). Augmentation pressure is influenced by ventricular 7:14441. doi: 10.1038/s41598-017-14844-5 Frontiers in Physiology | www.frontiersin.org 14 February 2021 | Volume 12 | Article 608098 Flores Gerónimo et al. cPP Estimation From Aortic Flow Ohno, Y., Kanno, Y., and Takenaka, T. (2016). Central blood pressure and Thijssen, D. H. J., Carter, S. E., and Green, D. J. (2015). Arterial structure and chronic kidney disease. World J. Nephrol. 5, 90–100. doi: 10.5527/wjn. function in vascular ageing: are you as old as your arteries? J. Physiol. 594, v5.i1.90 2275–2284. doi: 10.1113/JP270597 O’Rourke, M. F. (1970). Influence of ventricular ejection on the relationship van der Heijden-Spek, J. J., Staessen, J. A., Fagard, R. H., Hoeks, A. P., Boudier, between central aortic and brachial pressure pulse in man. Cardiovasc. Res. 4, H. A. S., and Bortel, L. M. V. (2000). Effect of age on brachial artery wall 291–300. doi: 10.1093/cvr/4.3.291 properties differs from the aorta and is gender dependent. Hypertension 35, Pauca, A. L., Wallenhaupt, S. L., Kon, N. D., and Tucker, W. Y. (1992). Does 637–642. doi: 10.1161/01.HYP.35.2.637 radial artery pressure accurately reflect aortic pressure? Chest 102, 1193–1198. Vennin, S., Li, Y., Willemet, M., Fok, H., Gu, H., Charlton, P., et al. (2017). doi: 10.1378/chest.102.4.1193 Identifying hemodynamic determinants of pulse pressure. Hypertension 70, Pichler, G., Martinez, F., Vicente, A., Solaz, E., Calaforra, O., and Redon, J. (2016). 1176–1182. doi: 10.1161/HYPERTENSIONAHA.117.09706 Pulse pressure amplification and its determinants. Blood Press 25, 21–27. Wilkinson, I. B., Franklin, S. S., Hall, I. R., Tyrrell, S., and Cockcroft, J. R. (2001). doi: 10.3109/08037051.2015.1090713 Pressure amplification explains why pulse pressure is unrelated to risk in young Pierce, G. L., Zhu, H., Darracott, K., Edet, I., Bhagatwala, J., Huang, Y., et al. (2012). subjects. Hypertension 38, 1461–1466. doi: 10.1161/hy1201.097723 Arterial stiffness and pulse-pressure amplification in overweight/obese African- Wilkinson, I. B., Mohammad, N. H., Tyrrell, S., Hall, I. R., Webb, D. J., Paul, V. E., American adolescents: relation with higher systolic and pulse pressure. Am. J. et al. (2002). Heart rate dependency of pulse pressure amplification and arterial Hypertens. 26, 20–26. doi: 10.1093/ajh/hps014 stiffness. Am. J. Hypertens. 15, 24–30. doi: 10.1016/S0895-7061(01)02252-X Pinto, E. (2007). Blood pressure and ageing. Postgrad. Med. J. 83, 109–114. Williams, B., Brunel, P., Lacy, P. S., Baschiera, F., Zappe, D. H., Kario, K., doi: 10.1136/pgmj.2006.048371 et al. (2017). Application of non-invasive central aortic pressure assessment Safar, M. E., Blacher, J., Pannier, B., Guerin, A. P., Marchais, S. J., Guyonvarc’h, P.- in clinical trials: clinical experience and value. Artery Res. 17, 1–15. M., et al. (2002). Central pulse pressure and mortality in end-stage renal disease. doi: 10.1016/j.artres.2016.10.154 Hypertension 39, 735–738. doi: 10.1161/hy0202.098325 Williams, B., Lacy, P. S., Thom, S. M., Cruickshank, K., Stanton, A., Segers, P., Mahieu, D., Kips, J., Rietzschel, E., Buyzere, M. D., Bacquer, Collier, D., et al. (2006). Differential impact of blood pressure; lowering D. D., et al. (2009). Amplification of the pressure pulse in the upper drugs on central aortic pressure and clinical outcomes. Principal limb in healthy, middle-aged men and women. Hypertension 54, 414–420. results of the conduit artery function evaluation (CAFE) study. doi: 10.1161/HYPERTENSIONAHA.109.133009 Circulation 113, 1213–1225. doi: 10.1161/CIRCULATIONAHA.105.5 Sharman, J. E., Avolio, A. P., Baulmann, J., Benetos, A., Blacher, J., Blizzard, C. 95496 L., et al. (2017). Validation of non-invasive central blood pressure devices: Williams, B., Mancia, G., Spiering, W., Agabiti Rosei, E., Azizi, M., Burnier, ARTERY Society task force consensus statement on protocol standardization. M., et al. (2018). ESC/ESH guidelines for the management of arterial Eur. Heart J. 38, 2805–2812. doi: 10.1093/eurheartj/ehw632 hypertension: the task force for the management of arterial hypertension Sharman, J. E., Marwick, T. H., Gilroy, D., Otahal, P., Abhayaratna, W. P., and of the European Society of Cardiology and the European Society of Stowasser, M. (2013). Randomized trial of guiding hypertension management Hypertension. J. Hypertens. 36, 1956–2041. doi: 10.1097/HJH.00000000000 using central aortic blood pressure compared with best-practice care. 01940 Hypertension 62, 1138–1145. doi: 10.1161/HYPERTENSIONAHA.113.02001 Shih, Y.-T., Cheng, H.-M., Sung, S.-H., Hu, W.-C., and Chen, C.-H. Conflict of Interest: The authors declare that the research was conducted in the (2011). Quantification of the calibration error in the transfer function- absence of any commercial or financial relationships that could be construed as a derived central aortic blood pressures. Am. J. Hypertens. 24, 1312–1317. potential conflict of interest. doi: 10.1038/ajh.2011.146 Sugawara, J., Tomoto, T., and Tanaka, H. (2018). Arterial path length estimation Copyright © 2021 Flores Gerónimo, Corvera Poiré, Chowienczyk and Alastruey. for heart-to-brachium pulse wave velocity. Hypertens. Res. 41, 444–450. This is an open-access article distributed under the terms of the Creative Commons doi: 10.1038/s41440-018-0019-3 Attribution License (CC BY). The use, distribution or reproduction in other forums Swamy, G., Xu, D., Olivier, N. B., and Mukkamala, R. (2009). An adaptive transfer is permitted, provided the original author(s) and the copyright owner(s) are credited function for deriving the aortic pressure waveform from a peripheral artery and that the original publication in this journal is cited, in accordance with accepted pressure waveform. Am. J. Physiol. Heart Circ. Physiol. 297, H1956–H1963. academic practice. No use, distribution or reproduction is permitted which does not doi: 10.1152/ajpheart.00155.2009 comply with these terms. Frontiers in Physiology | www.frontiersin.org 15 February 2021 | Volume 12 | Article 608098