Relation between blood pressure and pulse wave velocity for human arteries

In Vitro Experiments.

An in vitro hemodynamic simulator is developed, as shown in Fig. 2A, to verify the theory. The simulator includes a pulsatile flow generator, an artificial blood vessel, strain sensors, pressure sensors, and a water reservoir to define the base pressure on the tube. Pressurized water controlled by solenoid valve provides pulsatile flow, while the height of the water reservoir determines the diastolic pressure in the tube. Thin PDMS tubes with various wall thicknesses and elastic properties (controlled by changing the base to curing agent mixing ratio) provide artificial blood vessels with linear elastic properties within the range of deformations studied (Fig. 2B). Two strain sensors made of carbon black-doped PDMS (CB-PDMS) (22) detect the pulses at two different positions along the tube. The time difference Δt between the arrival of the pulse at each sensor position together with the distance L between the two sensors allows calculation of PWV (Fig. 2C). Fig. 2D shows the voltage signal of the two sensors with the distance L, which gives the PWV (23)PWV=LΔt.[5]PDMS (base polymer: curing agent = 15:1, 17:1, and 19:1) is used to fabricate the tube for the in vitro experiment. SI Appendix, Fig. S1 shows the storage and loss moduli (E′ and E″) of the 17:1 PDMS measured by dynamic mechanical analysis at frequencies between 0.1 Hz and ∼10 Hz. The dynamic modulus is E′2+E″2. The moduli of the 15:1, 17:1, and 19:1 PDMS at 10 Hz are 650, 540, and 420 kPa, respectively, for the samples used in the experiment shown in Fig. 2F. SI Appendix, Fig. S2 shows the relation between the true stress and logarithmic strain of the 17:1 PDMS measured by tensile testing, which displays good linearity for strain less than 30%. The tube in the in vitro experiment is, therefore, linear elastic with the modulus of E (i.e., the dynamic modulus E′2+E″2 at 0 Hz).

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Fig. 2.

(A) Schematic diagram of the in vitro experimental setup. (B and C) Experimental image of (B) the PDMS tube and (C) the strain sensor. (D) Signals from the two sensors. (E) Comparison of normalized pressure versus normalized PWV for linear elastic behavior of the tube between the present model and the MK Equation. (F and G) Comparison between the present model and the in vitro experiment, without any parameter fitting, for (F) different materials and (G) different tube thickness.

The Relation Between Pressure and PWV for Linear Elastic Tube Walls.

The linear stress−strain relation for the PDMS tubes, together with Eq. 4, gives the relation between the pressure P and inner area A as (see SI Appendix, Note 1 for details)P=E¯4[dilog(A+AwallA0+Awall)−dilog(AA0)]+E¯8[ln(A+AwallA0+Awall)2−ln(AA0)2],[6]where E¯=E/(1−ν2) is the plane strain modulus; ν = 0.5 is the Poisson’s ratio for PDMS; A0=πR02 and Awall=π(R0+h0)2−πR02 are the inner area of the artery and the area of artery wall, respectively, without pressure; and dilog is the dilogarithm function (24). Substitution of Eq. 6 into Eq. 2 gives the PWV asPWV=E¯A4ρ[A0A(A−A0)lnAA0−A0+Awall(A+Awall)(A−A0)ln(A+AwallA0+Awall)].[7]Eqs. 6 and 7 are parametric equations for the relation between the pulse wave velocity PWV and pressure P; elimination of the intermediate variable A yields the following scaling law between the normalized PWV and pressure P:PWVE¯ρ=g(PE¯,h0R0),[8]where g is a nondimensional function shown in Fig. 2E. It is clear that PWV displays a strong dependence on P. For comparison, the MK Equation [1a] predicts a constant PWV (independent of the pressure), and is also shown in Fig. 2E. Fig. 2F indicates that, without any parameter fitting, the relation between PWV and P obtained from Eq. 8 agrees well with the in vitro experiments for 15:1, 17:1, and 19:1 PDMS and fixed R0 = 6.3 mm, h0 = 0.63 mm, and ρ = 1,000 kg/m³ for water. The effect of liquid viscosity is shown in SI Appendix, Note 2 and Fig. S3. Similarly, Fig. 2G shows excellent agreement with experimental results for two thicknesses (h0 = 0.63 and 0.29 mm) of the tube made of 19:1 PDMS and fixed R0 = 6.3 mm, and ρ = 1,000 kg/m³, without any parameter fitting. The experimental data all display strong dependence on the pressure, which clearly do not support the MK + Hughes Equations.

The Relation Between Blood Pressure and PWV for Human Artery Walls.

The human artery walls are well characterized by the Fung hyperelastic model (21), which has the strain energy densityW=C2ea1Eθθ2+a2Ezz2−C2,[9]where Eθθ and Ezz are the Green strains in the circumferential and axial directions of the artery, respectively, and a1, a2, and C are the material parameters, which are related to the elastic modulus (at zero pressure) by E0=Ca1. Following the same analysis, but with the linear elastic model replaced by the Fung hyperelastic model for human arteries, yields parametric equations for the relation between the pulse wave velocity and pressure, similar to Eqs. 6 and 7, as (see SI Appendix, Note 1 for details)P=14Cea2Ezz2πa1{erfi(A−A02A0a1)−erfi[A−A02(A0+Awall)a1]},[10]PWV=Cea2Ezz2a1A4ρ[1A0ea1(A−A0)24A02−1A0+Awallea1(A−A0)24(A0+Awall)2].[11]where erfi is the imaginary error function (25). Elimination of the intermediate variable A in Eqs. 10 and 11 yields the following scaling law between the normalized pulse wave velocity PWV and blood pressure P:PWVCea2Ezz2ρ=f(PCea2Ezz2,a1,h0R0),[12]where f is a nondimensional function, and is shown in Fig. 3A for a1 = 0.97 (26) and h0/R0 = 0.15 (19) for the human artery. Fig. 3B examines the effect of artery stretching Ezz by comparing the limit Ezz = 0 of Eq. 12, which takes the formPWVCρ=f(PC,a1,h0R0),[13]to the scaling law in Eqs. 10 and 11 for a representative a2 = 2.69 (21) and Ezz = 0.1 and 0.2. The effect of artery stretching is negligible even for 20% stretching.

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Fig. 3.

(A) Normalized blood pressure P versus normalized PWV for the human artery characterized by the Fung hyperelastic model. (B–D) Normalized P versus normalized PWV for (B) different axial stretching of the artery, (C) different thickness-to-radius ratio h0/R0 of the artery, and (D) different SD σ for a normal distribution of h0/R0.

The scaling law in Eq. 13 degenerates to the MK Equation [1a] in the limit of low blood pressure, which gives A→A0; therefore ea1(A−A0)2/(4A02)=∼1 and ea1(A−A0)2/[4(A0+Awall)2]=∼1. Eq. 11, at the limit Ezz = 0, then becomes PWV=Ca1Awall/[4ρ(A0+Awall)], which is identical to the MK Equation [1a] for a thin artery wall [i.e., Awall/(A0+Awall)=∼2h0/R0] at zero blood pressure.

The artery thickness, in general, is not a constant even for the same human artery; the thickness-to-radius ratio h0/R0 has an average of 0.15 and a variation of 40% (19). Fig. 3C shows the normalized pressure P/C versus PWV/C/ρ for h0/R0 = 0.09, 0.12, 0.15, 0.18, and 0.21, corresponding to ±20% and ±40% variations of h0/R0 = 0.15. Even for 40% variations, the curves in Fig. 3C are different by only ∼6%. For a1 = 0.97 and a normal distribution of h0/R0 with the mean 0.15 and SD σ, the mean PWV is obtained asPWVCρ=f[PC,a1=0.97,h0R0∼N(0.15,σ2)],[14]and is shown in Fig. 3D for several values of σ. The curve based on the mean h0/R0 gives an accurate relation between the PWV and blood pressure.

Fig. 4 A and B compares the present model (in Eq. 13) to the classical MK + Hughes Equations (in Eqs. 1a and 1b) for a human artery characterized by the Fung hyperelastic model with C = 39 kPa, a1 = 0.97, and h0/R0 = 0.15 (19, 26). The arterial stiffness, or the equivalent tangent modulus E, is shown in Fig. 4A versus the blood pressure P. In the range of human blood pressure (5 kPa to ∼20 kPa), the arterial stiffness is used to determine the material parameters in the Hughes Equation [1b] as E0 = 563 kPa and ζ = 0.121 kPa⁻¹, which yields good agreement between the Hughes Equation and the present model. However, for the same range of blood pressure, Fig. 4B shows that the MK + Hughes Equations overestimate the PWV by a factor of ∼2 compared with the present model. This large discrepancy results from the large change of radius and thickness of the artery wall (>50%), which is neglected in the MK Equations (due to assumption ii) but is accounted for in the present model.

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Fig. 4.

(A) The arterial stiffness (equivalent modulus) E versus the blood pressure P for a human artery characterized by the Fung hyperelastic model; the Hughes Equation is also shown, where its parameters E0 and ζ are determined by fitting the arterial stiffness within the range of human blood pressure (5 kPa to ∼20 kPa). (B) The blood pressure P versus the PWV of the human artery, given by the present model and by the MK + Hughes Equations, where the parameters E0 and ζ are determined from A. (C) The blood pressure P versus the PWV for the human artery characterized by the Fung hyperelastic model; the MK + Hughes Equations are also shown, where the parameters E0 and ζ in the Hughes Equation are determined by fitting within the range of human blood pressure (5 kPa to ∼20 kPa). (D) The artery stiffness (equivalent modulus) E versus the PWV of the human artery, given by the present model and by the Hughes Equation, where the parameters E0 and ζ are determined from C.

Another important clinical application of PWV is to determine the arterial stiffness (equivalent tangent modulus) of the artery wall as the elastic properties of arteries are affected by aging and cardiovascular diseases, therefore providing useful prognostic information (27). The blood pressure P is shown in Fig. 4C versus the PWV. In the range of human blood pressure (5 kPa to ∼20 kPa), the pressure−PWV relation is used to determine the material parameters in the Hughes + MK Equations (Eqs. 1a and 1b) as E0 = 145 kPa and ζ = 0.117 kPa⁻¹, which yields good agreement between the MK + Hughes Equations and the present model. However, for the same range of PWV, Fig. 4D shows that the MK + Hughes Equations significantly underestimate the equivalent tangent modulus by a factor of ∼3 compared with the present model. The main reason for this large discrepancy is the same as that shown in Fig. 4 A and B.

For the human artery characterized by the Fung hyperelastic model with C = 39 kPa, a1 = 0.97, and h0/R0 = 0.15 (19, 26), the range of human blood pressure (5 kPa to ∼20 kPa) gives A/A0 = 2.46 to ∼3.55, which is relatively large such that the function erfi in Eq. 10 can be approximated by erfi(x)≈ex2/(πx) (28). Eqs. 10 and 11, at the limit Ezz = 0, have the asymptotes for large A/A0 (see SI Appendix, Note 3 for details),P≈C2A0A−A0ea1(A−A0)24A02,[15]PWV2≈Ca14ρAA0ea1(A−A0)24A02.[16]Eliminating the variable A yields the following relation (see SI Appendix, Note 3 for details):lnPC+ln(1+8ρa1PWV2P−1)=a116(1+8ρa1PWV2P−1)2,[17]which, as shown in Fig. 5A, is in reasonable agreement with the P versus PWV curve at large pressure, and their difference is approximately a constant (i.e., a shift along the vertical axis). The above equation suggests that the blood pressure scales with PWV², i.e., P≈αPWV2, and the scaling coefficient α is approximately a constant since the logarithmic term ln(P/C) has a very weak dependence on the pressure. Accordingly, the relation between P and PWV can be represented byP=αPWV2+β,[18]where β represents the constant shift between the two curves in Fig. 5A, and α and β depend on the material properties and geometry of the artery (C, a1, ρ, R0, and h0) and are to be determined from the experiments.

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Fig. 5.

(A) The normalized blood pressure P versus the normalized PWV given by the present model (in Eqs. 10 and 11) and its asymptote (in Eq. 17). (B and C) Comparison of P=αPWV2+β to (B) the present model (Eqs. 10 and 11) and (C) literature data (29). (D) Comparison of P=αL2/Δt2+β to the experiments of blood pressure P versus 1/Δt, where Δt is pulse arrival time.

For the human artery characterized by the Fung hyperelastic model with C = 39 kPa, a1 = 0.97, and h0/R0 = 0.15 (19, 26), the constants are α = 0.18 kPa·s²⋅m⁻² and β = 2.7 kPa, which show excellent agreement (Fig. 5B) with the P versus PWV relation obtained from Eqs. 10 and 11 in the range of human blood pressure (5 kPa to ∼20 kPa). Fig. 5C further compares Eq. 18 with literature data (29) of the experimental diastolic blood pressure (DBP, measured by an invasive method) versus PWV (obtained from the ear and toe pulses) during and after anesthesia for surgery. For α = 0.046 kPa·s²⋅m⁻² and β = 5.1 kPa, Eq. 18 agrees reasonably well with the experimental data.

SI Appendix, Fig. S4A, Inset shows a multimodal wearable sensor called the BioStamp (MC10 Inc.) (30), which contains electrodes and an optical sensor for collection of electrocardiogram (ECG) (31) and photoplethysmograph (PPG) (32) signals simultaneously. The BioStamp mounts on the torso in the subclavicular region (SI Appendix, Fig. S4A) or, alternatively, on the posterior side overlaying the scapula (SI Appendix, Fig. S4B). These two positions allow collection of ECG and PPG signals concurrently, and exploit the temporal relationship between these two signals to compute pulse arrival times (PVW=L/Δt: pulse arrival distance, L, pulse arrival time, Δt). Evaluations of the correlations between these Δt measurements and blood pressure rely on episodic measurements of blood pressure with a conventional cuff device with the subject instrumented with the BioStamp. The Δt and DBP versus time are measured during the postexercise period. The 1/Δt and DBP both decrease during the first 300 s (after sprinting), then approach a stable resting value as shown in SI Appendix, Fig. S5. Their relation can be described by substituting Eq. 5 into Eq. 18,P=αL2Δt2+β.[19]The 1/Δt and DBP response during the first 300 s are shown in Fig. 5D, together with Eq. 19 and αL2 = 0.064 kPa·s² and β = 5.8 kPa, which shows reasonable agreement with the experimental data. In general, once β and α in Eq. 18 (or αL2 in Eq. 19) are determined, then the continuous, cuffless, and noninvasive blood pressure can be monitored by measuring the PWV.