The experimental studies all of the display strong importance of the stress, and this obviously don’t hold the MK + Hughes Equations

This new linear stress?strain relation on the PDMS tubes, in addition to Eq. cuatro, gives the relation involving the tension P and you may internal city A great once the (get a hold of Lorsque Appendix, Notice step one to have details) P = Elizabeth ? cuatro [ dilog ( Good + A beneficial w a beneficial l l A great 0 + A good w a l l ) ? dilog ( Good An effective 0 ) ] + Elizabeth ? 8 [ ln ( An effective + A good w a beneficial l l A beneficial 0 + Good w a good l l ) dos ? ln ( A An effective 0 ) 2 ] , in which Elizabeth ? = Elizabeth / ( step one ? ? 2 ) ‘s the plane strain modulus; ? = 0.5 ‘s the Poisson’s proportion having PDMS; A 0 = ? Roentgen 0 dos and you can Good w a good l l = ? ( R 0 + h 0 ) 2 ? ? R 0 2 are definitely the internal part of the artery plus the part of artery wall structure, respectively, instead of pressure; and you will dilog is the dilogarithm function (24). Substitution of Eq. six for the Eq. dos provides the PWV given that PWV = E ? An effective 4 ? [ A great 0 An effective ( A great ? A great 0 ) ln An effective An effective 0 ? A beneficial 0 + A beneficial w good l l ( Good + Good w good l l ) ( A good ? A 0 ) ln ( An excellent + A good w a beneficial l l A good 0 + A beneficial w a l l ) ] . Eqs. 6 and you can eight is actually parametric equations on the loved ones involving the pulse revolution speed PWV and you can tension P; elimination of the brand new advanced variable An efficiency next scaling rules between your normalized PWV and pressure P: PWV E ? ? = grams ( P Age ? , h 0 R 0 ) , where grams try a great nondimensional form shown inside Fig. 2E. It is clear that PWV displays a robust need for P. For testing, the fresh new MK Equation [1a] forecasts a reliable PWV (in addition to the pressure), and it is shown from inside the Fig. 2E. Fig. 2F implies that, with no factor installing, the new relation between PWV and P obtained from Eq. 8 agrees better into within the vitro experiments to possess fifteen:step one, step one7:1, and you will 19:step 1 PDMS and you may fixed R 0 = six.step three mm, h 0 = 0.63 mm, and you may ? = 1,100000 kilogram/m step three to own drinking water. The outcome away from drinking water viscosity is revealed from inside the Au moment ou Appendix, Note 2 and you may Fig. S3. Also, Fig. 2G reveals sophisticated agreement that have fresh results for a few thicknesses ( h 0 = 0.63 and you may 0.31 mm) of the tubing produced from 19:step one PDMS and you may fixed Roentgen 0 = six.3 mm, and you can ? = 1,100000 kilogram/m step 3 hitch giriÅŸ yap , without any parameter fitted.

The new Family relations Anywhere between Blood pressure and you can PWV getting Person Artery Wall space.

The human artery walls are well characterized by the Fung hyperelastic model (21), which has the strain energy density W = C 2 e a 1 E ? ? 2 + a 2 E z z 2 ? C 2 , where E ? ? and E z z are the Green strains in the circumferential and axial directions of the artery, respectively, and a 1 , a 2 , and C are the material parameters, which are related to the elastic modulus (at zero pressure) by E 0 = C a 1 . 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 = 1 4 C e a 2 E z z 2 ? a 1 < erfi>, PWV = C e a 2 E z z 2 a 1 A 4 ? [ 1 A 0 e a 1 ( A ? A 0 ) 2 4 A 0 2 ? 1 A 0 + A w a l l e a 1 ( A ? A 0 ) 2 4 ( A 0 + A w a l l ) 2 ] . 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: PWV C e a 2 E z z 2 ? = f ( P C e a 2 E z z 2 , a 1 , h 0 R 0 ) , where f is a nondimensional function, and is shown in Fig. 3A for a 1 = 0.97 (26) and h 0 / R 0 = 0.15 (19) for the human artery. Fig. 3B examines the effect of artery stretching E z z by comparing the limit E z z = 0 of Eq. 12, which takes the form PWV C ? = f ( P C , a 1 , h 0 R 0 ) , to the scaling law in Eqs. 10 and 11 for a representative a 2 = 2.69 (21) and E z z = 0.1 and 0.2. The effect of artery stretching is negligible even for 20% stretching.