The latest experimental data all of the monitor solid dependence on the pressure, and therefore clearly do not support the MK + Hughes Equations
The latest linear fret?strain relation into PDMS hoses, also Eq. 4, gives the relatives between the tension P and you can internal city A beneficial given that (discover Quand Appendix, Mention step 1 getting info) P = E ? 4 [ dilog ( A + A good w a good l l A beneficial 0 + A good w an effective l l ) ? dilog ( An excellent A 0 ) ] + E ? 8 [ ln ( Good + A great w a beneficial l l Good 0 + A great w a l l ) dos ? ln ( An excellent An excellent 0 ) dos ] , where Age ? = E / ( step one ? ? dos ) is the flat strain modulus; ? = 0.5 is the Poisson’s proportion to possess PDMS; Good 0 = ? Roentgen 0 dos and you will A great w a good l l = ? ( Roentgen 0 + h 0 ) 2 ? ? R 0 dos certainly are the interior a portion of the artery and part of artery wall, respectively, in the place of tension; and you may dilog is the dilogarithm setting (24). Replacement out-of Eq. 6 towards Eq. dos supplies the PWV since the PWV = Age ? A great 4 ? [ Good 0 A beneficial ( A beneficial ? Good 0 ) ln Good A great 0 ? A good 0 + A w an excellent l l ( A + A good w a great l l ) ( A ? A beneficial 0 ) ln ( A great + A great w an excellent l l A 0 + A great w a l l ) ] . Eqs. six and you can eight is actually parametric equations on the family members between your heart circulation revolution acceleration PWV and you may stress P; elimination of the brand new intermediate variable An output the next scaling rules within stabilized PWV and you may stress P: PWV E ? ? = g ( P E ? , h 0 R 0 ) , where g is actually a nondimensional means found into the Fig. 2E. It is obvious you to PWV displays a strong need for P. Getting evaluation, new MK Equation [1a] forecasts a reliable PWV (independent of the tension), and is also shown into the Fig. 2E. Fig. 2F implies that, without having any parameter fitted, the new loved ones between PWV and P taken from Eq. 8 believes well into the during the vitro tests having fifteen:step 1, step 17:step one, and 19:1 PDMS and you may repaired Roentgen 0 = six.3 mm, h 0 = 0.63 mm, and you can ? = step one,000 kg/meters step 3 for drinking water. The end result out-of drinking water viscosity is found from inside the Si Appendix, Mention dos and you will Fig. S3. Also, Fig. 2G reveals expert contract that have fresh outcomes for several thicknesses ( h 0 = 0.63 and you will 0.30 mm) of your tubing produced from japan cupid mobil sitesi 19:step 1 PDMS and repaired Roentgen 0 = 6.step 3 mm, and you can ? = 1,100000 kilogram/yards step 3 , without having any parameter fitting.
The fresh Loved ones Anywhere between Blood circulation pressure and you will PWV to have 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.