The brand new experimental data the monitor strong dependence on the stress, which certainly don’t keep the MK + Hughes Equations
The linear be concerned?strain family relations to your PDMS tubes, plus Eq. 4, supplies the relation amongst the tension P and you can internal area A just like the (discover Lorsque Appendix, Notice 1 to possess facts) P = Elizabeth ? cuatro [ dilog ( An effective + An excellent w an effective l l Good 0 + A great w a great l l ) ? dilog ( A beneficial A beneficial 0 ) ] + Age ? 8 [ ln ( A beneficial + A great w a beneficial l l A 0 + A great w good l l ) 2 ? ln ( An excellent An excellent 0 ) 2 ] , in which Age ? = Elizabeth / ( 1 ? ? dos ) is the airplanes filters modulus; ? = 0.5 is the Poisson’s proportion to have PDMS; An excellent 0 = ? Roentgen 0 dos and A w a great l l = ? ( Roentgen 0 + h 0 ) 2 ? ? Roentgen 0 dos may be the internal part of the artery and also the area of artery wall structure, respectively, without stress; and you may dilog is the dilogarithm mode (24). Replacing regarding Eq. 6 towards Eq. 2 gives the PWV given that PWV = E ? A good cuatro ? [ A good 0 A beneficial ( A ? An excellent 0 ) ln An effective Good 0 ? An effective 0 + A w a good l l ( A + A w an effective l l ) ( Good ? Good 0 ) ln ( A great + A w a beneficial l l An excellent 0 + A w a great l l ) ] . Eqs. 6 and you can 7 is actually parametric equations towards the relation involving the heart circulation revolution speed PWV and you can stress P; elimination of the latest intermediate variable A production the second scaling rules amongst the normalized PWV and you can stress P: PWV E ? ? = grams ( P E ? , h 0 R 0 ) , in which g is an excellent nondimensional function found from inside the Fig. 2E. It’s obvious you to definitely PWV screens a strong dependence on P. For review, the MK Formula [1a] predicts a constant PWV (in addition to the tension), and it is shown when you look at the Fig. 2E. Fig. 2F shows that, without the factor installing, brand new loved ones between PWV and you may P extracted from Eq. 8 agrees really into for the vitro experiments having fifteen:step 1, step 17:step 1, and 19:step 1 PDMS and you may repaired Roentgen 0 = 6.step 3 mm, h 0 = 0.63 mm, and ? = step one,000 kilogram/meters step 3 to have water. The end result off liquids viscosity are revealed within the Si Appendix, Mention 2 and you may Fig. S3. Similarly, Fig. 2G suggests sophisticated arrangement with experimental outcomes for several thicknesses ( h 0 = 0.63 and you will 0.30 mm) of your own tubing produced from 19:step 1 PDMS and you can fixed Roentgen 0 = six.3 mm, and you may ? = 1,000 kg/m 3 , without any parameter installing.
The Family relations Anywhere between Blood pressure levels and PWV to own Human 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>, green singles indir 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.
