# Modèle de zener équation différentielle

Le comportement élastique observé pour l`ECM et le tissu natif, en mode traction, s`est révélé corrélé (R2 > 0,97) avec le modèle Hook (printemps idéal), alors que le comportement viscoélastique, couvrant à la fois les modes de traction et de relaxation, apparié (R2 > 0,94) le Zener modèle (Fig. 3A). Aucune différence (p > 0,05) n`a été constatée entre le module de Young (E), dérivé du modèle Hook, et les modules élastiques Zener (E1 et E2) calculés pour les mêmes matériaux, avec une moyenne de 1,3 ± 0,6 pour l`ECM décellularisé et de 0,3 ± 0,1 (MN/m2) pour les tissus indigènes (Fig. 3b). Le module statique équivalent, qui devrait être équivalent au module du jeune et qui est censé refléter l`élasticité globale, s`est révélé être plus petit que le module E1, E2 ou Young pour les deux matériaux (p<0.01). the="" zener="" elastic="" moduli="" (e1="" and="" e2)="" as="" well="" as="" the="" young`s="" modulus="" and="" the="" equivalent="" static="" modulus="" (e*)="" were="" found="" to="" be="" larger="" (="">quadrufold, p<0.05) for="" the="" decellularized="" ecm="" than="" for="" the="" native="" tissue.="" the="" viscosity="" (fig.="" 3c)="" was="" also="" found="" to="" be="" almost="" three="" times="" larger=""> <0.05) for="" the="" ecm="" (1.7×102±50%)="" than="" for="" the="" native="" tissue,="" averaging="" 0.6×102±50%="" (mn·s/m2).="" the="" resulting="" relaxation="" times="" ,="" calculated="" according="" to="" the="" zener="" viscosity="" and="" elastic="" moduli="" (fig.="" 3d),="" were="" found="" to="" be="" significantly=""><0.05), yet not substantially longer for the native tissue than for the ECM. To further validate the applicability of our analysis method, samples of the acellular ECM were seeded with MSCs, cultured, and compared with the native and acellular ECM. As reported in the Supplementary Data, the cells grew sixfold in 2 weeks, uniformly covering the myocardial surface with a layer of aligned cells that also changed the overall appearance of the ECM, making it seem more like the native tissue. Tensile-failure analysis of the reseeded ECM has shown that most viscoelastic and failure properties were either partially restored (E1, E2, and the integrity loss) to their native values, or unchanged (η and τrx) if they had not initially differed between the decellularized ECM and native tissue. These results, achieved through tensile-failure analysis that also utilizes the basic implementation of the Zener model, further validate the model, and increase the prospects of this more biologically relevant reseeded scaffold to be suitable for regenerative medicine applications. We believe that regardless of these results, the comparison between the native and decellularized pcECM alone, which occupies the greater part of this article, is enough to prove the validity of our model for soft tissue analysis.

Further, when taken together with our previously reported results, the mechanical similarity between the native and decellularized pcECM strengthens the applicability of the decellularized pcECM as a scaffold material for cardiac tissue engineering. In comparison to the ideal spring model, SRT models, which are simple serial and/or parallel combinations of springs and dampers40 (Fig. 1a), provide more information regarding material viscosity and elasticity. By setting varying boundary conditions into their constitutive differential equations, SRT models may accommodate several mechanical testing modes or response regions, which are depicted in a single time-continuous stress–strain curve. Such combinations of several modes and/or regions are not only common in many experiments, but also better resemble the physiological conditions in which these biomaterials are believed to operate.41,42 Much like multiple relaxation time (MRT) models, SRT models are expected to correlate to a large extent with empirical data and may provide better assessments of viscoelastic properties than the Young`s modulus alone,43–45 with an advantage of avoiding the information redundancy commonly incurred by MRT models. SRT models may also facilitate the analysis of soft biomaterials, not as monolithic solids, but as a bundle of strings, each having the same elastic moduli and viscosity, but with different failure points, thereby enabling the modeling of such solids beyond their apparent linear viscoelastic region and into their failure region. yet=”" not=”" substantially=”" longer=”" for=”" the=”" native=”" tissue=”" than=”" for=”" the=”" ecm.=”" to=”" further=”" validate=”" the=”" applicability=”" of=”" our=”" analysis=”" method,=”" samples=”" of=”" the=”" acellular=”" ecm=”" were=”" seeded=”" with=”" mscs,=”" cultured,=”" and=”" compared=”" with=”" the=”" native=”" and=”" acellular=”" ecm.=”" as=”" reported=”" in=”" the=”" supplementary=”" data,=”" the=”" cells=”" grew=”" sixfold=”" in=”" 2=”" weeks,=”" uniformly=”" covering=”" the=”" myocardial=”" surface=”" with=”" a=”" layer=”" of=”" aligned=”" cells=”" that=”" also=”" changed=”" the=”" overall=”" appearance=”" of=”" the=”" ecm,=”" making=”" it=”" seem=”" more=”" like=”" the=”" native=”" tissue.=”" tensile-failure=”" analysis=”" of=”" the=”" reseeded=”" ecm=”" has=”" shown=”" that=”" most=”" viscoelastic=”" and=”" failure=”" properties=”" were=”" either=”" partially=”" restored=”" (e1,=”" e2,=”" and=”" the=”" integrity=”" loss)=”" to=”" their=”" native=”" values,=”" or=”" unchanged=”" (η=”" and=”" τrx)=”" if=”" they=”" had=”" not=”" initially=”" differed=”" between=”" the=”" decellularized=”" ecm=”" and=”" native=”" tissue.=”" these=”" results,=”" achieved=”" through=”" tensile-failure=”" analysis=”" that=”" also=”" utilizes=”" the=”" basic=”" implementation=”" of=”" the=”" zener=”" model,=”" further=”" validate=”" the=”" model,=”" and=”" increase=”" the=”" prospects=”" of=”" this=”" more=”" biologically=”" relevant=”" reseeded=”" scaffold=”" to=”" be=”" suitable=”" for=”" regenerative=”" medicine=”" applications.=”" we=”" believe=”" that=”" regardless=”" of=”" these=”" results,=”" the=”" comparison=”" between=”" the=”" native=”" and=”" decellularized=”" pcecm=”" alone,=”" which=”" occupies=”" the=”" greater=”" part=”" of=”" this=”" article,=”" is=”" enough=”" to=”" prove=”" the=”" validity=”" of=”" our=”" model=”" for=”" soft=”" tissue=”" analysis.=”" further,=”" when=”" taken=”" together=”" with=”" our=”" previously=”" reported=”" results,=”" the=”" mechanical=”" similarity=”" between=”" the=”" native=”" and=”" decellularized=”" pcecm=”" strengthens=”" the=”" applicability=”" of=”" the=”" decellularized=”" pcecm=”" as=”" a=”" scaffold=”" material=”" for=”" cardiac=”" tissue=”" engineering.=”" in=”" comparison=”" to=”" the=”" ideal=”" spring=”" model,=”" srt=”" models,=”" which=”" are=”" simple=”" serial=”" and/or=”" parallel=”" combinations=”" of=”" springs=”" and=”" dampers40=”" (fig.=”" 1a),=”" provide=”" more=”" information=”" regarding=”" material=”" viscosity=”" and=”" elasticity.=”" by=”" setting=”" varying=”" boundary=”" conditions=”" into=”" their=”" constitutive=”" differential=”" equations,=”" srt=”" models=”" may=”" accommodate=”" several=”" mechanical=”" testing=”" modes=”" or=”" response=”" regions,=”" which=”" are=”" depicted=”" in=”" a=”" single=”" time-continuous=”" stress–strain=”" curve.=”" such=”" combinations=”" of=”" several=”" modes=”" and/or=”" regions=”" are=”" not=”" only=”" common=”" in=”" many=”" experiments,=”" but=”" also=”" better=”" resemble=”" the=”" physiological=”" conditions=”" in=”" which=”" these=”" biomaterials=”" are=”" believed=”" to=”" operate.41,42=”" much=”" like=”" multiple=”" relaxation=”" time=”" (mrt)=”" models,=”" srt=”" models=”" are=”" expected=”" to=”" correlate=”" to=”" a=”" large=”" extent=”" with=”" empirical=”" data=”" and=”" may=”" provide=”" better=”" assessments=”" of=”" viscoelastic=”" properties=”" than=”" the=”" young`s=”" modulus=”" alone,43–45=”" with=”" an=”" advantage=”" of=”" avoiding=”" the=”" information=”" redundancy=”" commonly=”" incurred=”" by=”" mrt=”" models.=”" srt=”" models=”" may=”" also=”" facilitate=”" the=”" analysis=”" of=”" soft=”" biomaterials,=”" not=”" as=”" monolithic=”" solids,=”" but=”" as=”" a=”" bundle=”" of=”" strings,=”" each=”" having=”" the=”" same=”" elastic=”" moduli=”" and=”" viscosity,=”" but=”" with=”" different=”" failure=”" points,=”" thereby=”" enabling=”" the=”" modeling=”" of=”" such=”" solids=”" beyond=”" their=”" apparent=”" linear=”" viscoelastic=”" region=”" and=”" into=”" their=”" failure=”" region.=”">

As reported in the Supplementary Data, the cells grew sixfold in 2 weeks, uniformly covering the myocardial surface with a layer of aligned cells that also changed the overall appearance of the ECM, making it seem more like the native tissue. Tensile-failure analysis of the reseeded ECM has shown that most viscoelastic and failure properties were either partially restored (E1, E2, and the integrity loss) to their native values, or unchanged (η and τrx) if they had not initially differed between the decellularized ECM and native tissue. These results, achieved through tensile-failure analysis that also utilizes the basic implementation of the Zener model, further validate the model, and increase the prospects of this more biologically relevant reseeded scaffold to be suitable for regenerative medicine applications. We believe that regardless of these results, the comparison between the native and decellularized pcECM alone, which occupies the greater part of this article, is enough to prove the validity of our model for soft tissue analysis. Further, when taken together with our previously reported results, the mechanical similarity between the native and decellularized pcECM strengthens the applicability of the decellularized pcECM as a scaffold material for cardiac tissue engineering. In comparison to the ideal spring model, SRT models, which are simple serial and/or parallel combinations of springs and dampers40 (Fig. 1a), provide more information regarding material viscosity and elasticity. By setting varying boundary conditions into their constitutive differential equations, SRT models may accommodate several mechanical testing modes or response regions, which are depicted in a single time-continuous stress–strain curve.

Such combinations of several modes and/or regions are not only common in many experiments, but also better resemble the physiological conditions in which these biomaterials are believed to operate.41,42 Much like multiple relaxation time (MRT) models, SRT models are expected to correlate to a large extent with empirical data and may provide better assessments of viscoelastic properties than the Young`s modulus alone,43–45 with an advantage of avoiding the information redundancy commonly incurred by MRT models. SRT models may also facilitate the analysis of soft biomaterials, not as monolithic solids, but as a bundle of strings, each having the same elastic moduli and viscosity, but with different failure points, thereby enabling the modeling of such solids beyond their apparent linear viscoelastic region and into their failure region. >