Herein we proposed a material and mathematical model for myocardium stiffness based on the concept of linear correlation between e/e’ and wedge pressure. Diastolic dysfunction is a common clinical manifestation in heart failure of geriatric patients^{ [1][2] }. With background variable like age, a reliable parameter to predict stiffness of myocardium and its relaxation under similar end-diastolic pressure remains developing^{[3]}.

**The clinical and echocardiographic characteristics of younger and older populations**

A total of 919 patients (male gender: 52.6 % (484/919)) were enrolled in this analysis. Compared with younger population, the elderly (>65 years of age) had more prevalence of hypertension, diabetes mellitus and coronary artery disease. Echocardiographic parameters to describe left ventricular function were measured accordingly (Fig.1). In contrast to left ventricular systolic ejection fraction, which was within the normal range in all subjects (LVEF 68.2 ± 8.4 vs. 70.0 ± 6.7, *p* = 0.09), the averaged early diastolic velocity of mitral annulus in tissue Doppler was significantly attenuated in the elders (e’ 0.09 ± 0.02 vs. 0.08 ± 0.02, *p* = 0.02), corresponding to the higher estimated wedge (e/e’) pressure (7.76 ± 2.44 vs. 8.35 ± 2.64, *p* = 0.02).

Figure 1. The measured echocardiographic parameters; IVSd: intraventricular septal width in diastole, LVIDd: left ventricular Internal dimension in diastole, LVPWd: left ventricualr posterior wall width in diastole and LVIDs: left ventricular Internal dimension in systole.

**The relationship between E value and echocardiographic intra-ventricular pressure in younger and older populations**

The E value (Young’s modulus) was calculated to describe the tensile elasticity of the myocardium according to the cylinder mathematic model (Fig.2).

Figure 2. The model of the cylinder in gross and cross section.

Our results showed that, even under the same intra-ventricular filling pressure, the E value was significantly higher in the geriatric subjects, especially with higher e/e’ value (>9) (Fig.3A). By regression analysis, an exponential correlation between age and E value could be summarized to delineate the soaring of E value in older ages (Fig. 3B)

Figure 3.A The relationship between e/e’ and E value in various age group. B. the exponential correlation between age vs. E value in the regression analysis.

**The relationship between the E value and echocardiographic wall thickness in younger and older populations**

In our mathematical model, positive associations between IVSd vs. E value were displayed, along with the increasing e/e' (Fig. 4). The result also indicated that in the same LV wall width, the higher intra-ventricular pressure correlated to the higher E value.

Figure 4. In the mathematical model, the positive association between IVSd vs. E value in low, moderate and high e/e' value. LVPWd= left ventricular posterior wall width in diastole.

We conclude that our material and mathematical geometric model successfully described the myocardial stiffer character in aging hearts under higher intra-ventricular pressures. Individual differences, especially in healthy status, will be validated its future application in detecting diastolic heart failure.

**Reference**

- Javed Butler AK VG, Rhonda Belue, Nicolas Rodondi, Melissa Garcia, Douglas C. Bauer, Suzanne Satterfield, Andrew L. Smith, Viola. “Incident heart failure prediction in the elderly : The health abc heart failure score.” Circ Heart Fail 1: 125-133 (2008).
- Owan TE HD, Herges RM, Jacobsen SJ, Roger VL, Redfield MM. “Trends in prevalence and outcome of heart failure with preserved ejection fraction.” N Engl J Med 355: 251–259 (2006).
- Yang T CY, Wang Y, et al. “Mathematical modeling of left ventricular dimensional changes in mice during aging. BMC Systems Biology” 6: 1-12 (2012).