San Antonio Cardiovascular Proteomics Center, The University of Texas Health Science Center at San Antonio, USA

Department of Electrical and Computer Engineering, The University of Texas at San Antonio, USA

Barshop Institute of Longevity and Aging Studies; Division of Geriatrics, Gerontology and Palliative Medicine, Department of Medicine, The University of Texas Health Science Center at San Antonio, USA

Department of Biochemistry, The University of Texas Health Science Center at San Antonio, USA

Department of Mechanical Engineering, The University of Texas at San Antonio, USA

Abstract

Cardiac aging is characterized by diastolic dysfunction of the left ventricle (LV), which is due in part to increased LV wall stiffness. In the diastolic phase, myocytes are relaxed and extracellular matrix (ECM) is a critical determinant to the changes of LV wall stiffness. To evaluate the effects of ECM composition on cardiac aging, we developed a mathematical model to predict LV dimension and wall stiffness changes in aging mice by integrating mechanical laws and our experimental results. We measured LV dimension, wall thickness, LV mass, and collagen content for wild type (WT) C57/BL6J mice of ages ranging from 7.3 months to those of 34.0 months. The model was established using the thick wall theory and stretch-induced tissue growth to an isotropic and homogeneous elastic composite with mixed constituents. The initial conditions of the simulation were set based on the data from the young mice. Matlab simulations of this mathematical model demonstrated that the model captured the major features of LV remodeling with age and closely approximated experimental results. Specifically, the temporal progression of the LV interior and exterior dimensions demonstrated the same trend and order-of-magnitude change as our experimental results. In conclusion, we present here a validated mathematical model of cardiac aging that applies the thick-wall theory and stretch-induced tissue growth to LV remodeling with age.

Introduction

Over 70% of 50 million Americans over 60 years of age have cardiovascular disease (CVD)

While most studies focus on the myocyte contribution to cardiac systolic function, indices for systolic function, such as ejection fraction, systolic velocities, and systolic isovolumic acceleration rate, have been shown to have little relation with age in both clinical and animal studies

We have shown diastolic dysfunction at the organ level in mice during cardiac aging

Different LV wall stress models, such as the Laplace law based thin-wall models, thick-wall shell models, and finite element models, have been established to describe LV mechanics and compute stress

Phenomenological models on geometric remodeling with aging have been developed to apply on arteries under hypertensive conditions

There are very few computational models available to study LV geometric adaptation with aging. Recently, we established a computational model of LV aging that incorporates Laplace law based stress model

Methods

With physiological aging, the LV undergoes tonic structural and functional changes including increased mean LV wall thickness, chamber diameter, mass, concentric remodeling, and decline in LV diastolic function in humans

1) The LV was modeled as a concentric spherical shell with interior and exterior radii

(2) Internal and external radii, ie., _{i }
_{o}
_{o }- R_{i}

Thick-wall LV model

Based on assumption 1), deformation of the LV wall obeys Hook's law, which can be expressed in tensor form: _{ij }

The stresses for elastic, isotropic, homogeneous, and spherical thick-wall contains the following analytical form

in which, P denotes pressure difference applied to the inner and exterior walls, and _{i}
_{o}

Integrate both sides of Eq. (4) from _{i }
_{o}

It's concluded that Poisson's ratios of all stable isotropic materials falls in the region (-1, 0.5) and only auxetic materials with honeycomb structures and networks have been found to have negative values _{o }- r_{i }
_{o }- R_{i}

The Young's modulus

in which _{c }
_{m }
_{c}

In equation (7), _{c}
_{m}
_{c}
_{m}
_{c }
_{m }
_{c }
_{m }

Due to the fact that both collagen and muscle mass evolve with aging, the volume of the wall changed accordingly. The mass evolution enforces the following equations:

Take the derivative of equations (8) and (9) with respect to time, we obtained

Add up Eqs. (10-11) and divide both sides by 4

Wall incompressibility yields:

Since we assumed that the material is isotropic and homogeneous in our model (assumption 3), the spatial remodeling can only occur in the radial direction. As a result, the spherical geometry is always maintained. Therefore, only radii, e.g., the free radii _{i }
_{o }

Stretch-induced tissue growth model

The dynamic equation that governs the temporal behavior of the exterior radius in the free state was defined as _{o }
_{o }
_{i }> R_{i}
_{o }
_{o}

in which, _{R }
_{o}
_{f }
_{R}
_{f }

Equations (5-7) and (12-14), together with the definitions of _{c}
_{m}

Input functions and parameters

_{c}
_{m}
_{LV}
_{LV }

Fitted temporal profile of the LV mass with age

**Fitted temporal profile of the LV mass with age**. Red rectangles denoted the average values of LV mass at the ages of 8, 15, 20, 24, and 29.5 months collected over 140 wild type C57BL6J mice. Mass was measured by necropsy. The solid curve was the fitted temporal profile based on our experimental data.

Constant parameters and initial values adopted in numerical calculation and the source of the data.

**Parameters (unit)**

**Values**

**References**

Initial value _{i}

2.0

Our experiments on over 148 mice

Muscle density _{m }^{-3})

1.06

Collagen density _{c }^{-3})

1.70

Muscle elastic modulus _{m }

5.0×10^{4}

Collagen elastic modulus _{c }

3.2×10^{7}

Poisson' ratio

0.47

Time step

3

Computational results for mass functions for LV, collagen, and muscle, as well as volume fraction for collagen

**Computational results for mass functions for LV, collagen, and muscle, as well as volume fraction for collagen**. These functions were used as inputs to our mathematical model. (A) The total LV mass fitted with data generated in autopsy experiments; (B) Collagen temporal profile fitted with known trend from experiments; (C) Temporal profile of muscle mass calculated as the difference between total and collagen mass; (D) Volume fraction of collagen, the values for young (6-9 months) and senescent (>25 months) were obtained from published results of mouse.

Time-dependent pressure difference between the interior and exterior surfaces of the LV was another input function. We collected the pressure measurement in the literature and derived its temporal progression as

Temporal profile of pressure difference of LV wall and predicted Young's modulus of the myocardium

**Temporal profile of pressure difference of LV wall and predicted Young's modulus of the myocardium**. (A) Time-dependent pressure difference across the LV. The curved is derived by fitting experimental data; (B) Elastic modulus of LV, which was calculated as isotropic and homogeneous composite made up of muscle and collagen. The modulus was calculated by taking account of temporal profiles of muscle and collagen in LV.

Further, from equations (2-3), the pressure difference

Discretization of equations

For convenience in the writing of equations, we defined

Given _{i}
_{o}
_{i}
_{o}

The radii ^{3}(

We studied the variation of free and deformed radii in the time domain [_{0}, _{0 }+ _{0}, _{0 }+

1. Read initial values _{i}
_{0}), _{o}
_{0}), calculate _{i}
_{0}) and _{o}
_{0}) and _{0}) = 1. Iterate steps 2-5 until the final step

2. According to discretized equation (14) using Euler method, perform the following compute:

3. Utilize discretized version of Eq. (15) taking account of Eq. (14) to calculate the _{i }

4. Update values for _{i }
_{o}

5. Compute new values for _{i}
_{o }

The numerical calculations were performed using in-house Matlab codes. In all simulations, parameter _{R }
_{f }

Results

Changes of EDD, wall thickness and stress

The numerical simulation on LV dimensional changes including EDD and wall thickness were shown in Figures

Temporal curves of LV dimensions and stresses derived by solving the simultaneous equations

**Temporal curves of LV dimensions and stresses derived by solving the simultaneous equations**. The parameters in Eq. (14) adopted are

The temporal profiles of the mean _{RR }
_{θθ }
_{φφ}

As is seen in Eq. (2),

Pressure-volume relationship with aging

End diastolic portion of pressure-volume loop in a cardiac cycle provides important information of passive filling for LV and therefore describes passive properties of myocardium. For example, the slope of end diastolic pressure-volume relationship (EDPVR) gives the measure of LV stiffness. We plotted end diastolic pressure-volume curve with age for C57/BL6J mice (see Figure

End diastolic pressure-volume relationship with age

**End diastolic pressure-volume relationship with age**. The arrow implies aging direction. Both pressure and end diastolic volume go up with age.

Analysis of constraints of remodeling equation

We also analyzed the candidates of the tissue growth function f(t) based on assumption 2). From Eq. (12), we had: _{o }

where _{o}
_{0}), i.e. initial value of _{o}
_{o}
_{i}
_{i}
_{0}). Use the mass of each component at the final time and corresponding density, the final volume was determined and Parameter _{o}

And the above inequalities do not rely on a concrete system. Both sides are known functions without parameter dependence. Therefore, it is a universal condition for the selection of the type of rate function.

As is shown in Figure

Boundaries for the LV growth rate function

**Boundaries for the LV growth rate function ****to generate growing trends for internal radius and wall thickness**. The dotted cyan line in the middle is our

In our simulations, parameters _{R }
_{f }
_{i }
_{i}
_{exp}
^{2 }+ (_{exp}
^{2}, in which WT means wall thickness. The growth function

Discussion

In this study, we established the first analytical mathematical model to quantify long-term LV geometric remodeling dynamics with aging by applying thick-wall theory and stretch-induced tissue growth postulate. This model addressed the temporal progression of cardiac aging and is an advancement from most current models that analyze the static equilibrium of LV remodeling and our previous thin-wall model of LV remodeling. In addition, we analytically determined the boundary for the tissue growth rate to guarantee the stability of the remodeling. This boundary might provide a reference for experimental measurement to examine the remodeling outcomes. Further, the parameters in the mathematical model were determined by using our experimental results from over 140 C57/BL6 mice. Some parameters and initial conditions were selected by minimizing the error between our computational predictions and experimental measurements. In addition, the model was validated by comparing the predictions of internal and external radii and wall thickness in diastole to experimental results. The predicted LV geometry trends were consistent with the experimental results. Thus, this study is a real integration of computational and experimental approaches for model establishment and validation using mice data. This approach and the proposed model can be applied to establish a cardiac aging model for human in the future.

The P-V curve in Figure

Though our computational results follow the trend of LV geometry changes, the proposed model has some limitations. First, we assumed the LV to be a spherical shell. With this assumption, all directions grow at the same pace to maintain the spherical symmetry. Sphere is a special geometry that requires minimum linear dimension for a constant volume. However, LV does not have a spherical geometry in reality. In addition, we assumed that LV was composed of elastic, isotropic, and homogeneous materials and applied the linear mixture theory to calculate the Young's modulus of the myocardium. Though the assumptions simplify the mathematical model and the computational analysis, such simplification might lead to prediction errors. From our computational results, the radii growth calculated are 3.43% for external, 2.37% for internal radius, and 5.72% for wall thickness collected from C57/BL6 mice. Some of these values were small compared to our echo experiment on C57/BL6 mice, in which EDD grows 2.60% and wall thickness is up 13.20% from young to senescent. In addition, our previous experimental work on CB6F1 mice reported EDD and wall thickness have 12.73 and 13.00% growth, respectively, in the LV between the groups of young and senescent

Conclusions

We have established the first mathematical model to study age-related temporal-spatial LV remodeling by adopting thick-wall theory and stretch-induced tissue growth theory. IInputs of the mathematical model were real experimental data including temporal profiles of LV mass, collagen content change, and pressure across LV, which were obtained by over 140 mice. The established model captured the major property of LV remodeling with age and yielded predicted results of LV geometry progressions in mice comparable to experimental results.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Y.F.J and M.L.L designed the research; Y.A.C, A.V, H.C.H, and M.L.L performed the experiment, Y.F.J, T.Y, and Y.W performed the computational analysis and simulation. T.Y, H.C.H, M.L.L, and Y.F.J analyzed the results and wrote the manuscript.

Acknowledgements

The authors acknowledge grant and contract support from NSF CAREER award #0644646 (to HCH), NHLBI HHSN268201000036C (N01-HV-00244), NIH R01 HL75360, Veteran's Administration Merit Award, and the Max and Minnie Tomerlin Voelcker Fund (to MLL), and NIH 1R03EB009496, and NIH SC2HL101430 (to YFJ.).

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