Volumetric growth of soft tissues evaluated in the current configuration

Abstract

The growth and remodelling of soft tissues plays a significant role in many physiological applications, particularly in understanding and managing many diseases. A commonly used approach for soft tissue growth and remodelling is volumetric growth theory, introduced in the framework of finite elasticity. In such an approach, the total deformation gradient tensor is decomposed so that the elastic and growth tensors can be studied separately. A critical element in this approach is to determine the growth tensor and its evolution with time. Most existing volumetric growth theories define the growth tensor in the reference (natural) configuration, which does not reflect the continuous adaptation processes of soft tissues under the current configuration. In a few studies where growth from a loaded configuration was considered, simplifying assumptions, such as compatible deformation or geometric symmetries, were introduced. In this work, we propose a new volumetric growth law that depends on fields evaluated in the current configuration, which is residually stressed and loaded, without any geometrical restrictions. We illustrate our idea using a simplified left ventricle model, which admits inhomogeneous growth in the current configuration. We compare the residual stress distribution of our approach with the traditional volumetric growth theory, that assumes growth occurring from the natural reference configuration. We show that the proposed framework leads to qualitative agreements with experimental measurements. Furthermore, using a cylindrical model, we find an incompatibility index that explains the differences between the two approaches in more depth. We also demonstrate that results from both approaches reach the same steady solution published previously at the limit of a saturated growth. Although we used a left ventricle model as an example, our theory is applicable in modelling the volumetric growth of general soft tissues.

An Updated Lagrangian Constrained Mixture Model of Pathological Cardiac Growth and Remodelling

“…The distributions of τ cc resulting from concentric and hybrid growth are consistent with literature-reported values, i.e. negative at the endocardium and positive at the epicardium within a range of several kPa [6,[52][53][54] . Negative τ cc at the middle wall and positive at the endocardium and epicardium in eccentric growth ( Fig.…”

“…As introduced in Section 2.2 , each growth cycle consists of a long-term G&R step and a short-term loading step. In the shortterm loading step, the growth tensor G i and the remodelling tensor P i r are assumed to be constant, then the passive stress response of the myocardium is only a function of the elastic deformation tensor F i e as suggested in Luo and Zhuan [6] , that is…”

“…In this study, average values of end-diastolic stretch and peak active tension have been used to represent the homeostatic states, similar to other modelling studies [18,46] . Specifically, we have chosen the average tr (σ a ) at peak systole as the only stimulus to trigger concentric growth, neither the Mandel stress [5] nor the trace of total Cauchy stress [6] . It is worthwhile to mention that future work shall consider the total stress tensor over cardiac cycles.…”

“…Since its first introduction, kinematic growth theory has been widely used to model cardiac G&R under physiological and pathological conditions. For example, the generic eccentric and concentric growth [5,6,[14][15][16] , reversal G&R [17,18] , explaining experimental observations [19,20] , as well as post myocardial infarction [21] .…”

“…Not only structural and functional adaption in myocytes, the extracellular matrix, mainly the collagen network, also experiences extensive G&R with altered cardiac fibroblast function, which is further dynamically regulated by biomechanical factors [4] . Consequently, systolic stress and diastolic stretch have commonly been considered to be the triggers of pathological cardiac G&R in previously published studies [5][6][7] .…”