Simultaneous presence of growth and remodeling in the bone adaptation theory

American Journal of Applied Sciences, Feb, 2009 by Seyyed Amir Hooshiar Ahmedi, Gholamreza Rouhi, Hamidreza Katouzian, Seyyed Ali Hooshiar Ahmedi

INTRODUCTION

The development, growth, and remodeling of skeletal structures is a highly regulated process beginning with mesenchymal stem cells condensations in the early embryo and finishing with the homeostatic skeleton of the adult. It is widely accepted that both genetic and epigenetic factors determine the final shape and strength of the skeleton, and many authors have specifically proposed an epigenetic role for mechanical forces (1), (2), (3). Equations have been proposed to describe how mechanical forces modulate growth where a mechanobiological remodeling rate is superimposed on a baseline biological growth rate (4).

In 1976 a rigorous mathematical bone remodeling model was proposed by Cowin and Hegedus which is so-called adaptive elasticity theory. In the adaptive elasticity theory, it is assumed that the rate of change in bone mass is correlated with the history of mechanical strain (5). Skalak et al. later formulated a continuum model of growth (6). Carter (7) proposed a bone remodeling model which was targeted to produce a homeostatic level of an effective stimulus and at the same time, Huiskes et al. proposed that the process of bone remodeling is aimed at producing a homeostatic value of the strain energy density in the tissue (8).

Bone growth models incorporating both biological and mechanobiological influences have been proposed by Van der Meulen and colleagues (9) for modeling the cross-sectional growth of long bones and by Stevens et al. for modeling endochondral growth using a finite element model (4). An alternative approach is that remodeling of bone is in response to microdamage, either to regulate microdamage to a homeostatic level or as the stimulus for the activation of the coupled responses of osteoclast and osteoblast cells (10), (11), (12).

Recently, Vahdati and co-workers incorporated a cellular accommodation effect into the Huiskes et al.'s semi-mechanistic model (13). They showed that the model is sensitive to temporal sequence in which loading is applied. In a very recent effort, Vahdati and Rouhi proposed another modification on the Huiskes et al's model (13). They included the effects of both microcracks and disuse on activation of resorption in one unifying formulation based on latest experimental findings besides considering cellular accommodation effect. These findings have not been published, yet. In 2004, Rouhi and co-workers proposed a modification on the adaptive elasticity theory by replacing volume fraction with free surface density in the constitutive equations (14). In another attempt, Rouhi et al. incorporated a microcrack factor in their first model and showed that not only mechanical stimuli, but also their rate and history are effective and at play in the bone remodeling process (15). Considering the great importance of bone resorption process in the osteoporotic cases, Rouhi et al. proposed a separate model of bone resoprtion by using mixture theory with chemical reactions. In their bi-phasic mixture model, they found that not only mechanical factors are at play in the resorption process, but also chemical and biological factors have crucial effects (16). Considering three different constituents of bone, i.e. bone matrix, bone fluid, and bone resorbing cells; Rouhi proposed a tri-phasic model of bone resorption using mixture theory with chemical reactions (17). It is concluded that rate of bone resorption is a function of apparent density of bone matrix and bone fluid, fluid velocity, momentum supply to the fluid phase, and internal energy densities of different constituents, in the former model.

Frost demonstrated that not only mechanoregulation diagram is not plateau in the lazy zone for growing bone, but also it has a considerable time dependent positive slope. This positive slope represents modeling process (Fig. 1) (18).

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MATERIALS AND METHODS

It is widely accepted that immature bone structure can be altered via two major mechanisms: (1) Metabolic base growth, which is dependent on genetic patterns, nutrition, etc., (2) Mechanoregulatory mechanisms. The former is called Genetic Factor while the latter is called Epigenetic Factor (19). Generally speaking, it can be applicable superimposing Genetic to Epigenetic effects (20).

Computational simulations are used to investigate the mechanoregulation of bone growth. The bone growth model employed here was proposed by Van der Meulen et al. (9). It simulates the growth of the cross-section of a long bone from an embryonic bone collar to maturity, where the rate of bone apposition, [[rho].sub.t], is equal to the sum of the baseline biological rate, [[rho].sub.m], and the rate due to mechanobiological effects, [[rho].sub.m], as defined in Eq. (1):

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The baseline biological growth rate is a decaying exponential function of time that reduces to approximately zero by the age of 18 years, defined as follows (9), (20):

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