Numerical test concerning bone mass apposition under electrical and mechanical stimulus
© Garzón Alvarado et al.; licensee BioMed Central Ltd. 2012
Received: 24 February 2012
Accepted: 11 May 2012
Published: 11 May 2012
This article proposes a model of bone remodeling that encompasses mechanical and electrical stimuli. The remodeling formulation proposed by Weinans and collaborators was used as the basis of this research, with a literature review allowing a constitutive model evaluating the permittivity of bone tissue to be developed. This allowed the mass distribution that depends on mechanical and electrical stimuli to be obtained. The remaining constants were established through numerical experimentation. The results demonstrate that mass distribution is altered under electrical stimulation, generally resulting in a greater deposition of mass. In addition, the frequency of application of an electric field can affect the distribution of mass; at a lower frequency there is more mass in the domain. These numerical experiments open up discussion concerning the importance of the electric field in the remodeling process and propose the quantification of their effects.
Bones provide mechanical stability to the human body and are a source of minerals for metabolism . Bones have been studied extensively from the mechanical and mineral standpoint, and in terms of functionality [1, 2]. From a mechanical point of view, bones can be adapted to loads on trajectories of stress through mineral apposition, which is due to the action of osteoblasts [1–4]. Furthermore, they reabsorb minerals when the mechanical stimulus is sufficiently low, as it is unnecessary to maintain structure . Reabsorption is directed by osteoclasts. Osteoblasts and osteoclasts are the primary cells involved during bone remodeling that are stimulated by the action of mechanical strain sensors, for example, osteocytes . These three cell types play an important role during the processes of replacement, maintenance and modeling of bones .
Following the work of Meyer during the nineteenth century, Wolff  proposed a theory of trabecular bone architecture, which assumes that trajectories of high mechanical stress form the trabecular bone. In 1987, Frost [6–8] suggested an adaptive mechanism of bone mass as a function of mechanical stress. Consequently, several bone remodeling algorithms have been developed including those proposed by Frost , Pauwels , Kummer , Cowin [11–13] and Hegedus , which predict the formation of bone structure from internal mechanical loads studied in terms of stress and strain.
From mechanical models of bone remodeling, sophisticated studies have been carried out concerning the processes of apposition and reabsorption during bone turnover, and particularly concerning the distribution of mass in the femur [15, 16], hip replacement [17, 18], and dental implants . Generally, these studies have been phenomenological. Therefore, researchers have made significant efforts to include mathematical models and the role of cell biology and biochemistry in the remodeling process, resulting in research that begins at the microscopic level, concerning the effects of basic cellular remodeling units (BMU, Basic multicellular units) during tissue replacement [20, 21]. From the perspective of BMU, important work was initiated at the biochemical and mechanical level concerning the effects of cracks , cell cycles throughout adult life , active molecules within each cell  and the spatial distribution of each BMU . With these important advances in the understanding of bone remodeling, researchers in the field increasingly turned to the study of other biophysical stimuli that can affect this process. Most models have not taken account of the physical-chemical phenomena of tissue mechano-transduction. For this reason, new investigations that allow the piezoelectric and electrokinetic behavior of the bone to be studied were undertaken .
A clinical study demonstrated that a local electromagnetic field accelerates the healing process after bone fracture . Therefore, an article by Demiray and Dost  began new research concerning the effect of the electromagnetic field on interior injury to bone. In another article, Ramtani  presented a mathematical model relating to the benefit of the electric field in the reparair and maintenance of the solid matrix of bone. Furthermore, there are studies concerning the electrical behavior of bone tissue during the production of electric fields, and external electrical flow. Fukada and Yasuda  demonstrated that bone exhibits piezoelectric behavior, i.e. mechanical stress creates electric polarization (the indirect effect) and an external electric field causes strain (the converter effect). In addition, the properties of bones that produce piezoelectric potentials have been determined [29–33]. These data led to the development of mathematical models that include the effect of electromagnetic fields during the repair [34, 35] and remodeling  of bone. For example, Qu and Yu  developed a mathematical model (no spatial dimension) of the remodeling process and healing under the effect of mechanical loads and the use of electric charges. In this model, the higher the voltage applied to a bone after fracture, the lower the percentage of bone damage and micro damage in the few days after the stimulus. Similarly, during osteoporosis an electric field increases bone density over time. Huang et al.  established a hypothesis concerning the biological and biochemical pathways that activate cells, particularly osteocytes, during the imposition of an electric field. Furthermore, Qu and Yu  proposed a mathematical model that included mechanical loads and electromagnetic effects during the process of bone remodeling.
To date, there have been no phenomenological models concerning bone remodeling that have been tested and compared with purely mechanical models. Therefore, this article proposes a new electro-mechanical model relating to bone remodeling. To test its performance, various numerical experiments were carried out and compared with previous mechanical models. The electric model constants were obtained from relevant literature and numerical experimentation. From these assumptions it was concluded that the electric field can affect the distribution of mass, which originates from the remodeling process, under mechanical effects only. Using the remodeling model of Nackenhorst  as a starting point, it was demonstrated that the electric field can increase bone density and accelerate the process of apposition. Therefore, the model proposed herein can be used as a basis for further work concerning electrical effects in the maintenance of bone.
The electro-mechanical model
Where is the well known mechanical stimulus described by Weinans , which depends on tissue density ( ), and the work carried out by mechanical stress ( ) and is the electrical stimulus that depends on density, frequency and the work carried out by the electric field ( ). During this first approach, we consider that the two stimuli are added to determine the bone remodeling process. However, we will develop each of the terms that determine the electro-mechanical model throughout this article.
The mechanical model
where is a constant and establishes a relationship of power density that has been uncovered through experimentation .
Where is the electrical permittivity of bone tissue and depends on the density, ( ) where frequency ( ), is the electrical energy per unit of volume, is a constant and is the electric energy (per unit volume) of reference.
where is a constant and establishes a relationship of power density that can be proved through experimentation, and will be developed in the following sections. Furthermore, is a function of frequency at which the electric field is applied.
We have chosen as the reference value of electric energy.
where and are mechanical and electrical constants that define the conversion rate of bone remodeling, dependent on the mechanical stress and the electrical potential, respectively.
Solution by the finite element method
where is the derivative operator (discrete version of operator of equation (6)) that converts the displacements into strains (see ). This system of equations is completed by applying the Neumann and Dirichlet conditions suitable for solving the elastic problem (see Figure 1).
where nod is the number of nodes of each element.
Solution to the equation of relative density
where is the integration time interval and refers to the evaluation of the variable at a specific time, i.e.: . This method has been used extensively in the prediction of bone density through remodeling . The forward Euler method is of the first order and has the disadvantage of being unstable for large time intervals .
The constants for the mechanical energy
For the mechanical case we used an initial Young’s modulus , a Poisson's modulus and an initial dimensionless density with values 64 MPa, 0.3 and 1.00, respectively . The dimensionless parameters of the density equation were obtained from Nackenhorst (1997) [39, 44] and were , and .
The mesh was produced using bilinear quadrilateral elements and four points of Gauss integration .
The constants for electric energy
Where and depends on the initial density that considers the computational model.
For the remaining constant changes we made variations of and used , as in the mechanical case.
Figure 9 presents the results for various values of K elect . In this case, a voltage of 100 V was imposed on the right side and a null voltage on the left side. Null Neumann conditions were imposed on remaining contours (see Figure 7b). It is noted that there were changes in topology; columns 2 and 3, from left to right in Figures 9c and 9d, are thicker and closer to one another. In addition, upper branches of greater density were created and an additional branch that begins from the last column to the right. In the case of Figure 9e, the formation of a non-defined region of the "chessboard" type on the right side of the domain is presented.
Figures 10 and 11 present results for various values of k elect . In these cases a voltage of 100 V was imposed at the top and bottom, respectively. Null Neumann conditions were imposed on the remaining contours. Note that in these cases the mass distribution increases as k elect increases. In the case of Figure 10e, the density increases in the upper part of the domain, so that the chessboard becomes more continuous in the central part of the domain. In Figures 10e and 11e the columns that are formed in each simulation have higher bandwidths than previous cases.
In the second example there is a variation of the frequency, while the value of k elect remains constant at 7.0x107. With the voltage configuration of Figure 7d, this is with a lower voltage of 100 V. Note in this case that at low frequencies the density and the amount of tissue deposited are greater than at high values. Figure 9a demonstrates that the frequency generates a total deposit of tissue. In Figure 12b the formation of a high density topology and continuing formation is apparent. Figures 12c, 12d and 12e present results comparable to those observed in previous cases. However, in Figure 12c, the columns are wider than in Figures 12d and 12e.
Discussion and conclusions
In this article several numerical examples were developed concerning bone remodeling of the plate during mechanical stress, assuming the imposition of an electric field in the domain. To calculate the mechanical and electrical stimulus of remodeling, and the evolution of density, the elemental approach was utilized. This pioneering article includes the electrical effect, previously designed by Weinans et al. , in a model of bone remodeling.
Comparable with previous articles, the results of the study presented herein demonstrate in the mechanical case formation of the columnar zone (of high density) in the area remote from the load and formation of the trabecular zone (of low density) in the area close to the load . The results are similar to those obtained by Weinans et al. , Fernandez et al.  and Chen et al. . When applying an electric field there is an increase in bone density and an alteration in the topology of the distribution of mass in the domain. In general, there is greater bone mass apposition in the domain. Therefore, the columns developed by the mechanical stress increase in size due to the electric field. In addition, a greater number of columns and localized compact zones may be observed. In the formulation the effect of electrical frequency has been included to allow increased apposition of mass at low frequency to be observed.
This work was financially supported by Division de Investigación de Bogotá, of Universidad Nacional de Colombia, under the modality: Fortalecimiento a grupos de investigación y creación artística con proyección nacional. ALIANZAS DE GRUPOS. The Project title is “Programa para la obtención y validación de parámetros numéricos utilizados en modelos matemáticos y simulación de sistemas y procesos biológicos mediante el diseño y caracterización de modelos celulares in vitro”, Hermes code 12873.
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