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## West Indian Medical Journal

*versión impresa* ISSN
0043-3144

### West Indian med. j. vol.61 no.8 Mona nov. 2012

**DENTAL FORUM**

**Mathematical model for osteobstruction in bone regeneration mechanisms a Headway in skeletal tissue engineering**

**Modelo matemático de la osteobstrucción en el mecanismo de regeneración del hueso: un avance en la ingeniería del tejido esquelético**

**C Ogunsalu ^{I}; FI Arunaye^{II}; C Ezeokoli^{III}; M Gardner^{I}; M Rohrer^{IV}; H Prasad^{IV}**

^{I}Section of Anatomy, Department of Basic Medical Sciences, Faculty of Medical Sciences, The University of the West Indies, Kingston 7, Jamaica

^{II}Department of Mathematics and Computer Science, Faculty of Science and Technology, The University of the West Indies, Kingston 7, Jamaica

^{III}School of Veterinary Medicine, Faculty of Medical Sciences, The University of the West Indies, St Augustine, Trinidad and Tobago

^{IV}Department of Pathology, University of Minnesota, Minnesota, United States of America

**ABSTRACT**

In this paper, we formulate a mathematical model for the evaluation of parameters responsible for the retardation and eventual acceleration of bone regeneration on the contralateral side of the mandible of experimental animals, following the discovery of a new mechanism of bone regeneration called the osteobstruction mechanism (a negative mechanism of bone regeneration as opposed to the well established and extensively documented positive mechanisms such as osteogenesis, osteoinduction and osteoconduction). This osteobstructive mechanism was demonstrated by episodes of overtaking and re-overtaking on single photon emission computed tomography (SPECT) following evaluation of osteoblastic activities in a sequential animal experiment to validate both the Ogunsalu sandwich technique (a double guided tissue technique; D-GTR) and the interceed membrane technique (a single guided tissue regeneration technique; S-GTR) utilizing SPECT, histological and histomorphometric evaluation. This work is now given special attention in terms of mathematical analysis because of limited experimental observations since experiments cannot be observed infinitely. Mathematical modelling is as such essential to generalize the results of this osteobstructive mechanism in bone regeneration. We utilize the Fisher's equation to describe bone cell mobilization during bone regeneration by two different techniques: the Ogunsalu sandwich bone regeneration technique (D-GTR) and the S-GTR.

**Keywords: **Bone regeneration, mathematical model, osteobstruction

**RESUMEN**

En este trabajo se formula un modelo matemático para la evaluación de los parámetros responsables del retardo y la posterior aceleración de la regeneración del hueso en el lado contralateral de la mandíbula de animales experimentales, tras el descubrimiento de un nuevo mecanismo de regeneración del hueso, denominado mecanismo de osteobstrucción de la regeneración del hueso (un mecanismo negativo de regeneración del hueso, el cual contrasta con mecanismos positivos bien establecidos y ampliamente documentados, tales como la osteogénesis, la osteoinducción y la osteoconducción). Este mecanismo osteobstructivo fue demostrado mediante una serie de fenómenos consecutivos en la tomografía computarizada por Emisión de Fotones Individuales (SPECT), tras la evaluación actividades osteoblásticas en experimentos secuenciales con animales, para validar tanto la técnica de sandwich de Ogunsalu (técnica de regeneración de tejidos guiada doble; RTG-D) como la técnica de membrana con interceed (técnica de regeneración de tejidos guiada simple; RTG-S) utilizando SPECT así como evaluaciones histológicas e histomorfométricas. Este trabajo recibe aquí especial atención en términos de análisis matemático, ya que en los experimentos las observaciones no puede realizarse infinitamente. Los modelos matemáticos son pues esenciales para generalizar los resultados de este mecanismo del osteobstructivo de la regeneración ósea. Se utiliza la ecuación de Fisher para describir la movilización de las células óseas durante la regeneración del hueso por dos técnicas diferentes: la técnica de Ogunsalu para la regeneración ósea por "sandwich" (D-GTR) y la técnica S-GTR.

**Palabras claves: **Regeneración del hueso, modelo matemático, osteobstrucción

**INTRODUCTION**

The osteobstruction mechanism in bone regeneration was coincidentally discovered during a sequential single photon emission computed tomography (SPECT), histological and histomorphometric analysis on animal model in the validation of the Ogunsalu sandwich bone regeneration technique (1–3). Mathematical models are essential for refining laboratory/clinical experimental data since such experimental observations are subject to time constraint. Bone regeneration, an important phenomenon in mathematical biology/tissue engineering, has attracted desirable attention in the literature (4–11). The common assumption in the literature is that bone or tissue can only be generated if the concentration of the growth factor exceeds a certain threshold value (7, 12). The expressions for the equation of motion and the distribution of the growth factor are usually related in a way that no bone formation (healing) occurs if the growth factor concentration is below a certain value (12). Mathematical models in the bone regeneration mechanism which incorporate cell mitosis, cell proliferation, cell death, capillary density, oxygen supply and growth factor regeneration coupled to a cell density, have all been recently studied and documented (13–16).

Generally, cell ingrowth may be retarded by various constraints *ie* pore size of porous biomaterials for bone graft, unfavourable surface characteristics, or may be counteracted by chemotactic movement towards external supplies of nutrients and other important biochemical factors (7). Tissue ingrowth can also be enhanced by applying surface modifications to the biomaterial (17–20). Systematic approach to cell motility can be studied in two-dimensional or three-dimensional cell adhesion (7, 21). Mathematical modelling is essential in biological systems/tissue engineering to predict, interpret and generalize the results of clinical experiments and also to derive the intrinsic parameters that can be used to predict cell behaviour (4, 22).

It is well known that in the formulation of mathematical model, based upon the interactions of individual cells, it is required that the position and velocity of each individual cell within the system be considered. Individual cell-based models comparatively provide comprehensive and detailed information about the dynamics of the cell population and as well offer distinct advantages over continuum-based models (7, 23). However, these models are relatively complex due to the fact that rules governing cell behaviour and the ability to track the movement of individual cells is extremely tedious and difficult (7, 23). It is also well known that the continuum-based model (23) allows mathematical expressions that exploit partial differential equations (PDEs), and hence are amenable to PDE theory for analytic or asymptotic solutions as well as numerical simulations.

General conceptualization of transport model in fluid mechanic, physical or biological mathematical modelling assumed the usual diffusion model [1]

where *X *represents concentration, *d* is the diffusion coefficient and *q* is the death and birth (consumption and synthesis) rate of the biological phenomenon that is on motion (migration); usually governed by [2]

and *X _{º}* is the maximum concentration,

*γ*is the proliferation rate. The predicted concentration

*X*is known to be in good qualitative agreement with laboratory experiment (4, 24).

The well known Fisher's equation [3] has been used to represent experimental data from bone regeneration (wound-healing, gene transportation *etc* as the case may be) experiment (4, 5, 7). This mathematical model [3] has very wide applications particularly in the area of cell transportation in bioengineering and has proved formidable (4, 5, 7). Equation 3 describes the behaviour of a cell population (continuum-based model) as a combination of random cell motion and logistic proliferation (proliferation up to a maximum cell density). The condition on cell motility takes the form of a travelling wave of fixed shape that propagates at constant speed (23–25). The velocity depends only on a simple combination of the motility coefficient and the proliferation rate. Thus with given motility front velocity determined experimentally, it is easy to derive the motility coefficient if the proliferation rate is known (7, 24). Recently, a series of mathematical models suitable for describing biological tissue growth *in vitro* was developed (23); these models investigated the influence of cell-cell and cell-scaffold interactions and the mechanical environment on tissue growth *in vitro* tissue engineering (23). In this paper, we formulate mathematical models for the Ogunsalu osteobstruction phenomenon.

**MATERIAL AND METHOD**

After the observation of the osteobstructive phenomenon/ mechanism and its cumulative SPECT, histological and histomorphometric evaluation (1–3), the Fisher's equation [3] was utilized to speculate the bone cell concentration variable and other parameters. We also assumed that the sandwich membrane bag of the Ogunsalu double guided tissue technique (D-GTR) is cubic of unit sides and the dimension of the single guided tissue technique (S-GTR) is unit side square membrane.

**Mathematical models for osteobstruction**

*Single GTR model*

The area of membrane of square width *L* is *L x L = L*^{2} unit square, osteobstruction parameter is *λ*_{1} unit square per second, per second is the measure of soft tissue layer at graft site, effect of membrane/tissue interaction is *ξ* (real variable).

The presence of *λ*_{1} activates negative flow of cells to graft site *ie* negative sign of diffusivity coefficient, thus we have *D*_{1} = –*D * where *D*_{1} is the coefficient of diffusivity when *l*1 is present; *D* is the coefficient of diffusivity of the standard Fisher's equation.

We modelled the relation between the rate of bone cell proliferation *R*_{1} at graft site when *λ*_{1} is present and that of the standard Fisher's equation (23) rate of bone cell proliferation *R* at graft site in the absence of osteobstructivity as [4].

The retardation effect on bone cell proliferation is obtained by ensuring that [5]. We define the general osteobstruction parameter in the two states of GTR by the relation* λ*_{m} *= surface area of membrane for GTR – 3m; m =* 1,2 [6]

*ie:* [7].

*Double GTR (Ogunsalu sandwich GTR) model*

In this case, we have *λ*_{2} = 6*L*^{2} – 6 = 0; *L* = 1 *ie* the value of osteobstruction parameter is zero when *m* = 2 in equation 6. So *D*_{2} = *D* and *R*_{2} = *R*. We observed from the SPECT that the effect of resorption of D-GTR membrane and some biochemical reaction produced a sudden drop in bone cell concentration which we denoted by ∈ *λ*_{º} (real number).

**RESULTS**

The sequential SPECT is shown in Figs 1–6 with evidence of osteobstruction between the 11^{th} week (Fig. 2) and the 13^{th} week (Fig. 3). The osteobstruction was positively correlated with the occurrence of foreign body reaction on histology at the 11^{th} week (Fig. 7a–c) which retarded and obstructed bone formation into the 13th week, what we now term as osteobstruction.

The governing mathematical equation for the osteobstruction in bone regeneration for the S-GTR model is the partial differential equation [8] where *B* is the bone cell concentration; the first term on the right hand side of equation 8 represents bone cell motility (migration) and the second term is the bone cell proliferation at the graft site.

The partial differential equation governing the D-GTR model is [9] where the last term on the right hand side of equation 9 represents the shock of resorption of membrane and some biochemical reactions.

**DISCUSSION **

The significance of the GTR membrane is basically to confine the bone graft to the graft site. Obviously the thickness/ type of membrane influences the osteoblastic activities and osseointegration of the graft material at the recipient site. Unfortunately, limitation exists because of limited observation in clinical, radiological, histological and histomorphometric evaluations, particularly because most experiments and observations are limited to twenty-four weeks and twenty-four months at most (or even less) despite the fact that osteoblastic potential may still remain at the site after this period. Osseointegration may be considered as a continuum since the graft material *eg* Bio-oss may still remain within the regenerative site after two years.

Fortunately, in an extensive sequential animal experiment to validate the Ogunsalu sandwich regeneration technique using SPECT with correlative histological and histomorphometric evaluation, this experiment led to a coincidental finding of osteobstruction based on an interesting and unexpected observation of overtaking and re-overtaking phenomenon on SPECT during the sequential animal experiment evaluation. We strongly suspect that this is membrane type dependent and also the technique by which the membrane is utilized (S-GTR, D-GTR) as the bone graft used for the experiment was identical in both sites.

We note that the osteoblastic activities of the S-GTR site were more than that in the D-GTR site during the eighth–eleventh weeks. However, the D-GTR site suddenly slightly overtook the S-GTR site at the thirteenth week. This was the week that succeeded the foreign body reaction events on histology in the S-GTR site at the 11^{th} week – a coincidental finding.

The histological event of foreign body reactions must have caused the retardation in osteoblastic activities in the D-GTR site. This sudden decreasing rate of bone formation in the D-GTR site from the 11^{th} week into the 13^{th} week and gradually fading out by the 17^{th} week to allow the interceed (S-GTR) site to re-overtake the D-GTR site was a consequence of a histological event of foreign body reactions.

The above discussions are in agreement with the mathematical formulations. *Vis-a-vis* the 1^{st} week to the 11^{th} week, there were more osteoblastic activities at the S-GTR site than at the D-GTR site. Specifically in the 1^{st} week of the experiment, the S-GTR site enabled a very high concentration of bone cell which allowed higher rate of osteoblastic activities, as the bone graft is not enclosed in a membrane sack; thus *R*_{1} >* R*_{2} = *R ie* equation 4 implies [10] hence very high rates of bone cell proliferation is attained. In weeks later than the 11^{th} week, there is the formation of soft tissue wall (foreign body reaction) at the S-GTR site which triggered reversal of bone cell migration from the S-GTR site; this is accounted for by the negative sign of diffusivity coefficient of bone cell motility in equation 8. This reversal of influx of bone cell to the S-GTR site led to retardation of osteoblastic activities at the site (osteobstruction). On the other hand, the diffusivity coefficient of bone cell motility in the case of the D-GTR site is always positive in equation 9, and we note that since bone graft is enclosed in the membrane sack, there is no rapid osteoblastic activity within the 1^{st} week to the 11^{th} week that could warrant accelerated osteoblastic activities at the D-GTR site. Hence, there is a slow and steady flux of bone cell into the D-GTR site which enabled slow and steady osteoblastic activities. Illustratively, S-GTR was 260 at the 8^{th} week, 250 at the 11^{th} week, and 240 at the 13^{th} week, while the D-GTR gradually increased from 170 at the 8th week to 200 at the 11th week and 270 at the 13th week (Table; Figs. 1–6,).

Consequently, while there is osteoblastic activities retardation at the S-GTR site, there is a simultaneous acceleration of osteoblastic activities at the D-GTR site. Equation 9 also accommodates the sudden shock which produced a small decrease in osteoblastic activities at the D-GTR site (decrease from 270 at the 13^{th} week to 160 at the 17^{th} week). This shock is suspected to be the result of resorption of large quantities of membrane (approximately six times that of the S-GTR site) with some biochemical reactions in this site (Table; Figs. 1–6).

We strongly assert in this paper that the model equations could be utilized to predict, interpret and generalize the osteobstruction phenomenon with desirable precisions, as solutions to the above equations are attainable.

**CONCLUSIONS **

Mathematical model has been utilized to explain the events that lead to osteobstruction (a newly identified bone regeneration mechanism) in the S-GTR side of the experimental animal in comparison with the D-GTR side. It is suggested that mathematical model be made mandatory in the explanation of bone regeneration mechanisms.

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**Correspondence:**

Dr C Ogunsalu

Section of Anatomy, Department of Basic Medical Sciences, Faculty of Medical Sciences, The University of the West Indies

Kingston 7, Jamaica

E-mail: chrisogun@yahoo.com

**Footnote**: The abstract of this paper was previously published in Oral Surg Oral Med Oral Pathol Oral Radiol Endod 2010; **110:** 315–16. It may be viewed at http://dx.doi.org/10.1016/j.tripleo.2010.05.024