Behavioral model of a basic trabecular-bone multi-cellular unit using cellular automaton submitted date: October 2013 received date: November 2013 Aldemar Fonseca Velásquez* accepted date: December 2013 Abstract Bone remodeling is the biological process through which old bone tis- sue is transformed into new bone through the action of a basic unit of bone remodeling (BMU). This paper presents a computational model to determine the variation of cell population caused by the presence of micro-fractures within the BMU during the remodeling process, also determines the spatial evolution at particular points of the trabecular bone surface. The remodeling process presented is governed by a two- dimensional cellular automaton that evolves according to a set of rules and states, based on biological processes that occur in bone remodeling. The simulation of the remodeling process, programmed in MATLAB® performed in time intervals of 0.3 seconds, in the model showed a ratio of 20 osteoclasts to 190 osteoblasts. On the other hand, an approximate resorption time of 34 days and 150 days for formation was obtained. Bone mass showed a maximum percentage loss of 19%. Key words bone remodeling, cellular automaton, osteoclasts, osteoblasts, resorption. * B.Sc. In Electronic Engineer, Universidad Distrital Francisco José de Caldas (Colombia). M.Sc. In Biomedical Engi- neering, Universidad Nacional de Colombia (Colombia). INTEGRA Research Group, Universidad Distrital Francisco José de Caldas (Colombia). Current position: professor at Universidad Distrital Francisco José de Caldas (Colombia). E-mail: [email protected]. 6 Universidad Distrital Francisco José de Caldas - Technological Faculty BEHAVIORAL MODEL OF A BASIC TRABECULAR-BONE MULTI-CELLULAR UNIT USING CEULAR AUTOMATON of the remodeling process caused by a micro- Introduction A RESEARCH fracture in normal bone can be verified. VISION Bone remodeling is the biological process through which old bone tissue is transformed The aim of this paper is to develop a computa- into new bone through the action of a basic unit tional model based on histomorphometric data of bone remodeling (BMU). In these units, two and so obtain quantitative estimates (in the types of cells are involved, namely osteoclast process of bone remodeling) of cell populations lineage resorbing bone cells and osteoblast and their activity within the BMU on the a tra- lineage cells, which form new bone. Some of becular surface. This model can be taken as most common bone-remodeling causes are: 1) the basis for future studies and simulations in- Bone mass increase to resist additional mecha- volving some factors that play a particular role nical loads, 2) Bones micro- cracks or micro- in the remodeling process and may cause an fissures and 3) fractures. imbalance between resorption and formation. This paper focuses on bone remodeling cau- Various models that explain the remodeling sed by micro-cracks or micro-fissures, which process at a specific trabecular-bone site takes place in the trabecular surface due to have been formulated. Some of the currently various types of efforts and also to natural at- known models include that of Pivonka [10], trition of the tissue [1]. Micro-cracks are de- who investigated the influence of the available tected by the osteocytes and so the remode- bone surface for remodeling using represen- ling process starts with a BMU formation that tative bone volume units as well as defining moves onto the surface of a trabecula seg- that, just as there is a mechanical control, ment [2] [3] [4]. Within the BMU formation, there may be a geometric regulation to the osteoclasts and osteoblasts coexist through a remodeling process that affects the porosity mechanism known as coupling, which is the of the bone tissue. The work of Fazzalari [11] balance between bone resorption and bone provided a model of the effect of the remode- formation [5]. This coupling is mediated by a ling process on the local structure of spongy set of interrelated factors that may be of ge- bone when there are several active BMUs. netic, mechanical, vascular, nutritional, hor- Meanwhile, Buenzli proposed a model, based monal and/or local nature. on an automaton, for explaining the resorp- tion phase in cortical bone from the osteo- Different studies have proposed models, with clasts part [12]. In another work, Buenzli [13] specific approaches, in order to explain, verify, also proposes a space-time continuum model analyze and make assumptions about the pro- based on cell populations, which integrates cess of bone remodeling. Some of these mo- some signs of interaction between the osteo- dels include those based on cell population dy- blast and osteoclast cell lineages involved in namics, or a set of static-mechanicals models, the remodeling process model; as results, the biochemical models and mechanical-biological model indicates that cell populations behave models, among others [2] [3] [4] [6] [7] [8] [9]. as traveling waves moving across the surface However, despite the studies available, it is of the cortical bone. In Wang’s work [14], a still required to integrate knowledge of cell po- new 3D simulation method to resemble the pulations along with their spatial and temporal trabecular-bone remodeling process is de- actions at a specific trabecular remodeling site, veloped to quantitatively study the dynamic so the biological, spatial and temporal aspects evolution of bone mass and of the trabecular Electronic Vision - year 7 number 1 pp. 6 - 18 january -june of 2014 7 ALDEMAR FONSECA VELÁSQUEZ microstructure as a response to different me- of phenomenological aspects such as cell po- chanical loading conditions. pulations and their part on the remodeling site, and is developed through a cellular au- This paper presents a computational model tomaton in a two-dimensional domain which to determine the variation of cell population represents a surface segment. Therefore, for caused by the presence of micro-fractures the automaton approach, the evolution rules within the BMU during the remodeling pro- and the set of states were defined based on cess. The model also determines the spatial the characterization of the biological process evolution at particular points of the trabe- found in previous studies and reports. cular bone surface. The remodeling process presented in this model is governed by a two- The definition, specification and development of dimensional cellular automaton that evolves the automaton are presented below. The whole according to a set of rules and states, based construction was completed by considering mo- on biological processes that occur in bone re- dels based on automata, especially those with modeling. The parameters used in the model applications to biological systems [17] [18] [19]. were derived from case studies, case reports and specialized literature. 1.1 Defining the cellular automaton for the model The simulation of the remodeling process in For the present model, a Moore neighborhood the model showed a ratio of 20 osteoclasts to was used as shown in Figure 1, where C(i,j) 190 osteoblasts. On the other hand, an appro- represents the state of a generic cell on which ximate resorption time of 34 days and 150 evolution rules that depend on the state of its days for formation was obtained. Bone mass eight neighbors are applied. showed a maximum percentage loss of 19%. The results indicate there is a tradeoff bet- Figure 1. Moore Neighborhood ween the spatial configuration of the trabecu- lar surface and how the BMU cell changes its cell-population spatial and temporal features. Ergo, these characteristics are dependent on the size of the trabecular segment and also on the depth of the crack. These features are known from research studies that show how dimensions of the micro-fracture behave when changing stress conditions and there- fore provide an insight into the remodeling device response [13], [15], and [16]. Source: own elaboration 1. Materials and Methods A two dimensional domain, which repre- The proposed model is an exemplification of sents a cross section of a trabecular segment, targeted remodeling on biological processes was defined. The domain dimensions are as a result of the existence of micro-fractures 250μmx125μm in a 50x25 cell array, as shown in the trabecular surface. The model is aware in Figure 2. 8 Universidad Distrital Francisco José de Caldas - Technological Faculty BEHAVIORAL MODEL OF A BASIC TRABECULAR-BONE MULTI-CELLULAR UNIT USING CEULAR AUTOMATON Figure 2. 2D Domain that represents a trabecula segment, cross-sectional view A RESEARCH VISION Source: own elaboration The cellular automaton allows the validation Then the cellular automaton model is defined of biological models because the evolution ru- by the quadruple array C= AQ, ,α , N , where A les can be modified according to the temporal is the two-dimensional domain (according to Fi- gure 2), Q is the set of states that each cell may evolution [19]. In this case, it is necessary to take (according to Table 1), α is the transition establish several stages in the bone remode- function defined by the set of evolution rules ling process that is being represented. To this (which will be presented next), and N defines end, each stage will have its own set of rules a neighborhood of eight neighbors (as shown in for the time evolution of the automaton. Figure 1), namely a Moore neighborhood. Table 1. Lists the set Q of possible automaton states that each cell can take, states C (i, j), at time t Q Cell State OCV Osteoclast with v days of life (Vm states) OBW Osteoblast with w days of life (Wm states) LC Lining cell MD Marrow TB Mineralized bone (Trabecular Bone) F Fissure WTB Recently formed bone (Woven Trabecular Bone) Source: own elaboration Electronic Vision - year 7 number 1 pp. 6 - 18 january -june of 2014 9 ALDEMAR FONSECA VELÁSQUEZ 1.1.1 Evolution rules Figure 3.