Growth of a Long Bone Section Based on Inorganic Hydroxyapatite Crystals as Cellular Automata

Abstract

This work explores the morphogenesis of the skeletal mineral component, with a specific emphasis on hydroxyapatite (HAp) crystal assembly. Bone is fundamentally a triphasic biomaterial, consisting of an inorganic mineral phase, an organic matrix, and an aqueous component. The inorganic phase (hydroxyapatite), is characterized by its hexagonal prismatic nanocrystalline structure. We leverage a cellular automata (CA) paradigm to computationally simulate the mineralization process, leading to the formation of the bone’s hydroxyapatite framework. This model exclusively considers the physicochemical aspects of bone formation, intentionally excluding the biological interactions that govern in vivo skeletal development. To optimize computational efficiency, a simplified anatomical segment of a long bone (e.g., the femur) is modeled. This geometric simplification encompasses an outer ellipsoidal cylindrical boundary (periosteal envelope), an inner ellipsoidal surface defining the interface between cortical and cancellous bone, and a central circular cylindrical lumen representing the medullary cavity, which accommodates the bone marrow and primary vasculature. The CA methodology is applied to generate the internal bone microarchitecture, while deliberately omitting the design of smaller, secondary vascular channels.

1. Introduction

The process of bone formation is known as osteogenesis, which begins between the sixth and seventh week of embryonic development and continues until approximately 25 years of age, although this varies slightly from individual to individual. Two types of bone ossification are reported in the literature: “intromembranous and endochondral; in both cases, these processes begin with mesenchymal precursor tissue. For intromembranous ossification, the mesenchymal tissue is directly converted into bone, giving rise to the flat bones of the skull, clavicle, hip, etc. Endochondral ossification begins with the transformation of mesenchymal tissue into an intermediate cartilage that is subsequently replaced by bone and forms the rest of the axial skeleton and long bones” [1].

“Bone formation requires a template for development. This template is mostly cartilage, derived from embryonic mesoderm, but also includes undifferentiated mesenchyme (fibrous membranes) in the case of intramembranous ossification. This framework determines where the bones will develop. By the time of birth, the majority of cartilage has undergone replacement by bone, but ossification will continue throughout growth and into the mid-twenties” [1].

Since the model we will use is aimed at the formation of long bones, we will use only the description of the endochondral ossification.

1.1. Endochondral Ossification

“This process involves the replacement of hyaline cartilage with bone. It begins when mesoderm-derived mesenchymal cells differentiate into chondrocytes. Chondrocytes proliferate rapidly and secrete an extracellular matrix to form the cartilage model for bone. The cartilage model includes hyaline cartilage resembling the shape of the future bone as well as a surrounding membrane called the perichondrium. Chondrocytes near the center of the bony model begin to undergo hypertrophy and start adding collagen X and more fibronectin to the matrix that they produce; this altered matrix allows for calcification. The calcification of the extracellular matrix prevents nutrients from reaching the chondrocytes and causes them to undergo apoptosis. The resulting cell death creates voids in the cartilage template and allows blood vessels to invade. Blood vessels further enlarge the spaces, which eventually combine and become the medullary cavity; they also carry in osteogenic cells and trigger the transformation of the perichondrium to the periosteum. Osteoblasts then create a thickened region of compact bone in the diaphyseal region of the periosteum, called the periosteal collar. It is here that the primary ossification center forms. While bone is replacing cartilage in the diaphysis, cartilage continues to proliferate at the ends of the bone, increasing bone length. These proliferative areas become the epiphyseal plates (physeal plates/growth plates), which provide longitudinal growth of bones after birth and into early adulthood. After birth, this entire process repeats itself in the epiphyseal region; this is where the secondary ossification center forms” [1].

The description of the endochondral ossification process shown in the previous paragraph is taken as is from [1], which allows us to justify the model of bone matrix growth using cellular automata, where the transition from mesenchymal tissue to cartilage and subsequently to bone is taken as a change of state for our purposes. All these elements are limited by a membrane (perichondrium) that contains the bone mineral matrix and other elements necessary for the development of the complete structure.

1.2. Components of a Bone

Vertebrate bone is a sophisticated composite biomaterial, typically characterized by an approximate composition of 60% inorganic matter, 30% organic matter and 10% water [2,3,4,5,6]. Its capacity to undergo elastic deformation in response to external mechanical stresses and to transmit these forces throughout the skeletal structure underscores its dynamic interaction with the physiological environment. This responsiveness triggers continuous internal remodeling [2]. The inorganic phase, primarily composed of hydroxyapatite (HAp), with the chemical formula ${\left[{\mathrm{Ca}}_3{\left({\mathrm{PO}}_4\right)}_2\right]}_3\mathrm{Ca}{\left(\mathrm{OH}\right)}_2$ exhibits a characteristic hexagonal prismatic morphology [2,3,4,5,6] as depicted in Figure 1. Initially, HAp crystal formation follows a physicochemical nucleation and growth process. Subsequently, in interaction with living tissue, this mineral matrix undergoes bioremodeling, driven by the antagonistic actions of osteoclasts (bone-resorbing cells) and osteoblasts (bone-forming cells) [4]. This highlights a two-stage process in bone structure formation: an initial mineral matrix generation phase and a subsequent chemical-biological interaction phase with the surrounding cellular environment. The latter phase, which is not investigated in this study, results in the modified mineral matrix of spongy (trabecular) bone. Encircling the spongy bone is compact (cortical) bone [2], a highly dense structure formed by the tessellation of these hexagonal HAp prisms.

A direct consequence of bone’s adaptive tissue response to mechanical loading is the potential for deformation within its inorganic structures. However, to constrain the scope of this investigation, our model focuses exclusively on the physicochemical genesis of the bone matrix. This process is simulated using a cellular automata (CA) methodology [7], derived from the thermodynamic conditions prevalent within vertebrate organisms. We deliberately exclude the influence of hormonal regulation on bone growth and do not account for mechanical stress-induced deformations of the bone matrix.

Our methodology adapts a thermodynamic growth model, akin to that employed for snowflake formation, but with parameters specifically calibrated to the physiological temperatures and pressures characteristic of vertebrate species, enabling the generation of a bone matrix from non-aqueous constituents. This three-dimensional (3D) cellular automata (CA) framework utilizes four distinct cell states. An F-cell (Fixed ossified cell) represents a fully mineralized and structurally integrated component of the bone matrix, whose state is immutable. Neighboring F-cells within a defined range (e.g., the first eight immediate neighbors) transition to a B-cell (Boundary cell or Cartilage cell) state, signifying proximity to the ossification front. Cells positioned further from F-cells (e.g., second or third nearest neighbors) are classified as N-cells (Non-receptive cells), indicating their potential for future state modification. Finally, E-cells (Edge cells or Perichondrium cells) delineate the simulation boundary; these include cells on the perimeter of each growth layer and those comprising the basal and apical boundary layers of the volumetric configuration, and their state remains fixed. Importantly, B, N, and E cells are not considered part of the mature bone matrix, as they represent hydroxyapatite molecules that have not attained the requisite thermodynamic conditions (temperature and pressure) for integration into the ossified structure (i.e., becoming an F-cell). The iterative progression of these four states is governed by a diffusion function, calculated via a graph-theoretic approach that considers the central tile and its eight immediate neighbors [7,8].

Our methodological approach for designing the bone section involved a deliberate geometric abstraction of a long bone (femur), allowing for focused investigation of the cellular automata’s spatial progression. The model’s architecture comprises three primary concentric features: an outer ellipsoidal cylinder serving as the external boundary; an inner ellipsoidal cylinder that robustly separates the cortical (compact) bone from the spongy (trabecular) tissue; and a central circular cylinder, concentric with the ellipses, which models the medullary cavity (facilitating marrow and primary vascular passage). The geometric center of these structures is defined at ($a,b$), where a is the major semi-axis of the outer ellipse and b is its minor semi-axis; the circular cylinder’s radius is set to $a/10$. These dimensions ($a,b$) are parametrized according to the specific vertebrate species. The two-dimensional cross-section is then extended into a three-dimensional volume by stacking n sequential planes, defining the overall height h of the simulated bone segment.

Our computational scheme initiates with a horizontal rectangular plane tessellated by regular hexagons. Within this hexagonal grid, the predefined elliptical and circular geometries are embedded. Hexagons situated within the rectangular boundary but external to the outer ellipse, as well as those falling within the central circular lumen (Figure 2), are designated as E-cells. Conversely, all hexagons located between the inner ellipse and the central circle are initially marked as N-cells; these constitute the potential region for spongy bone formation.

To establish nucleation centers for ossification, two distinct processes are implemented. First, starting from layer 1 and proceeding with alternating layers, all hexagons immediately adjacent to the perimeter of the inner circle are designated as F-cells. Second, in every layer, a stochastic process is employed to randomly select additional F-cells, ensuring that each newly designated F-cell is surrounded by six B-cell in the same plane.

The cortical (compact) bone zone is subsequently defined by an elliptical region between the inner and outer ellipses. All hexagons situated within this specific annulus are pre-assigned as F-cells, with the thickness of this cortical region being a species-specific parameter.

2. Definition of the Long Bone Section to Be Developed

The ellipsoidal cylindrical surface (ECS), depicted in the first octant of the coordinate system (Equation (1)), is geometrically characterized by its major semi-axis (a) and minor semi-axis (b). These axes define the shape and scale of the elliptical cross-section. The surface is generated by sweeping this ellipse along a heigh h. The angular sweep parameter $\phi$ covers the full 360° range, and the vertical parameter h extends from 0 up to the maximum number of layers intended for the model’s construction.

$$f\left(h,\phi \right)=\left({\displaystyle \frac{abcos\phi }{\sqrt{a^2{sin}^2\phi +b^2{cos}^2\phi }}}+a,{\displaystyle \frac{absin\phi }{\sqrt{a^2{sin}^2\phi +b^2{cos}^2\phi }}}+b,h\right)$$

(1)

To segregate distinct bone regions, an internal ellipsoidal surface is established within the previously defined ECS, utilizing the same mathematical formulation as Equation (1). This interior surface is characterized by dimensions corresponding to a scale factor between 90% and 95% of the product of the outer ellipse’s major and minor semi-axes ($ab$). The volumetric domain situated between these two concentric ellipsoidal surfaces is specifically allocated to cortical bone. Concurrently, a function $g\left(h,\phi \right)$ (Equation (2)) defines a circular cylindrical surface (CCS) of radius r and height h. This CCS is also centered at $\left(a,b\right)$ and serves to delineate the medullary canal, which would physiologically contain bone marrow and principal vascular networks.

$$g\left(h,\phi \right)=\left(rcos\phi +a,rsin\phi +b,h\right)$$

(2)

The value of r is a biologically derived parameter, contingent upon the specific skeletal element and the vertebrate classification. Subsequently, the dimensions of the horizontal flat surface (HFS), quantified by its number of rows and columns of tessellated cells (Equation (3)), are directly coupled to the previously defined major and minor semi-axes $\left(a,b\right)$ of the ellipsoidal geometry.

$$\left\{\begin{array}{ccc}\mathrm{number}\mathrm{of}\mathrm{columns}& n=& {\displaystyle \frac{b}{\sigma sin\left(60\frac{\pi }{180}\right)}}\\ \mathrm{number}\mathrm{of}\mathrm{rows}& m=& {\displaystyle \frac{2a}{\sigma }}\end{array}\right.$$

(3)

The horizontal flat surface (HFS) is composed of tessellated hexagonal tiles (HT), each with a side length of $\sigma$. This HFS is then projected across various heights along the ellipsoidal cylindrical surface (ECS). The intersection of the HFS with the ECS defines an elliptical perimeter, which is used to ascertain whether a given tile is internal or external to this boundary. Specifically, tiles located within the inner elliptical perimeter but outside the central circular perimeter are designated as capable of transitioning between the N, B or F states. Conversely, tiles situated outside the outer elliptical perimeter or inside the central circular perimeter can only exist in the E state.

3. Hexagonal Prism Based Growth Model

The growth of a crystal from a nucleation point, involving the aggregation of molecules into a structured lattice, is governed by the transport of thermodynamic variables within the system. This process is fundamentally described by the diffusion equation (Equation (4)) [9,10,11,12,13,14,15,16].

$${\displaystyle \frac{\partial \phi }{\partial t}}=D{\nabla }^2\phi$$

(4)

with D as the diffusion constant, the hexagonal tessellation forming the hydroxyapatite molecule upon crystallization is conceptualized as a graph (Figure 3). Each hexagonal tile (TE) constitutes a vertex connected to its eight immediate neighbors, which are also hexagonal prisms. The spatial component of the diffusion equation (Equation (4)), expressed as ${\nabla }^2\phi \approx {\Delta }_{\gamma }\phi$ for a given position v and time t, with equal weights ${\gamma }_{uv}$ on all edges, acts upon the function $\phi \left(t,v\right)$ and is explicitly defined by Equation (5).

$$\left({\Delta }_{\gamma }\phi \right)\left(t,v\right)=\sum _{u:d\left(u,v\right)=1}{\gamma }_{uv}\left[\phi \left(t,v\right)-\phi \left(t,u\right)\right]$$

(5)

where $d\left(u,v\right)$ represents the constant distance between connected vertices u and v in the graph. Consequently, this summation is applied to the nearest neighbors u of vertex v. For a hexagonal lattice, the discrete approximation of the Laplacian (Equation (5)) is extended to three dimensions by including two additional neighbors: one directly above and one directly below the central cell under consideration, in addition to the base hexagonal neighbors:

$$\phi \left(t+1,v\right)\approx \phi \left(t,v\right)+{\displaystyle \frac{\alpha }{12}}\left[-8\phi \left(t,v\right)+\sum _{N\in nn\left(v\right)=1}^N\phi \left(t,N\right)\right]$$

(6)

Here, the relationship $D{\gamma }_{uv}=\alpha /12$ is applied. This coefficient determines the relative contributions of the central cell and its neighbors to the diffusion calculation. Specifically, the central vertex carries a weight of $2/3$, while each nearest neighbor contributes $1/2$. This weighting becomes explicit when $\alpha =1$, impacting the calculated transport of thermodynamic variables within the system.

Each rectangular horizontal plane in the model is entirely filled with a hexagonal tessellation. Figure 4 provides a schematic representation of the initial stages of this tessellation process. The systematic enumeration shown in Figure 4a, with alternating rows of even and odd numbers, facilitates the identification of the six immediate (first-order) neighbors for any given cell. For instance, in the tessellated surface depicted, with $m=5$ rows, a set of two distinct rules is applied to determine the complete neighborhood of any cell.

• For even numbers, the rule for finding the first six neighbors is:
  lpcp = {p − 2, p − 1, p + 1, p + 2, p − (2m + 1) − 2, p − (2m + 1)}
• For odd numbers, the rule for finding the first six neighbors is:
  lpci = {p − 2, p − 1, p + 1, p + 2, p + (2m + 1), p + (2m + 1) + 2}

Therefore, if it is wanted to find the first six neighbors of cell 39, using the lpci rule, this set of first neighbors consists of {37, 38, 40, 41, 50, 52}. Now, if it is wanted to find the set of first six neighbors of cell 30, we use the lpcp rule, which shows {28, 29, 31, 32, 17, 19}.

Each cell in the model is characterized by a continuous state function $s_t\left(z\right)$, which ranges from 0 to 1 for non-fixed automata and represents its degree of fixation. This state function updates its value based on the influence of thermodynamic factors $\alpha$, $\beta$, $\gamma$.

Initially, one or more cells are designated as nucleators, with their state set to $s_0\left(z\right)=1$. These nucleators can be positioned at predefined locations, within a single layer, across multiple planes, or at randomly determined coordinates. All other remaining cells initially take on a value of $\beta$.

The classification of cell states proceeds as follows:

• An automaton becomes a fixed cell (F-cell) if its state function $s_t\left(z\right)⩾1$.
• If $s_t\left(z\right)<1$ and at least one of its immediate neighbors is an F-cell, it is categorized as a border cell (B-cell).
• A cell where $s_t\left(z\right)<1$ and none of its immediate neighbors are F-cell is defined as a non-receptive cell (N-cell).
• Finally, any cell located at the boundary of the defined surface is classified as an edge cell (E-cell).

The tile-marking procedure begins with the complete lower and upper layers of the simulated volume being designated as edge cells (E-cells). For the intermediate “interior” layers, two primary classifications are applied: tiles located on or outside the defined perimeter within each layer are identified as E-cells, while the remaining tiles situated within that perimeter are initially assigned as non-receptive cells (N-cells).

A nucleator, defined as an ossified cell, is established to initiate the growth process. This nucleator replaces an existing N-cell with an F-cell at a specific point, which can be in the central layer, any other defined or randomly selected layer. Subsequently, the first eight neighbors of this newly formed F-cell, which were previously N-cells, are re-designated as B-cells. Thus, a nucleator acts as an externally imposed boundary condition that compels the system to initiate the ossification process at a discrete spatial location.

Once all automata have been initialized with their respective states, the dynamic simulation of the process can commence. The initial conditions for this three-dimensional model, as established by [7], are:

$$s_0\left(z\right)=\left\{\begin{array}{ccc}1& \mathrm{si}& z=\mathcal{O}\\ \beta & \mathrm{si}& z\ne \mathcal{O}\end{array}\right.$$

(7)

where $\mathcal{O}$ denotes the set of spatial positions occupied by F-cells. Furthermore, the following functions are defined for a cell z:

• $u_t\left(z\right)$: A measure of the cell’s fixation that actively participates in the diffusion process. This function evaluates the state of N or edge E cells. Notably, E-cells contribute to the diffusion but maintain a constant state, such that $u_t\left(z\right)=\beta$.
• $v_t\left(z\right)$: A distinct measure of cellular fixation that does not participate in the diffusion process. This function assesses the state of B-cells) or F-cells.

Consequently, the overall state function for a cell z is defined as:

$$s_t\left(z\right)=u_t\left(z\right)+v_t\left(z\right)$$

(8)

In which if z is a receptive cell (B or F cells), we have $u_t\left(z\right):=0\to v_t\left(z\right):=s_t\left(z\right)$. Now if z is a non-receptive cell (N cells), we have $v_t\left(z\right):=0\to u_t\left(z\right):=s_t\left(z\right)$. It is worth mentioning that no punctual operations are performed on the edge tiles (E), but they do contribute to the diffusion of temperature when operations are performed on their neighboring tiles, whether they are receptive (B or F) or non-receptive (N).

In this model [7], the state of a cell evolves as a function of the states of its neighboring tiles and it has the following possibilities:

For any receptive cell (B or F) its posterior state $v_{t+1}\left(z\right)$ is updated via:

$$v_{t+1}\left(z\right):=v_t\left(z\right)+\gamma +{\displaystyle \frac{2\alpha }{3}}\left[\overline{u_t\left(z\right)}\right]$$

(9)

Figure 4 and Equation (9) illustrate that B-cells, characterized by $0<v_t\left(z\right)<1$ have N-cells as their immediate neighbors, where $u_t\left(z\right)\ne 0$. Consequently, applying Expression (10) to a B-cell, which inherently has $u_t\left(z\right)=0$, results in an average derived solely from its N-cell neighbors. This effectively means that boundary cells also exhibit diffusion (Equation (6)). In contrast, F-cells are completely surrounded by B-cells, leading to an average of zero $\overline{u_t\left(z\right)}=0$. Therefore, ossified cells are not affected by diffusion.

For any non-receptive cell (N) its posterior state $u_{t+1}\left(z\right)$ is updated through:

$$u_{t+1}\left(z\right):=u_t\left(z\right)+{\displaystyle \frac{2\alpha }{3}}\left[\overline{u_t\left(z\right)}-u_t\left(z\right)\right]$$

(10)

In Equations (10) and (11), subscripts denote the updated functions reflecting a cell’s state before and after a completed step. $\overline{u_t\left(z\right)}$ represents the average of $u_t\left(z\right)$ for the eight nearest neighbors of cell z, with this average only considering values from N-type cells (Equations (10) and (11)). Cells located at the model’s periphery, referred to as edge or marginal cells (E-cells), have their state updated to $u_{t+1}\left(z\right):=\beta$. Consequently, these marginal cells play a role in the diffusion of energy into the system.

The updated state of a cell at the culmination of a stage is obtained by combining the two intermediate states that have undergone updates, as shown in Equation (8) [7,8,9]. This integration reflects the culmination of the diffusion and fixation processes within the cell.

$$s_{t+1}\left(z\right)=u_{t+1}\left(z\right)+v_{t+1}\left(z\right)$$

(11)

By manipulating the parameters $\alpha ,\beta$ and $\gamma$ within this model, the geometric configurations of the bone matrix are generated on each surface. This generation is facilitated by the diffusive interaction between a fixed cell and its neighbors, which drives the evolution of the tiles within that specific layer, as well as the layers directly above and below it, all under the same state functions. This process ultimately generates the geometrized 3D patterns of spongy bone. It is important to emphasize that this entire process is exclusively physicochemical, intentionally excluding any interaction with osteoblasts, osteoclasts, or other living tissues.

4. Bone Section According to Parameter Values

A three-dimensional framework was constructed, comprising 51 horizontal planes (the model represents a volume that is 2 cm long, 1.74 cm wide, and 1.4 cm high). Each plane contains 40,602 hexagonal cells (201 rows × 202 columns) that intersect the spatial ellipsoidal cylindrical surface (ECS). Only the cells located within the ECS at each plane constitute the bone section. The parameters $\alpha ,\beta$ and $\gamma$ were systematically varied to explore their influence on bone formation.

To generate the cancellous bone section, nucleation centers were strategically placed. These centers were arranged around the inner circular cylindrical surface (CCS) in alternating layers, starting from the first layer. Each cellular automaton in close proximity to this CCS was designated as a nucleation site. Additionally, nucleation centers were randomly distributed within the region between the CCS and the inner ECS.

The development of the bone section is an iterative, multi-stage process. Each stage represents a distinct time increment, beginning with a state change in the bottommost non-edge layer (Layer 1, as Layer 0 is defined as an edge layer). A stage concludes with the penultimate upper layer, as the final layer is also an edge layer. Upon completion of a stage, the process continues, re-initiating from Layer 1. Typically, around 1500 stages were employed to generate the bone section, with the exact number being dependent on the values of $\alpha ,\beta$ and $\gamma$. It was observed that an excessive number of stages leads to overly dense structures, obscuring the characteristic morphology of spongy bone.

Considering the defined tile states (F, B, N and E) and the three limiting spatial surfaces, coupled with the influence of localized nucleation centers (as illustrated in previous figures), it’s evident that both the resulting morphology of the bone structure and the number of stages required for saturation are dependent on the parameters $\alpha ,\beta$ and $\gamma$. Specifically, $\alpha$ and $\gamma$ govern the internal thermodynamics of the bone section, mediated through the Laplacian operator (represented as a graph). Conversely, the parameter $\beta$ encapsulates the external temperature and pressure information, which is propagated into the system via the edge tiles and the two boundary layers.

5. Conclusions

This study established a procedure utilizing three perpendicular limiting surfaces intersected by horizontal planes. These planes, composed of hexagonal tiles, ultimately form the internal structures of the bone’s mineral matrix through a physicochemical process. We developed a simplified bone model, focusing exclusively on the development of the mineral matrix’s internal structures. However, this methodology could be extended to other anatomical forms, such as a vertebral bone model (which collectively form the spinal column), a phalanx, or other large bones.

These cellular automata represent hydroxyapatite molecules, existing in two forms: crystallized (F-cells) and non-crystallized (B, N, or E-cells). The initial F-cell(s), acting as nucleators, interact with their local environment, influencing a finite spatial domain. The extent of this influence is self-regulated by the growth of the mineral matrix (B and N-cells), governed by the thermodynamic parameters $\alpha$ and $\gamma$. Conversely, E-automata are responsible for transmitting the influence of the external environment via the parameter $\beta$. While $\beta$ remains fixed throughout the current calculation, it could be modified to simulate changes in the external conditions affecting the mineral matrix.

The thermodynamics of the process is modeled by considering a central tile and its eight nearest neighbors. Six of these neighbors are hexagonal tiles located within the same horizontal plane, while the remaining two are positioned directly in the immediately superior and inferior planes. This entire computational process, involving the application of Equations (8)–(11), is executed using a continuous indexing scheme for each tile. Our indexing method treats these first neighbors as a specific subset of six indices associated with the numbering of each tile, with the calculations further complemented by incorporating the tiles directly above and below the particular cell currently being processed.

The morphology of the bone mineral matrix, while entirely deterministic for a given set of $\alpha$, $\beta$ and $\gamma$ values, exhibits considerable richness in form. Any alteration to these parameters yields a distinct structure, indicating a strong dependence on initial conditions or, more broadly, a chaotic behavior. For visualization purposes, only half of the data points were plotted to reveal the resulting internal structures, and the chosen color scheme serves solely to highlight the changes within the mineral matrix.

The numerical simulations of the cellular automata process, presented in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, illustrate a geometrized spongy bone section. This outcome is attributable to our exclusive focus on the physicochemical interactions governed by the mathematical model (Equations (6)–(11)). Currently, our model lacks the necessary components to simulate the complex chemical-biological interactions that would yield a complete composite bone matrix. Another unaddressed aspect is the determination of the process termination point. This is crucial as the final density of the mineral matrix is a characteristic unique to each vertebrate type and individual within a species, and is typically regulated by hormonal processes.

A future second stage of this process will involve modeling the dynamic interaction between osteoclasts and osteoblasts with the bone matrix. This interaction will be governed by a force vector applied to specific skeletal regions. This will trigger a directed, non-random response from the organism’s living tissue mechanisms, promoting osteoblast proliferation in areas requiring reinforcement and osteoclast activity in regions under less mechanical stress. This intricate interplay would be simulated using a prey-predator model, acting exclusively on the bone matrix and aligning in the direction of the applied force vector.