The authors state that they have no conflicts of interest.

Research-Article

# Mathematical Modeling of Spatio-Temporal Dynamics of a Single Bone Multicellular Unit^{†}

Article first published online: 8 DEC 2008

DOI: 10.1359/jbmr.081229

Copyright © 2009 ASBMR

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#### How to Cite

Ryser, M. D., Nigam, N. and Komarova, S. V. (2009), Mathematical Modeling of Spatio-Temporal Dynamics of a Single Bone Multicellular Unit. J Bone Miner Res, 24: 860–870. doi: 10.1359/jbmr.081229

^{†}

#### Publication History

- Issue published online: 4 DEC 2009
- Article first published online: 8 DEC 2008
- Manuscript Accepted: 5 DEC 2008
- Manuscript Revised: 5 NOV 2008
- Manuscript Received: 2 MAY 2008

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### Keywords:

- mathematical modeling;
- bone remodeling;
- microfracture;
- targeted remodeling;
- osteoclast;
- osteoblast

### Abstract

During bone remodeling, bone-resorbing osteoclasts and bone-forming osteoblasts are organized in bone multicellular units (BMUs), which travel at a rate of 20–40 μm/d for 6–12 mo, maintaining a cylindrical structure. However, the interplay of local BMU geometry with biochemical regulation is poorly understood. We developed a mathematical model of BMU describing changes in time and space of the concentrations of proresorptive cytokine RANKL and its inhibitor osteoprotegerin (OPG), in osteoclast and osteoblast numbers, and in bone mass. We assumed that osteocytes surrounding a microfracture produce RANKL, which attracted osteoclasts. OPG and RANKL were produced by osteoblasts and diffused through bone, RANKL was eliminated by binding to OPG and RANK. Osteoblasts were coupled to osteoclasts through paracrine factors. The evolution of the BMU arising from this model was studied using numerical simulations. Our model recapitulated the spatio-temporal dynamics observed in vivo in a cross-section of bone. In response to a RANKL field, osteoclasts moved as a well-confined cutting cone. The coupling of osteoclasts to osteoblasts allowed for sufficient recruitment of osteoblasts to the resorbed surfaces. The RANKL field was the highest at the microfracture in front of the BMU, whereas the OPG field peaked at the back of the BMU, resulting in the formation of a RANKL/OPG gradient, which strongly affected the rate of BMU progression and its size. Thus, the spatial organization of a BMU provides important constraints on the roles of RANKL and OPG as well as possibly other regulators in determining the outcome of remodeling in the BMU.

### INTRODUCTION

Repairing structural microdefects during bone remodeling is critical for maintaining mechanical properties of bone. Bone remodeling proceeds asynchronously at multiple sites of the skeleton in the form of organized bone multicellular units (BMUs), which are spatially and temporally controlled teams of bone-resorbing osteoclasts and bone-forming osteoblasts.^{(1–3)} Osteoclasts are the first cells recruited to the sites of mechanically unsound bone (e.g., to a microfracture), where they start resorbing bone as a team of 10–20 cells. The exact molecular mediators of osteoclast recruitment are not known; however, osteocytes and bone-lining cells play critical roles in this process.^{(4)} The life span of an individual osteoclast is 9–10 days, after which it dies, primarily by apoptosis.^{(5)} Osteoblasts follow as a team of 1000–2000 cells and fill the resorbed space with osteoid, which subsequently mineralizes. The life span of an individual osteoblast is 14–16 days, after which it dies by apoptosis or differentiates into an osteocyte enclosed in the bone matrix or into a bone-lining cell covering the surface of the bone tissue.^{(6)} As time progresses, osteoclasts continue to move in form of a cutting cone along the damage lines, or, after the damage is repaired, along the strain field in the bone tissue. Osteoblasts follow the osteoclasts replacing the resorbed bone tissue. As an entity, a BMU exists for 100–300 days and reaches a length of 2–7 mm.^{(7)} In cortical bone, BMUs are lengthy units digging tunnels across the bone matrix, aligning themselves with the main axis of the bone.^{(2,7)} In trabecular bone, the BMUs move across the surfaces of trabeculae in the form of half-trenches.^{(2)} Thus, a BMU exhibits a complex spatial organization, which exists on a temporal scale considerably exceeding the lifespan of individual cells (Fig. 1).

To coordinate their actions, osteoclasts and osteoblasts communicate by means of autocrine and paracrine factors. Among numerous messengers involved in these communications, RANKL and its counterpart osteoprotegerin (OPG) have been shown to play critical roles in both physiological bone remodeling^{(8,9)} and in diseases associated with abnormal bone remodeling such as osteoporosis, rheumatoid arthritis, and periodontitis.^{(8,10–12)} RANKL and OPG are produced by cells of osteoblastic origin, including mature osteoblasts and their precursors, osteocytes, bone-lining cells, and stromal cells.^{(13–16)} RANKL stimulates osteoclast formation and activity and prevents osteoclast death by acting through its receptor RANK, which is expressed on osteoclast precursors and mature osteoclasts.^{(17)} RANKL and macrophage colony-stimulating factor (M-CSF), also produced by cells of the osteoblastic lineage, are necessary and sufficient for osteoclast differentiation.^{(15)} OPG is a soluble decoy receptor that binds to RANKL, thus preventing its interaction with RANK and effectively acting as a negative osteoclast regulator. Different systemic modulators of bone remodeling, such as PTH, estrogen, and calcitriol, have been shown to regulate the expression of RANKL and OPG and their abundance relative to each other.^{(18–22)} The expression of RANKL and OPG dynamically changes during bone remodeling. RANKL production by osteocytes^{(16,23,24)} is likely to contribute to initiation and progression of osteoclastic bone resorption. It has also been shown that, whereas immature osteoblasts produce mainly RANKL, in mature osteoblasts, production of OPG prevails.^{(25–27)} Thus, the structure of the BMU implies that RANKL and OPG change in distinct temporal and spatial patterns, with RANKL produced early by osteocytes that guide the cutting cone, and OPG generated at later times in the back of the remodeling space. In this study, we investigated the functional consequences of taking into account these differences in temporal and spatial profiles of RANKL and OPG.

The use of mathematical modeling in the field of bone biology is becoming more and more widespread. Models describing osteoclast/osteoblast interactions in BMUs based on nonlinear ordinary differential equations have been developed by several authors.^{(28–32)} On the opposite side of the spectrum, finite element methods are used to analyze strain fields induced by external loading.^{(33)} However, to our knowledge, there are currently no mathematical models that capture the complete spatio-temporal dynamics of a single BMU. In this study, we introduce a spatial extension of the temporal model suggested by Komarova et al.,^{(30)} resulting in a novel nonlinear model comprising of a system of partial differential equations, which describe the role of RANK/RANKL/OPG pathway in attracting and promoting the BMU, as well as the autocrine and paracrine interactions between osteoclasts and osteoblasts, the main constituents of the unit. The model consists of five state variables: densities of osteoclasts and osteoblasts, concentrations of OPG and RANKL, and the local bone mass. The goal of the study was to create a model that captures experimentally observed BMU dynamics and to assess how taking into account different temporal and spatial dynamics of RANKL and OPG affects the progression of BMUs.

### MATERIALS AND METHODS

#### Model assumptions

We developed a mathematical model describing the spatio-temporal evolution of a single BMU with the following goals: (1) to describe the dynamics (as a set of driving forces as well as consequent changes) of the key biochemical factors RANKL and OPG together with the population dynamics of osteoclasts and osteoblasts and (2) to describe the distinctive spatial and temporal features of the cutting cone and the BMU movement across the bone surface. The model is 2D in space and particularly suited for the description of trabecular remodeling, or it may also be interpreted as a model for the cross-section of a cortical BMU. We assume that remodeling occurs in the domain of 2.4 × 1.6 mm (denoted by Ω), representing a flat section of bone. We assume that the depth of the resorbed trench (∼10 μm) is small in comparison with its length (∼100–1000 μm), that the BMU evolves along the surface of the trabecula, and that local curvature effects are unimportant. We only consider the dynamics of two cell types, osteoclasts and osteoblasts, and assume that the vasculature established during bone remodeling acts as a reservoir for precursors. We only explicitly describe the dynamics of RANKL and OPG. However, the function of many other factors such as TGF-β, IGFs, and M-CSF, is also captured by the parameters describing the combined effectiveness of autocrine and paracrine interactions. We will start with a brief review of the temporal model^{(30,31)} and then will introduce the spatial extension of the model and the RANKL and OPG fields, and finally will complete the model by adding appropriate initial and boundary conditions.

#### Adaptation of a previously constructed temporal model

The model suggested by Komarova et al.^{(30,31)} is a local and purely temporal model describing the population dynamics of bone cells at a single site within the BMU. Denoting the number of osteoclasts and osteoblasts by *u*_{1} and *u*_{2}, the cell dynamics are given by the system of ordinary differential equations:

- (1)

where α_{i} and β_{i} are activities of cell production and death and the four parameters *g*_{ij} represent the effectiveness of the autocrine and paracrine interactions between the constituent cells, as described below. Also see Table 1 for the definition of all symbols used in this paper and Table 2 for the parameter values and units. equation 1 has well-defined steady-state solutions, denoted by ū_{1} and ū_{2}, respectively. It is assumed that cells below the steady-state conditions are precursors, which are less differentiated and therefore not actively involved in the processes of resorption and production of bone matrix but involved in the paracrine and autocrine signaling. Increases in *u*_{1} and *u*_{2} above steady-state values are regarded as proliferation and differentiation of precursors into mature osteoclasts and osteoblasts that participate actively in the remodeling process. This model does not describe the initiation of bone remodeling, which is manually induced by choosing initial values *u*_{1}(*t*_{0}) > ū_{1} We now denote the number of active osteoclasts and osteoblasts by *y*_{1} and *y*_{2}, respectively, where, for *i* = *1,2*:

- (2)

Let us now briefly discuss the various autocrine and paracrine mediators included in equation 1, thereby making some restrictions appropriate to the spatio-temporal model we are developing. The factor *g*_{11} represents the effectiveness of the osteoclast-derived autocrine factors (combined action of factors such as TGF-β, interferon β, annexin II, etc.^{(34)}). The analysis of the temporal model showed that this interaction plays a critical role in controlling the dynamic behavior of remodeling when acting as a positive feedback.^{(30)} Moreover, we have recently shown that positive autocrine regulation of osteoclasts is necessary to describe the complex behavior observed in osteoclast cultures in vitro.^{(35)} Therefore, *g*_{11} is assumed to be positive and equal to 0.5 in this paper. The factor *g*_{12} represents an osteoclast-derived paracrine regulation of osteoblasts. Although the exact nature of osteoclast-osteoblast coupling mediators is not known yet, the analysis of the temporal model showed that *g*_{12} has to be strictly positive^{(30)} and that it is crucial for the coupling of osteoclasts and osteoblasts.^{(31)} To achieve functional coupling we assume *g*_{12} = 1. The factor *g*_{22} represents an osteoblast-derived autocrine feedback. However, this factor did not influence the dynamical behavior of the temporal model^{(30)}; therefore, we assumed here that *g*_{22} = 0. Finally, osteoblast-derived paracrine regulation of osteoclasts is dominated by the RANKL/OPG pathway^{(17,36)}; therefore, in this study, we set *g*_{21} = 0 and explicitly described RANKL and OPG dynamics.

#### Construction of a spatio-temporal model

Using temporal equation 1 as the basis for the spatial extension, we now switch to space-dependent state variables: , where , and *u*_{i} have the units of a surface density (cells/mm^{2}). At the same time, we introduced two new state variables: the RANKL field and the OPG field , which have the dimensions of a surface concentration (mol/mm^{2}). We assumed that osteoclast movement is proportional to the RANKL gradient and is described in the following form: . Here, *ζ* indicates the effectiveness of migration, which has units (mm^{6}/d/mol) and ∇ is the differential operator . In addition, osteoclast formation is governed by the amount of RANKL bound to RANK receptors. We describe this effect in the following form: , the reaction rate *k*_{1} has units (d^{−1}), RANK receptors have a saturation threshold, resulting in the sigmoid function with *λ* denoting the concentration of half-saturation, and θ(*y*_{1}) is the Heaviside-function defined as {*θ*(*x*) =0 *if x* < 0, *θ*(*x*) = 1 *if x* > 0}. Finally, the resulting changes in osteoclast numbers are described by osteoclast formation (which depends on the concentration of RANKL and autocrine signals released by osteoclasts), osteoclast death (proportional to the number of osteoclasts), and movement of osteoclasts (in response to the RANKL field):

- (3)

We next describe the evolution of osteoblasts. We assume that osteoblasts are recruited by active osteoclasts and do not move by themselves and therefore are governed by the equation from model 1. The explicit solution formula for the osteoblast equation can be given in the following form:

- (4)

We next describe the spatio-temporal evolution of RANKL and OPG fields. The rate of change of both RANKL and OPG is governed by three contributions: a source term, describing production of these factors by osteoblasts and osteocytes, a diffusion term, describing the movement of soluble forms of these factors, and a reaction term, describing interaction of RANKL with OPG and RANK. RANKL is produced by osteocytes and by active osteoblasts, it spreads across the bone by porous diffusion, and it binds to OPG and to RANK receptors on osteoclasts. We assume that RANKL production by osteocytes takes places before the initiation of the BMU (and as such is described in initial conditions). During the BMU evolution, new RANKL is produced only by osteoblasts as described in the following term: , where *a*_{R} is the corresponding rate constant and has units (mol/cell/d) and

- (5)

Equation 5 incorporates the idea that it takes a certain time *t*_{R} before the precursor cells differentiating into osteoblasts start to produce RANKL. The porous diffusion can vary between very low for membrane bound RANKL and high for soluble RANKL and is described as where diffusion constant *κ*_{R} has units (mm^{2(1+ϵ)}/mol^{1−ϵ}/d], and Δ = ∂^{2}_{x} + ∂^{2}_{y} is the Laplace operator in two dimensions. The dimensionless exponent ϵ reflects the porosity of the medium for RANKL; in particular ϵ = 1 yields the usual diffusion equation. The diffusion of RANKL would technically happen in all directions, allowing RANKL to diffuse into the bone marrow and into the bone matrix of the trabecula. However, at this time, we made an assumption that the trabecula is very thin relatively to its length and that the permeability of the canopy of bone-lining cells for RANKL is low, and therefore we neglect the vertical diffusion loss, thus restricting the model to a two dimensional diffusion along the surface of the trabecula. The receptor-ligand binding of RANKL to RANK receptors is described as It is almost identical to the term used in equation 2 to describe the effect of RANKL of osteoclasts with the difference that now we have *k*_{2} with units (mol/d) instead of *k*_{1}. Note that these rate constants are independent because we have to allow for partial reversibility of RANKL-RANK binding (i.e., RANKL molecule bound to RANK on an osteoclast contributes to the stimulation of the osteoclast and subsequently can detach with a certain probability). Finally, the reaction term for the binding of RANKL and OPG is described as *k*_{3}ϕ_{R}ϕ_{o}, where *k*_{3} is the rate constant having units (mm^{2}/mol/d). Thus, the changes in RANKL are described by the following equation:

- (6)

OPG production by mature osteoblasts is described as where *a*_{O} is the corresponding rate constant with units (mol/cell/d), and is given by equation 5, except the time delay *t*_{o} is strictly greater than *t*_{R}, reflecting the idea that immature osteoblasts produce primarily RANKL, whereas osteoblasts maturation leads to increase in OPG production.^{(25–27)} The porous diffusion of OPG is described similarly to RANKL as *κ*_{o}Δ(ϕ^{δ}_{O}), where *κ*_{O} is the diffusion constant with units (mm^{2(1+δ)}/mol^{1−δ}/d), Δ = ∂^{2}_{x} + ∂^{2}_{y} is the Laplace operator in two dimensions, and the dimensionless exponent δ reflects the porosity of the medium for OPG. The equation describing RANKL-OPG binding is equivalent to the corresponding term in equation 6. The resulting changes in OPG are described by the following equation:

- (7)

Finally, the interplay of the various parameters in equations 2–7 determines the final bone mass of the remodeled osteon. To keep track of the bone mass evolution, we add a fifth state variable, describing the local bone mass varying from 0% to 100%. We assume that only active cells, *y*_{i}, are able to resorb and produce bone matrix. Therefore, the rate of change of bone mass is governed by the following equation:

- (8)

Equations 2–8 constitute the nonlinear partial differential equation model describing the evolution of the BMU, which can be reduced to a system of four integro-differential equations and one dependent equation (describing changes in bone mass).

- (9)

#### Initial and boundary conditions

For mathematical completeness of the model, we have to add initial and boundary conditions to the system (equation 9). In the general case, the initial conditions were stated as follows:

- (10)

Let us comment on these conditions. Initial conditions for osteoclasts, reflect the fact that our model does not describe the process of initiation of bone remodeling. We initiate remodeling manually by choosing the function to equal everywhere except for a confined region U, on the side of the microfracture path, where we place a few active cells: for *x* ∈ *U*. Initial conditions for osteoblasts, are based on the assumption that no active osteoblasts are present initially and that the osteoblast density equals its steady-state value everywhere. The initial condition for the RANKL field, represents the assumption that RANKL is expressed by osteocytes along the microfracture. Because physiological RANKL is mainly membrane bound, we design the initial RANKL field to be zero everywhere except for the near proximity of the microfracture path. The initial condition for the OPG field is zero, based on the assumption that no active osteoblasts are present initially. The initial value for the bone mass (z) is set to 100%.

#### Numerical simulations

Computationally, this system is quite challenging to simulate. Ongoing work is currently focused on developing schemes that are both accurate and stable for this problem. In this paper, we implemented a fixed time step, fourth-order Runge-Kutta scheme with a second order centered differencing in space. We first introduced initial perturbation-a localized increase in osteoclast numbers and allowed the simulation to run for several simulation-days until the cutting cone reached stable regime. For generation of figures, at this point, we defined the time to be *t* = 0; therefore, we only show the evolution of the BMU and not the initiation phase.

#### Parameter estimations

Our model has 21 parameters for which we had to find appropriate values. We proceeded as follows: first, we considered the purely temporal system (equation 1) and followed the reasoning in Reference 30 to obtain meaningful values. In particular, the values for *β*_{i} can be deduced from experimental findings of the corresponding life spans of bone cells. In addition, we aimed for a ratio of as observed experimentally.^{(1–3)} The time delays *t*_{R} and *t*_{O} were estimated according to the literature that indicates that in immature osteoblast RANKL production dominates, whereas mature osteoblasts produce mainly OPG.^{(25–27)} The remaining parameters could not be matched explicitly with experimental findings, and we chose them in such a way that the simulations coincide spatially and temporally with in vivo observations. The values of all parameters used in the model are given in Table 2.

### RESULTS

#### Numerical simulation of microfracture remodeling

We have first considered the situation where a BMU is recruited to remodel a microfracture represented by a red line on Fig. 2. We assumed that damaged and apoptotic osteocytes surrounding the microfracture produce mainly membrane-bound RANKL, resulting in an initial RANKL distribution in the form of a path surrounding a microfracture (Fig. 2; RANKL at *t* = 0). Remodeling process is initiated at the left end of the path. Initially, there are no active osteoblasts or OPG, and the bone mass is at 100%.

The spatio-temporal evolution of the model is shown in the snapshots at 0, 90, and 160 days (Fig. 2). The RANKL field gradually becomes annihilated as a result of binding to RANK expressed on osteoclasts, which move along the RANKL path in the form of a well-confined cutting cone. Note that the kink in the microfracture is easily negotiated by the osteoclasts in the model. Osteoblasts, constantly recruited by active osteoclasts, build up a closing cone, gradually replacing resorbed bone. OPG is produced by mature osteoblasts in the closing cone, and it diffuses through the substrate. The bone mass reaches its minimum just behind the cutting cone, gradually increasing toward the very left of the initial path. At 160 days, once the BMU has almost reached the end of the RANKL path, the bone mass on the first segment has completely recovered. We note that the numbers of both osteoclasts and osteoblasts decrease over time. This is because the RANKL on the microfracture path has very low but nonzero porous diffusion and therefore slowly spreads in time. Reduced concentration of RANKL on the path leads to a slight decrease in the number of recruited osteoclasts, which in turn results in a decrease in the number of osteoblasts, because their recruitment is proportional to the current number of osteoclasts. Note that the spatio-temporal scales involved in this simulation correspond very well with experimental findings.^{(1–3)}

#### Role of OPG in controlling BMU branching and progression

We next considered the ability of BMUs to branch in different directions. We designed situations in which an additional microfracture deviates from the original RANKL path both at an angle of <90° (forward branching; Fig. 3, red line) and at an angle of >90° (backward branching; Fig. 4, red line). The remodeling is initiated at the left end of the path; there are initially no osteoblasts or OPG. The spatio-temporal evolution of the model is shown in the snapshots at 90 and 160 days. We found that when the microfracture branches in the forward direction, the BMU divides and successfully remodels both branches (Fig. 3). In contrast, when the microfracture branches in the backward direction, the BMU only remodels a very short section of the secondary branch, and the cutting cone dies out (Fig. 4). The reason for the BMU to abandon the backward facing branch is that the OPG field produced by mature osteoblasts in the back of the closing cone has diffused sidewise as well as forward, and by the time *t* = 90 days, OPG has already bound and removed RANKL on the secondary branch. Deprived of RANKL stimulation, osteoclasts of the secondary cutting cone disappear, and the damaged site is not remodeled. Thus, the OPG field has prevented the BMU from branching backward. These branching dynamics are in good agreement with experimental results obtained using 3D μCT,^{(7)} which strongly suggest that BMUs only branch at acute angles.

To assess the effect of OPG on BMU progression, we repeated the simulation presented in Fig. 2, but increased OPG production by mature osteoblasts (Fig. 5). We found that, in this situation, OPG eventually diffuses ahead of the cutting cone, where it binds and eliminates RANKL associated with the microfracture, thus depriving osteoclasts of stimulation. As a result, the cutting cone disappeared in the third snapshot at *t* = 155, leading to premature termination of microfracture remodeling (Fig. 5). Thus, we found that excessive OPG production by mature osteoblasts does not drastically affect the numbers of osteoclasts and osteoblasts involved in the remodeling process, but has an ability to terminate the BMU progression through the bone.

#### Effect of soluble RANKL on BMU progression

Thus far, we only considered situations where RANKL was membrane bound and therefore had very low diffusion. This seems to be an adequate assumption in the case of physiological remodeling.^{(37)} However, there are pathological settings where the soluble form of RANKL predominates.^{(38,39)} Therefore, we performed an experiment where we applied a point source of soluble RANKL with a high diffusion rate compared to the previous experiments. The corresponding RANKL field is depicted in Fig. 6 at time *t* = 0, with the red star representing the point source of RANKL. In this situation, we zoom in on the zone of interest, because close to the boundaries, the field loses radial symmetry because of the chosen boundary conditions. The remodeling is initiated at the left end of the field. As evident in the subsequent snapshots, the cutting cone, although still confined, now develops as a progressively opening semicircle, and therefore the BMU remodels a larger area of the bone (Fig. 6A; *t* = 75 and 145). The osteoblasts (data not shown) follow the osteoclasts and produce OPG. The spreading of the cutting cone forward and sidewise but not backward is because of the fact that the RANKL field behind the cone is essentially zero because of RANKL binding to OPG and RANK.

We next repeated the same experiment but switched off the production of OPG by mature osteoblasts. In this case, the RANKL field persists much longer during the BMU evolution and is eliminated only by binding to RANK at the sites of osteoclast activation. The combination of continuous production and diffusion of RANKL with its now limited removal results in the appearance of a radial gradient evident in Fig. 6B at time *t* = 145 days. This RANKL gradient causes the cutting cone of resorbing osteoclasts to start spreading in all directions. Because of the effective coupling assumed in the model, the number of recruited osteoblasts is proportional to the number of active osteoclasts. Therefore, this wider and likely deeper resorption space is completely filled with new bone matrix. However, in situations when osteoprogenitors are limited in the adjacent bone marrow, excessive resorption could lead to an overall negative bone mass balance. Interestingly, comparing the progression of the cutting cone in the absence of OPG to that observed in the previous experiment in the presence of OPG (depicted by the green semicircle in Fig. 6B), we observed that the absence of OPG results in a significant decrease in the rate of BMU progression. As evident from the RANKL field, the diffusion of OPG maintains a sharper gradient of RANKL and thus results in faster movement of the BMU as observed in Fig. 6A. Thus, in situations of excessive production of soluble RANKL, the presence of OPG maintains a more physiological spatial organization of the traveling BMU and, somewhat counterintuitively, also results in faster progression of the BMU.

### DISCUSSION

In this study, we present a mathematical model describing the spatio-temporal evolution of a single BMU. The model is based on the following three conceptual cornerstones. First, RANKL produced by osteocytes plays a critical role in steering the BMU. Second, strong autocrine and paracrine interactions among bone cells, particularly the coupling of osteoblasts to osteoclasts and the autocrine stimulation of osteoclasts, are responsible for the appearance of a well-confined and coordinated BMU entity. Third, location and timing, not only the magnitude of expression, are critical for the ability of RANKL and OPG to control the BMU progression. Combined, these three principles led us to develop a system of nonlinear partial differential equations that describes the spatio-temporal evolution of the BMU. The model was able to capture salient physiological features of remodeling, including the confinement of the cutting cone to a compact region as it moves across the bone, and the ability of the RANKL field to influence the direction of BMU steering. Most interestingly, taking into account the spatial distribution of the RANKL and OPG production, we found that their interplay is more complex than would be anticipated simply considering their ratio. In our model, the formation and movement of osteoclasts is most strongly affected by RANKL produced by cells resident to quiescent bone, such as lining cells and osteocytes. In contrast, osteoblasts, which emerge subsequent to osteoclasts, act mainly as a source of diffusible inhibitor of RANKL, OPG. Together, this results in RANKL appearing early and in front of the BMU and OPG coming into play later and in the back of the remodeling path. Such spatial and temporal differences in the RANKL and OPG fields lead to formation of RANKL/OPG gradients, which strongly affect both the rate and the direction of the BMU progression. In particular, we showed the emergence of nonobvious phenomena such as inhibition of backward branching and increase of BMU speed in the presence of OPG.

At this stage, we believe that our model is applicable to both cortical and trabecular bone remodeling for the following reasons. Histomorphometric studies indicate similar appearance, composition, and progression of osteonal and hemiosteonal remodeling.^{(2)} Even though hemiosteonal remodeling seems to be more accessible for interactions with bone marrow, the fact that this does not result in significant difference in the progression of remodeling in cortical and trabecular bone suggests that either the effects of these interactions are minor or that bone marrow is sufficiently compartmentalized to minimize these interactions during normal bone remodeling. The latter situation appears more likely in light of the current concepts of specific “niches” existing in bone marrow,^{(40)} as well as direct evidence presented in a study that showed that hemiosteonal remodeling occurs under a canopy of bone-lining cells.^{(41)} Thus, histomorphometric and biological evidence suggests that these two types of remodeling progress similarly. The model we have presented is 2D and thus particularly suited for the description of trabecular remodeling; however, it may also be interpreted as a model for the cross-section of a cortical BMU. One of the limitations of a 2D model is that questions related to bone volume and geometry cannot be addressed with this model. In applied mathematics, it is a well-established methodology to study lower-dimensional versions of the model in question, because geometric complexity, difficulty of analysis, and computational costs make the study of higher-dimensional models much more involved. 1D or 2D simplifications not only render the problem more tractable but also yield reasonable approximations to the original problem and help in an eventual study of the higher-dimensional case. Future development of a 3D model will allow to directly study potential differences in the progression of hemiosteonal and osteonal remodeling and to address questions related to bone geometry and architecture. In addition, the model could be reformulated to account for the highly irregular surfaces that make up trabecular bone.

Another important direction for future model development is the explicit incorporation of the effects of mechanical strains and stresses into the model. An eventual coupling of our equations to a model describing the strains to external loading would yield a fruitful tool in the investigation of responses to loading in physiological and pathological conditions. In this regard, the branching dynamics observed in our model can be interpreted in terms of different mechanisms of BMU navigation through the bone matrix. There are two major mechanisms of BMU steering proposed in the literature, namely strain-derived steering and targeted steering. The former mechanism has been proposed by Burger et al.^{(42)} and states that strain-derived canalicular fluid flow is responsible for osteoclast activity and motility of the cutting cone of the BMU. On the other hand, there is an established evidence that microdamage leads to activation of BMUs close to the damage site,^{(43,44)} resulting in the targeted steering of BMU toward microfracture. The relative contribution of strain-derived and targeted steering to the BMU progression is not known; however, a recent study showed that there is more damage removal than could be expected from BMUs that are navigated exclusively by strain fields.^{(45)} Osteocytes are known to be affected both by the mechanical loading of bone^{(46)} and by the appearance of microfractures,^{(47)} suggesting a unification of the two mechanisms at the level of mechanotransduction. From this prospective, our branching experiments can be interpreted as microfracture (branch) deviating from the principal stress direction (original path), thus confirming that already existing BMUs are affected in their evolution by the presence of microfractures and also showing that the ability of BMU to deviate at large angles is limited.

The focus of this model was on the RANK/RANKL/OPG pathway, which is well established as a critical regulator of osteoclast activation.^{(48)} Multiple other factors are known to play important roles in regulation of osteoclasts, osteoblasts, and osteocytes.^{(49)} To name just a few, TGF-β, IGF1/2,^{(50)} Wnt ligands,^{(51)} and sclerostin, which was recently identified as an important regulator of bone remodeling produced by osteocytes,^{(52)} without doubt play critical roles in regulating BMU initiation and progression. Of these factors, our model indirectly incorporated the actions of paracrine factors produced or activated by osteoclasts that have stimulatory effect on osteoblasts, such as TGF-β released and activated during bone resorption^{(53)} or putative soluble factors produced by osteoclasts,^{(45)} as well as autocrine stimulatory factors produced by osteoclasts, such as annexin-II,^{(34,54,55)} and Adam8.^{(56,57)} Nevertheless, explicit modeling of actions of these factors and incorporation of other factors known to be critical in bone physiology, such as sclerostin, will be an important future direction for model development.

In summary, the model presented in this study shows the necessity of taking into account spatial and temporal information about RANKL and OPG expression and not only the ratio of these cytokines. Our model represents a significant advance compared with previous temporal models; however, in future studies, several limitations still need to be addressed, including (1) the lack of description of other cells types in addition to osteoclasts and osteoblasts, (2) the lack of explicit formulation for the actions of coupling factors and osteoclast autocrine factors, and (3) the 2D nature of the model. However, even in the current form, our model provides a new tool for the in silico analysis of regulation of bone remodeling and, in the future, it may contribute to the evaluation of the impact of other cytokines, growth factors, potential therapies, and biomaterials on the process of bone remodeling.

### Acknowledgements

The study was supported by NSERC grants to S.V.K. and N.N. S.V.K. holds the Canada Research Chair in osteoclast biology.

### REFERENCES

- 11990 Skeletal structural adaptations to mechanical usage (SATMU): 1. Redefining Wolff's law: The bone modeling problem. Anat Rec 226: 403–413.
- 21994 Osteonal and hemi-osteonal remodeling: The spatial and temporal framework for signal traffic in adult human bone. J Cell Biochem 55: 273–286.
- 32006 Biomechanical and molecular regulation of bone remodeling. Annu Rev Biomed Eng 8: 455–498., ,
- 42008 Osteocytes, mechanosensing and Wnt signaling. Bone 42: 606–615.,
- 52004 Regulatory mechanisms operative in osteoclasts. Crit Rev Eukaryot Gene Expr 14: 255–270.
- 62003 Control of osteoblast function and regulation of bone mass. Nature 423: 349–355.,
- 72006 Three-dimensional microcomputed tomography imaging of basic multicellular unit-related resorption spaces in human cortical bone. Anat Rec A Discov Mol Cell Evol Biol 288: 806–816., , ,
- 81999 Activated T cells regulate bone loss and joint destruction in adjuvant arthritis through osteoprotegerin ligand. Nature 402: 304–309., , , , , , , , , , , , , , , , , , , ,
- 91998 Osteoprotegerin-deficient mice develop early onset osteoporosis and arterial calcification. Genes Dev 12: 1260–1268., , , , , , , , , , ,
- 102000 Functional human T-cell immunity and osteoprotegerin ligand control alveolar bone destruction in periodontal infection. J Clin Invest 106: R59–R67., , , , , , ,
- 112000 Osteoprotegerin blocks bone cancer-induced skeletal destruction, skeletal pain and pain-related neurochemical reorganization of the spinal cord. Nat Med 6: 521–528., , , , , , , , ,
- 122001 Multiple myeloma disrupts the TRANCE/osteoprotegerin cytokine axis to trigger bone destruction and promote tumor progression. Proc Natl Acad Sci USA 98: 11581–11586., , , , , , , ,
- 132001 Connection between B lymphocyte and osteoclast differentiation pathways. J Immunol 167: 2625–2631., , , , , , , , , ,
- 141998 Osteoprotegerin ligand is a cytokine that regulates osteoclast differentiation and activation. Cell 93: 165–176., , , , , , , , , , , , , , , , , , , , , , ,
- 151998 Identity of osteoclastogenesis inhibitory factor (OCIF) and osteoprotegerin (OPG): A mechanism by which OPG/OCIF inhibits osteoclastogenesis in vitro. Endocrinology 139: 1329–1337., , , , , , , , , , , , , ,
- 162002 MLO-Y4 osteocyte-like cells support osteoclast formation and activation. J Bone Miner Res 17: 2068–2079., , , ,
- 17 , ,
- 181998 Osteotropic agents regulate the expression of osteoclast differentiation factor and osteoprotegerin in osteoblastic stromal cells. Endocrinology 139: 4743–4746., , ,
- 192000 Protein expression and functional difference of membrane-bound and soluble receptor activator of NF-kappaB ligand: Modulation of the expression by osteotropic factors and cytokines. Biochem Biophys Res Commun 275: 768–775., , , , , ,
- 201999 Modulation of osteoclast differentiation and function by the new members of the tumor necrosis factor receptor and ligand families. Endocr Rev 20: 345–357., , , , ,
- 212005 Estrogen and testosterone use different cellular pathways to inhibit osteoclastogenesis and bone resorption. J Bone Miner Res 20: 2224–2232., , ,
- 222005 Interleukin-11 receptor signaling is required for normal bone remodeling. J Bone Miner Res 20: 1093–1102., , , , , , , ,
- 231995 Time-lapse microcinematography of osteocytes. Miner Electrolyte Metab 21: 189–192., , , , , , , ,
- 242006 Bone marrow cell differentiation induced by mechanically damaged osteocytes in 3D gel-embedded culture. J Bone Miner Res 21: 616–625., , ,
- 252000 The expression of osteoprotegerin and RANK ligand and the support of osteoclast formation by stromal-osteoblast lineage cells is developmentally regulated. Endocrinology 141: 4768–4776., , , , ,
- 262001 Changing RANKL/OPG mRNA expression in differentiating murine primary osteoblasts. J Endocrinol 170: 451–460., , ,
- 272006 Vitamin D action and regulation of bone remodeling: Suppression of osteoclastogenesis by the mature osteoblast. J Bone Miner Res 21: 1618–1626., , , , , , , , ,
- 282000 Parathyroid hormone temporal effects on bone formation and resorption. Bull Math Biol 62: 163–188.
- 292004 Modeling the interactions between osteoblast and osteoclast activities in bone remodeling. J Theor Biol 229: 293–309., , , ,
- 302003 Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling. Bone 33: 206–215., , , ,
- 312005 Mathematical model of paracrine interactions between osteoclasts and osteoblasts predicts anabolic action of parathyroid hormone on bone. Endocrinology 146: 3589–3595.
- 322008 Model structure and control of bone remodeling: A theoretical study. Bone 43: 249–263., , , , , , ,
- 332007 Reliable simulations of the human proximal femur by high-order finite element analysis validated by experimental observations. J Biomech 40: 3688–3699., ,
- 342006 Regulation of osteoclast differentiation. Ann N Y Acad Sci 1068: 100–109.
- 352008 Complex dynamics of osteoclast formation and death in long-term cultures. PLoS One 3: e2104., , , , ,
- 362001 Minireview: The OPG/RANKL/RANK system. Endocrinology 142: 5050–5055.
- 372006 Mechanical loading differentially regulates membrane-bound and soluble RANKL availability in MC3T3-E1 cells. Ann N Y Acad Sci 1068: 568–572., , , , , , , , , ,
- 382002 Involvement of receptor activator of NFkappaB ligand and tumor necrosis factor-alpha in bone destruction in rheumatoid arthritis. Bone 30: 340–346., ,
- 392004 RANKL/RANK/OPG: New therapeutic targets in bone tumours and associated osteolysis. Biochim Biophys Acta 1704: 49–57., , , , , ,
- 402008 The bone marrow niche: Habitat to hematopoietic and mesenchymal stem cells, and unwitting host to molecular parasites. Leukemia 22: 941–950., , ,
- 412001 Cancellous bone remodeling occurs in specialized compartments lined by cells expressing osteoblastic markers. J Bone Miner Res 16: 1575–1582., , , ,
- 422003 Strain-derived canalicular fluid flow regulates osteoclast activity in a remodelling osteon-a proposal. J Biomech 36: 1453–1459., ,
- 431997 Muscle strength, bone mass, and age-related bone loss. J Bone Miner Res 12: 1547–1551.
- 441993 Remodeling and the repair of fatigue damage. Calcif Tissue Int 53 (Suppl 1): S75–S80.
- 452007 Are nonresorbing osteoclasts sources of bone anabolic activity ? J Bone Miner Res 22: 487–494., , , ,
- 462003 Mechanical loading: Biphasic osteocyte survival and targeting of osteoclasts for bone destruction in rat cortical bone. Am J Physiol Cell Physiol 284: C934–C943., , , , , , , ,
- 472002 Spatial distribution of Bax and Bcl-2 in osteocytes after bone fatigue: Complementary roles in bone remodeling regulation ? J Bone Miner Res 17: 907–914., , ,
- 482008 Functions of RANKL/RANK/OPG in bone modeling and remodeling. Arch Biochem Biophys 473: 139–146.,
- 492008 The cell biology of bone metabolism. J Clin Pathol 61: 577–587., , , ,
- 50 , , ,
- 512008 Wnt signaling: A win for bone. Arch Biochem Biophys 473: 112–116., ,
- 522008 Osteocyte-derived sclerostin inhibits bone formation: Its role in bone morphogenetic protein and Wnt signaling. J Bone Joint Surg Am 90 (Suppl 1): 31–35., , , ,
- 532003 Molecular mechanisms underlying osteoclast formation and activation. Exp Gerontol 38: 605–614.
- 541999 Annexin II increases osteoclast formation by stimulating the proliferation of osteoclast precursors in human marrow cultures. J Clin Invest 103: 1605–1613., , , , ,
- 551994 Cloning and identification of annexin II as an autocrine/paracrine factor that increases osteoclast formation and bone resorption. J Biol Chem 269: 28696–28701., , , , , ,
- 562001 ADAM8: A novel osteoclast stimulating factor. J Bone Miner Res 16: 814–822., ,
- 572006 Alpha9beta1: A novel osteoclast integrin that regulates osteoclast formation and function. J Bone Miner Res 21: 1657–1665., , , , , , , , , , ,