
Ayati et al., (2010). A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease.
January 2012, model of the month by Vijayalakshmi Chelliah
Original model: BIOMD0000000401, BIOMD0000000402, BIOMD0000000403
Bone is a dynamic tissue that is constantly being remodeled in order to maintain a healthy skeleton, which serves to support and protect vital internal organs. Besides its structural function, it has an essential metabolic function serving as a reserve of calcium and phosphate needed for the maintenance of the serum homeostasis.
Bone remodeling requires the coordinated action of four major types of bone cells: bone lining cells, osteocytes, osteoclasts (boneresorping cells) and osteoblasts (boneforming cells), organized in bone multicellular units (BMU). The process involves the removal of old or damaged bone by osteoclasts followed by the formation of new bone matrix by osteoblasts that subsequently become mineralized. The remodeling cycle consists of three consecutive phases: resorption, during which osteoclasts digest old bone; reversal, when mononuclear cells appear on the bone surface; and formation, when osteoblasts lay down new bone until the resorped bone is completely replaced. The coupling between bone resorption and formation, mediated by osteoclasts and osteoblasts respectively, are tightly regulated. Dysregulation in the above cycle causes various metabolic bone diseases such as osteoporosis, renal osteodystrophy, osteopetrosis, Paget's disease and rickets (or osteomalacia) [1].

Figure 1: The effect of myeloma on the autocrine and paracrine signaling in the osteoclast and osteoblast cell populations in the presence of tumour is illustrated schematically. Figure taken from [2].

Figure 2: Simulation of normal bone remodeling (a) Single event initiated by a momentary increase in the osteoclast population. (b) Series of internally initiated regular cycles of oscillation resulted by increasing the osteoclast autocrine parameter g_{11}. Figure obtained by simulating BIOMD0000000401. Figures (a) and (b) corresponds to Figures 2 and 3, respectively of [2].

Mathematical modeling of bone remodeling has focused on various aspects, taking into account several key pathways that are involved in this process. Ayati et al. (2010) [2,BIOMD00000004013], have modelled the influence of tumour growth on bone remodeling. In particular, the influence of tumour growth on the autocrine and paracrine signaling of osteoclast and osteoblast cell population is well explored in this article. Patients with multiple myeloma have abnormal bone remodeling, i.e. the resorption and formation become uncoupled, with the end being an increase in bone resorption and a decrease in bone formation (causing weaker bones). Figure 1 shows schematically the effects of myeloma on the autocrine and paracrine signaling in the osteoclast and osteoblast cell populations in the presence of tumour.
Ayati et al. (2010), have used the already existing mathematical models of bone remodeling described by Komarova et al., 2003 [3, BIOMD0000000148] and Komarova 2005 [4, BIOMD0000000279], extended it further, to include the dysregulation caused due to multiple myeloma and the effect of drug treatment on bone remodeling. The paper presents the mathematical models of 1) the normal bone remodeling dynamics as described in [3], 2) the dysregulated bone remodeling dynamics caused due to the presence of tumour cells and 3) the dysregulated bone remodeling dynamics caused due to the presence of tumour cells and the effect of drug treatment on it.

1) Normal Bone Model [BIOMD0000000401]: The main parameters of the model are g_{ij} that describes the net effectiveness of autocrine and paracrine factors. The parameters g_{ij} denotes osteoclast autocrine signaling (g_{11}), osteoclast stimulation of osteoblast production (g_{12}), osteoblast inhibition of osteoclast production (g_{21}) and osteoblast autocrine signaling (g_{22}). The variables of the model are the density of osteoclasts C(t), the density of osteoblasts B(t) and bone mass z(t) at time t. Figure 2a shows the single event of normal bone remodeling initiated by a momentary increase in the number of osteoclasts. With the increase in the value of osteoclast autocrine parameter g_{11}, Figure 2b shows a series of internally initiated regular cycles over an extended period of time.
2) Dysregulated Bone Model with Tumour [BIOMD0000000402]: The regular cycles of the normal bone model are perturbed due to the presence of myeloma cells. This perturbation is illustrated in the model by the introduction of T(t), the density of tumour cells, L_{T} maximum tumour size and the tumour parameters r_{11}, r_{12}, r_{21}, r_{22}. Figure 3a shows the damped oscillation of osteoclast and osteoblast converging to nontrivial steady state. Figure 4a shows the depletion in bone mass converging to zero and the growth of tumour to the maximum capacity caused due to the imbalance between the osteoclasts and osteoblasts signaling.

Figure 3: (a) Tumour Model: The damped oscillation of osteoclast and osteoblast converging to nontrivial steady state, due to the presence of Tumour. Figure obtained by simulating BIOMD0000000402. (b) Tumour and Drug Treatment Model: The effect of drug treatment reverses the disruption of osteoclasts and osteoblasts interactions induced by tumour. Figure obtained by simulating BIOMD0000000403. Figures (a) and (b) correspond to Figures 4 and 9, respectively of [2].

Figure 4: (a) Tumour Model: The growth of tumour to the maximum capacity and the depletion in bone mass converging to zero, due to the imbalance between the osteoclasts and osteoblasts signaling. Figure obtained by simulating BIOMD0000000402. (b) The effect of drug treatment reduces the tumour size and recovers bone mass. Figure obtained by simulating BIOMD0000000403. Figures (a) and (b) correspond to Figures 5 and 10, respectively of [2].

3) Dysregulated Bone Model with Tumour and Drug Treatment [BIOMD0000000403]: Proteasome inhibitors are known to have direct antimyeloma effect. So, the effect of proteasome inhibition in myeloma bone disease, has been included in the tumour model. Time dependent treatment functions V1(t) and V2(t) are introduced into the tumour model, to promote osteoblast production and inhibit tumour growth. This serves as the tumour + drug treatment model. The effect of treatment that reverses the disruption of osteoclasts and osteoblasts interactions induced by tumour are shown in Figure 3b. Figure 4b shows the reduction in the tumour size and the recovery of the bone mass, as the effect of drug treatment.
This model reflects myeloma bone disease and illustrates how treatment procedures may be investigated using modeling approaches. Abnormalities that happen in bone remodeling cycle can develop numerous complications. Modeling approaches targeting key signaling pathways can possibly provide insights to better understand the causes of the dysregulation.

Bibliographic References

Feng X, McDonald JM. Disorders of bone remodeling. Annu Rev Pathol Feb 28;6:12145, 2011. [CiteXplore]

Ayati BP, Edwards CM, Webb GF, Wikswo JP. A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease. Biol Direct Apr 20;5:28,2010. [CiteXplore]

Komarova SV, Smith RJ, Dixon SJ, Sims SM, Wahl LM. Mathematical model predicts a critical role for osteoclast autocrine regulation in the control of bone remodeling. Bone. Aug; 33(2):20615, 2003. [CiteXplore]

Komarova SV. Mathematical model of paracrine interactions between osteoclasts and osteoblasts predicts anabolic action of parathyroid hormone on bone. Endocrinology. Aug;146(8):358995, 2005. [CiteXplore]
