A mathematical model of bone remodeling dynamics for normal bone cell populations and myeloma bone disease

Abstract

Background: Multiple myeloma is a hematologic malignancy associated with the development of a destructive osteolytic bone disease.

Results: Mathematical models are developed for normal bone remodeling and for the dysregulated bone remodeling that occurs in myeloma bone disease. The models examine the critical signaling between osteoclasts (bone resorption) and osteoblasts (bone formation). The interactions of osteoclasts and osteoblasts are modeled as a system of differential equations for these cell populations, which exhibit stable oscillations in the normal case and unstable oscillations in the myeloma case. In the case of untreated myeloma, osteoclasts increase and osteoblasts decrease, with net bone loss as the tumor grows. The therapeutic effects of targeting both myeloma cells and cells of the bone marrow microenvironment on these dynamics are examined.

Conclusions: The current model accurately reflects myeloma bone disease and illustrates how treatment approaches may be investigated using such computational approaches.

Reviewers: This article was reviewed by Ariosto Silva and Mark P. Little.

Figure 1

Schematic of the effects of myeloma on the autocrine and paracrine signaling in the osteoclast and osteoblast cell populations in the presence of tumor. The model of bone remodeling without tumor is taken from [22], including the meaning of the parameters g₁₁ (osteoclast autocrine signaling), g₁₂ (osteoclast stimulation of osteoblast production), g₂₁ (osteoblast inhibition of osteoclast production), and g₂₂ (osteoblast autocrine signaling). The tumor cells alter these interactions through modifications of the parameters g₁₁, g₁₂, g₂₁, g₂₂.

Figure 2

A single event of normal bone remodeling initiated by a momentary perturbation of the osteoclast nontrivial steady state population by an increase of 10 cell units. System of equations (1): The osteoclast population first decreases with a consequent increase in osteoblast population and a resorption of bone mass, followed by a return to steady-state levels. The blue line represents the steady-state solution. The parameters are α₁ = 3.0, α₂ = 4.0, β₁ = 0.2, β₂ = .02, g₁₁ = .5, g₂₂ = 0.0, g₁₂ = 1.0, and g₂₁ = -0.5. The nontrivial steady state with these parameters is = 1.06 and = 212.13. The initial conditions are C(0) = 11.06 and B(0) = 212.13. The bone mass parameters are k₁ = .24, k₂ = .0017, as in [22].

Figure 3

Simulation of oscillatory changes in osteoclast and osteoblast populations during normal bone modeling for which the model solutions are periodic. System of equations (1): The oscillations are stimulated by an initial increase in the number of osteoclasts by 10 cell units above the nontrivial steady state. The parameters are the same as in Fig. 2, except that g₁₁ = 1.1. The nontrivial steady state is = 1.16, = 231.72, the initial conditions are C(0) = 11.16, B(0) = 231.72, and the bone mass parameters are k₁ = .0748, k₂ = .0006395. The bone mass oscillates about a normalized value of 100.

Figure 4

The osteoclast and osteoblast populations in the presence of tumor stimulated by an initial osteoclast population elevated above the nontrivial steady state by 10 cell units. System of equations (7): The osteoclast population first increases as the osteoblast population decreases. The solutions converge to the nontrivial steady state = 5.0 and = 316.0 with damped oscillations. The parameters are α₁ = 3.0, α₂ = 4.0, β₁ = 0.2, β₂ = .02, g₁₁ = 1.1, g₂₂ = 0.0, g₁₂ = 1.0, g₂₁ = -0.5, γ_T = .005, L_T = 100, r₁₁ = .005, r₂₁ = 0.0, r₁₂ = 0.0, r₂₂ = 0.2. The initial conditions are C(0) = 15.0, B(0) = 316.0, T(0) = 1.

Figure 5

Bone mass and tumor response to the oscillations in Fig. 4. System of equations (7): The bone mass converges with oscillations to 0.0 and the tumor converges to maximum capacity L_T. The parameters are as in Fig. 4. The bone mass parameters are k₁ = .0748, k₂ = .0006395 as in Fig. 3.

Figure 6

The osteoclast and osteoblast populations in the presence of tumor exhibit unstable oscillations. System of equations (7): The parameters are as in Fig. 4 except r₁₁ = .02. The nontrivial steady state is = 5.46 and = 340.52. The initial conditions are C(0) = 8.46, B(0) = 340.52, T(0) = 1.

Figure 7

The effect of the unstable oscillations in Fig. 6 on bone mass. System of equations (7): The bone mass decreases with oscillations to 0.0. The tumor converges to maximum capacity L_T as in Fig. 5. The parameters are as in Fig. 4, except that r₁₁ = .02. The bone mass parameters are k₁ = .0748, k₂ = 0006395 as in Fig. 3.

Figure 8

The behavior of the solutions as a function of the tumor parameters r₁₁ and r₂₂. System of equations (7): The red surface is the plot of Φ = β₁(g₁₁(1 + r₁₁) 1) + β₂(g₂₂ - r₂₂ - 1) as a function of r₁₁ and r₂₂ (as in (11)). If the point on the surface corresponding to (r₁₁, r₂₂) is negative, then the solutions have decreasing amplitude oscillations converging to the nontrivial steady state; if positive, then the solutions have increasing amplitude and unstable oscillations. The values r₁₁ = .005 and r₂₂ = 0.2 in Fig. 4 and Fig. 5 correspond to -.00145 on the red surface, and the solutions converge slowly to the nontrivial steady state = 5.0, = 316.0. The values r₁₁ = .02 and r₂₂ = 0.2 in Fig. 6 and Fig. 7 correspond to .0002 on the red surface, and the solutions are unstable. The other parameters are r₁₂ = 0, r₂₁ = 0, α₁ = 3.0, α₂ = 4.0, β₁ = 0.2, β₂ = .02, g₁₁ = 1.1, g₂₂ = 0.0, g₁₂ = 1.0, g₂₁ = -0.5, γ_T = .005, L_T = 100.

Figure 9

Treatment of the tumor model in Fig. 4 starts at t_start = 600 with intensity values v₁ = .001 and v₂ = .008. System of equations (7): Treatment reverses the disruption of the osteoclasts' and osteoblasts' interaction induced by the tumor (compare to Fig. 4). The parameter values are as in Fig. 4 and the initial conditions are C(0) = 13.0 and B(0) = 300.0.

Figure 10

The tumor is extinguished and the bone mass begins to recover (compare to Fig. 5). System of equations (7): The parameters are as in Fig. 9 and the bone mass parameters are k₁ = .0748, k₂ = 0006395 as in Fig. 3.

Figure 11

The graph of the initial distribution C(0, x) for the bone model with an additional spatial dimension. System of equations (14): Osteoclast numbers are initially elevated above the normal nontrivial steady state (x) ≡ 1.16 at multiple sites. The initial distribution B(0, x) = (x) ≡ 231.72 is taken as the normal constant nontrivial steady state.

Figure 12

Graphs of the solutions C(t, x) and B(t, x) of the bone model with an additional spatial dimension. System of equations (14): We take C(0, x) as in Fig. 11 and B(0, x) = (x) ≡ 231.72. The solutions sustain regular spatial and temporal cycles characteristic of normal bone remodeling.

Figure 13

Left side: Graph of the bone mass z(t, x) for the spatially dependent normal bone model with C(0, x) and B(0, x) as in Fig. 11 and B(0, x) = (x) ≡ 231.72. Right side: Density plot of the bone mass. System of equations (14): The bone mass sustains regular spatial and temporal cycles uctuating about a normalized value of 100 dimensionless cell units.

Figure 14

Graphs of the solutions C(t, x) and B(t, x) of the spatially dependent bone model with tumor and with C(0, x) and B(0, x) as in Fig. 11. System of equations (15): The solutions lose regular spatial and temporal cycles and converge to the nontrivial steady states = 5.0, = 316.0 (compare to Fig. 4 and Fig. 12).

Figure 15

The graph (left side) and density plot (right side) of the bone mass z(t, x) for the spatially dependent bone model with tumor. System of equations (15): C(0, x) and B(0, x) are as in Fig. 11 (compare to Fig. 13).

Figure 16

Left side: Graph of the untreated tumor population T(t, x) for the spatially dependent bone model with tumor and with C(0, x) and B(0, x) as in Fig. 11. System of equations (15): The tumor is initially small and located on the right side of Ω = [0, 1]. The tumor density T(t, x) converges to capacity L_T for all x ∈ Ω as time increases. Right side: Graph of the treated tumor population T(t, x). The tumor is extinguished as time increases.

Figure 17

Graphs of the solutions C(t, x) and B(t, x) for the spatially dependent bone model with tumor and treatment. System of equations (16): The solutions recover regular spatial and temporal cycles after treatment begins at t_start = 600 (compare to Fig. 12 and Fig. 14).

Figure 18

The graph (left side) and density plot (right side) of the bone mass z(t, x) for the spatially dependent bone model with tumor and treatment (compare to Fig. 15). System of equations (16): Treatment stops the loss of bone mass from the advance of the tumor from the right side to the left side of the spatial region Ω = [0, 1] after initiation at t_start = 600. Bone mass already lost is not recovered on the right side of Ω as treatment continues.