Mathematical model for bone mineralization

Abstract

Defective bone mineralization has serious clinical manifestations, including deformities and fractures, but the regulation of this extracellular process is not fully understood. We have developed a mathematical model consisting of ordinary differential equations that describe collagen maturation, production and degradation of inhibitors, and mineral nucleation and growth. We examined the roles of individual processes in generating normal and abnormal mineralization patterns characterized using two outcome measures: mineralization lag time and degree of mineralization. Model parameters describing the formation of hydroxyapatite mineral on the nucleating centers most potently affected the degree of mineralization, while the parameters describing inhibitor homeostasis most effectively changed the mineralization lag time. Of interest, a parameter describing the rate of matrix maturation emerged as being capable of counter-intuitively increasing both the mineralization lag time and the degree of mineralization. We validated the accuracy of model predictions using known diseases of bone mineralization such as osteogenesis imperfecta and X-linked hypophosphatemia. The model successfully describes the highly nonlinear mineralization dynamics, which includes an initial lag phase when osteoid is present but no mineralization is evident, then fast primary mineralization, followed by secondary mineralization characterized by a continuous slow increase in bone mineral content. The developed model can potentially predict the function for a mutated protein based on the histology of pathologic bone samples from mineralization disorders of unknown etiology.

Keywords: X-linked hypophosphatemia; bone histomorphometry; matrix mineralization; mineralization inhibitors; nucleating centers; osteogenesis imperfecta; osteomalacia; rickets.

Figure 1

Schematic representation of bone mineralization described by the model. Thick lines represent the processes occurring during mineralization. Dotted lines represent the regulatory effects of different components on the mineralization process.

Figure 2

Changes in time in different players in the mineralization process in healthy bone. (A) The concentrations of naïve (x₁, light green) and mature (x₂, dark green) collagen matrix. (B) The concentration of the mineralization inhibitor (I, orange) and the nucleation centers (N, purple). (C) The concentration of mineral (y). Indicated are the mineralization lag time, measured as a time delay between time 0 and the onset of mineralization, and mineralization degree, measured as the amount of mineral at time = 100 days. For the simulation of healthy bone, the mineralization lag time is 10 days, and the mineralization degree is 1.

Figure 3

The effect of parameter affecting formation of hydroxyapatite crystals k₃ on the mineralization outcome. (A) The effect of decreasing k₃ 3-fold. (B) Comparison of the mineralization lag time and degree following decrease in k₃ to healthy mineralization. (C) The effect of increasing k₃ 3-fold. (D) Comparison of the mineralization lag time and degree following increase in k₃ to healthy mineralization. The same color scheme is used as in Figure 2.

Figure 4

The effect of parameters affecting nucleator production and removal on the mineralization outcome. (A–C) The effect of decreasing 3-fold (A) or increasing 3-fold (B) the number of nucleators per crosslinked collagen (k₂). (C) Comparison of the mineralization lag time and degree in conditions affecting k₂ to healthy mineralization. (D–F) The effect of decreasing 3-fold (D) or increasing 3-fold (E) the rate of use of nucleators by mineralized bone (r₂). (F) Comparison of the mineralization lag time and degree in conditions affecting r₂ to healthy mineralization. The same color scheme is used as in Figure 2.

Figure 5

The effect of parameters affecting inhibitor production and degradation on the mineralization outcome. (A–C) The effect of decreasing 10-fold (A) or increasing 10-fold (B) the rate of inhibitor production (v₁). (C) Comparison of the mineralization lag time and degree in conditions affecting v₁to healthy mineralization. (D–F) The effect of decreasing 3-fold (D) or increasing 3-fold (E) the rate of inhibitor degradation (r₁). (F) Comparison of the mineralization lag and degree in conditions affecting r₁ to healthy mineralization. The same color scheme is used as in Figure 2.

Figure 6

The effect of parameters affecting collagen maturation on the mineralization outcome. (A–C) The effect of decreasing 3-fold (A) or increasing 3-fold (B) the amount of naïve collagen deposited by osteoblasts at time = 0 (x₁(0)). (C) Comparison of the mineralization lag time and degree in conditions affecting x₁(0) to healthy mineralization. (D–F) The effect of decreasing 3-fold (D) or increasing 3-fold (E) the rate of collagen maturation (k₁). (F) Comparison of the mineralization lag and degree in conditions affecting k₁ to healthy mineralization. The same color scheme is used as in Figure 2.