Coping with time scales in disease systems analysis: application to bone remodeling

Abstract

In this study we demonstrate the added value of mathematical model reduction for characterizing complex dynamic systems using bone remodeling as an example. We show that for the given parameter values, the mechanistic RANK-RANKL-OPG pathway model proposed by Lemaire et al. (J Theor Biol 229:293-309, 2004) can be reduced to a simpler model, which can describe the dynamics of the full Lemaire model to very good approximation. The response of both models to changes in the underlying physiology and therapeutic interventions was evaluated in four physiologically meaningful scenarios: (i) estrogen deficiency/estrogen replacement therapy, (ii) Vitamin D deficiency, (iii) ageing, and (iv) chronic glucocorticoid treatment and its cessation. It was found that on the time scale of disease progression and therapeutic intervention, the models showed negligible differences in their dynamic properties and were both suitable for characterizing the impact of estrogen deficiency and estrogen replacement therapy, Vitamin D deficiency, ageing, and chronic glucocorticoid treatment and its cessation on bone forming (osteoblasts) and bone resorbing (osteoclasts) cells. It was also demonstrated how the simpler model could help in elucidating qualitative properties of the observed dynamics, such as the absence of overshoot and rebound, and the different dynamics of onset and washout.

Fig. 1

Schematic illustration of the bone-cell interaction model. R _u uncommitted osteoblast progenitor, R responding osteoblast, B active osteoblast responsible for bone formation, C _p osteoclast progenitor, C active osteoclast responsible for bone resorption, PTH parathyroid hormone, TGF-β transforming growth factor-β, OPG osteoprotegerin, RANK receptor activator of NF-κB, RANKL receptor activator of NF-κB ligand. RANK-RANKL-OPG regulatory pathway: RANKL binds to RANK and promotes osteoclast differentiation, while OPG inhibits this differentiation by binding RANKL. Definitions and values of the rate constants are provided in Tables 1 and 2. This figure and its legend are taken from Ref. [13] and were slightly modified

Fig. 2

Effect of slowly decreasing endogenous estrogen production on bone turnover. Top panel Impact on responding osteoblasts (R, red), active osteoblasts (B, blue), and active osteoclasts (C, green); Bottom panel Impact on the active osteoclasts/osteoblast (C/B) ratio. An increase in the C/B ratio results in bone loss, whereas a decrease results in bone gain. The solid lines represent the simulated change in bone cells using the full model, whereas the dashed lines represent the respective changes using the mathematically reduced model

Fig. 3

Effect of estrogen replacement therapy on the dynamics of bone cells (I) prior to the start of treatment (disease progression due to estrogen deficiency), (II) during treatment, and (III) after treatment cessation. Top panel Impact on responding osteoblasts (R, red), active osteoblasts (B, blue), and active osteoclasts (C, green); Bottom panel Impact on the active osteoclasts/osteoblast (C/B) ratio. The solid lines represent the simulated change in bone cells using the full model, whereas the dashed lines represent the respective changes using the mathematically reduced model. Treatment starts after 1 year (t = 365 days) and is discontinued after 4 years (t = 1460 days) and is depicted by a black solid arrow

Fig. 4

Effect of seasonal changes in Vitamin D exposure on bone turnover. Top panel: Impact on responding osteoblasts (R, red), active osteoblasts (B, blue), and active osteoclasts (C, green); Middle panel Impact on responding osteoblasts (R, red), active osteoblasts (B, blue), and active osteoclasts (C, green) with focus on R and B; Bottom panel Impact on the active osteoclasts/osteoblast (C/B) ratio. The solid lines represent the simulated change in bone cells using the full model, whereas the dashed lines represent the respective changes using the mathematically reduced model. The simulation starts at the highest Vitamin D exposure in the summer and peaks during the winter

Fig. 5

Effect of ageing on bone turnover. Top panel Impact on responding osteoblasts (R, red), active osteoblasts (B, blue), and active osteoclasts (C, green); Bottom panel Impact on the active osteoclasts/osteoblast (C/B) ratio. The solid lines represent the simulated change in bone cells using the full model, whereas the dashed lines represent the respective changes using the mathematically reduced model

Fig. 6

Effect of chronic glucocorticoid treatment on bone turnover. Top panel Impact on responding osteoblasts (R, red), active osteoblasts (B, blue), and active osteoclasts (C, green); Bottom panel Impact on the active osteoclasts/osteoblast (C/B) ratio. The solid lines represent the simulated change in bone cells using the full model, whereas the dashed lines represent the respective changes using the mathematically reduced model

Fig. 7

Effect of glucocorticoid treatment on bone turnover before (I) and after (II) treatment cessation. Top panel Impact on responding osteoblasts (R, red), active osteoblasts (B, blue), and active osteoclasts (C, green); Bottom panel Impact on the active osteoclasts/osteoblast ratio (C/B) during treatment/washout. The solid lines represent the simulated change in bone cells using the full model, whereas the dashed lines represent the respective changes using the mathematically reduced model. Treatment with glucocorticoids is discontinued after 6 years (t = 2190 days) and is depicted by a black solid arrow

Fig. 8

Effect of rapidly changing estrogen concentrations on bone cell dynamics (I) at normal (non-deficient) estrogen levels, (II) following a step-decrease to a constant, deficient estrogen level, and (III) following a step-increase back to normal (non-deficient) estrogen levels. Solid red lines represent simulated changes in responding osteoblasts (x), solid blue lines those in active osteoblasts (y), and solid green lines those in active osteoclasts (z) based on the full model, whereas dashed lines represent the respective changes based on the mathematically reduced model. The duration of the step-change is depicted by a black solid arrow. Note that the time t can be computed as t = τ/k _b

Fig. 9

Orbit of the reduced system (34) (in red) in the (z,y)-plane. At normal estrogen levels, the system is at baseline (y,z) = (1,1), which is characterized by the intersection point of the solid blue line (Г_y) and the solid green line (Г_z (normal)). Once estrogen levels change, Г_z changes and the system starts moving towards a new steady-state (y _ss,z _ss). In case of a sudden drop in estrogen levels, realized here by a step-decrease in β, y _ss,z _ss is now determined by the intersection point of Г_y and the new Null Cline Г_z(decreased) (dashed green line). As a result, the system starts moving from (1,1) towards (y _ss,z _ss). Once estrogen concentrations return to their baseline levels, the system moves back to its original baseline (1,1)