Coping with time scales in disease systems analysis: application to bone remodeling
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.
KeywordsDisease progression modeling Mathematical model reduction Osteoporosis Bone cell interaction model RANK-RANKL-OPG
The objective of disease system analysis is to characterize and predict the status of biological systems under physiological and pathophysiological conditions as well as the impact of therapeutic interventions [1, 2, 3]. Models characterizing this dynamic behavior can be established at different levels of complexity, ranging from data driven and descriptive to completely mechanistic approaches (systems pharmacology) [3, 4]. Descriptive approaches usually start at a clinical observation level and become increasingly more complex in order to understand the system better, whereas systems pharmacology approaches start at the molecular level and provide a full description of the pathways involved. While descriptive models may not predict the clinical response beyond the data on which they were established, completely mechanistic approaches may face problems with the identifiability of model parameters [3, 4]. To obtain a sufficient understanding of a biological system, its dynamics, and the impact of therapeutic interventions, a compromise between the descriptive and the systems approach is frequently needed. This compromise results in mechanisms-based disease system models, which strive to characterize a system’s behavior rather than its complexity .
One important challenge to be met when developing mechanism-based disease system models is the appropriate handling of the different time scales present in biological systems. While processes on the molecular level, such as receptor binding or enzymatic reactions, are usually fast (within milliseconds), it can take months or even years before clinical signs and symptoms of chronic, progressive diseases become manifest. The design of mechanism-based disease system models consequently relies on a sufficient understanding of the relative speeds of the underlying (patho)physiological processes. Acquiring this information requires familiarity with various mathematical analysis techniques including dimensional analysis, dynamical systems analysis, and mathematical model reduction approaches (i.e., singular perturbation theory (see, e.g. ). When applying these techniques for the analysis of complex dynamic systems, the relative importance and speed of the different processes involved can be determined. Information on the system’s dynamic properties thus obtained can then be used to derive simpler models. Such reduced models yield dynamic properties that are very similar to those of completely mechanistic models but require the identification of fewer parameters. They also yield important insights into the impact of different parameters on the full system and often explicit expressions and quantitative estimates for drug-, system-, and/or disease-specific characteristics, such as clearance or the area under the curve of different compounds [6, 7].
The objective of this article is to demonstrate the added value of mathematical model reduction for establishing mechanism-based disease system models, using bone remodeling as an example. Bone remodeling is a physiological process that allows continuous renewal and repair of bone structure . It is accomplished by groups of osteoblasts (bone forming cells) and osteoclasts (bone resorbing cells), which closely collaborate in so-called basic multicellular units (BMU) [9, 10]. The interaction between osteoblasts and osteoclasts is highly regulated and provides the basis for a temporally and spatially coordinated bone remodeling process. Disturbances in the regulation of these cell–cell interactions can result in pathophysiological conditions, such as osteoporosis .
The RANK-RANKL-OPG signaling pathway is one of the key players involved in the osteoblast-osteoclast regulation . This regulatory pathway consists of three main components: (i) the receptor activator of nuclear factor κB (RANK), which is expressed on the surface of osteoclasts, (ii) the RANK ligand (RANKL), a polypeptide expressed on the surface of osteoblasts, and (iii) osteoprotegerin (OPG), a soluble decoy receptor for RANKL released by osteoblasts . To date, multiple conceptual bone cell interaction models have been established [10, 13, 14, 15, 16, 17, 18] some of which specifically incorporate the RANK-RANKL-OPG pathway [10, 13, 16]. Of these conceptual frameworks, Lemaire et al. were the first to propose a model, where the interaction between the different types of bone cells within a BMU (responding osteoblasts (R), active osteoblasts (B), and active osteoclasts (C)) is mediated by the RANK-RANKL-OPG regulatory pathway .
It will be shown how the mechanistic bone cell interaction model proposed by Lemaire et al.  may be mathematically reduced for the parameter values quoted in  and for physiologically and therapeutically relevant time scales. The dynamic properties of the full and the reduced model will then be compared using simulations, in which the response of both models to changes in physiological states and/or therapeutic interventions will be evaluated using physiologically meaningful scenarios. Estrogen (deficiency and replacement therapy) will be used as the primary example. In addition, the effects of Vitamin D, ageing, and chronic glucocorticoid treatment on the bone cell dynamics will be evaluated. The reduced model will then be used to obtain answers to questions about qualitative properties of response curves, such as the possibility of overshoot and rebound. Finally, we will conclude with a discussion of the advantages and limitations of mathematical model reduction as well as its implications for clinical situations.
Materials and methods
Parameter values provided by Lemaire et al. 
7 × 10−4
Differentiation rate of osteoblast progenitors
Differentiation rate of responding osteoblasts
Elimination rate of active osteoblasts
2.1 × 10−3
Differentiation rate of osteoclast precursors
Osteoclast apoptosis rate due to TGF-β
Positive constant characterizing the minimum TGF-β receptor occupancy
5 × 10−3
About half the value of C to get maximum TGF-β receptor occupancy
Rate of OPG-RANKL binding
Rate of OPG-RANKL dissociation
5.8 × 10−4
Rate of RANK-RANKL binding
1.7 × 10−2
Rate of RANK-RANKL dissociation
Rate of PTH binding to its receptor
Rate of PTH dissociation from its receptor
Fixed concentration of RANK
3 × 106
Maximum number of RANKL attached to the cell surface of each active osteoblast
Elimination rate of OPG
pM day−1/pM cells
2 × 105
Minimum OPG production rate per responding osteoblast
Rate of PTH administration
Rate of PTH synthesis
Rate of PTH elimination
Mathematical model reduction
For the other scenarios we obtain the same Reduced Model. For these scenarios the derivation is very similar and we shall not reproduce it here.
Thus, we have shown that for the parameter values used in Lemaire et al. , after a brief initial period we may put the right-hand side of the equation for dR/dt to zero and use the resulting equation to express R in terms of C, which allows one to reduce the original system involving the three dependent variables R, B, and C to one of two with the dependent variables B and C. We refer to the latter system as the Reduced System.
For the other three scenarios we arrive at the same Reduced System (19) and (20). However, different parameters may vary with time. Thus, in the Vitamin D scenario, both α and β vary with time, in the ageing scenario it is πc(C) that changes and in the glucocorticoid scenario DR changes with time.
The reduced system (19), is of a type recently discussed by Zumsande et al. . However, in their study they focused on the stability of steady states. As we shall see, this is no issue in our study because for the parameter values from Lemaire et al. the baseline is stable, and remains so when it slowly changes under the impact of disease progression and therapeutic interventions.
Reduction to a two-dimensional system opens the way for a transparent discussion of its dynamics. The state of a system at a given time t, is given by the pair (B(t), C(t)), which can be represented by a point in the (B, C)-plane (in terms of dimensionless variables this is the (y, z)-plane), often referred to as the Phase Plane. In Appendix B we describe how the state (B(t), C(t)) moves through the phase plane as time progresses and show what information about the system we can derive from it.
Evaluation of the model behavior
Value of β at time zero
Maximum inhibition of OPG production
Rate at which estrogen production declines
Rate at which estrogen production increases during estrogen replacement therapy
Maximum increase in β
Value of α at time zero
Maximum value of α at maximum deficiency (6 months)
Value of β at time zero
Value of β at maximum deficiency (6 months)
5 × 10−3
Value of Cs at time zero
Factor by which Cs increases
6 × 10−4
Rate at which Cs increases
7 × 10−4
Differentiation rate of osteoblast progenitors at time zero
1.7 × 10−4
Differentiation rate of osteoblast progenitors at time infinity
7.8 × 10−4
Rate of onset of glucocorticoid-induced side effects
7.8 × 10−3
Rate at which glucocorticoid-induced side effects wash out
Estrogen promotes its action, at least in part, through the RANK-RANKL-OPG pathway by stimulating the production of OPG . As estrogen production increases significantly during menarche, the RANKL/OPG ratio decreases resulting in a relative decrease in osteoclast activity leading to a substantial increase in longitudinal and radial bone growth as well as rapid skeletal mineralization . On the other hand, a decrease in estrogen production by 85–90% during menopause results in rapid bone loss and subsequently in an increased risk of bone fracture . The decrease in estrogen production during menopausal transition does not occur instantaneously but slowly evolves over a period of several years.
In this equation, ∆β represents the maximum increase in β due to estrogen replacement, kint the rate at which the corresponding OPG production increases, t1 the time at which treatment with exogenous estrogen starts, t2 the time at which treatment is discontinued (t1 = 1 year and t2 = 4 years), and H the Heaviside function.1
Vitamin D plays an important role in maintaining the body’s calcium and phosphate homeostasis and is consequently important for the formation as well as the maintenance of bone . While recent morphogenetic studies also suggest a direct effect on the osteoblastic phenotype expression , Vitamin D promotes its main effect on bone by regulating PTH levels and thus the RANKL/OPG ratio. At physiological levels, Vitamin D decreases the synthesis and secretion of PTH  as well as the number of PTH receptors [30, 31] resulting in a decrease in RANKL expression and an increase in OPG secretion. In case of Vitamin D deficiency, this inhibiting effect on PTH diminishes leading to an increased RANKL/OPG ratio and increased bone resorption.
Calcitriol, the bioactive form of Vitamin D, is formed from mainly two biologically inert precursors, cholecalciferol and ergocalciferol, via biotransformation in the liver and the kidneys . Cholecalciferol is formed in the skin when 7-dehydrocholesterol is exposed to ultraviolet B light (UVB, 290–320 nm), whereas ergocalciferol is produced by plants and taken up by diet. Assuming that the dietary intake of ergocalciferol does not significantly change during the course of 1 year, changes in Vitamin D levels, and thus changes in bone mineral density, are correlated with seasonal differences in sunlight exposure .
Ageing is associated with significant bone loss in both men and women . The extent of this loss can differ between the different bone sites and has been associated with a decrease in TGF-β production as well as its release from bone [34, 35, 36]. Once TGF-β levels decrease, their stimulating effect on osteoclast apoptosis decrease resulting in increased osteoclast activity and increased bone resorption. In addition, the differentiation of OPG-secreting responding osteoblasts to RANKL-expressing active osteoblasts is no longer inhibited. This gradual loss of regulatory feedback leads to an increased RANKL/OPG ratio and further stimulation of osteoclasts.
Bone loss and increased fracture risk due to long-term glucocorticoid therapy is the most common cause of drug-induced osteoporosis . The extent of this drug-induced side effect seems to be dependent on the cumulative glucocorticoid dose and affects trabecular bone more than cortical bone [37, 38]. Although glucocorticoid receptors are present in almost every vertebrate cell, glucocorticoids seem to primarily affect bone formation by decreasing the expression of osteoblastic differentiation factors, such as core binding factor A1 [39, 40, 41].
Here, T represents the time at which treatment with glucocorticoids was discontinued and kwash represents the first-order rate constant characterizing the offset of the glucocorticoid effect.
Simulations were performed in MatLab version R2011a. In light of the stiffness of the system, the ode-solver “ode23s” was used.
Application of dimensional analysis to the conceptual bone cell interaction model by Lemaire et al.  allowed us to evaluate (i) the relative importance of its model terms, (ii) the relative speeds of the processes involved, and (iii) the critical dimensionless numbers (often combinations of parameters), which determine the qualitative character of the dynamics of the system. In particular, we were able to show mathematically that responding osteoblasts (R) rapidly reach a quasi-steady state with active osteoclasts (C) for the model parameters provided in . Thus, for R and C the quasi-steady state assumption was shown to hold and the original three-dimensional system containing R, B, and C could be reduced to a simpler, two-dimensional system, whose dynamics is determined by B and C. Reduction to a two-dimensional system further allowed for a graphical representation of its dynamics in the planar State Space. While the state of the system can be depicted as a point in the state space, its evolution is characterized by a respective curve parameterized by the time t (the orbit, cf. Fig. 9 in Appendix B). Representation in the state space also enables a transparent discussion of the system’s dynamics and readily reveals qualitative properties, such as the absence of overshoot and rebound.
When evaluating the performance of both the full Lemaire model and the reduced model following rapid interventions, such as a sudden decrease or increase in estrogen levels (Appendix B), our findings show that the dynamic properties of both models are very similar but not identical (Fig. 8 in Appendix B). Small discrepancies between the dynamic properties of the two models exist during the first 10–20 days after the rapid intervention. Once the speed of the onset and/or offset of these interventions decreases to more (patho)physiological/therapeutic levels, the profiles of both models become more and more similar. On the time scale of disease progression and therapeutic intervention both the full Lemaire model and the mathematically reduced model show negligible differences in their ability to characterizing the dynamic interaction between osteoclasts and osteoblasts.
The effects of both local and systemic control mechanisms on the regulation of bone remodeling result in the establishment of a complex framework that contains multiple spatial and temporal levels. To obtain a sufficient understanding of this framework, its dynamics, and the impact of therapeutic interventions and disease processes, the use of mathematical models is required (for a more elaborate conceptual discussion of the role of mathematical modeling for characterizing bone turnover see also ). Mathematical modeling provides a powerful tool as it allows incorporation of information from different in vitro and in vivo experiments into a single approach. Once developed and validated, these models can be used in silico to explore the cause-effect relationship and to assist the formulation of new hypotheses as well as the design of new experimental studies. However, as these frameworks become more complex, problems with identifying the key mechanisms that cause a system to undergo pathophysiological changes may arise [3, 4]. To identify these key components, sufficient understanding of a system’s dynamic properties is often more informative than characterizing its complexity. One way of exploring a system’s dynamic properties is to mathematically reduce completely mechanistic models in order to evaluate (1) the relative importance of the various model components and (2) the relative speed of the processes involved for the overall performance of the system.
To demonstrate the benefits and limitations of model reduction, we analyzed the well-known bone-cell interaction model proposed by Lemaire et al. , which is based on the RANK-RANKL-OPG signaling pathway. By performing a dimensional analysis, we identified critical properties, such as overall and relative time scales, on the basis of the parameter values quoted in . We found that for these parameter values the dynamics of the responding osteoblasts was relatively fast compared to that of active osteoblasts and osteoclasts. The dynamics of the system were thus primarily driven by changes in osteoclasts and active osteoblasts, whilst responding osteoblasts follow their lead. Although not all of the parameter values provided in  seem to have been previously validated, corresponding model-predicted bone cell dynamics are in agreement with clinical observations , where rapid changes in bone resorption markers during/after treatment with conjugated estrogen and/or alendronate are followed by respective changes in bone formation markers.
Based on these findings, the conceptual bone cell interaction model by Lemaire et al.  could be reduced from a three- to a two-dimensional system. Reducing the model’s complexity allowed for a transparent discussion of its dynamics and also opened the way for a geometric, two-dimensional analysis. This approach added significantly to the transparency of the system as it allowed its representation in the Phase Plane. Results of this geometric analysis indicate that there can be no overshoot at onset and no rebound at washout for the reduced model. Given the proximity of the concentration curves of the reduced and full model this implies that any overshoot or rebound the full model might exhibit will be very small (cf. Appendix B).
When simulating the response of both the full Lemaire model and the mathematically reduced model to rapid changes, such as a sudden onset/offset of effect, we showed that there is overall a good match between the two models (cf. Fig. 8). Small discrepancies in their dynamic properties were only observed during the first 10–20 days after the onset/offset of the effect. However, once the relative speeds of the underlying (patho)physiological processes and therapeutic interventions were taken into account, both models were at any time at quasi-equilibrium (Figs. 2, 3, 4, 5, 6, 7). Consequently, both models can be used interchangeably for characterizing bone cell dynamics on the time scale of disease progression and therapeutic intervention. From a data analysis point of view, the use of the simpler model is advantageous as fewer parameters have to be identified and estimated. This aspect becomes particularly important for the analysis of clinical data, where usually only few samples per subject are available. On the other hand, the development and validation of disease system models heavily depends on current knowledge about the biological system, the availability of sufficient data on different spatial/temporal levels, and the availability of appropriate software tools that allow running and visualizing these models based on widely accepted modeling standards [4, 45].
The application of advanced mathematical and statistical tools, such as mathematical model reduction, can guide the development of disease system models as it allows one to identify the rate-limiting steps within complex, dynamic systems. The joint use of systems pharmacology and mathematical model reduction approaches provides, therefore, a powerful combination as it can guide the identification of drug-, system-, and disease-specific parameters, informative biomarkers as well as the generation of data, where such information can be obtained from. In particular, knowledge on the system’s dynamics and the time scales involved in the establishment of disease and drug effects can guide clinical trial design as it allows to identifying its maximal susceptibility to changes in the underlying physiology and/or therapeutic interventions. For example, the response of the reduced model to a step-decrease in estrogen suggests that in this case a washout design would be superior to a delayed start design for characterizing the impact of this physiological change on bone remodeling. This is due to the fact that in this case equilibrium is reached much faster after washout of the intervention than following its onset (cf. Appendix B; Figs. 8, 9). These findings are in agreement with those of Ploeger and Holford, who found a washout design superior to a delayed start design for characterizing and distinguishing treatment effect types in Parkinson’s disease .
In conclusion, mathematical model reduction is a valuable approach for analyzing disease systems and simplifying complex models while maintaining their dynamic properties. A significant decrease in the number of parameters to be identified and estimated in addition to an increased system transparency qualifies reduced models as tools to evaluate the impact of changes in physiological states and/or therapeutic interventions with respect to the different time scales involved.
The Heaviside function H(s) is defined as follows: H(s) = 0 for s < 0 and H(s) = 1 for s > 0. Thus, for any time T > 0, H(t−T) = 0 for t < T and H(t−T) = 1 for t > T.
The authors would like to thank Drs. Oscar E. Della-Pasqua, Rik de Greef and Thomas Kerbusch for their valuable comments and suggestions. This study was performed within the framework of Dutch Top Institute Pharma, “Mechanism-based PK/PD modeling platform (project number D2-104)”.
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
- 4.Schmidt S, Post TM, Boroujerdi MA, van Kesteren C, Ploeger BA, Della Pasqua OE, Danhof M (2010) Disease progression analysis: towards mechanism-based models. In: Kimko HC, Peck CC (eds) Clinical trial simulations, 1st edn. Springer, New York, pp 437–459Google Scholar
- 5.Post TM (2009) Disease system analysis: between complexity and (over)simplification. Dissertation, Leiden UniversityGoogle Scholar
- 19.Bender CM, Orzag SA (1978) Advanced mathematical methods for scientists and engineers: asymptotic methods and perturbation theory. Mc-Graw Hill, Inc, New YorkGoogle Scholar
- 23.Soules MR, Sherman S, Parrott E, Rebar R, Santoro N, Utian W, Woods N (2001) Executive summary: stages of reproductive aging workshop (STRAW) July, 2001. Menopause 8:402–7, Park City, UtahGoogle Scholar
- 24.Anderson GL, Limacher M, Assaf AR, Bassford T, Beresford SA, Black H, Bonds D, Brunner R, Brzyski R, Caan B, Chlebowski R, Curb D, Gass M, Hays J, Heiss G, Hendrix S, Howard BV, Hsia J, Hubbell A, Jackson R, Johnson KC, Judd H, Kotchen JM, Kuller L, LaCroix AZ, Lane D, Langer RD, Lasser N, Lewis CE, Manson J, Margolis K, Ockene J, O’Sullivan MJ, Phillips L, Prentice RL, Ritenbaugh C, Robbins J, Rossouw JE, Sarto G, Stefanick ML, Van Horn L, Wactawski-Wende J, Wallace R, Wassertheil-Smoller S (2004) Effects of conjugated equine estrogen in postmenopausal women with hysterectomy: the Women’s Health Initiative randomized controlled trial. JAMA 291:1701–1712PubMedCrossRefGoogle Scholar
- 26.Rossouw JE, Anderson GL, Prentice RL, LaCroix AZ, Kooperberg C, Stefanick ML, Jackson RD, Beresford SA, Howard BV, Johnson KC, Kotchen JM, Ockene J (2002) Risks and benefits of estrogen plus progestin in healthy postmenopausal women: principal results from the women’s health initiative randomized controlled trial. JAMA 288:321–333PubMedCrossRefGoogle Scholar
- 30.Titus L, Jackson E, Nanes MS, Rubin JE, Catherwood BD (1991) 1,25-dihydroxyvitamin D reduces parathyroid hormone receptor number in ROS 17/2.8 cells and prevents the glucocorticoid-induced increase in these receptors: relationship to adenylate cyclase activation. J Bone Miner Res 6:631–637PubMedCrossRefGoogle Scholar
- 32.Holford NHG, Baathe S and Karlsson M (2001) Auckland bones and summer sun. In: Annual meeting of the population approach group in Europe, BaselGoogle Scholar
- 34.Kahn A, Gibbons R, Perkins S and Gazit D (1995) Age-related bone loss. A hypothesis and initial assessment in mice. Clin Orthop Relat Res 313: 69–75Google Scholar
- 36.Nicolas V, Prewett A, Bettica P, Mohan S, Finkelman RD, Baylink DJ, Farley JR (1994) Age-related decreases in insulin-like growth factor-I and transforming growth factor-beta in femoral cortical bone from both men and women: implications for bone loss with aging. J Clin Endocrinol Metab 78:1011–1016PubMedCrossRefGoogle Scholar
- 41.Komori T, Yagi H, Nomura S, Yamaguchi A, Sasaki K, Deguchi K, Shimizu Y, Bronson RT, Gao YH, Inada M, Sato M, Okamoto R, Kitamura Y, Yoshiki S, Kishimoto T (1997) Targeted disruption of Cbfa1 results in a complete lack of bone formation owing to maturational arrest of osteoblasts. Cell 89:755–764PubMedCrossRefGoogle Scholar
- 44.Greenspan SL, Emkey RD, Bone HG, Weiss SR, Bell NH, Downs RW, McKeever C, Miller SS, Davidson M, Bolognese MA, Mulloy AL, Heyden N, Wu M, Kaur A, Lombardi A (2002) Significant differential effects of alendronate, estrogen, or combination therapy on the rate of bone loss after discontinuation of treatment of postmenopausal osteoporosis. A randomized, double-blind, placebo-controlled trial. Ann Intern Med 137:875–883PubMedGoogle Scholar
- 47.Blanchard P, Devaney RJ, Hall GR (1998) Differential equations. Brooks/Cole, BelmontGoogle Scholar