A semi-mechanistic model of bone mineral density and bone turnover based on a circular model of bone remodeling

Abstract

Development of novel therapies for bone diseases can benefit from mathematical models that predict drug effect on bone remodeling biomarkers. Therefore, a bone cycle model (BCM) was developed that takes into consideration the concept of the basic multicellular unit and the dynamic equilibrium of bone remodeling. The model is a closed form cyclical model with four compartments representing resorption, formation, primary mineralization, and secondary mineralization. Equations describing the time course of bone turnover biomarkers were developed using the flow rate of bone cycle units (BCU) between the compartments or the amount of BCU in each compartment. A disease progression model representing bone loss in osteoporosis, a vitamin D and calcium supplementation (placebo) model, and a drug model for antiresorptive treatments were added to the model. Initial model parameter values were derived from published bone turnover data. The BCM accurately described biomarker-time profiles in postmenopausal women receiving either placebo or bisphosphonate treatment. The slow continual increase in bone mineral density (BMD) observed after 1 year of treatment was accurately described when changes in bone turnover were combined with increases in mineralization. For this purpose, the secondary mineralization compartment was replaced by three catenary chain compartments representing increasing mineral content. The refined BCM satisfactorily predicted biomarker profiles after long-term (10-year) bisphosphonate treatment. Furthermore, the model successfully described individual bone turnover markers and BMD results following treatment with denosumab in postmenopausal women. Analyses with this model could be used to optimize dosing regimens and to predict effects of novel osteoporotic treatments.

Appendices

Appendix 1: Basic bone cycle model

Model assumptions

• Lifespan can be converted to a first-order rate constant.

• Origination frequency is proportional to BCUs in the mineralization compartments.

• Bone from both fast and slow mineralization stages can be resorbed.

• Fast and slow mineralization contributes to the same extent to BMD.

• No mineralization lag-time.

• Slow mineralization has a duration of approximately 3 years.

• BMD at one specific site (e.g. lumbar spine) can be correlated to the overall changes of BTM in plasma such as CTX or osteocalcin.

• There is no seasonal fluctuation in bone remodeling.

Total amount of BCU is based on a distribution of the BCU over the various compartments

$$\sum {BCU = BCU_{RC}}\,+\,BCU_{CF} + BCU_{FM} + BCU_{SM}$$

(7)

Model equations of the various compartments

$$\frac{{dBCU_{RC} }}{dt} = f \cdot BCU_{FM} \cdot k_{FM} + BCU_{SM} \cdot k_{SM} - BCU_{RC} \cdot k_{RC}$$

(8)

$$\frac{{dBCU_{CF} }}{dt} = BCU_{RC} \cdot k_{RC} - BCU_{CF} \cdot k_{CF}$$

(9)

$$\frac{{dBCU_{FM} }}{dt} = BCU_{CF} \cdot k_{CF} - BCU_{FM} \cdot k_{FM}$$

(10)

$$\frac{{dBCU_{SM} }}{dt} = \left( {1 - f} \right) \cdot BCU_{FM} \cdot k_{FM} - BCU_{SM} \cdot k_{SM}$$

(11)

Baseline or steady state conditions

$$BCU_{RC,0} = \sum {BCU/\left( {1 + \frac{{k_{RC} }}{{k_{CF} }} + \frac{{k_{RC} }}{{k_{FM} }} + \frac{{k_{RC} \cdot (1 - f)}}{{k_{SM} }}} \right)}$$

(12)

$$BCU_{CF,0} = \sum {BCU/\left( {1 + \frac{{k_{CF} }}{{k_{RC} }} + \frac{{k_{CF} }}{{k_{FM} }} + \frac{{k_{CF} \cdot (1 - f)}}{{k_{SM} }}} \right)}$$

(13)

$$BCU_{FM,0} = \sum {BCU/\left( {1 + \frac{{k_{FM} }}{{k_{RC} }} + \frac{{k_{FM} }}{{k_{CF} }} + \frac{{k_{FM} \cdot (1 - f)}}{{k_{SM} }}} \right)}$$

(14)

$$BCU_{SM,0} = \sum {BCU/\left( {1 + \frac{{k_{SM} }}{{(1 - f) \cdot k_{RC} }} + \frac{{k_{SM} }}{{(1 - f) \cdot k_{CF} }} + \frac{{k_{SM} }}{{(1 - f) \cdot k_{FM} }}} \right)}$$

(15)

Lifespan parameters are transferred to first-order rate constants

k _SM  = 1/lifespan of BCU in slow mineralization process, k _RC  = 1/life span of BCU in resorption cavities, k _CF  = 1/life span of BCU in matrix formation and  k _FM  = 1/lifespan of BCU in fast mineralization phase.

The biomarker CTX is assumed to be determined by the amount of BCU in the mineralization compartments and the rate of transfer of the BCUs from the mineralization compartment to the resorption compartment.

$$CTX = \frac{{\left( {f \cdot BCU_{FM} \cdot k_{FM} + BCU_{SM} \cdot k_{SM} } \right) - \left( {f \cdot BCU_{FM,0} \cdot k_{FM} + BCU_{SM,0} \cdot k_{SM} } \right) \cdot 100}}{{f \cdot BCU_{FM,0} \cdot k_{FM} + BCU_{SM,0} \cdot k_{SM} }}$$

(16)

where BCU _FM and BCU _SM represent the amount of BCUs in the fast and slow mineralization compartments, and the parameters BCU _FM,0 and BCU _SM,0 represent the amounts of BCU at baseline. In this way, the CTX parameters are dimensionless and described as percentage change from baseline.

For the biomarker osteocalcin, the output rate of BCUs from the collagen formation compartment was used, as this biomarker is exclusively released by osteoblasts and is deposited into bone matrix. Since osteocalcin had an apparent residual concentration not related to bone turnover, an empirical correction factor of 1.44 was used.

$$OC = \frac{{\left( {BCU_{CF} \cdot k_{CF} - BCU_{CF,0} \cdot k_{CF} } \right) \cdot 100}}{{1.44 \cdot BCU_{CF,0} \cdot k_{CF} }}$$

(17)

For BMD, the amounts of BCUs in the fast and slow mineralization compartments were used.

$$BMD = \frac{{\left( {BCU_{FM} + BCU_{SM} } \right) - \left( {BCU_{FM,0} + BCU_{SM,0} } \right) \cdot 100}}{{BCU_{FM,0} + BCU_{SM,0} }}$$

(18)

Appendix 2: Final bone cycle model (Fig. 6)

Model assumptions

• Lifespan can be converted to a first-order rate constant.

• Origination frequency is proportional to BCUs in the mineralization compartment.

• Resorption rate constant (k_SM) is the same whatever the mineralization compartment.

• Disease progression is due to an increase in bone turnover only.

• Disease progresses with time and is independent of BMD.

• Magnitude of the placebo effect is constant over time and does not impact the underlying disease progression.

• Bisphosphonates inhibit the resorption pathways similarly (same constant whatever the mineralization compartment).

• Bisphosphonates do not impact the underlying disease progression and placebo effect.

• BMD at one specific site (e.g. lumbar spine) can be correlated to the overall change in biomarkers such as CTX or osteocalcin.

• Relationship between CTX, osteocalcin, and BMD under placebo is the same as under bisphosphonates.

Total amount of BCU is based on a distribution of the BCU over the various compartments

$$\sum {BCU = BCU_{RC}} + BCU_{CF} + BCU_{FM} + BCU_{SM1} + BCU_{SM2} + BCU_{SM3}$$

(19)

Model equations of the various compartments

$$\frac{{dBCU_{RC} }}{dt} = \left( {BCU_{FM} + BCU_{SM1} + BCU_{SM2} + BCU_{SM3} } \right) \cdot k_{SM} \cdot DP \cdot PL \cdot INH - BCU_{RC} \cdot k_{RC}$$

(20)

$$\frac{{dBCU_{CF} }}{dt} = BCU_{RC} \cdot k_{RC} - BCU_{CF} \cdot k_{CF}$$

(21)

$$\frac{{dBCU_{FM} }}{dt} = BCU_{CF} \cdot k_{CF} - BCU_{FM} \cdot k_{FM} - BCU_{FM} \cdot k_{SM} \cdot DP \cdot PL \cdot INH$$

(22)

$$\frac{{dBCU_{SM1} }}{dt} = BCU_{FM} \cdot k_{FM} - BCU_{SM1} \cdot k_{TR} - BCU_{SM1} \cdot k_{SM} \cdot DP \cdot PL \cdot INH$$

(23)

$$\frac{{dBCU_{SM2} }}{dt} = BCU_{SM1} \cdot k_{TR} - BCU_{SM2} \cdot k_{TR} - BCU_{SM2} \cdot k_{SM} \cdot DP \cdot PL \cdot INH$$

(24)

$$\frac{{dBCU_{SM3} }}{dt} = BCU_{SM2} \cdot k_{TR} - BCU_{SM3} \cdot k_{SM} \cdot DP \cdot PL \cdot INH$$

(25)

Baseline or steady state conditions

$$BCU_{RC,0} = \sum {BCU/\left( {1 + k_{RC} \cdot \left( {Sub1a + Sub1b + Sub1c} \right)} \right)}$$

(26)

$$Sub1a = \frac{1}{{k_{CF} }} + \frac{1}{{k_{FM} + k_{SM} }} + \frac{{k_{FM} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)}}$$

(26a)

$$Sub1b = \frac{{k_{FM} \cdot k_{TR} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(26b)

$$Sub1c = \frac{{k_{FM} \cdot \left( {k_{TR} } \right)^{2} }}{{k_{SM} \cdot \left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(26c)

$$BCU_{CF,0} = \sum {BCU/\left( {1 + k_{CF} \cdot \left( {Sub2a + Sub2b + Sub2c} \right)} \right)}$$

(27)

$$Sub2a = \frac{1}{{k_{RC} }} + \frac{1}{{k_{FM} + k_{SM} }} + \frac{{k_{FM} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)}}$$

(27a)

$$Sub2b = \frac{{k_{FM} \cdot k_{TR} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(27b)

$$Sub2c = \frac{{k_{FM} \cdot \left( {k_{TR} } \right)^{2} }}{{k_{SM} \cdot \left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(27c)

$$BCU_{FM,0} = \sum {BCU/\left( {1 + \left( {k_{FM} + k_{SM} } \right) \cdot \left( {Sub3a + Sub3b + Sub3c} \right)} \right)}$$

(28)

$$Sub3a = \frac{1}{{k_{RC} }} + \frac{1}{{k_{CF} }} + \frac{{k_{FM} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)}}$$

(28a)

$$Sub3b = \frac{{k_{FM} \cdot k_{TR} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(28b)

$$Sub3c = \frac{{k_{FM} \cdot \left( {k_{TR} } \right)^{2} }}{{k_{SM} \cdot \left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(28c)

$$BCU_{SM1,0} = \sum {BCU/\left[ {1 + (k_{TR} + k_{SM} ) \cdot \left[ {\frac{{k_{FM} + k_{SM} }}{{k_{FM} }}} \right] \cdot \,(Sub4a + Sub4b + Sub4c)} \right]}$$

(29)

$$Sub4a = \frac{1}{{k_{RC} }} + \frac{1}{{k_{CF} }} + \frac{1}{{\left( {k_{FM} + k_{SM} } \right)}}$$

(29a)

$$Sub4b = \frac{{k_{FM} \cdot k_{TR} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(29b)

$$Sub4c = \frac{{k_{FM} \cdot \left( {k_{TR} } \right)^{2} }}{{k_{SM} \cdot \left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(29c)

$$BCU_{SM2,0} = \sum {{{BCU} \mathord{\left/ {\vphantom {{BCU} {\left( {1 + \left( {k_{TR} + k_{SM} } \right) \cdot \left( {\frac{{k_{FM} + k_{SM} }}{{k_{FM} }}} \right) \cdot \left( {\frac{{k_{TR} + k_{SM} }}{{k_{TR} }}} \right) \cdot \left( {Sub5a + Sub5b + Sub5c} \right)} \right)}}} \right. \kern-0pt} {\left( {1 + \left( {k_{TR} + k_{SM} } \right) \cdot \, \left( {\frac{{k_{FM} + k_{SM} }}{{k_{FM} }}} \right) \cdot\, \left( {\frac{{k_{TR} + k_{SM} }}{{k_{TR} }}} \right) \cdot \left( {Sub5a + Sub5b + Sub5c} \right)} \right)}}}$$

(30)

$$Sub5a = \frac{1}{{k_{RC} }} + \frac{1}{{k_{CF} }} + \frac{1}{{\left( {k_{FM} + k_{SM} } \right)}}$$

(30a)

$$Sub5b = \frac{{k_{FM} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)}}$$

(30b)

$$Sub5c = \frac{{k_{FM} \cdot \left( {k_{TR} } \right)^{2} }}{{k_{SM} \cdot \left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(30c)

$$BCU_{SM3,0} = \sum {BCU/\left( {1 + k_{SM} \cdot \left( {\frac{{k_{FM} + k_{SM} }}{{k_{FM} }}} \right) \cdot \,\left( {\frac{{k_{TR} + k_{SM} }}{{k_{TR} }}} \right)^{2} \cdot \, \left( {Sub6a + Sub6b + Sub6c} \right)} \right)}$$

(31)

$$Sub6a = \frac{1}{{k_{RC} }} + \frac{1}{{k_{CF} }} + \frac{1}{{\left( {k_{FM} + k_{SM} } \right)}}$$

(31a)

$$Sub6b = \frac{{k_{FM} }}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)}}$$

(31b)

$$Sub6c = \frac{{k_{FM} \cdot k_{TR} \left( {k_{TR} } \right)}}{{\left( {k_{FM} + k_{SM} } \right) \cdot \left( {k_{TR} + k_{SM} } \right)^{2} }}$$

(31c)

Lifespan parameters are transferred to first-order rate constants

k _SM  = 1/lifespan of BCU in slow mineralization process, k _RC  = 1/lifespan of BCU in resorption cavities, k _CF  = 1/lifespan of BCU in matrix formation, k _FM  = 1/lifespan of BCU in fast mineralization phase and k _TR  = 1/lifespan of BCU in transition in slow mineralization.

Placebo effect

$$PL = 1 - (PLE \cdot (1 - \exp ( - 0.154 \cdot T)$$

(32)

Disease progression

$$DP = 1 + T \cdot DPE$$

(33)

Drug inhibitory function

$$INH = 1 - \frac{{C_{drug}^{'} \cdot I_{\hbox{max} } }}{{C_{drug}^{'} + EDK_{50} }}$$

(34)

The biomarker CTX is assumed to be determined by the amount of BCU in the mineralization compartments and the rate of transfer of the BCUs from the mineralization compartment to the resorption compartment.

$$CTX = \left( {BCU_{FM} + BCU_{SM1} + BCU_{SM2} + BCU_{SM3} } \right) \cdot k_{SM} \cdot DP \cdot INH$$

(35)

$$CTX_{0} = \left( {BCU_{FM,0} + BCU_{SM1,0} + BCU_{SM2,0} + BCU_{SM3,0} } \right) \cdot k_{SM}$$

(36)

CTX as change from baseline

$$CTX = \frac{{CTX_{i} - CTX_{0} }}{{CTX_{0} }} \cdot 100$$

(37)

where BCU _FM and BCU _SM represent the amount of BCUs in the fast and slow mineralization compartments, and the parameters BCU _FM,0 and BCU _SM,0 represents the amounts of BCU at baseline. In this way, the CTX parameters are dimensionless and described as percentage change from baseline.

For the biomarker osteocalcin, the output rate of BCUs from the collagen formation compartment was used, as this biomarker is exclusively released by osteoblasts and is deposited into bone matrix. Since osteocalcin had an apparent residual concentration not related to bone turnover, an empirical correction factor of 1.44 was used.

$$OC_{i} = BCU_{CF} \cdot k_{CF}$$

(38)

$$OC_{0} = BCU_{CF,0} \cdot k_{CF}$$

(39)

OC as change from baseline

$$OC = \frac{{OC_{i} - OC_{0} }}{{1.44 \cdot OC_{0} }} \cdot 100$$

(40)

For BMD, the amounts of BCUs in the fast and slow mineralization compartments were used.

$$BMD_{i} = BCU_{FM} + 1.1 \cdot BCU_{SM1} + 1.2 \cdot BCU_{SM2} + 1.3 \cdot BCU_{SM3}$$

(41)

$$BMD_{0} = BCU_{FM,0} + 1.1 \cdot BCU_{SM1,0} + 1.2 \cdot BCU_{SM2,0} + 1.3 \cdot BCU_{SM3,0}$$

(42)

BMD as change from baseline

$$BMD = \frac{{\left( {BMD_{i} } \right) - \left( {BMD_{0} } \right)}}{{BMD_{0} }} \cdot 100$$

(43)