Physiologically based mathematical model of ... - Springer Link

Biomech Model Mechanobiol (2012) 11:83–93 DOI 10.1007/s10237-011-0294-2

ORIGINAL PAPER

Physiologically based mathematical model of transduction of mechanobiological signals by osteocytes Ridha Hambli · Romain Rieger

Received: 21 December 2010 / Accepted: 2 February 2011 / Published online: 20 February 2011 © Springer-Verlag 2011

Abstract Developing mathematical models describing the bone transduction mechanisms, including mechanical and metabolic regulations, has a clear practical applications in bone tissue engineering. The current study attempts to develop a plausible physiologically based mathematical model to describe the mechanotransduction in bone by an osteocyte mediated by the calcium-parathyroid hormone regulation and incorporating the nitric oxide (NO) and prostaglandin E2 (PGE2) effects in early responses to mechanical stimulation. The inputs are mechanical stress and calcium concentration, and the output is a stimulus function corresponding to the stimulatory signal to osteoblasts. The focus will be on the development of the mechanotransduction model rather than investigating the bone remodeling process that is beyond the scope of this study. The different components of the model were based on both experimental and theoretical previously published results describing some observed physiological events in bone mechanotransduction. Current model is a dynamical system expressing the mechanotransduction response of a given osteocyte with zero explicit space dimensions, but with a dependent variable that records signal amplitude as a function of mechanical stress, some metabolic factors release, and time. We then investigated the model response in term of stimulus signal variation versus the model inputs. Despite the limitations of the model, predicted and experimental results from literature have the same trends. Keywords Bone mechanotransduction · Osteocytes · Shear stress · Ca-PTH · NO-PGE2 interactions · Mechanobiology R. Hambli (B) · R. Rieger PRISME Institute, University of Orleans, 8 rue Léonard deVinci, 45072 Orléans cedex 2, France e-mail: [email protected]; [email protected]

1 Introduction Bone remodeling process occurs as a response of bone multicellular units (BMUs) activities that are directly triggered by the mechanotransduction phase within bone. Mechanical loading in the form of shear stress is clearly involved in mechanosensation and transduction by osteocytes (Bonewald 2006; Bonewald and Johnson 2008). In addition, there may be no single mechanoreceptor in osteocytes, but instead a combination of different mechanical and biological events that has to be triggered for mechanosensation of signal to occur (Bonewald 2006; Bonewald and Johnson 2008). Several published studies were developed by different authors to describe mathematically some cellular aspects of bone remodeling process based on description of BMUs activities without considering the mechanotransduction phase (Rattanakul et al. 2003; Lemaire et al. 2004; Martin and Buckland-Wright 2004; Komarova 2005; Moroz et al. 2006; Maldonado et al. 2006; Wimpenny and Moroz 2007; Pivonka et al. 2008; Ryser et al. 2009). Moreover, bone adaptation models based on considerations of the physiological and biological processes of bone were proposed by (Hart et al. 1984; Heaney 1994; Martin 1995). Additionally, a number of publications indicate the importance of the level of osteocyte regulation (Burr and Martin 1993; Martin 2000; You et al. 2004; Bonewald 2006; Bonewald and Johnson 2008), the role of the osteocyte apoptosis as a part of the mechanotransduction control mechanism (Noble et al. 2003; Bonewald 2006), and role of stress (Nakamura et al. 2003). The full potential of BMUs models describing the remodeling process can only be fully realized when the bone mechanotransduction process is considered. Normal loading strains applied to bone generate mechanical stretch and pressure gradients in bone canaliculi that drive

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extracellular fluid flow, resulting in stress on the membranes of osteocytes (Cowin 1995). The results of in vitro experiments suggest that fluid shear is a major factor affecting bone cell metabolism (Fox et al. 1996; Chow et al. 1998; Bakker et al. 2003; McGarry et al. 2005). Although it is clear that both biological and mechanical factors influence bone adaptation (Vedi and Compston 1996; Jódar Gimeno et al. 1997), a review of literature shows that most mathematical proposed stimuli functions are phenomenological and are not developed based on physiological considerations (Prendergast and Taylor 1994; Huiskes et al. 2000; Hernandez et al. 2001; Hambli et al. 2009, 2010). These models assumed that biologic factors remain constant so that only the mechanical aspects of the model were subject to change. In addition, events triggering bone remodeling are not only based on mechanical stimulus. Several investigations showed that factors like Ca, PTH, NO, and PGE2 play a major roles in the bone cells transduction (Marotti et al. 1990 Chow et al. (1993, 1998); Chow and Chambers 1994; Fox et al. 1996; Bakker et al. 2003; Bonewald 2006; Bonewald and Johnson 2008). Other signaling factors have been implicated in the sensing cascades (Bonewald 2006; Bonewald and Johnson 2008). Nevertheless, local messenger pathways (Wnt/β-catenin, SOST, PPAR, DKK, OPN, …) in bones are much less understood, and recent studies have indicated that load transduction involves complex interactions between several messenger pathways rather than one specific mechanism (Bonewald 2006). A review of literature shows that despite the progress in transduction comprehension, there is still a lack of mathematical models combining the mechanotransduction function and the biological responses occurring at a given osteocyte. Therefore, the current study attempts to develop a plausible physiologically based mathematical model to describe the mechanotransduction in bone by an osteocyte mediated by the calcium-parathyroid hormone regulation and incorporating the nitric oxide (NO) and prostaglandin E2 (PGE2) effects in early responses to mechanical stimulation. The inputs are mechanical stress and calcium concentration, and the output is a stimulus function corresponding to the stimulatory signal to osteoblasts. The focus here will be on the development of the mechanotransduction model rather than investigating the bone remodeling process that is beyond the scope of this study. We then investigated the model response in term of stimulus signal variation versus the model inputs. Despite the limitations of the model, predicted and experimental results from literature have the same trends. Current model is a dynamical system with zero explicit space dimensions, but with a dependent variable that records signal amplitude as a function of mechanical stress, Ca-PTH regulation, NO, PGE2 release, and time.

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R. Hambli, R. Rieger

2 Materials and methods The model proposed in this work formulates the mechanotransduction model in terms of local responses of an osteocyte subjected to a fluid shear stress induced by mechanical stress at bone matrix scale. The general framework for model development is based on four steps (Fig. 1; Table 1): (i) Applied mechanical stress as model input: Events (1 and 2). (ii) Reception of the external signals by an osteocyte: Events (3 and 4). (iii) Integration (combination) and amplification of the signals: Events (5–10). (iv) Stimulatory mechanotransduction signal sent to osteoblasts which potentially can trigger bone remodeling (not modeled here): Event (11). Therefore, the model can be represented by a diagram composed of serial and parallel events connected by lines that show the relationships between the events (Fig. 2). Each event represents a specific observed mechanism in relation to osteocyte transduction. Such flow chart diagram (FCD) approach provides information about the timing of processes and about whether processes will operate in sequence or in parallel. Changing the amplitude value of an input should automatically force dynamical adaptation of the diagram output (transduction signals) by recalculation of the internal values of the whole model variables.

2.1 The mathematical model development The mathematical description of many molecular messengers is very difficult to investigate due to experimental difficulties in the measurement of the activities of molecular messengers, which are subjected to considerable binding constants to receptors (Moroz et al. 2006). These difficulties forces researchers to develop cellular-level models with limited number of molecular messengers considering the earlier responses based on experimental observations or assumptions. Therefore, because NO and PGE2 production appear to be essential in early responses to mechanical stimulation in vivo, we used their production to develop a mathematical model of the responsiveness of an osteocyte to applied stress to bone tissue mediated by Ca-PTH regulation. Development of the present model is based upon available literature reported experimental and theoretical results. The four following phases in relation to Fig. 1 and Table 1 summarize the mathematical development of the model. In Table 1 are

Physiologically based mathematical model

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Fig. 1 Mechanotransduction model within an osteocyte and cascades of events based on Bonewald and Johnson (2008): (1) Mechanical load applied to bone generates interstitial fluid flow in the canaliculus (2) Interstitial fluid flow generates fluid velocity (3) Fluid velocity generates shear stress (4) Ca-PTH homeostasis mediate mechanical response in the form of mediating of messenger pathways implicated in load transduction (5) and (6) Perception of shear stress triggers a number of intracellular responses including the release of NO and PGE2, through a poorly understood mechanism into the lacunar–canalicular fluid where it can act in an autocrine and/or paracrine fashion. Connexin-43 hemichannels (Cx43HC) in the PGE2 and integrin proteins appear to be involved (7) Binding of NO and PGE2 to their receptors generates a

cascades of events leading to the nuclear translocation of β-catenin which leads to changes in the expression of a number of key target genes (8) One of the apparent consequences is the increase in Wnt and the reduction in SOST, Dkk, OPN, and PPAR (9). The net result of these changes is to create a permissive environment for amplification of the load signal (10). Amplification of the load signal mediated by Ca-PTH regulation in the form of stimulatory signal to osteoblasts (). Solid black lines represent early osteocytes responses to shear stress. Dashed lines represent secondary messenger pathways (not modeled in the current study). Solid red bolt lines represent the amplified mechanotransduction signals mediated with Ca-PTH regulation

reported the different model components, their descriptions, their mathematical expressions, their parameters, and their sources.

Phase 2 (Reception):

Phase 1 (Input signals): Applied stress to bone matrix (Fig. 1: event#1 (P)) generates interstitial fluid flow in the lacuno-canalicular network (Fig. 1: event#2 (VF )). According to You et al. (2001), the idealized canaliculus model can be considered as an individual circular hollow cylinder with its central cell process containing a centrally positioned osteocyte process and its surrounding fluid annulus. Therefore, standard Darcy law can be used to describe the fluid velocity (Lemaire et al. 2006) (Table 2): VF = −K p dP dx

dP dx

(1)

denotes the pressure gradient in the canaliculus, and x denotes the axial coordinate of the canaliculus. K P is the coefficient of permeability.

You et al. (2001) suggested that fluid flow within the canaliculus is amplified in the actin cytoskeleton of osteocytes due to fluid drag on pericellular matrix. Therefore, we consider in the present work that fluid movement in the canaliculus resulting from mechanical loading induces shear stress (Fig. 1: event#3 (τ )) on the osteocyte membrane. Hence, the shear stress (τ ) can be expressed as a function of the fluid velocity in the form: τ = μVFθ

(2)

μ is a model parameter which can be considered as the amplification factor (You et al. 2001), and θ is a model parameter expressing a nonlinear relation between the fluid flow velocity and the shear stress (You et al. 2001). It has been observed that shear stress (τ ) effect on the osteocyte sensitivity responsiveness is mediated by Ca-PTH regulation (Fig. 1: event#4) (Chow et al. 1993, 1998). Enhanced mathematical model was developed recently by Peterson

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Table 1 Cascades of events describing the mechanotransduction model Events

Bone matrix deformation: Fluid flow Shear stress Ca-PTH

Order

1

2

3 4

• NO secretion

5

• PGE2 secretion

6

Description

Mathematical model

Model parameters

Source

Mechanical load applied to bone generates Instertitial fluid flow in the canaliculus

Input: pressure gradient applied to bone matrix

Instertitial fluid flow generates fluid velocity.

Eq. (1)

KP, γ

Lemaire et al. (2006)

Fluid velocity generates shear stress

Eq. (2)

μ, θ

Cowin (1995)

Regulation of Ca-PTH homeostasis

Eqs. (3)–(4)

α1 , α2 , α3 , α4 , Ca0

Perception of shear stress triggers a number of intracellular responses including the release of NO and PGE2

Eqs. (5)–(7)

K1, K2, K3,

–

Mayer and Hurst (1978), Aguilera-Tejero et al. (1996) Maldonado et al. (2006)

K 4 , K 5 , α5 , α6 , X NO , X PGE2

Secondary messenger pathways Signals integration

Signals

7

10

11

to

9

β-cat, Sost, Wnt, Dkk, …

Not modeled here due to lack of information

Not modeled

Not modeled

Integration of the load signal mediated by Ca-PTH regulation in the form of signals

Eqs. (8)–(11)

M , xM , xM τ M , xPTH NO PGE2

Proposition

Mechanotransduction signal:

Eq. (12)

W M , WPTH , WNO , WPGE2

Proposition

The description of the events, their mathematical models, parameters, and sources are reported. Note that the model is not aimed to detail all the observed events. The focus was on the description of the early events (Ca-PTH, NO, PGE2)

and Riggs (2010) to describe the integrated components governing bone and Ca homeostasis. Nevertheless, mechanotransduction phase was not taken into account by the authors. Present model considers the Ca concentration as an input varying in time resulting from a previously homeostasis loop. Therefore, the Ca concentration (xCa) is represented here as an input data during the reception phase in the form: xCa = Ca0 (t)

(3)

where Ca0 (t) denotes the level of Ca serum of the blood which can vary with time. The extracellular ionized Ca concentration is the primary determinant of PTH secretion by the parathyroid gland with an experimental inverse sigmoidal relationship between Ca concentrations and PTH (xPTH ) secretion (Mayer and Hurst 1978; Aguilera-Tejero et al. 1996) expressed by:

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Signals

Integration

Reception

Target

Amplification

Mechanical

Fluid flow o o

Biological

Production of NO & PGE2 PTH regulation

Signal to BMUs

Ca

Fig. 2 Flow chart diagram describing the transduction model composed of a given number of parallel and serial events during transduction. Inputs are mechanical and some biological factors. A given osteocyte receives and integrates the input signals and reacts by generating a coupled mechanobiological signal

xPTH = α1 +

α2 1+e

xCa −α3 α4

(4)

α1 , α2 , α3 , and α4 are the parameters of the inverse sigmoidal relationship. In the present work, PTH production by is assumed to be continuous and does not include pulsatile secretion.


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Table 2 Model’s parameters Symbol

Parameter value

Description

KP

7.5e-14 mm2 Pa−1 s−1

Coefficient of permeability

Lemaire et al. (2006)

μ

10 MPa s mm−1

Shear stress coefficient

Based on You et al. (2001)

θ

0.4

Shear stress exponent

α1

9.92 pM

PTH equation coefficient

α2

136.4 pM


α3

2.317 mM


α4

3.097e-2 mM


MPa−1

α5

44.e12

α6

1.

Osteocytes influence coef.

K1

3.45e-8 pM day1 MPa−1

NO release coefficient

K2

1.day−1

NO elimination coefficient

K3

1.73e-9 pM day−1 MPa−1

PGE2 release coefficient

K4

1.e-3 day−1

PGE2 coef. increased by NO PGE2 elimination coefficient

Fluid flow influence coef.

Source

Aguilera-Tejero et al. (1996)

Noble et al. (2003) Maldonado et al. (2006)

K5

1.e-2 day−1

Ca0

2.– 2.8 mM

Basal value of Calcium

Aguilera-Tejero et al. (1996)

X NO

0.20 pM day−1

NO initial value

Maldonado et al. (2006)

day−1

X PGE

0.22 pM

PGE2 initial value


M X PTH

150 pM

Maximum PTH level

Mayer and Hurst (1978)

M X NO

2.86 pM

Maximum NO level


M X PGE M τ

1.199 pM

Maximum PGE2 level


150 MPa

Maximum shear stress level

McGarry et al. (2005)

Phase 3 (Integration): Shear stress causes NO (Fig. 1: event#5) and PGE2 (Fig. 1: event#6) releases through signaling downstream of the P2X7 receptor. PGE2 binds and signals through one of the EP receptors, (EP4 and/or EP2) and results in enhanced bone formation (Komarova 2005). Also, Wnt signaling (Fig. 1: event#7) through the Lrp5 receptor, which acts through β-catenin (Fig. 1: event#8), appears to be important in mechanically induced bone formation. Activation of Wnt/βcatenin signaling suppresses PPARγ expression, shifting mesenchymal cell differentiation toward the osteoblast lineage (David et al. 2007; Kang et al. 2007; Bonewald and Johnson 2008). Also, Dickkopf (Dkk) proteins are highly expressed in osteocytes. Dkk inhibits Wnt activation of the pathway by binding to Lrp5/6. Osteocytes can also control the osteoblastic differentiation of mesenchymal precursors in response to mechanical loading by modulating Wnt signaling pathways through the protein sclerostin (SOST). SOST is found nearly exclusively in the osteocytes. SOST provides a potential mechanism for the osteocytes to control mechanotransduction, by adjusting their SOST (Wnt inhibitory) signal output to modulate Wnt signaling in the effector cell population (Bonewald 2006).

However, it is unclear how SOST might physically exert its action in mechanotransduction (Potter et al. 2005). Furthermore, mechanical signals do increase the number of osteoblasts while simultaneously decreasing the expression of peroxisome proliferator-activated receptor (PPARγ ). Also, PPARγ negatively regulates osteoblast differentiation of bone marrow stromal cells in unloading, resulting in bone loss (David et al. 2007). Osteocytes may also direct the removal of bone due to disuse or matrix damage through mechanisms linked to their own apoptosis or via the secretion of specialized cellular attachment proteins such as osteopontin (OPN) (Heinegard et al. 1995; Gross et al. 2005). Other signaling pathways that are important in response mechanical loading may also crosstalk with the Wnt/β-catenin signaling pathway. For example, activation of integrins leading to the stimulation of integrin-linked kinase (Marshal 1995) and potentially any pathway that activates Akt could cross talk with the Wnt/β-catenin pathway (Bonewald 2006). Effects of these factors are not direct and are mediated by a long sequence of other molecular intermediates. Furthermore, with respect to NO and PGE2, it remains uncertain as to what upstream events control their release.

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In the current investigation, only NO and PGE2 effects are directly modeled in the integration phase as an essential events occurring in early responses to mechanical stimulation. Nevertheless, effects of SOST, PPARγ , OPN (Fig. 1: event#9), and Wnt/β-catenin signaling pathways effects can be simulated in a first step by appropriately modifying the value of the associated parameters in the integration phase by the osteocytes. The release rates of NO and PGE2 by a given osteocyte depend on the applied shear stress on its membrane (Noble et al. 2003). Based on this Maldonado et al. (2006), have developed a formulation of the production rates of NO and PGE2 by the osteocytes expressed in relation to a stress function ( f τ ) defining the sensitivity of the osteocytes to release NO and PGE2. The relationship between bone cells responses and the daily stress stimulus is typically characterized by a sigmoidal curve (Huiskes et al. 2000; Hernandez et al. 2001). Therefore, the proposed stress function can be expressed by the sigmoidal form: τ (5) fτ = 1 + exp [− (α5 τ + α6 )] α5 and α6 are sigmoidal shear stress function parameters. Recently, experimental results performed by Rath et al. (2010) showed a linear increase in the NO release versus the applied strain to a single osteocyte. Also, NO secretion have been observed to exhibit a rate-dependent effect (KleinNulend et al. 1995). Maldonado et al. (2006) suggested that the NO rate growth within a given osteocyte d xdtNO is influenced by the shear stress function f τ with constant K 1 , a degradation term with constant K 2 and an external input X NO (initial value). It is expressed by (Maldonado et al. 2006): d xNO = K 1 f τ − K 2 xNO + X NO dt

(6)

PGE2 Also, the PGE2 rate growth within a given osteocyte d xdt is influenced by the shear stress function f τ with constant K 3 . In addition, PGE2 is increased by NO with constant K 4 . A degradation term with constant K 5 and an external input X PGE2 (initial value) The PGE2 rate growth is given by (Maldonado et al. 2006):

d xPGE2 = K 3 f τ + K 4 xNO − K 5 xPGE2 + X PGE2 dt

(7)

Phase 3 (Responses calibration): Equations (1)–(7) describe some observed events occurring during transduction at a given osteocyte. Since we combine mechanical dimension ( MPa) and biological concentrations (pM), the difference in nature and magnitude between the model variables (inputs, intermediates variables, and outputs) imposes that prior integration of the several responses, these data must be calibrated to be compatible with each other.

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To integrate the cascades of responses (Fig. 1: event#10), a way to consider the calibration problem is to perform normalization of individual responses in the form of signals. Therefore, the intermediate responses and the outputs are rescaled within a range of 0 to 1 given by: τ τM xPTH f PTH = M xPTH xNO f NO = M xNO xPGE2 f PGE2 = M xPGE2

fM =

(8) (9) (10) (11)

M , x M , and x M where τ M , xPTH NO PGE2 denote, respectively, the maximum fluid shear stress applied on the osteocyte membrane, the maximum physiological PTH level produced by parathyroid gland, the maximum physiological NO level, and the maximum physiological PGE2 level.

Phase 4(Stimulus signal): Remodeling is a process where resorption by osteoclasts precedes formation by osteoblasts that fills the resorption cavity. Mechanical loading stimulates proliferation of osteoblast precursors, which then differentiate into osteoblasts. It is well known that osteoclasts development and activities are under the control of the osteoblasts (Rodan and Martin 1981). Unlike modeling, which involves either resorption or formation (but not both) at different bone sites, bone remodeling always follows an activation → resorption → formation sequence (A → R → F) (Parfitt 2000). During remodeling cycles, osteoblast-lineage cells in addition to regulating bone formation also regulate bone resorption via a coupling mechanism (RANK/RANKL/OPG pathway) that controls osteoclasts generation and activities. Therefore, the stimulus function ψ (Fig. 1: event#11) sent to BMUs can be viewed as a signal that can triggers downstream osteoblasts activities to express cytokines to stimulate osteoclasts resorption. Finally, a coupled mechanotransduction signal ψ can be expressed in term of weighted cumulative effects of the different normalized mechanobiological signals (Eqs. 8–11). Finally, the effects of mechanical and the biological signals (Eqs. 8–11) can be weighted and summed to generate a unique normalized stimulus function representing the coupled cumulative effects of the different mechanobiological signals given by: ψ = W M f M + WPTH f PTH + WNO f NO + WPGE2 f PGE2 Wi (i = M, P T H, N O, P G E2) with 0 ≤ ψ ≤ 1;

Wi = 1 and 0 ≤ Wi ≤ 1.

(12)


89

Wi are weighting factors expressing the relative importance of each signal. The weighting factors Wi can be viewed as the relative physiological importance of each messenger on the signal transmission amplitude to the BMUs. There are numerous schemes for assigning their values. All requires the use of judgment based on observed results or experiments. In the present work and in order to investigate the mechanobiological effects of stimulus signal, an average weighting-factor was assigned (Wi = 0.25).

3 Results To explore the relationships between the transduction variables, several response surfaces (RS) was plotted which consists in 3D graphical representations of a response plotted between two independent variables (inputs) and one transduction signal (output). The use of 3D RS plots allows understanding of the behavior of the system and investigates the interactions between the mechanical and the biological effects. The predicted stimulus results are investigated versus the levels of the effecting factors to check the model validity in relation to observed theoretical and experimental results. The aim was not to investigate the effects of the factors doses through time. Time-varying responses are provoked by time-varying stimulus. These dependencies vary with stimulus frequency, amplitude, and the characteristic sensitization and desensitization times of the ligand–receptor on- and offrates and the bound-species conversion rates (Segel et al. 1986). Figure 3 shows the stimulus signal evolution versus the Ca concentration and the interstitial fluid velocity. Our model predicts an increase in the stimulatory signal due to increase in the fluid velocity which generate an increase in the shear stress. The signal amplitude is mediated by the Ca concentration value with three regimes:

Fig. 3 RS of stimulus signal

Fig. 4 RS of stimulus signal versus Ca and PTH concentrations (VF = 5.e − 07) m/s

(i) The stimulus remains constant for a Ca concentration ranging between 0 and 2.25 mM. (ii) A rapid increase in the stimulus for a Ca concentration ranging between 2.25 and 2.40 mM. (iii) A decrease in the slope of the stimulus for Ca concentration greater then 2.40 mM. This result is consistent with Heaney (1995) conclusions. Ca accumulation in bone is a linear function of dietary Ca up to an optimum threshold; however, above the threshold, a further increase in dietary Ca causes no further increase in bone Ca. In the case of low fluid velocity (low or nonexistent mechanical stress), the stimulus signal remains active for Ca concentration values greater than 2.25 mM. This result is in conformity with Komarova (2005) one suggesting that bone continues to remodel in sites where the level of mechanical stress is low or nonexistent. The two most important hormones for maintaining Ca level in the body are PTH and 1, 25(OH)2 D3 (the active form of vitamin D). To restore falling plasma Ca levels, PTH released by parathyroid gland promotes Ca liberation from bone. PTH stimulates the release of Ca from bone, stimulating bone resorption. Effects of both Ca and PTH concentration are plotted in Fig. 4. Observation of the 3D response reveals strong dependance between Ca and PTH in stimulus (ψ) amplitude. High concentration of PTH combined with low concentration of Ca leads to an immediate decrease in the signal (ψ). Continuous PTH administration is catabolic to the skeleton, resulting in increased osteoclastic bone resorption (Lotinun et al. 2002; Blair and Athanasou 2004), whereas pulsatile doses of PTH result in net bone formation (Parfitt 1996; Hodsman 2000; Lotinun et al. 2002; Rubin et al. 2002). Despite many attempts to identify the source of these differential dosing effects on bone turnover, the precise mechanism remains

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Fig. 5 RS of stimulus signal versus Ca and PGE2 concentrations (VF = 5.e − 07) m/s

elusive (Parfitt). This lack of sufficient detail adds to the difficulty in modeling the specific mechanism(s) for the differential effects of PTH dosing on the skeleton. The NO and PGE2 kinetics are based on Maldonado et al. (2006) model and correspond to experimental observations of NO and PGE2 production. Predicted stimulus variation versus Ca and PGE2 concentrations plotted in Fig. 5 shows a strong dependency between both factors. Nonlinear interactions can be observed on the signal response with a linear increase in the stimulus amplitude with increase in PGE2 concentration. Also, a decrease in the stimulus for a combined (low dose of Ca with low dose of PGE2) is predicted. This finding corroborates with well-established results that PGE2 is potent modulators of bone metabolism since they can increase in vitro both bone resorption and bone formation for human (Ueda et al. 1980; Ringel et al. 1982) and animals (Jee et al. 1985; Marks and Miller 1993; Raisz 1993). Gao et al. (2009) examined the effects of varying doses, schedules, and routes of administration of PGE2 on bone in mice. Mice treated with a high dose of PGE2 (6 mg/kg/day) showed decreased trabecular bone volume (BV/TV) indicating increased bone resorption. Additional experiments using a higher dose or longer exposure did not increase bone mass. The authors concluded that exposure to high doses of PGE2 in mice may be anabolic but is balanced by catabolic effects. One explanation for the ability of PGE2 to exert both anabolic and catabolic effects, depending upon dose and cell type, may relate to PGE2 receptor usage. Machwate et al. (2001) examined in vivo the effect of prostanoid receptor (EP4), one of the PGE2 receptors, on bone formation induced by PGE2 in young rats in the presence or absence of EP4. They found that treatment with EP4 suppresses the increase in trabecular bone volume induced by PGE2, suggesting that the bone anabolic effect of PGE2 in rats is mediated by the EP4 receptor.

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Fig. 6 RS of stimulus signal versus Ca and NO concentrations (VF = 5.e − 07) m/s

Combined Ca and NO concentrations’ effect on stimulus is plotted in Fig. 6. The model predicts strong gradient of the response for high dose of the NO. A deficit in NO production combined with low dose of Ca lead to a decrease in the transmitted signal. This result agrees with Burger et al. (2003). The authors suggested that low concentration of NO is a potential factor to trigger osteocyte apoptosis. In their study Pitsillides et al. (1995), examined the changes in NO production in response to applied mechanical strains in a series of well characterized, strain-sensitive organ (cores of cancellous canine bone, rat ulnae, and vertebrae), and cell culture systems in vitro. The obtained results indicate rapid, transient increases in both production and release of NO by bone cells. This occurs almost immediately upon application of physiological levels of mechanical strain in organ cultures of bone. In the fluid flow experiment on bone cells by Klein-Nulend et al. (1995), it was found that fluid flow regulated NO and PGE2 release relative to cells that were not stimulated by fluid flow. By correlating predicted results with experimental Klein-Nulend et al. (1995) ones, both results suggest that fluid flow generating fluid shear stress leads to increased NO and PGE2 release. 4 Discussion In the present study, a biologically plausible physiologically based mathematical model to describe the mechanotransduction in bone by an osteocyte mediated by the Ca-PTH regulation by the parathyroid gland and incorporating the NO and PGE2 effects in early responses to mechanical stress. The inputs are mechanical stress and Ca concentration that can very over time, and the output is a stimulus function that can downstream trigger the remodeling process (remodeling process modeling is beyond the scope of this study). The different components of the model were based on both experimental and theoretical previously published results


describing some observed physiological events in bone mechanotransduction. Our model predicts a strong dependence between the computed signal, the applied stress, Ca-PTH regulation, and NO/PGE2 concentrations in conformity with experimental results from published literature. Nevertheless, the model presents some simplifications and limitations. First, we have considered some observed events during the transduction mechanisms in bone. Transduction can be also influenced by factors such as age, sex, growth factors, and different clinical treatments. All these factors can be simply analyzed in a first step by appropriately modifying the value of the associated parameters in the model. In fact, theoretical model has been used to study the influence of bisphosphonate treatment by modifying some parameters values (Nyman et al. 2004). PTH has also been shown to cause net bone loss when given continuously, so that the action of PTH on bone is dependent on the dosing pattern. Pulsatile PTH dosing has been shown to induce net bone formation in both animals and humans. While both anabolic and catabolic effects of PTH result directly from PTH receptors signaling (PTH1R), the model does not explicitly account for the PTH1R kinetics. An extended version of the transduction model will explicitly incorporate the kinetics of PTH1R response to PTH dosing developed by Potter et al. (2005). The integrity of the system depends critically on vitamin D status. Vitamin D plays a major role in calcium homeostasis and calcium absorption. There is evidence that hip fracture rates are a function of vitamin D intake (Baker et al. 1979). A model describing Ca-PTH-vitamin D homeostasis is needed to enhance our current model. Nevertheless, vitamin D effects can be also simulated in a preliminary step by appropriately modifying the value of the associated parameters in the CaPTH regulation. A number of secondary messenger pathways have been implicated in load transduction by osteocytes. Due to lack of precise mathematical description of the different events occurring during mechanotransduction within the osteocyte (Bonewald and Johnson 2008), it is very difficult to describe in precise way the behavior of secondary messengers and their interactions. These parameters are not completely characterized, and extensive experiments are needed to highlight the role of every model component prior mathematical modeling attempts. However, mathematical modeling provides a useful approach to integrate existing knowledge of the osteocytes transduction phase and to predict and test possible interactions between the different factors. Another important hormone involved in bone adaptation is estrogen (Frost 1988; Smalt et al. 1997; Luo et al. 2000). Frost (1988) suggested that estrogen regulates the mechanical set point for the mechanical responsiveness. Deficiency of estrogen shifts the set point for bone mass adaptation to mechanical loads.

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It is important to note that experiments are required to determine many parameters required for the proposed model and to check its validity quantitatively. Nevertheless, in spite of these limitations, this model is consistent and has physiological bases. While incorporating these biological processes adds mathematical complexity, most of the additional variables are histologically measurable. This makes the model amenable to experimental verification. Application of this model in understanding bone transduction physiology may be significant. Through an iterative simulation, the model has the ability to predict effects of variation of each physiological factor. The model is not complete and will be subjected to expansion and modifications in accordance with additional development and with new experimental results. Acknowledgments This work has been supported by French National Research Agency (ANR) through TecSan program (Project MoDos, n◦ ANR-09-TECS-018).

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