computer methods biomechanics & biomedical ...

7 downloads 0 Views 428KB Size Report
The aim of this work is to develop a novel mechanobiological coupled model to describe the Osteocyte transduction phase with a physical and metabolic ...
BONE REMODELING MODEL BASED ON MECHANOBIOLOGICAL STIMULUS R. Rieger1, R. Hambli1 and R. Jennane1

1. ABSTRACT The aim of this work is to develop a novel mechanobiological coupled model to describe the Osteocyte transduction phase with a physical and metabolic description of some observed mechanism during transduction step. The proposed model is described as a cascade of serial and parallel mechanical and metabolic observed reactions. The inputs are mechanical and biologic factors and the output is a scalar coupled mechanical-metabolic stimulus function which is used to regulate the bone remodeling process. The model shows good qualitative results in comparison to clinical observation.

2. INTRODUCTION Bone adaptation to its environment is influenced by both mechanical and metabolic stimuli. Indeed, mechanical strain, hormones like parathyroid, estrogen and mineral elements like Calcium or Vitamin D influence the rate of bone remodeling and the bone cells activities [Goltzman, D., 1999; Heldring et al., 2007]. Review of literature shows that proposed stimuli functions are phenomenological and are not developed based on physical/metabolic considerations. In general published works dealing with remodeling process [García-Aznar J. M., et al., 2005; McNamara, 2007] retained mechanical stimuli functions without taking into account the metabolic aspects of the bone adaptation. Under the control of mechanical agents such as body weight, gravity, muscular strength and biological agents such as hormones, vitamins, cytokynes, growth factors, bone cells regulate bone remodeling in response to mechanical demands and set free calcium and phosphate ions. A comprehensive model of bone remodeling process should describe the above mentioned mechanisms cascades including both mechanical and metabolic factors. We propose in the next section to provide a mathematical basis for describing the bone stimulus as it is mediated by metabolic factors such as Calcium (Ca), Parathyroid Hormone (PTH), Nitric Oxyde (NO), Prostaglandin E2 (PGE2) and mechanical strain.

3. METHODS The proposed model is based on the idea that the osteocyte’s network wiring system is the operating center which generates the interaction between both mechanical stimulus and the biochemical factors. After combining both stimuli (mechanical and biological/chemical), the osteocyte’s network triggers the signal to active Bone 1

PRISME Institute, University of Orléans, 8 rue Léonard de Vinci, 45072 Orléans cedex 2, France

1

Modeling Units (BMUs) to form or resorb bone. Several authors [Weinbaum et al., 1994; Cowin, 1995; Knothe Tate, 2003] have suggested that when a whole bone is deformed, the deformation induced pressure gradient. This causes bone fluid to flow in the Pericellular Matrix (PCM) space of the lacunar-canalicular system and induces a drag force on the matrix fibers. The flow is considered as a stimulus for osteocyte response. The osteocyte then produces and secretes Nitric Oxide (NO) and Prostaglandin E2 (PGE2) [Pitsihides,A. A et al., 1995; Bakker A.D. et al., 2001]. These local factors have an influence on the BMU activities via the RANK/RANKL/OPG cytokine system, altering the microenvironment where the highly coupled activity among bone cells takes place [Maldonado 2006; Lemaire V. 2004]. From a biological point of view, bone can be viewed as the calcium‘s reservoir of body. It is under the hormonal control of Parathyroid Hormone (PTH) [Kroll, 2000]. To restore falling plasma calcium levels, PTH promotes calcium liberation from bone. The proposed model is aimed to describe the role of PTH in normal bone and calcium homeostasis during a timespan of 200 days which largely covers a biological bone remodeling sequence of 4 months [Frost H.M., 1997]. Based on the mentioned remarks, we developed a framework to describe a bone stimulus function coupling both effects; mechanical and biological. Diagram of figure 1 summarizes the model. The upper part represents the mechanical stimulus due to interstitial fluid flow generated by stress. The lower part represents the biological stimulus due to PTH release as a consequence of calcium concentration change when NO & PGE2 production come from the osteocyte’s reaction to applied stress. Before signaling phase: Sensation After signaling phase: Transduction V Stress VC

fM  VTM   VT 

VT

VE

Mechanical part

C

+

Ocy



Signals to BMU

fPTH

Metabolic factors Ca

PTH

NO, PGE2

fB

fNO, fPGE Metabolic part

Figure 1. Block diagram describing the events cascade for bone stimulus with coupled mechanical and metabolic effects. Dashed lines represent the mechanical effects and the solid lines represent the biological effects. Double arrow denotes biological reaction due to mechanical stimulation represented by NO & PGE2 production by the osteocyte. A part of the set of equations which describe the cascade of reactions during osteocytes transduction stages are based on well admitted experimental and theoretical results presented in the next sections (Table 1), the rest can be found in the literature and will be mentioned later. The proposed model describes the transduction in 2 stages; mechanisms before and after osteocytes activities. Firstly, Lemaire T.et al., 2006 studied the coupled phenomena that govern the interstitial fluid movement. Based on averaged description of fluid velocity in the canaliculi due to mechanical pressure, chemical and electro-chemical motions have been proposed by the authors. Investigation of Lemaire T.et al., 2006 concluded that the fluid 2

movement VP generated by pressure gradient is the most important driving effect in fluid movement. The contribution of deformation driving velocity due to fluid pressure exceeds 95% of the total fluid velocity. Based on these finding, we neglect the effects of VC and VE (equ. (1) of table 1). The metabolic signaling part is assumed to be triggered by Calcium demands. The calcium level is considered as an input to the model (equ. (2)) which leads to PTH release. To describe the mathematical relation between PTH release rate and level of calcium, we propose a fitted model obtained by experiments of [Haden et al., 2000; Houillier P. et al., 2000] expressed by relation (equ. (3)). Then, two steps are considered and described as follow. Step1 describes the release of NO and PGE2 expressed in Maldonado, 2006. Step2 ensures a variable variation i between 0% and 100% of its maximum value by the relation X in  XMax , (equ. 4-7). It Xi enables to leave aside dimensional consideration of the variables since we combine mechanical dimension (MPa) and biological concentration (mM) [Brazel C.S., N.A. Peppas, 1999; Coatanéa E. et al., 2003]. The biological signals are finally summed up (equ. 9,10). And finally we propose the relation expressing the stimulus signal generated by the cascade of mechanical and biological events (equ. 11). Table 1. Model’s equations Mechanical part Before signaling phase by Osteocyte

VP

(1)

Biological part

Ca  Ca0  t ; x PTH   1 

2 xCa  3

1 e After signaling phase by Osteocyte Signals

Stimulus

fM 

fM 

VP V PM VP V PM

4

(4)

x f PTH  xPTHM ; f NO  xNOM ; f PGE 2  PGE 2M xNO xPTH x PGE 2

(8)

f MB  WNO f NO  WPGE 2 f PGE 2 f B  WPTH f PTH  WMB f MB

  WM f M  WB f B

x  x if x0

(2,3)

(5-7) (9) (10)

x 0 if x0

(11)

Notations : xCa : Calcium level; xPTH : PTH level; xNO : Nitric oxide level; xPGE : Prostaglandin E2 level; VP : Interstitial fluid velocity generated by pressure. Finally, the developed set of equations which describe the different transductions reactions mentioned above, are summarized in table 1. Application of serial and parallel blocks models leads to the coupled stimulus function (equ. 11) which takes into account both mechanical and biological parts of figure 1, with Wi , i= M, B, MB, PTH, NO, PG2E : are mechanical and biological weight factors verifying: 0  Wi  1 and WM  WB  WPTH  WMB  WNO  WPGE 2  1. This stimulus will be used to drive BMUs [Komarova S. 2005] considering the bone volume formed/resorbed by osteoblasts/osteoclasts respectively.

3

4. RESULTS Figure 2 shows the stimulus versus the calcium concentration in blood for different stresses applied onto the bone. For figure 2.A, we have considered the stimulus as composed of 50% of the mechanical part and 50% of the biological part (   0.5 f M  0.5 f B ). First, we notice on each curves of the figure 2 an important evolution of the stimulus due to the increase of calcium concentration for low level of stress. Next, when we increase the stress applied, the stimulus is more influenced by the mechanical part but still conserves the curvature of calcium’s concentration decrease. This curvature becomes bigger since we decide to define the stimulus with a dominant biological component (Figure 2.B   0.2 f M  0.8 f B ), whereas it lowers the impact of calcium concentration for a stimulus with high mechanical influence (Figure 2.C   0.8 f M  0.2 f B

A

B

C

Fig. 2. Mechanobiological stimulus in function of the calcium concentration in blood for a stress ranged from 1MPa to 50MPa. A,   0.5 f M  0.5 f B . B,   0.2 f M  0.8 f B . C,   0.8 f M  0.2 f B . On the two following curves we can observe for a specific stress applied (5MPa) the evolution of the biomechanical stimulus versus the calcium concentration. The curvature due to the increase of calcium concentration is still noticeable even if it is reduced but the important aspect is the increase of the stimulus as the weight of the PTH and the production rate coefficient of the PGE2 increases. It shows very well the multiple influences of the different biological components in the stimulus for low 4

stresses applied and a dominant biological component (  0.2 f M  0.8 f B ). A

B

Fig. 3. Mechanobiological stimulus in function of the calcium concentration in blood for a stress applied of 5MPa and   0.2 f M  0.8 f B . A, PTH weight factor ranged from 0.1 to 0.9. B, PGE2 production rate coefficient ranged from 4 to 40. The table 2 brings together all the parameters of the model we have calculated or get from different sources. Table 2. Model’s parameters Parameter value

Cao  1 2 3 4 M X PTH M X NO M X PGE

Source

Description

Symbol 2.1 mM

Basal value of Calcium

Based on P.Houillier et al., 2009

mM.day-1

Caclium evolution rate

Based on P.Houillier et al., 2009

pg.ml-1

PTH equation coefficient

Calculation based on P.Houillier et al., 2009

pg.ml-1

PTH equation coefficient

Calculation based on P.Houillier et al., 2009

2.317 mM

PTH equation coefficient

Calculation based on P.Houillier et al., 2009

3.097e-2 mM

2e-3

9.92

136.4

PTH equation coefficient

Calculation based on P.Houillier et al., 2009

pg.ml-1

Maximum PTH level

Based on P.Houillier et al., 2009

15.e-3 pM

Maximum NO level

Calculation based on Maldonado et al., 2006

15.e-3 pM

Maximum PGE2 level

Calculation based on Maldonado et al., 2006

150

5. DISCUSSION This work has been developed as a first stage in a full mechanobiologic bone remodeling model. The originality consists on describing the physical behavior of biomechanical transduction by the osteocytes. In this sense the model is able to define a certain biological stimulus due to bed-rest or some metabolic dysfunction. We have developed a bone mechanobiological transduction model which contrast with the actual purely mechanical and phenomenological transduction models. We have recently used this work on the BMUs model of Komarova et al., 2005 to see the effect of the stimulus and it confirms the pertinence of the model. Actually we can observe the influence on bone formation/resorption when the stimulus is applied on osteoblasts/osteoclasts. The results are qualitatively relevant with what we can observe in clinical data; unfortunately we weren’t able to add more results in this paper.

5

6. REFERENCES 1. Goltzman D., Interactions of PTH and PTHrP with the PTH/PTHrP Receptor and with Downstream Signaling Pathways: Exceptions That Provide the Rules, Journal of Bone and Mineral Research, 1999, Vol. 14, 2, 173-177. 2. Heldring Nina, et al., Estrogen Receptors: How Do They Signal and What Are Their Targets, Physiol. Rev., 2007, 87: 905-931. 3. García-Aznar J. M., et al., A bone remodelling model coupling microdamage growth and repair by 3D BMU-activity, Biomechanics and Modeling in Mechanobiology, 2005, Vol.4, 2, 147-167. 4. McNamara, Laoise M., et al., Bone remodelling algorithms incorporating both strain and microdamage stimuli. Journal of Biomechanics, 2007, 40, 6, 1381-1391. 5. Weinbaum S., et al., A model for the excitation of osteocytes by mechanical loading-induced bone fluid shear stresses, Journal of Biomechanics, 1994, Vol. 27, 3, 339-360. 6. Cowin S. C., et al., A case for bone canaliculi as the anatomical site of strain generated potentials, Journal of Biomechanics, 1995, Vol. 28, 11, 1281-1297. 7. Knothe Tate Melissa L., 'Whither flows the fluid in bone?' An osteocyte's perspective, Journal of Biomechanics, 2003, Vol. 36, 10, 1409-1424. 8. Pitsillides AA, et al., Mechanical strain-induced NO production by bone cells: a possible role in adaptive bone (re)modeling?, FASEB J., 1995,Vol. 9: 1614-1622. 9. Bakker Astrid D., et al., The production of nitric oxide and prostaglandin E2 by primary bone cells is shear stress dependent, Journal of Biomechanics, 2001, Vol. 34, 5, 671-677. 10. Maldonado S, Borchers S, Findeisen R, Allgöwer F., Mathematical modeling and analysis of force induced bone growth, In: Proc 28th Int Conf IEEE-EMBC, NewYork, 2006, p. 3154-3160. 11. Lemaire V., et al.., Modeling the interactions between osteoblast and osteoclast activities in bone remodeling, J. Th. Biol., 2004, 229(3):293-309, 2004. 1(9):15-23. 12. Kroll, M.H., Parathyroid hormone temporal effects on bone formation and resorption, Bull. Math. Bio., 2000, 62, 163–188. 13. Frost H. M., Why do marathon runners have less bone than weight lifters?, A vitalbiomechanical view and explanation. Bone, 1997, Vol. 20, 3, 183-189. 14. Lemaire T., et al., Multi-scale analysis of the coupled effects governing the movement of interstitial fluid in cortical bone Biomechan, Model. Mechanobiol., 2006, 5 39-52. 15. Haden et al., The effects of age and gender on parathyroid hormone dynamics, Clinical Endocrinology 2000, 52,329-338. 16. Houillier P., Calcium-sensing receptor: A central role in calcium metabolism, Medecine Nucleaire, 2009, Vol. 33, 1, 39-45. 17. Brazel C.S., N.A. Peppas, Dimensionless analysis of swelling of hydrophilic glassy polymers with subsequent drug release from relaxing structures, Biomaterials 1999, 20, 721-732. 18. Coatanéa E., et al., Applying dimensionless indicators for the analysis of multiple constraints and compound objectives in conceptual design, 16ème Congrès Français de Mécanique (2003). 19. Komarova S., Mathematical Model of Paracrine Interactions between Osteoclasts and Osteoblasts Predicts Anabolic Action of Parathyroid Hormone on Bone, Endoc., 2005, 146(8):3, 589-3595. 6