Simulation of bone remodeling at the bone cell level can predict changes in bone microarchitecture and density due to bone diseases and drug treatment. Their clinical application, however, is limited since bone microarchitecture can only be measured in the peripheral skeleton of patients and since the simulations are very time consuming. To overcome these issues, we have developed an analytical model to predict bone density adaptation at the organ level, in agreement with our earlier developed bone remodeling theory at the cellular level. Assuming a generalized geometrical model at the microlevel, the original theory was reformulated into an analytical equation that describes the evolution of bone density as a function of parameters that describe cell activity, mechanotransduction and mechanical loading. It was found that this analytical model can predict changes in bone density due to changes in these cell-level parameters that are in good agreement with those predicted by the earlier numerical model that implemented a detailed micro-finite element (FE) model to represent the bone architecture and loading, at only a fraction of the computational costs. The good agreement between analytical and numerical density evolutions indicates that the analytical model presented in this study can predict well bone functional adaptation and, eventually, provide an efficient tool for simulating patient-specific bone remodeling and for better prognosis of bone fracture risk.

References

References
1.
Eriksen
,
E.
,
2010
, “
Cellular Mechanisms of Bone Remodeling
,”
Rev. Endocr. Metab. Disord.
,
11
(
4
), pp.
219
227
.10.1007/s11154-010-9153-1
2.
Firoozbakhsh
,
K.
, and
Aleyaasin
,
M.
,
1989
, “
The Effect of Stress Concentration on Bone Remodeling: Theoretical Predictions
,”
ASME J. Biomech. Eng.
,
111
(
4
), pp.
355
360
.10.1115/1.3168391
3.
Cowin
,
S. C.
,
Moss-Salentijn
,
L.
, and
Moss
,
M. L.
,
1991
, “
Candidates for the Mechanosensory System in Bone
,”
ASME J. Biomech. Eng.
,
113
(
2
), pp.
191
197
.10.1115/1.2891234
4.
Huiskes
,
R.
,
Ruimerman
,
R.
,
van Lenthe
,
G. H.
, and
Janssen
,
J. D.
,
2000
, “
Effects of Mechanical Forces on Maintenance and Adaptation of Form in Trabecular Bone
,”
Nature
,
405
(
6787
), pp.
704
706
.10.1038/35015116
5.
Kameo
,
Y.
,
Adachi
,
T.
, and
Hojo
,
M.
,
2011
, “
Effects of Loading Frequency on the Functional Adaptation of Trabeculae Predicted by Bone Remodeling Simulation
,”
J. Mech. Behav. Biomed. Mater.
,
4
(
6
), pp.
900
908
.10.1016/j.jmbbm.2011.03.008
6.
Carter
,
D. R.
,
Orr
,
T. E.
, and
Fyhrie
,
D. P.
,
1989
, “
Relationships Between Loading History and Femoral Cancellous Bone Architecture
,”
J. Biomech.
,
22
(
3
), pp.
231
244
.10.1016/0021-9290(89)90091-2
7.
Beaupré
,
G. S.
,
Orr
,
T. E.
, and
Carter
,
D. R.
,
1990
, “
An Approach for Time-Dependent Bone Modeling and Remodeling—Application: A Preliminary Remodeling Simulation
,”
J. Orthop. Res.
,
8
(
5
), pp.
662
670
.10.1002/jor.1100080507
8.
Fernandes
,
P.
,
Rodrigues
,
H.
, and
Jacobs
,
C.
,
1999
, “
A Model of Bone Adaptation Using a Global Optimisation Criterion Based on the Trajectorial Theory of Wolff
,”
Comput. Methods Biomech. Biomed. Eng.
,
2
(
2
), pp.
125
138
.10.1080/10255849908907982
9.
Coelho
,
P. G.
,
Fernandes
,
P. R.
,
Rodrigues
,
H. C.
,
Cardoso
,
J. B.
, and
Guedes
,
J. M.
,
2009
, “
Numerical Modeling of Bone Tissue Adaptation—A Hierarchical Approach for Bone Apparent Density and Trabecular Structure
,”
J. Biomech.
,
42
(
7
), pp.
830
837
.10.1016/j.jbiomech.2009.01.020
10.
Andreaus
,
U.
,
Colloca
,
M.
, and
Iacoviello
,
D.
,
2012
, “
An Optimal Control Procedure for Bone Adaptation Under Mechanical Stimulus
,”
Control Eng. Pract.
,
20
(
6
), pp.
575
583
.10.1016/j.conengprac.2012.02.002
11.
Doblare
,
M.
, and
Garcia
,
J. M.
,
2001
, “
Application of an Anisotropic Bone-Remodelling Model Based on a Damage-Repair Theory to the Analysis of the Proximal Femur Before and After Total Hip Replacement
,”
J. Biomech.
,
34
(
9
), pp.
1157
1170
.10.1016/S0021-9290(01)00069-0
12.
Hambli
,
R.
,
2010
, “
Application of Neural Networks and Finite Element Computation for Multiscale Simulation of Bone Remodeling
,”
ASME J. Biomech. Eng.
,
132
(
11
), p.
114502
.10.1115/1.4002536
13.
Ruimerman
,
R.
,
Hilbers
,
P.
,
van Rietbergen
,
B.
, and
Huiskes
,
R.
,
2005
, “
A Theoretical Framework for Strain-Related Trabecular Bone Maintenance and Adaptation
,”
J. Biomech.
,
38
(
4
), pp.
931
941
.10.1016/j.jbiomech.2004.03.037
14.
Tsubota
,
K.-I.
,
Suzuki
,
Y.
,
Yamada
,
T.
,
Hojo
,
M.
,
Makinouchi
,
A.
, and
Adachi
,
T.
,
2009
, “
Computer Simulation of Trabecular Remodeling in Human Proximal Femur Using Large-Scale Voxel FE Models: Approach to Understanding Wolff's Law
,”
J. Biomech.
,
42
(
8
), pp.
1088
1094
.10.1016/j.jbiomech.2009.02.030
15.
Komarova
,
S. V.
,
2006
, “
Bone Remodeling in Health and Disease: Lessons From Mathematical Modeling
,”
Ann. N.Y. Acad. Sci
,
1068
(
1
), pp.
557
559
.10.1196/annals.1346.052
16.
Buenzli
,
P. R.
,
Pivonka
,
P.
,
Gardiner
,
B. S.
, and
Smith
,
D. W.
,
2012
, “
Modelling the Anabolic Response of Bone Using a Cell Population Model
,”
J. Theor. Biol.
,
307
, pp.
42
52
.10.1016/j.jtbi.2012.04.019
17.
Pivonka
,
P.
,
Buenzli
,
P. R.
,
Scheiner
,
S.
,
Hellmich
,
C.
, and
Dunstan
,
C. R.
,
2013
, “
The Influence of Bone Surface Availability in Bone Remodeling—A Mathematical Model Including Coupled Geometrical and Biomechanical Regulations of Bone Cells
,”
Eng. Struct.
,
47
, pp.
134
147
.10.1016/j.engstruct.2012.09.006
18.
Christen
,
P.
,
Ito
,
K.
,
Müller
,
R.
,
Rubin
,
M. R.
,
Dempster
,
D. W.
,
Bilezikian
,
J. P.
, and
van Rietbergen
,
B.
,
2012
, “
Patient-Specific Bone Modelling and Remodelling Simulation of Hypoparathyroidism Based on Human Iliac Crest Biopsies
,”
J. Biomech.
,
45
(
14
), pp.
2411
2416
.10.1016/j.jbiomech.2012.06.031
19.
van Rietbergen
,
B.
,
Huiskes
,
R.
,
Eckstein
,
F.
, and
Rüegsegger
,
P.
,
2003
, “
Trabecular Bone Tissue Strains in the Healthy and Osteoporotic Human Femur
,”
J. Bone Miner. Res.
,
18
(
10
), pp.
1781
1788
.10.1359/jbmr.2003.18.10.1781
20.
Martin
,
R. B.
,
1984
, “
Porosity and Specific Surface of Bone
,”
Crit. Rev. Biomed. Eng.
,
10
(
3
), pp.
179
222
.
21.
Currey
,
J. D.
,
1988
, “
The Effect of Porosity and Mineral Content on the Young's Modulus of Elasticity of Compact Bone
,”
J. Biomech.
,
21
(
2
), pp.
131
139
.10.1016/0021-9290(88)90006-1
22.
Burger
,
E. H.
, and
Klein-Nulend
,
J.
,
1999
, “
Mechanotransduction in Bone-Role of the Lacuno-Canalicular Network
,”
FASEB J.
,
13
(
9001
), pp.
S101
S112
.
23.
Martin
,
R. B.
,
2000
, “
Toward a Unifying Theory of Bone Remodeling
,”
Bone
,
26
(
1
), pp.
1
6
.10.1016/S8756-3282(99)00241-0
24.
Fazzalari
,
N. L.
,
Kuliwaba
,
J. S.
, and
Forwood
,
M. R.
,
2002
, “
Cancellous Bone Microdamage in the Proximal Femur: Influence of Age and Osteoarthritis on Damage Morphology and Regional Distribution
,”
Bone
,
31
(
6
), pp.
697
702
.10.1016/S8756-3282(02)00906-7
25.
Eriksen
,
E. F.
, and
Kassem
,
M.
,
1992
, “
The Cellular Basis of Bone Remodeling
,”
Triangle
,
31
(
2
), pp.
45
57
.
26.
Mullender
,
M. G.
,
Huiskes
,
R.
,
Versleyen
,
H.
, and
Buma
,
P.
,
1996
, “
Osteocyte Density and Histomorphometric Parameters in Cancellous Bone of the Proximal Femur in Five Mammalian Species
,”
J. Orthop. Res.
,
14
(
6
), pp.
972
979
.10.1002/jor.1100140618
27.
Garnero
,
P.
,
Sornay-Rendu
,
E.
,
Chapuy
,
M.-C.
, and
Delmas
,
P. D.
,
1996
, “
Increased Bone Turnover in Late Postmenopausal Women is a Major Determinant of Osteoporosis
,”
J. Bone Miner. Res.
,
11
(
3
), pp.
337
349
.10.1002/jbmr.5650110307
28.
Rubin
,
M. R.
,
Dempster
,
D. W.
,
Kohler
,
T.
,
Stauber
,
M.
,
Zhou
,
H.
,
Shane
,
E.
,
Nickolas
,
T.
,
Stein
,
E.
,
Sliney
,
J.
, Jr.
,
Silverberg
,
S. J.
,
Bilezikian
,
J. P.
, and
Muller
,
R.
,
2010
, “
Three Dimensional Cancellous Bone Structure in Hypoparathyroidism
,”
Bone
,
46
, pp.
190
195
.10.1016/j.bone.2009.09.020
29.
Recker
,
R. R.
,
Lappe
,
J. M.
,
Davies
,
K. M.
, and
Kimmel
,
D. B.
,
1992
, “
Change in Bone Mass Immediately Before Menopause
,”
J. Bone Miner. Res.
,
7
(
8
), pp.
857
862
.10.1002/jbmr.5650070802
30.
Morita
,
M.
,
Ebihara
,
A.
,
Itoman
,
M.
, and
Sasada
,
T.
,
1994
, “
Progression of Osteoporosis in Cancellous Bone Depending on Trabecular Structure
,”
Ann. Biomed. Eng.
,
22
(
5
), pp.
532
539
.10.1007/BF02367089
31.
Li
,
X. J.
,
Jee
,
W. S. S.
,
Chow
,
S.-Y.
, and
Woodbury
,
D. M.
,
1990
, “
Adaptation of Cancellous Bone to Aging and Immobilization in the Rat: A Single Photon Absorptiometry and Histomorphometry Study
,”
Anat. Rec.
,
227
(
1
), pp.
12
24
.10.1002/ar.1092270103
32.
Fyhrie
,
D. P.
, and
Schaffler
,
M. B.
,
1995
, “
The Adaptation of Bone Apparent Density to Applied Load
,”
J. Biomech.
,
28
(
2
), pp.
135
146
.10.1016/0021-9290(94)00059-D
33.
Parfitt
,
A. M.
,
1994
, “
Osteonal and Hemi-Osteonal Remodeling: The Spatial and Temporal Framework for Signal Traffic in Adult Human Bone
,”
J. Cell. Biochem.
,
55
(
3
), pp.
273
286
.10.1002/jcb.240550303
34.
Colloca
,
M.
,
van Rietbergen
,
B.
,
Blanchard
,
R.
,
Hellmich
,
C.
, and
Ito
,
K.
,
2012
, “
From Cell Level to Organ Level: A Multiscale Approach for Bone Remodeling Simulation
,”
J. Biomech.
,
45
, p.
S470
.10.1016/S0021-9290(12)70471-2
35.
Vuong
,
J.
, and
Hellmich
,
C.
,
2011
, “
Bone Fibrillogenesis and Mineralisation: Quantitative Analysis and Implications for Tissue Elasticity
,”
J. Theor. Biol.
,
287
, pp.
115
130
.10.1016/j.jtbi.2011.07.028
36.
Malandrino
,
A.
,
Fritsch
,
A.
,
Lahayne
,
O.
,
Kropik
,
K.
,
Redl
,
H.
,
Noailly
,
J. R. M.
,
Lacroix
,
D.
, and
Hellmich
,
C.
,
2012
, “
Anisotropic Tissue Elasticity in Human Lumbar Vertebra, by Means of a Coupled Ultrasound-Micromechanics Approach
,”
Mater. Lett.
,
78
, pp.
154
158
.10.1016/j.matlet.2012.03.052
37.
Mulder
,
L.
,
van Rietbergen
,
B.
,
Noordhoek
,
N. J.
, and
Ito
,
K.
,
2012
,”
Determination of Vertebral and Femoral Trabecular Morphology and Stiffness Using a Flat-Panel C-Arm-Based CT Approach
,“
Bone
,
50
(
1
), pp.
200
208
.10.1016/j.bone.2011.10.020
38.
Rauch
,
F.
,
Plotkin
,
H.
,
Zeitlin
,
L.
, and
Glorieux
,
F. H.
,
2003
, “
Bone Mass, Size, and Density in Children and Adolescents With Osteogenesis Imperfecta: Effect of Intravenous Pamidronate Therapy
,”
J. Bone Miner. Res.
,
18
(
4
), pp.
610
614
.10.1359/jbmr.2003.18.4.610
39.
Glorieux
,
F. H.
,
Bishop
,
N. J.
,
Plotkin
,
H.
,
Chabot
,
G.
,
Lanoue
,
G.
, and
Travers
,
R.
,
1998
, “
Cyclic Administration of Pamidronate in Children With Severe Osteogenesis Imperfect
,”
New Engl. J. Med.
,
339
(
14
), pp.
947
952
.10.1056/NEJM199810013391402
40.
Boivin
,
G. Y.
,
Chavassieux
,
P. M.
,
Santora
,
A. C.
,
Yates
,
J.
, and
Meunier
,
P. J.
,
2000
, “
Alendronate Increases Bone Strength by Increasing the Mean Degree of Mineralization of Bone Tissue in Osteoporotic Women
,”
Bone
,
27
(
5
), pp.
687
694
.10.1016/S8756-3282(00)00376-8
41.
Weinstein
,
R. S.
,
Nicholas
,
R. W.
,
Manolagas
,
S. C.
,
2000
, “
Apoptosis of Osteocytes in Glucocorticoid-Induced Osteonecrosis of the Hip
,”
J. Clin. Endocrinol. Metab.
,
85
(
8
), pp.
2907
2912
.10.1210/jc.85.8.2907
42.
Lin
,
B. Y.
,
Jee
,
W. S. S.
,
Ma
,
Y. F.
,
Ke
,
H. Z.
,
Kimmel
,
D. B.
, and
Li
,
X. J.
,
1994
, “
Effects of Prostaglandin E2 and Risedronate Administration on Cancellous Bone in Older Female Rats
,”
Bone
,
15
(
5
), pp.
489
496
.10.1016/8756-3282(94)90272-0
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