Journal of Biosciences and Medicines, 2013, 1, 19-22 JBM Published Online October 2013 (
Animal bone growth exp eriment of rapid-growing rats in
different stress environment and its mathematical model*
Jun Zhang1, Wenzhi Zhao2
1Advanced Technology of Transportation Vehicle Key Laboratory of Liaoning Province, Dalian Jiaotong University, Dalian, China
2Department of Orthopaedics, The Second Affiliated Hospital of Dalian Medical Univer s ity, Dalian, China
Received 2013
The aim of this study is to explore a way th a t qua nt ify
the qualitative equation of bone growth and remode-
ling which was based on the animal Experiment of
rapid-growing Rats in Different Stress Environment.
These results were proved to be of good stability and
identification precision with the numerical method of
inversion. It suggested that the growing coefficient
and the threshold in function were variables changing
with ti me and spa ce. The idea an d method u sed in th e
research of bone growth and remodeling adaptation
in this paper also provided clue and reference to es-
tablish other models for living system.
Keywords: Bone Growth Equation; Parameter
Identification; Animal Experiment; Bone Mineral
In the 21st century, bioengineering research, which is
integrated with biology, mathematics, mechanics and
computing science, is gradually to be a hot spot. Since
last century, many researchers have been working in this
field on biological experiments and theoretical dedu ction
[1]. So far, few quantificational research findings about
the life process were put forward because of the difficul-
ty of combination of experimental results and theoretical
studies. In this paper, biological parameters of bone
growth equation were identified using the inversion
theory by the computing numerical method. The investi-
gation was based on the combination of animal experi-
ment and its mathematical representation and provided
chance of deeply discussing the relationship between life
phenomenon and environmental stimulation.
In this study, on the basis of biological experiment on
rapid growing rats, the effects of overloading and un-
loading on bone growth and remodeling were investi-
gated. Different stress environments were created so that
the rats’ left femurs beared dissimilar stress stimulus
from right femurs. The numerical value was obtained of
proximal femur bone mineral density (BMD) at equal
interval time within two months. The loading of each
femur was calculated by the combination of the BMD
value and the same rat individual body weight data.
Based on the identification theory [2], we designed
computer program and calculated the unknown coeffi-
cients in bone growth equation in different periods and
moreover, the changing trend of rat femur BMD several
weeks in the future was predicted.
After the “law of bone transformation [3]” put fo rwarded
by Wolff in 1892, researchers characterized this theory as
bone self-optimization process with mathematical me-
thod (as shown in Eq.1). In order to identify the un-
known parameters in the equation, we carried out two
experiments on rats within two years and got the data of
bone mineral density (BMD) and load. The differences
between the two experiments on the aspects of rats’ se-
lection, experiment instruments, data acquisition and
analysis were controlled to be a minimum.
2.1. Anim al Selec tion, Grouping and Feeding
60 six-week-old female Sprague Dawley rats (body mass
160 ± 12 g) obtained from the experimental animal cen-
ter of Dalian Medical University were randomly divided
into two groups: 45 animals as experiment group (E), the
other 15 animals without any treatment were served as
comparison group (C); In the experiment group, the ani-
mals were performed sciatic nerve resection of right hin-
dlimb through a small incision in the gluteal area, under
anesthesia (mebubarbital 30 mg/kg intraperitoneally in-
jected). The right femurs which lost the ability of load
were defined as unloading group (U) of experiment
group and the left should bear more loads defined as
overloa di ng group (O) of e xperime nt gro u p.
*Project of Liaoning Province Education Department, LS2010030.
J. Zhang, W. Z. Zhao / Journal of Biosciences and Medicines 1 (2013) 19-22
Copyright © 2013 SciRes. OPEN ACCESS
After the operation, the rats were housed for 8 weeks
individually in temperature-controlled room (220˚C ±
20˚C) with a 12:12 hour light/dark cycle. The animals
have free access to water and a standard rat laboratory
diet and were encouraged to move for 30 minutes in the
morning , noon and eveni n g respective l y.
2.2. Bone Mineral Density Data Obtaining and
After the animals had been anesthetized and weighed, the
bone mineral density(BMD) of rat femur was measured
once a week, the value of area BMD of the same sit in
the proximal Femur was obtained (as shown in Figure 1).
The adopted instrument is the Challenger double energy
x-ray absorption instrument from France DMS Company,
the applied software is the rat analysis softwa re provided
by DMS Company. Data of rat mean weights and BMD
during experiment were shown in Tables 1 and 2.
3.1. Bone Growth Equation
“Stress and growth is the key problem in the study of
Biomechanics [4].” The stress stimulus and the adapta-
bility towards the stimulus are the basic rules of growth
and remodeling of bone. Studying the relationship be-
tween growth and stress, we can understand the remode-
ling mechanism of bone from the point of foundation.
Mathematical description of bone self-optimization
process based on Wolff’s law was as follows (as shown
Figure 1. Measurement for bone mineral density.
Tab le 1. Data of rat mean weights during experiment (g).
Week 1 Week 2 Week 3 Weed 4
Experiment 181.0 ± 16.0 186.2 ± 17.9 194.9 ± 17.4 200.8 ± 18.0
Comparison 209.1 ± 19.0 215.1 ± 19.7 220.6 ± 20.0 223.9 ± 20.0
Week 5 Week 6 Week 7 Week 8
Experiment 177.5 ± 14.7 188.9 ± 15.6 199.0 ± 16.2 204.5 ± 16.4
Comparison 211.5 ± 17.6 217.0 ± 17.0 224.4 ± 18.6 226.7 ± 19.6
Table 2. Data of mean BMD of rat proximal femur during ex-
periment (g/cm2).
Week 1 Week 2 Week 3 Week 4
Overloading 0.2378 ±
0.0473 0.2584 ±
0.0362 0.2746 ±
0.0440 0.2902 ±
unloading 0.2508 ±
0.0476 0.2584 ±
0.0410 0.2618 ±
0.0471 0.2649 ±
comparison 1 0.2328 ±
0.0333 0.2518 ±
0.0412 0.2665 ±
0.0485 0.2805 ±
comparison 2 0.2538 ±
0.0294 0.2757 ±
0.0475 0.2908 ±
0.0517 0.3064 ±
Week 5 Week 6 Week7 Week 8
Overloading 0.3002 ±
0.0600 0.3086 ±
0.0453 0.3155 ±
0.0512 0.3179 ±
unloading 0.2635 ±
0.0616 0.2609 ±
0.0472 0.2583 ±
0.0580 0.2529 ±
comparison 1 0.2888 ±
0.0432 0.2941 ±
0.0575 0.2977 ±
0.0509 0.2980 ±
comparison 2 0.3155 ±
0.0459 0.3214 ±
0.0552 0.3249 ±
0.0427 0.3248 ±
in Eq.1) [3].
= −
where ρ is the apparent density of the trabecular bone
structure; t is time; S represents mechanical stimulus,
here it is equivalent stress; B represents the growth coe f-
ficient; K represents a goal stress state. Among them, the
selection and quantification of B and K are the keys of
establishing bone gr owth equation.
For the time interval was constant as 1 week, Eq.1
could be simplified as:
()BS K t
∆=− ∆
We defined:
min ()
=∆ −∆
Then, the object function for identifying B and K was
3.2. Parameter Identification and Analysis
Lots of theories and Algorithms towards inversion have
been studied [2,5]. Thus far, there are so many popular
optimization algorithms such as ant group algorithm,
genetic algorithm and analogue anneal algorithm. BFGS
algorithm was explored by Broyden, Fletcher, Goldfarb
and Shanno in 1970, which was a kind of variable metric
method. In this paper, BFGS method was put forward to
serve the inversion problem.
BFGS procedure was designed under Microsoft Visual
Basic 6.0; it was debugged successfully under both
Windows 2000 and Windows XP system.
J. Zhang, W. Z. Zhao / Journal of Biosciences and Medicines 1 (2013) 19-22
Copyright © 2013 SciRes. OPEN ACCESS
The BMD value of rat proximal femur obtained every
week was categorized into four groups: left hindlimb of
comparison group, right hindlimb of comparison group,
left hindlimb of experiment group and right hindlimb of
the experiment group. The variation quantity per week of
BMD-∆BMD was regarded as dρ/dt and was input into
program. The outer load P of each group was defined as
body mass/4, body mass/4, body mass/3 and body mass/
googol, Cowin considered that the stimulus of bone
growth equation was linear or quadric with strain in 1976
and 1981 respectively. Later, Huiskes [6] introduced stain
energy density as the stimulus in 1987. According to
Huiskes, we defined stimulus as p2/BMD2. Th e stimulus
S and goal stress K are same in dimension, which unit is
*mN cm
, and the uni t of growth coe fficient B is
From the results (as shown in Figure 2), we can see that
the bone growth coefficient B diminished rapidly with
time, the changing trend can be simulated with index
function. The threshold value K showed an ascend trend
with time, it can be simulated with polynomial function
(as shown in Figure 3). Inversion results must be proved
by comparing with positive analysis [7], so these results
of B and K together with weight of comparison group
were substituted into bone gro wt h equation. The obtai ned
BMD-Time simulated curve was compared with the one
of comparison group of the experiment in the other year
(as shown in Figure 4).
The inversion results in this paper indicate that bone
growth coefficient B grows small with time. At week 10,
it approaches zero approximately, whereas factor K in-
creased remarkably at week 1, 2, 3, 4, and 5, at the week
6, 7, 8, 9 and 10, the increasing speed begins to decrease.
10 weeks later, it approaches a plane line. This illumi-
nated that the rapid growing period of bone went to an
Figure 2. Result and growing tendency of B.
y = A + B *x + C*x^2 + D*x^3
A4295.98531 ±1295. 62802
B-2014.40355 ±1307.21692
C750.08274 ±367.62169
D-52.47166 ±30.36522
parameter K
time (W)
Figure 3. Result and variation tendency of K.
BMD (g/cm^2)
time (W)
group of experiment
group of analysis
Figure 4. Comparison between analysis and experiment.
end and the bone entered a slow-growing stage. That is,
bone formation and absorption had reached a balanced
state and rat femur BMD had stopped rising basically.
The above inversion outcome is coincidence with the
actual growth condition of rats (as shown if Figure 4)
By now, human beings understand themselves far more
than enough, studies on the causes of diseases, treat-
ments and relationships between the human body and the
surrounding environments are greatly confined for the
lack of accurately-quantified model. In this paper, the
unknown parameters in the bone growth equation by
using inversion method based on animal experiments
were provided, the relationships between environmental
stress and bone growing in rats were also quantified. In
addition, the bone growth coefficient B was the function
of the time as well as the space. B and K were usually
regarded as constants [9] for easy calculation reason, but
it’s more reasonable to be regarded as functions of time
because bone growth coefficient and stimulus threshold
could vary during different growing periods. In clinics,
parameters B and K play important roles an d were closely
Equation: y = y0 + A1*exp(-(x-x0)/t1)
y0 0± 0
x0 1± 0
A1 6. 62E-7±3.9274E-8
t1 3.3043±0.36619
parameter B
time (W)
J. Zhang, W. Z. Zhao / Journal of Biosciences and Medicines 1 (2013) 19-22
Copyright © 2013 SciRes. OPEN ACCESS
related with treatments and disease recovery [10,11].
In this paper, some assumption and predigestion about
the environmental stimulus were investigated . The shape
of proximal femur is inalterable and BMD has uniform
distribution; Overloading group, underloading group and
comparison group were defined as equal parts of weight
respectively, outer stimulus was defined as a function of
weight and BMD. Actually, the anatomy of rat proximal
femur was irregular and the stress distribution was com-
plex. In order to get the data of objective and specific
condition of stimulus, we had to work on numerical re-
sult by creating a three -dimensional model of bone based
on CT images with the analysis of finite element method
software. This will be left to further research. The em-
phasis of this study is the idea of applying parameter
identification method to the quantification of bone
growth equation. Though outer stimulus was simplified,
but the result of both animal experiments and data analy-
sis was acceptable compared with that of reference [8].
The differences may exist during the whole experiments,
but the idea of creating numerical biological model by
combining animal experiments and inversion theory was
novel and it gave a perfect illustration of the creating of
the self-adaptation model of human body.
Nowadays, the inversion method is being applied ex-
tensively in natural sciences, engineering and many other
fields [6]. As known to all, in normal problems, we know
the reasons to seek the results, whereas in inversion, we
know the results to seek the reasons. The Human body is
a complex system and there exists extensive coupled
biomechanical phenomena, which is still largely immea-
surable and unknown. Therefore, a new requirement was
brought forward for modern medicine: by inversion me-
thod, namely according to some phenomena, measure-
ment results and pre-enacted mathematical mode, we can
deduce the unknown physical parameters and changing
rule by reversed process, modify it by mathematical me-
thod and investigate it in depth. With the development of
human society, life science is a new area that we must
encounter. If chemistry has driven the development of
life science in 20th century, the study of combining of
biochemics and biomechanics based on the inversion and
modern computing techniques will gain the achieve-
ments that exceed human’s imagination. This study is
one of the steps to create quantitative biomechanical
model of vivo bone and it could also provide clues and
references to establish biomechanical model of other
organs or t i s s u e s in humans.
[1] Feng, Y.Z. (1983) Biomechanics brief. Science Press,
Beijing. (in Chines e)
[2] Wang, D.G. (2001) Nonlinear inversion algorithms and
applications, doctoral dissertation of Dalian University of
Technology. Dalian University of Technology Press, Da-
lian. (in Chinese)
[3] Wolff, J. (1884) Das Gesetz der transformation der inne-
ren architectur der knochen bei pathologischen verände-
rungen der äusseren knochenform. Sitzg. Physik-Math,
Klausuren, Sitz. Ber. Pres s, Berlin.
[4] Tao, Z.L. (1996) Relationship between growth and stress-
one prospect of biomechanics, advances of biomechanics.
5th Science Congress of Biomechanics, Chengdu Univer-
sity of Technique Press, Chengdu, 15-17. (in Chinese)
[5] Wang, D.G., Liu, Y.X. and Li, S.J. (2003) Summaries of
elastic mechanics inversion method. Advances of Me-
chanics, 2, 166-174. (in Chinese)
[6] Huiskes, R., Weinans, H.J., Grootenboer, H.J., et al. (1987)
Adaptive bone remodeling rheory applied to prosthetic-
design analysis. Biomechanics, 20, 1135.
[7] Chinese Academy of Sciences (2000) Report on science
development. Science Press, Beijing. (in Chinese)
[8] Danielsen, C.C. (1993) Cortical bone mass, composition,
and mechanical properties in female rats in relation to age,
long-term ovariectomy, and estrogen substitution. Calci-
fied Tissue International, 52, 26.
[9] Zhu, X.H., He, G., Zhu, D., et al. (2002) A study of the
effect of non-linearities in the equation of bone remode-
ling. Journal of Biomechanics, 7, 951-960
[10] Zhao, W.Z., Liu, Y.X., Zhang, J., et al. (2003) Biomecha-
nical analysis of internal implant failure after internal fix-
ation of compressing steel plate. Journal of Medical
Biomechanics, 18, 50-53. (in Chinese)
[11] Lin, Y.J., Yang, J.W., Chen, J., et al. (2000) Kinesiatrics
for osteoporosis. Physical Medicine and Rehabilitation of
Foreign Medicine, 20, 97-101. (in Chinese)