Animal bone growth experiment of rapid-growing rats in different stress environment and its mathematical model

doi:10.4236/jbm.2013.12005

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Journal of Biosciences and Medicines, 2013, 1, 19-22 JBM

http://dx.doi.org/10.4236/jbm.2013.12005 Published Online October 2013 (http://www.scirp.org/journal/jbm/)

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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

Email: [email protected]

Received 2013

ABSTRACT

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

Density

1. INTRODUCTION

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.

2. METHOD OF ANIMAL EXPERIMENTS

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

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

Analysis

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. METHOD OF PARAMETER

IDENTIFICATION FOR BONE

GROWTH EQUATION

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

Group

Experiment 181.0 ± 16.0 186.2 ± 17.9 194.9 ± 17.4 200.8 ± 18.0

Group

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

Group

Experiment 177.5 ± 14.7 188.9 ± 15.6 199.0 ± 16.2 204.5 ± 16.4

Group

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

Group

Overloading 0.2378 ±

0.0473 0.2584 ±

0.0362 0.2746 ±

0.0440 0.2902 ±

0.0418

Group

unloading 0.2508 ±

0.0476 0.2584 ±

0.0410 0.2618 ±

0.0471 0.2649 ±

0.0469

Group

comparison 1 0.2328 ±

0.0333 0.2518 ±

0.0412 0.2665 ±

0.0485 0.2805 ±

0.0425

Group

comparison 2 0.2538 ±

0.0294 0.2757 ±

0.0475 0.2908 ±

0.0517 0.3064 ±

0.0441

Week 5 Week 6 Week7 Week 8

Group

Overloading 0.3002 ±

0.0600 0.3086 ±

0.0453 0.3155 ±

0.0512 0.3179 ±

0.0578

Group

unloading 0.2635 ±

0.0616 0.2609 ±

0.0472 0.2583 ±

0.0580 0.2529 ±

0.0599

Group

comparison 1 0.2888 ±

0.0432 0.2941 ±

0.0575 0.2977 ±

0.0509 0.2980 ±

0.0484

Group

comparison 2 0.3155 ±

0.0459 0.3214 ±

0.0552 0.3249 ±

0.0427 0.3248 ±

0.0430

in Eq.1) [3].

()

dBS K

= −

(1)

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

∆=− ∆

(2)

We defined:

min ()

ρρ

=∆ −∆

∑

exp

(3)

Then, the object function for identifying B and K was

generated.

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

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

Week mNcm

∆

4. CONCLUSIONS

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.

02468

2000

3000

4000

5000

6000

7000

8000

9000

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.

02468

0.24

0.25

0.26

0.27

0.28

0.29

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)

[8].

5. DISCUSSIONS

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

02468

0.0000000

0.0000001

0.0000002

0.0000003

0.0000004

0.0000005

0.0000006

0.0000007

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

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.

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