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Vol. 46 No. 3 SCIENCE IN CHINA (Series G) June 2003

Biomechanical properties of soft tissues

ZENG Yanjun (mAX)1,2, XU Chuanqing (0)1,2, YANG Jian (M )2

& XU Xiaohu ()1

1. Medical School, Shantou University, Shantou 515031, China;

2. Biomedical Engineering Center, Beijing Polytechnic University, Beijing 100022, China

Correspondence should be addressed to Zeng Yanjun (email:[email protected])

Received September 20, 2002

Abstract Viscoelasticity is the primary mechanical property of bio-soft tissues. It has been widely

applied in basic research of biological tissues including cornea, lung, heart and blood vessels.

Along with the development of tissue engineering research, the evaluation of soft tissue viscoelas-

ticity is becoming more and more important. In this paper, using the Whittaker function, we give an

approximate power series of the exponential integral E1(x) and the parameters c, 1 and 2 of the

generalized relaxation function G(t) and generalized creep function J(t). With expanded skin as an

example, the relationship between stress relaxation, creep and stress-strain finite deformation are

studied.

Keywords: bio-soft tissue, viscoelasiticity, creep, stress relaxation, stress-strain relationship.

All biosolids are bio-viscoelastic bodies. However, there are differences in the viscous and

elastic properties among different tissues and organs. Biosolids can be divided into hard tissues

(such as cartilage and bone) and soft tissues (such as blood vessels, nerves, muscle and skin). The

hysteresis curves of soft tissues have a salient feature, i.e. the hysteresis loop is almost independ-

ent of the strain rate, even if the variation of the rate enlarges a thousand times. This insensitivity

is incompatible with any viscoelastic model that consists of a finite number of springs and damp-

ers because such a model will have a discrete relaxation spectrum. If the vibrations and dampers

reach a maximum at certain frequencies, the insensitivity is incompatible with the bio-material

properties. Therefore to describe soft tissues, one must resort to the concept of continuity. Fung et

al. have established the generalized reduced relaxation function G(t) and reduced creep function

J(t)[1]

. In the equation of G(t) and J(t) there are three parameters 1, 2 ,3/.. According to the

generalized stress relaxation function, using G(t) we can obtain the expressions of1, 2and c[2,3]

.

However, in the generalized creep function J(t) given by Fung, E1(t) is an exponential integral

function. It is impossible to obtain its precise elementary functional expression. In this paper, us-

ing the Whittaker function, we obtain an approximate power series of the exponential integral E1(t)

and the generalized creep function t equal to the value of J(t) at 1 s and the expressions of pa-

rameters 1, 2,.. A bio-soft tissue belongs to finite deformation except that its stress-strain rela-

tionship has viscoelasticity. In a previous paper[5]

, by means of Lagrange stress and Green strain

we analyzed the biomechanics of cornea[5]

. In order to analyze finite deformation, here we adopt

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No. 3 BIOMECHANICAL PROPERTIES OF SOFT TISSUES 285

Euler stress and Green strain methods. Bio-soft tissues, such as artery, skin and muscle have simi-

lar biomechanical properties. They all have hysteresis, relaxation, creep and anisotropism proper-

ties and nonlinear stress-strain relationships. Skin is a bioviscoelastic material. This property in

expanded skin frausplants will change with time after grafting. Skin expansion has become a

widely utilized technique in plastic surgery. By subcutaneously implanting a balloon, the skin

overlying the balloon is mechanically expanded and its surface area increased. The objective of

this technique is to gain extra soft tissues for reconstructive purposes[610]

. Clinical experience

tells us that if the expander is left beneath the skin for a period after expansion, the stretch-back

ratio of the expanded skin can be reduced significantly, and the maximum expansion can be

achieved. However, these observations have not been studied theoretically. Using a dog model

(whose skin is similar to human skin in biomechanical properties) as a typical example of soft tis-

sue biomechanical property analysis, this paper compares the biomechanical propertiesin vitro

with different stretch maintaining durations.

1 Material and methods

Six adult dogs weighing over 20 kg were utilized. Six 5 8 cm2

skin areas were marked with

an electric tattoo instrument on each animal on the rib cage. The specimens were divided into dif-

ferent groups: rapidly expanded group (E = expanded, every other day for two weeks);

sham-operated group; (the balloons are inserted but not inflated); and control. Expanded groups

are fruter divided into subgroups according to the maintaining time after expansion: m1 (main-

tained for one week), m2 (maintained for two weeks), m4 (maintained for 4 weeks). Cut 4 strips

with the same size area of 1 4 cm2. Then the stress-strain relationship, creep characteristics,

stress relaxation and tensile strength of the specimens were tested on an INSTRON material ma-

chine. The experiment was performed at ambient temperature.

To obtain the stress-strain relationship, the specimens were loaded and unloaded at a constant

rate of 20 mm/min for three cycles. The hysteresis loop decreased between successive cycles. Af-

ter three cycles, the specimens were regarded as preconditioned. In mechanical testing, soft tissues

need to be preconditioned because only after an initial period of adjustment following a large dis-

turbance, will the physical properties be stable. If a certain load is suddenly exerted on the tissue

and the stress is kept constant afterwards, the tissue will continually deform. This phenomenon is

called creep. In this test, a constant load of 20 N was applied for over 20 min. In the test of stress

relaxation, the tissue was loaded to a finite strain and its length was kept constant, and the stress

decreased with time. In this study, the elongation speed was 125mm/min, and the stretched length

was maintained for 20 min.

2 Result and analysis

2.1 Creep

In order to give a uniform description of different states of strain, we normalize the creep

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286 SCIENCE IN CHINA (Series G) Vol. 46

curves, namely consider the stretch length as 1, and

the other length as the relevant value. Shown in fig.

1 are the generalized creep curves of specimens in

vitro with mean values of all the experimental re-

sults and various maintaining durations.

Fig. 1 shows that the curves of expanded

specimens are higher than the curves of their con-

trols and sham controls; the sham specimens are

between the control and the expanded specimens.

However the maintaining time has great effect on

the extent of creep. Significant difference exists in

the groups with one week maintaining duration and

its control, but this difference reduces with the

maintaining time prolonged to 2 weeks or 4 weeks.

Like other soft tissues, the hysteresis loop of skin is

almost independent of the strain rate. This insensitivity is incompatible with any viscoelastic

model that consists of a finite number of springs and dashpots. That is to say, it is improper to de-

scribe skin with discrete creep spectrum. One should consider a continuous creep spectrum.

The generalized reduced creep function is[1]

1 12 1

2

1

1

( ) .

1 ln

t tc E Ec c

J tc

cc

+ + =

+

+

(1)

Here1

e( ) .

t

x E x dt

t

= Because 1( )E x is an exponential integral function, we cannot get the

elementary expression. Use

1

2 21 1

,02

( ) e ( )

x

E x x W x

= , where 1

,02

( )W x

is Whittaker function.

When x is much small,1

1

( )( ) ln ,

!

k

k

x E x x

k k

=

= where is an Euler constant, and

1 1 11 lnlim

2 3nn

n

= + + + +

; when x is very large,

1

1

e !( ) ~ 1 .

x

k

k

kE x

x x

=

+

Because 2 is

much larger than 1 and 1 is much smaller than 1,1

t

c +is relatively large, so 1

1

tE

c

+ is

approximately approaching 0; however,

2+c

t is relatively small, so

Fig. 1. Reduced creep curves J(t) of expanded skin and

its control. E2, Expanded every other day for two weeks,

mi, maintained for i week; e.g. E2m4, expanded every

other day and maintained for 4 weeks.

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21

12 2

ln!

k

k

t

ct tE

c c k k

=

+ = + +

. (2)

For convenience in calculation, using the first order linear equation to approximate 12

,t

Ec

+

namely the value t of the first order of 1 s, we obtain

2

2

1

1 ln

( )

1 ln

tc c

cJ t

cc

c

+ ++

=+

+

, (3)

2

1

1

( )1 ln

J cc

c

= +

+

, (4)

2

1

ln1 ln

dJ c

cd tc

c

=+

+

. (5)

Using (3)(5), we get

2, ,

ln ( ) 1 ( )exp( )

dJ tc c

d t JJ t

c J c

= =

+

i

1 2

1 1( ) exp .

( )

c ccJ c

= +

The creep parameters c, 1 and 2 in the experimental test are listed in table 1.

Table 1 Parameter c, 1 ,3/2 of continuous creep spectrum of the expanded skin and control specimens

Specimen c 1/s 2/s

Em1 0.0032B0.0011 A 0.0432B0.0080 A 52.3622B7.1322 A

Em2 0.0025B0.0008 B 0.0548B0.0062 A 55.2890B4.2680 A

Em4 0.0024B0.0009 B 0.0597B0.0075 A 62.8166B7.9236 B

sham 0.0022B0.0006 B 0.0684B0.0071 B 75.6169B9.1142 B

Control 0.0020B0.0004 B 0.0880B0.0098 B 85.0273B9.8365 B

This result shows that creep levels rely on the maintaining time. Significant difference exists

between the groups with one week maintaining duration and its control, but this difference reduces

when the maintaining time is prolonged to 2 or 4 weeks.

2.2 Stress relaxation

In order to give a uniform description of different states of stress, we normalize the relaxation

curves. The generalized relaxation curves of specimens in vitro with mean values of all the ex-

perimental results and various maintaining durations are shown in fig. 2. Fig. 2 shows that the

curves of their controls and sham controls are higher than the curves of expanded specimens, and

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the sham specimen is between the control and the

expanded specimens, which means that the ex-

panded skin easily gets relaxed. The curves for one

week maintaining are far from those of the control

specimen, but with the prolonging of maintaining

time, the curves approach those of the sham and

control specimens. The curves of 4 weeks maintain-

ing is almost near that of the sham specimen.

The generalized reduced relaxation function

G(t) is

1 2 1 1

2 1

{1 [ ( / ) ( / )]}( ) .

[1 ln( / )]

c E t E t G t

c

+ =

+(6)

According to ref. [1], for tu , E1(t/2), E1(t/1)

approaches to 0. Then G( ) is obtained. For values

t of the order of 1 s we have the value of G(t). The

slope of the relaxation versus the logarithm of time is dG/dlnt. Then 1, 2 and c can be obtained

2 1 2

( ) 1 ( ) 1 1, exp , exp

( ) ( )( ) ln( )

dG t G tc t

c cG cG cG d t

= = =

.

All the relaxation parameters c, 1 and 2 in the experimental test are listed in table 2.

Table 2 Parameters c, 1 and 2 of continuous relaxation spectrum of the expanded skin and control specimens

Specimen c 1/s 2/s

Control 0.0590B0.0067 B 0.0273B0.0075 B 8.9835B0.9389 B

sham 0.0756B0.0052 B 0.0560B0.0041 B 9.7445B0.8735 B

Em4 0.0820B0.0078 B 0.0880B0.0089 B 12.7507B0.9917 A

Em2 0.0954B0.0056 A 0.1331B0.0097 A 13.4371B1.0049 A

Em1 0.0993B0.0091 A 0.1791B0.0103 A 18.8236B1.3394 A

This result shows that the maintaining time has great effect on relaxation levels. Significant

difference exists in the groups with maintaining duration of one week and its control, but this dif-

ference reduces when the maintaining time is 2 or 4 weeks.

2.3 Stress-strain relationship of finite deformation

Because skin is a soft tissue, in the calculation it belongs to finite deformation. Therefore us-

ing the stress-strain relationship of finite deformation. The stretch-back ratio is = L/L0, Euler

stress0

,F F

TA A

= = = T = F/A0 is Lagrange stress, Green strain is21 ( 1),

2 F is the

force on the specimen, A is the sectional area, A0 the original area, L the length of specimen, and

L0 the original length of the specimen.

Fig. 2. Reduced relaxation curves G(t) of expandedskin and its control. E2, expanded every other day for

two weeks; mi, maintained for i week; E2m4; expanded

every other day and maintained for 4 weeks.

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The curves of control, sham and ex-

panded specimens after 6 weeks in vitro with mean

values of all the experimental result and various

maintaining durations are shown in fig. 3.

Fig. 3 shows that the stress-strain curve of the

expanded specimen deviates from that of its control

most with one week maintaining duration, but the

deviation reduces with the increase in maintaining

duration. The following power function:

= (m)

(7)

is used to fit the stress-strain curves of the speci-

mens. m is the original stretch of a specimen; and are coefficients to be determined. The fitted values

of,about the curve of the in vitro specimens in experimental test are listed in table 3.

Table 3 The fitted parameters about the stress-strain curves of expanded skin

Specimen /MPa

control 227.03f18.21 2.61f0.73

sham 186.39f31.47 2.62f0.52

Em4 130.24f27.39 2.62f0.49

Em2 105.05f10.84 2.63f0.64

Em1 85.22f

11.73 2.63f

0.42

From table 3 we found that the value of for an expanded specimen is smaller than that of

controls, which is in accordance with the figures. But there is no apparent change in the value of

.

3 Conclusion

Viscoelasticity such as hysteresis in stress-strain, stress relaxation and creep are very com-

mon in soft tissues. The study of these characters can be very rewarding. In this paper, we

provided an effective method of quantifying the time-dependent change in creep and stress

relaxation, and used it to evaluate the biomechanical change in expanded skin.

Skin expansion technique in the plastic surgery is to utilize the viscoelastic characteristics of

soft tissue to gain the additional tissue. After the implanted balloon is inflated to a specific pres-

sure, the stress of the expanded skin begins to relax. After some time, the internal pressure in bal-

loon reduces to a specific level, and new inflation can be done again. This cycle is just a new ex-

erted force and stress-relaxation cycle. In some other skin expansion regimens, the balloon pres-

sure is kept constant by some devices: in this case, a typical creep is going on the skin.

One aim of this study is to explore how the biomechanical properties change after different

skin expansion regimens. The changes were quantified in terms of some parameters in

Fig. 3. Stress-strain relationship curves of the expanded

skin and its control specimen. E2: expanded once in a

week for 6 weeks; mi, maintained for i weeks; E6m4,

expanded once for a week and maintained for 4 weeks.

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stress-strain, stress-relaxation, and creep analysis. We found that the expanded skin became softer,

as is characterized by right shifted strain-stress relation curves, more relaxation and more creeping.

However, rapid expansion and conventional expansion method do not show significant difference

in these parameters. Another aim of this study is to investigate whether it is of any good to main-

tain the balloon in the tissue for some time after expansion stops. Our study shows that a pro-

longed maintaining time helped the expanded skin to retain its normal biomechanical properties.

In conclusion, rapid expansion and prolonged maintenance time are desirable in the plastic sur-

gery practice.

Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No.

10072006).

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