XXVI Szkola Tribologiczna

5-2003
TRIBOLOGIA
Krzysztof WIERZCHOLSKI*
369
POWRÓT
TRIBOLOGY OF BONE TISSUE CULTURE
IN BIOREACTOR
TRIBOLOGIA HODOWLI TKANKI KOSTNEJ
W BIOREAKTORZE
Key words:
friction forces,perfusion, grow of tissue
6áRZDNOXF]RZH
VLá\WDUFLDSROHZDQLHZ]URVWWNDQNL
Summary
This paper describes the tribological parameters during the perfusion process of tissue in the bioreactor. We perfuse cells of tissue by the nutrition
liquids and other biologically tolerable media with oxygen carrying fluorocarbons. There are two different kinds of flows in the bioreactor to be
expected: The first one is the Newtonian potential flow in some distance
from tissue surface and is caused by horizontal perfusion. The second is the
viscous non-Newtonian flow in boundary layer direct near the motionless
tissue surface. Liquid velocity components and pressure values will be de*
Gdynia Maritime University ,Faculty of Marine Eng. and Technical University
*GDVN)DFXOW\ RI 2FHDQ (QJ6KLS 7HFK XO 0RUVND *G\QLD
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termined. Analytical calculations deliver results for flow parameter as flow
components and friction forces to the optimisation of tissue culture in bioreactor and to optimise the growth of cells. Present paper shows analytical
formulae for velocity components of nutrient fluids in thin boundary layer
near to the tissue surface to obtain necessary friction forces. Here are derived analytical formulae of tribological parameters in bioreactor and the
necessary volume flow rate of stream of media in horizontal direction for
good protection of the perfusion of tissue surface. The friction forces occurring in the thin layer of nutrition liquids resting on the superficial tissue
layer are examined by numerical way and are illustrated in the pictures.
AIM OF THE PAPER AND SKETCH OF COMPUTATIONAL
METHOD
The transplantation of cartilage in human hip joint is more profitable than
transplantation of total artificial joint. The culture of tissue is performed
in bioreactor. A bioreactor was designed and developed with the aim of
perfusing the scaffolds with the culture medium through their fibre
structure in order to expose the cells located in the entire scaffold
thickness both to convective solute transport, to a flow-induced shear
stress and friction forces. Fig. 1 depicts a schematic picture of device.
Fig. 1.
Cell culture reactor and boundary conditions of single layer potential flow
around motionless tissue body during the perfusion with the biological
liquid, a≤xs≤b, c≤zs≤d, 0≤ys≤∞
Rys. 1. Reaktor hodowli komórkL RUD] ZDUXQNL EU]HJRZH GOD SU]HSá\ZX SRWHQFMDOQHJR
ZDUVWZ\ SRMHG\QF]HM ZRNyá QLHUXFKRPHJR FLDáD WNDQNL SRGF]DV SROHZDQLD FLe-
F] ELRORJLF]Q
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371
The co-ordinate system xs,ys,zs is stiffly fixed with the plane
(xs0szs).It is the lower surface of potential flow. Between the lower surface of potential flow and the tissue surface occurs the thin boundary layer.
The distance between these surfaces is ε≈ 30µm -70µm. The motionless
external tissue surface F∗≈F is continually under horizontal perfusion by
the biological liquid, which is supplied by the source with flow rate Qo
distant H from xs,zs plane see Fig. 1.
The flow of biological nutrient fluid around the tissue surface creates
the very small friction forces, which have significant meaning during the
growth process i bioreactor. The control of such small friction forces during the growth process is necessary. The measurement of mentioned
small friction forces is very expensive, moreover introducing of measurement devices into the bioreactor has negative consequences on the growth process of tissue. Therefore in this paper is presented the computational model of simulation and determination of tribological friction forces in bioreactor during the growth process. Presented model enables the
control operation and optimization of proper friction forces on the tissue
surface. Friction forces in bioreactor are caused by the flow rate of perfusion of nutrient fluid and by the suction and boosted squeezing of the
biological fluid into porous structure of the tissue.
A computational model consists of two parts. A first part presents the
computational model of potential flow of nutrient fluid in the some distance from the tissue surface.
In the sufficient large distance from the lower surface, all fluid velocity components are equal zero [L. 1–4]. The fluid velocity component
Vx,Vz tangential to the lower surface of potential flow, in horizontal
directions are different from zero. The vertical component of fluid velocity Vy on the level ys = 0 is equal zero.
here a in m, b in m, c in m, d in m are the limits of region F. Simple
potential function φ in m2/s determines fluid velocity components in the
form [L. 7]:
Vx(xs, ys,zs) =
∂φ
∂φ
∂φ
Q
, Vy(xs, ys,zs) = , Vz(xs, ys,zs) = , φ ≡
, ∇2φ = 0
2 2 2
∂xs
∂ys
∂zs
xs + ys +zs
whereas: Q – integration constant in m3/s .
(1)
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We perfuse the total surface of tissue. Therefore absolute value of the
resultant vector of velocity on the tissue surface multiplied by the value
of surface equals flow rate Qo in m3/s of supplied fluid. Hence we have
condition:
2
2
2
∫∫ Vx (xs, ys = 0, zs ) +Vy (xs, ys = 0, zs ) +Vz (xs, ys = 0, zs )dxsdzs = Qo
F
(2)
where: F(xs,zs): {a≤xs≤ b, c≤zs≤ d} is the perfuse region of tissue surface in m2.
STUDY OF COMPUTATIONAL MODEL OF FRICTION
FORCES
For a new coordinate system (x,y,z) following relations are valid: ys=y-ε,
xs=x, zs=z. Axis x is parallel to the axis xs and axis z to the axis zs. Axis y
coincides with ys. Continuity, conservation of momentum equations and
stress-strain relations have the form [L. 4], [L. 5], [L. 6], [L. 7]:
div(ρv)=0, DivS=0, S=-pI+ηA1+αA1A1+βA2,
(3)
where:v (vx,vy,vz) – velocity vector of the fluid in boundary layer, I-unit
tensor, S-stress tensor in the biological fluid in Pa, A1, A2 strain tensors,
η-dynamic, viscosity in Pas, α, β-empirical coefficient, which describes
viscoelastic properties in Pas2.
Gap is limited by the inequalities: a≤x≤b, 0≤y≤ε, c≤z≤d. For Newtonian fluid (α=0,β=0). Boundary conditions are graphically presented in
Fig.2 for fluid flow near the tissue body caused by horizontal perfusion,
vertical grow and suction effect of porous tissue. Lower thin layer surface
coincides with upper tissue surface. We have following boundary conditions:
*
vx(x = 0, y = 0, z = 0) = 0, vy(x = 0, y = 0, z = 0) = ∂ε /∂t +
(-ρg + ∂pp/∂y)ck/η, vz(x = 0, y = 0, z = 0) = 0
(4)
where: pp – pressure in porous, ck – coefficient in m2 of penetration of
biological fluid to the porous external surface of tissue, g – acceleration
of gravity in m/s2, t – time in s, ε*– height of superficial layer of tissue in
m,ε - height of boundary layer in m (see Fig. 2). Upper surface of bounda-
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373
ry layer coincides with the lower surface of region of potential flow. In
this place fluid particles move in x and z directions and values of these
velocities we can determine from solutions (1). The horizontal velocity
component in potential flow on the lower surface of potential region must
have the same values as horizontal velocity component of viscous flow in
upper surface of boundary layer. Hence:
vx(x,y=ε,z)=Vx(xs=x,ys=0,zs=z),
vz(x,y=ε,z)=Vz(xs=x,ys=0,zs=z).
(5)
Fig. 2. Boundary conditions for horizontal perfusion with suction, grow, weeping
Rys. 2. Warunki brzegowe dla poziomego polewania ze ssaniem, wzrostem i przesczaniem
Now we assume dimensionless values:x1,y1,z1, ψ1,ψ,a1,b1,c1,d1 and
dependencies between dimensional and dimensionless values respectively:
x≡Lx1, y≡εy1, z≡Dz1, κ1≡D/L, ψ1≡ε/L, ψ≡ε/D, a≡La1, b≡Lb1, c≡Dc1, d≡Dd1 (6)
We assume following notations: L = b – a length of tissue surface
under perfusion, D = d – c wide of tissue surface under perfusion, F = LD
area of tissue surface under perfusion, whereas L in m and D in m and
F1{[a1≤x1≤b1]× [c1≤z1≤d1]}, 0≤y1≤1.We denote: x1 – dimensionless coordinate in length direction, y1 – dimensionless coordinate in gap height
direction, z1 – dimensionless coordinate in wide direction, a1 – first value
in length direction, b1 – second value in length direction, c1 – first value
in wide direction,d1 – second value in wide direction. We integrate equ-
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ations (3) with respect to the variable y for α=0,β=0,ψ≡ε/(d – c)≈0,0001,
after boundary layer simplifications,. Continuity equation we integrate
once with respect to the variable y. Afterwards we impose boundary conditions (7),(5) on the fluid velocity components. Hence velocity component in boundary layer near to the tissue surface have the following form:
(
)
(
)
v x (x1 , y1 , z1 ) =
ρε 2 Q 2
Q
X 1 (x1 , z1 )× y12 − y1 −
X 2 (x1 , z1 )× y1 , (7)
5
2η L
L2
v z (x1, y1 , z1 ) =
ρε 2 Q 2
Q
X 4 (x1, z1 )× y1 , (8)
X 3 (x1 , z1 )× y12 − y1 −
2η L5
L2
v y (x1, y1, z1 ) =
(
)
εQ
ρε3 Q2
X5 (x1, z1 ) × 2 y13 − 3y12 −
X 6 (x1, z1 ) × y12 −
6
3
12η L
2L
c
− k
η
X 1 (x1 , z1 ) ≡ 2 x1Y13 ,

 ∂ε *(t)
 ρg − ∂p p
+

∂y y=0 
∂t


(9)
X 2 (x1, z1 ) ≡ x1Y11,5 , X 3 ≡ 2κ 1z1Y13 ,
X 2 (x1, z1 ) ≡ κ 1z1Y11,5
X 5 (x1 , z1 ) ≡ 8Y13 ,
(10)
(
X 6 (x1 , z1 ) ≡ Y11,5 , Y1 ≡ x12 + κ 12 z12
)−1
(11)
where: 0≤y1≤1, a1≤x1≤b1, c1≤z1≤d1,κ1 = D/L, Q – constant flow ratio in
m3/s.
By virtue of solutions (7), (8), (9) follows, that if flow ratio Q tends
to zero i.e. if we have not external perfusion, then flow in boundary layer
in horizontal plane not exists i.e. vx = 0, vz = 0 except the vertical velocity
component which is different from zero vy ≠ 0 because this component
depends on suction and vertical grow of porous tissue. For the fluid flow
around the tissue surface in boundary thin layer we have that vertical
component vy of fluid velocity vector different from zero in lower surface
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375
of the thin layer and equals zero in the upper surface of boundary thin
layer i.e. in plane y = ε ⇔ y1 = 1, hence we write:
1
∫∫ v y (x 1 , y1 = 1, z1 )dx1dz1 = 0
F1 F
(12)
1
Imposing condition (12) on the solution (9) we obtain following
equation:
c
ρε 3 Q 2
εQ
−
X 5c −
X 6c − k
6
3
η
12 η L
2L


*
∂p p
 ∂ε ( t )

+
ρ
= 0 (13)
−
g


∂t
∂y
=
y
0


X ic ≡ ∫∫ X i (x1, z1 )dx1dz1 ; for i = 1, 2, 3, 4, 5, 6,
F1
where: F1{a1≤x1≤b1,c1≤z1≤d1} (14)
From (13) follows, that unknown constant flow ratio Q in m3/s has
the following form:
U L2
Q=−
Γc Sηk ,
2
X
6ηL
Γc ≡ 6c , U ≡
,
X 5c
ρε 2


X ε
Ga  ∂p p
Sηk ≡ 1 + 1 + 8(Str) 5c  o sest1 +
− 1

6Str  ρg∂y
X 62c  ε
y =0



(15)
where we assume following assumptions for dimensionless time t1, dimensionless superficial layer ε1, dimensionless grow coefficient s,
Strouhal Number, Galileus Number:
t1 = t/to ε1(t1) ≡ exp(st1), ε* = εoε1, 0 < s < 1, Str ≡ L(Uto)-1,
Ga ≡ ckεgρ2/(η2)
(16)
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If we put constant (15),(16) in solutions (7), (8), (9), hence we obtain
velocity components of nutrient fluid in bioreactor in following final
form:
v x (x1 , y1 , z1 ) = Uv x1 = U 0,75 × Γc2 × Sη2k × X 1 (x1 , z1 )× y12 − y1 +
{
(
}
+ 0,50 × Γc × Sηk × X 2 (x1 , z1 )× y1 ,
)
(17)
{
(
)
v z (x 1 , y1 , z1 ) = Uv z1 = U 0,75 × Γc2 × Sη2 k X 3 (x1 , z1 )× y12 − y1 +
)+ 0,50× Γc × Sηk X 4 (x1, z1 )× y1},
[
(18)
(
)
1
1
v y (x1, y1, z1 ) = Uψv y1 = Uψ1 0,25× Γc2 ×Sη2k X5 (x1, z1 )× 2y13 − 3y12 +
2
2
+ 0,50× Γc × X6 (x1, z1 )Sηk × y12 +
(19)



ε o st 1 
1
1 ∂p p


+ 2 (Str )
− ( Ga )  1 −
se
,

ε
ρ g ∂y y = 0 
3




where: 0≤y1≤1 a1≤x1≤b1, c1≤z1≤d1,ψ1≡ε/L and U denotes dimension velocity.
Friction forces in x and z directions occurring in the upper surface of
boundary layer we obtain from the following formulae:
 ∂v
FRx = ∫∫  η x
F  ∂y

 ∂v

dxdz, FRz = ∫∫  η z
 y=ε
F  ∂y


dxdz,
 y=ε
(20)
Putting solutions (17), (18) in formulae (20), thus for F1{a1≤x1≤b1,
c1≤z1≤d1}.we obtain:

FRx = FRo  0,75 × Γc2 × ∫∫ S η2 k X 1 ( x 1 , z 1 )dx 1 dz 1 +

F1

+
×
Γ
×
0
,
50
S
X
(
x
,
z
)
dx
dz
(21)
∫∫
ηk 2 1 1
c
1 1
1

F1
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377

FRz = FRo  0 , 75 × Γ c2 × ∫∫ S η2 k X 3 ( x 1 , z 1 ) dx 1 dz 1 +

F1

+ 0 ,50 × Γ c × ∫∫ S η k X 4 ( x 1 , z 1 ) dx 1 dz 1 

F1
(22)
If pressure in porous pp is neglected and Ga=O(10-2), then from equation (15) follows:
X ε
Sηk ≡ 1+ 1+ 8(Str ) 5c o sest1
X62c ε
(23)
and friction forces (21),(22) have the following form:
FRx = FRo FRx 1 , FRz = FRo FRz 1 ,
FR Σ ≡ FRo
F2
Rx 1
+ F2
Rz 1
,
FRo ≡
6 η 2 L2 D
ρε 3
(24)
FRx1 ≡ 0,75 × Γc2 × X1cSη2 k + 0,50 × Γc × X 2 c × Sηk ,
FRz1 ≡ 0,75 × Γc2 × X 3c × Sη2 k + 0,50 × Γc × X 4 c × Sηk
(25)
FRICTION FORCES CALCULATIONS
Now we examine numerically time depended friction forces during the
time of growth of tissue. In this example we are neglecting pressure in
porous. In calculations we take following data: height of fluid boundary
layer flowing around a tissue: ε=30×10-5m, height value of superficial
layer of tissue:εo=10-3m.Nutrient dynamic viscosity has value: η=10-3Pas.
Fluid density magnitude equals ρ=1000kg/m3.We assume following dimensionless growth coefficients s=0,08;0,10;0,12;Average time value of
tissue growth to=1000s for dimensionless time interval of tissue
growth:0<t1≤40. Length of tissue equals L = 0,10m, wide of perfuse tissue is D = 0,10m,hence κ1=D/L=1,0. We assume that the region of tissue
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surface determine following inequalities: a1≤x1≤b1, c1≤z1≤d1 for
a1=1,b1=2, c1=1,d1=2.Hence from (14) we obtain: X1c=0,0422,
X2c=0,1621, X3c=0,0422 X4c=0,1621, X5c=0,1274, X6c=0,1147.From
equations (15),(16),(23),(24),(25) we obtain following formulae and values:
U=
6ηL
ρε
Str =
FRo =
2
=
6 × 10 −3 Pas × 10 −1 m
1000 kg / m × (30 × 10
3
−5
m)
2
=
2
× 10 m / s;
3
(26)
L
0,10 m
=
− 1,500 × 10 − 5 ,
Ut o ( 2 / 3) × 10 m / s × 1000 s
6η 2 L2 D
ρε 3
=
6 × (10 −3 m) 2 × (10 −1 m) 3
1000kg / m 3 × (30 × 10 − 5 m) 3
2
= 0,22 N
(27)
( )
3  0,1147 
0,1147
× 0,16212 × Sηk (28)
FRx1 = FRz1 = 
 × 0,0422 × Sηk 2 +
4  0,1274 
2 × 0,1274
thus
FRx 1 = FRz 1 = 0 , 0255 S η2 k + 0 , 0729 S η k ,
S ηk = 1 + 1 + 8 ×
(29)
3
0,1274
10 − 3 m
1
× 10 − 5 ×
×
×
× e s× t1
−
2
5
2
10
(30)
( 0,1147 )
30 × 10 m
= 1 + 1 + 0,00391 × e s × t 1 .
Dimensionless velocity distribution in dimensionless height y1=0,50
of thin layer has by virtue of solution (17) the following form:
v x 1 ( x 1 , z 1 , t 1 ) = v x / U = − 0 , 3039 × S η2 k ×
+ 0 , 225 × S η k ×
x1
(x 12 + z 12 )1,5
x1
(x 12 + z 12 )3
(31)
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379
1≤x1≤2, 1≤z1≤2. Fig. 3 and Fig. 4 shows dimensionless velocity distributions for s = 0,10 and s = 0,08 for dimensionless times: t1 = 10; t1 = 20;
t1 = 30; t1 = 40 respectively.
To obtain dimensional velocity values we must multiply values obtained from the formula (31) or values presented in Fig. 3, by the dimensional velocity factor U=6,67 m/s.
Vx 1
z1
Vx 1
z1
s = 0.10; t1=10
s = 0.10; t1=20
2.0
2.0
0.038
0.038
0.038
0.038
1.8
1.8
0.042
0.042
0.042
0.042
0.047
0.047
1.6
1.6
0.047
0.047
0.051
1.4
1.4
0.056
0.051
0.051
0.056
0.051
0.056
0.056
1.2 0.038
0.060
0.060
0.033
1.0
1.2 0.038
x1
1.2
1.0
1.4
1.6
1.0
2.0
1.8
Vx1
z1
s = 0.10; t1=30
2.0
0.046
0.042
s = 0.10; t1=40
0.051
0.046
0.056
0.051
x1
1.8
0.056
1.2 0.035
0.060
1.6
0.056
0.051
0.056
0.060
1.4
0.046
0.040
1.4
1.2 0.037
0.032
0.040
1.6
0.051
0.046
2.0
1.8
1.8
1.6
1.2
1.6
Vx1
z1
0.042
1.0
1.4
0.035
1.8
1.0
x1
1.2
1.0
2.0
0.037
0.037
1.4
0.060
0.060
0.033
2.0
0.061
0.061
0.030
1.0
1.0
x1
1.2
1.4
1.6
1.8
2.0
Fig. 3.
Dimensionless velocity distribution in the dimensionless distance y1 = 0,50
from the external tissue surface for grows coefficient s = 0,10 in dimensionless times t1 = 10; 20; 30; 40
Rys. 3. %H]Z\PLDURZH UR]NáDG\ SU
GNRFL Z EH]Z\PLDURZHM RGOHJáRFL \1 = 0,50 od
]HZQ
WU]QHM SRZLHU]FKQL WNDQNL GOD ZVSyáF]\QQLND Z]URVWX V Z EH]ZyPLDURZ\FK RVW
SDFK F]DVX t1 = 10; 20; 30; 40
For dimensionless time: t1 = 0, t1 = 10, t1 = 20, t1 = 30, t1 = 40 and
s = 0,10, we obtain: from the formula (30) following dimensionless values Sηk = 2,000; 2,005: 2,014; 2,038; 2,100 respectively, from the formulas (24), (25) we obtain dimensionless values of friction forces:
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FRΣ/FRo= 0,3509; 0,3518; 0,3540; 0,3600; 0,3759, from the formula (16)
we have following dimensionless values of the height of superficial
layer:ε1 = 1,000; 2,7183; 7,3891; 20,0855; 54,5982 respectively. To obtain realistic values of friction forces we must multiply dimensionless
values of friction forces by the factor FRo = 0,22 N. To obtain realistic
time we must multiply dimensionless time values t1 by the factor
to=1000s.To obtain realistic values of the height of superficial layer we
must multiply dimensionless values ε1 by the factor εo = 0,001 m. Hence
for times: t=0s, 10 000s, 20 000s, 30 000s, 40 000s we have values of
friction forces: 0,0770 N, 0,0772 N, 0,778 N, 0,789 N, 0,0825 N respectively. In above mentioned times the superficial layer has following realistic heights ε*: 0,00100m; 0,00271m; 0,00739m; 0,02008m;0,05459m.
CONCLUSIONS
In this paper are derived analytical formulae for friction force components caused by the nutrient fluid flow in boundary thin layer resting on
the tissue surface in bioreactor during the perfusion, growth and suction
of porous occurring in superficial layer. Typical nutrient fluid has density
value ρ = O(1000kg/m3). Height of fluid boundary layer in bioreactor has
values from ε = 3×10-5m to ε = 30×10-5m. Characteristic height value of
superficial layer of human cartilage has value εo = 10-3 m. Ambient
growth coefficient of joint cartilage has dimensionless value s = O(10-1)
for dimensionless time interval 0<t1≤40,where characteristic time value
of growth of tissue in bioreactor equals to = O(1000 s).Dynamic viscosity
of typical nutrient fluid equals η = O(10-3 Pas). Typical length of tissue
in bioreactor has values L from 10-2 m, to 10 -1m. The coefficient of
penetration of nutrient fluid to the porous superficial layer of normal cartilage of human hip joint has value ck = O(10-12 m2). For above data the
Strouhal Number has values from Str = 0(10-5) to Str = 0(10-4), Galileus
Number has value Ga = 0(10-2).Typical volume flow rate of perfusion of
nutrient fluid in bioreactor has value Qo = 0(10-8m3/s).By virtue of presented computational model we obtain following quantities of velocity
components: vx = 0(10-2 m/s), vz = O(10-2 m/s), vy = O(10-4 m/s), and following quantities of components of friction forces FRx, FRz in interval
from O(10-2 N) to O(10-1 N) .
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381
Acknowledgement
Author thanks for cooperation with Central Institute Department of
Biomedical Engineering in University Ulm in Germany and cooperation
with Clinic of Georg August University Göttingen, Department of Jaw
Orthopedics This paper was financed by KBN Founds in years 20032006 as Scientific Project KBN 411E-030-25
APPENDIX
V x1
z1
s = 0 .0 8 ; t 1 = 1 0
Vx1
z1
2 .0
s = 0 .0 8 ; t 1 = 2 0
2 .0
0 .0 3 8
0 .0 3 8
0 .0 3 8
1 .8
0 .0 3 8
1 .8
0 .0 4 2
0 .0 4 2
0 .0 4 2
0 .0 4 2
0 .0 4 7
1 .6
0 .0 4 7
1 .6
0 .0 4 7
1 .4
0 .0 4 7
0 .0 5 1
0 .0 5 1
1 .4
0 .0 5 6
0 .0 5 1
0 .0 5 1
0 .0 5 6
0 .0 5 6
1 .2
0 .0 3 8
1 .0
0 .0 3 8
0 .0 6 0
0 .0 3 4
1 .0
0 .0 5 6
1 .2
0 .0 6 0
1 .2
1 .4
0 .0 6 0
0 .0 3 3
1 .6
Vx1
z1
0 .0 6 0
2 .0
1 .8
x1
1 .0
1 .0
1 .2
1 .4
V x1
z1
s = 0.08; t 1 = 30
1 .6
0 .0 38
0 .0 3 7
0 .0 3 7
0 .0 38
0 .0 4 2
1 .8
1 .8
0 .0 42
0 .0 42
1 .6
0 .0 47
1 .4
0 .0 4 6
0 .0 4 2
0 .0 47
1 .6
0 .0 4 6
0 .0 5 1
0 .0 51
0 .0 51
1 .4
0 .0 56
0 .0 5 6
0 .0 5 1
0 .0 5 6
0 .0 56
1 .2 0 .0 38
1 .2
0 .0 60
0 .0 60
0 .0 33
1 .0
x1
s = 0 .0 8 ; t 1 = 4 0
2 .0
2 .0
2 .0
1 .8
1 .2
1 .4
1 .6
0 .0 6 1
0 .0 6 1
0 .0 3 2
x1
1 .0
0 .0 3 7
1 .8
2 .0
1 .0
x1
1 .0
1 .2
1 .4
1 .6
1 .8
2 .0
Fig. 4.
Dimensionless velocity distribution in the dimensionless distance y1 = 0,50
from the external tissue surface for grows coefficient s = 0,08 in dimensionless times t1 = 10; 20; 30; 40
Rys. 4. %H]Z\PLDURZH UR]NáDG\ SU
GNRFL Z EH]Z\PLDURZHM RGOHJáRFL
y1 RG ]HZQ
WU]QHM SRZLHU]FKQL WNDQNL GOD ZVSyáF]\QQLND
Z]URVWX V
Z EH]Z\PLDURZ\FK RVW
SDFK F]DVX t1 = 10; 20; 30; 40
REFERENCES
[1] Fung Y.C.: Biomechanics:Mechanical Properties of Living Tissues,ed.Springer Verlag New York,220-241,1993
[2] http://www.boku.ac.at/iam/ifa/mbr.gif
[3] http://bvt.gbf.de/zkt/tissue-engineering.html
382
TRIBOLOGIA
5-2003
[4] Mow, V.C., Guilak, F., Cell Mechanics and Cellular Engineering, Springer
Verlag Berlin, Heidelberg, New York 1994.
[5] Maurel,W.,Wu,Y.,Thalmann,D:Biomechanical Modells for Soft Tissue
Simulation, Springer, Verlag Berlin/Heidelberg 1998.
[6] Pioletti D.P: Viscoelastic Properties of Soft Tissues: Application to Knee
Ligaments and Tendons, ph.D.Thesis Epfl-Lausanne Univ. 1997.
[7] Truesdell C.:A First Course in Rational Continuum Mechanics, John Hopkins University/Baltimore,Maryland 1972.
Recenzent:
Jan BURCAN
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