CHAPTER 10
HEAT TRANSFER IN LIVING TISSUE
10.1 Introduction
 Examples
 Hyperthermia
 Cryosurgery
 Skin burns
 Frost bite
 Body thermal regulation
 Modeling
Modeling heat transfer in living tissue requires
the formulation of a special heat equation
1
 Key features
(1) Blood perfused tissue
(2) Vascular architecture
(3) Variation in blood flow rate and tissue properties
10.2 Vascular Architecture and Blood Flow
s
Vessels
s
v
a
s
Artery/vein
P
c
Aorta/vena cava
t
Supply artery/vein
v
Primary vessels
S
c c
, 5- 1 md
a
Secondary vessels P  p
a
a
v
1
-3
m d
Arterioles/venules SA m s
a
a v 30 - 1 md
Capillaries
F 1
c
V
2
10.3 Blood Temperature Variation
b
t
t
i
Tb
t
Bloodtemperature
T
a0
t
b
a

t
b
m
i
t
f
l
p
p
l
c
s
p
v
a
F 1
S
ob
t
v
o
c
ev
i v
i
0
c
• Blood leaves heart at Ta 0
 Equilibration with tissue: prior to arterioles and capillaries
 Metabolic heat is removed from blood near skin
• Blood mixing from various sources brings temperature to Ta 0
3
10.4 Mathematical Modeling of Vessels-Tissue
Heat Transfer
10.4.1 Pennes Bioheat Equation (1948)
(a) Formulation
Assumptions:
(1) Equilibration Site: Arterioles, capillaries & venules
(2) Blood Perfusion: Neglects flow directionality. i.e.
isotropic blood flow
(3) Vascular Architecture: No influence
(4) Blood Temperature: Blood reaches capillary bed at
body temperature Ta 0 , leaves at tissue temperature T
4
Conservation of energy for the
v
c
element shown in Fig. 10.3:
E& in + E& g  E& out  E&
b
y
(1.6)
Treat energy exchange due to
blood perfusion as energy
generation
q m
x
a
r
F 1
Let
q b  net rate of energy added by the blood per unit
volume of tissue
  rate of metabolic energy production per unit
qm
volume of tissue
E& g  q dx dy dz  ( q b + q m )dx dy dz
(a)
5
Formulation of q b : Blood enters at body temperature Ta 0
and exists at the tissue temperature T
qb  b cb w& b (Ta 0  T )
(10.1)
c b  specific heat of blood
w& b  blood volumetric flow rate per unit tissue volume
 b  density of blood
Eq. (10.1) into (a)
 + b cb w& b (Ta 0  T )
q  qm
(10.2)
Eq. (1.6) leads to (1.7). Modify eq. (1.7): set U  V  W  0,
and use eq. (10.2),
T
(10.3)
   c
  kT +  b c b w& b (Ta 0  T ) + q m
t
c  specific heat of tissue
6
k  thermal conductivity of tissue
  density of tissue
  kT  conduction terms, form depends on coordinates:
Cartesian coordinates:

T

T

T
  kT 
( k ) + ( k ) + (k ) (10.3a)
 x  x  y  y z z
cylindrical coordinates:
1 
T
1 
T

T
  kT 
( k r ) + 2 ( k ) + ( k ) (10.3b)
r r
r

z
z
r 
spherical coordinates:
1 
1

T
1

T
2 T
  kT  2 ( k r
)+ 2
(k sin ) + 2 2
(k )
r

r r
r sin  
r sin   
(10.3c)
7
Notes on eq. (10.3):
  kT +  b c b w& b (Ta 0
T
   c
 T ) + qm
t
(10.3)
(1) This is known as the Pennes Bioheat equation
(2) The blood perfusion term is mathematically identical to
surface convection in fins, eqs. (2.5), (2.19), (2.23) and (2.24)
(3) The same effect is observed in porous fins with coolant flow
(see problems 5.12, 5.17, and 5.18)
(b) Shortcomings of the Pennes equation
(1) Equilibration Site:
• Does not occur in the capillaries
8
 Occurs in the thermally significant pre-arteriole and
post-venule vessels (dia. 70-500  m )
 Thermally significant vessels: Le  1
L

Le = Equilibration length: distance blood travels for
its temperature to equilibrate with tissue
(2) Blood Perfusion:
 Perfusion in not isotropic
 Directionality is important in energy interchange
(3) Vascular Architecture :
•
•
•
Local vascular geometry not accounted for
Neglects artery-vein countercurrent heat exchange
Neglects influence of nearby large vessels
9
(4) Blood Temperature:
Blood does not reach tissue at body core temperature
Ta 0
 Blood does not leave tissue at local temperature T
(c) Applicability
 Surprisingly successful, wide applications
 Reasonable agreement with some experiments
10
Example 10.1: Temperature Distribution
in the Forearm
f
w
&b
Model forearm as a cylinder
Blood perfusion rate w& b

Metabolic heat production qm
Convection at the surface
r
R
Heat transfer coefficient is h
Ambient temperature is T
Use Pennes bioheat equation to
determine the 1-D
temperature distribution
(1)
0
q
m

h, T
w
&b

qm
m
F 1
Observations
11
•
Arm is modeled as a cylinder with uniform energy
generation
 Heat is conduction to skin and removed by convection
 In general, temperature distribution is 3-D
(2) Origin and Coordinates. See Fig. 10.4
(3) Formulation
(i) Assumptions
(1)
(2)
(3)
(4)
(5)
(6)
(7)
Steady state
Forearm is modeled as a constant radius cylind
Bone and tissue have the same uniform propert
Uniform metabolic heat
Uniform blood perfusion
No variation in the angular direction
Negligible axial conduction
12
(8) Skin layer is neglected
(9) Pennes bioheat equation is applicable
(ii) Governing Equations
Pennes equation (10.3) for 1-D steady state radial heat transfer

qm
1 d dT  b cb w& b
(a)
(r
)+
(Ta 0  T ) +
0
r dr dr
k
k
(iii) Boundary Conditions:
dT (0)
 0, or T(0) = finite
dr
(b)
dT ( R)
k
 hT ( R)  T 
dr
(c)
(4) Solution
13
Rewrite (a) in dimensionless form. Define
T  Ta 0
r

, 
T  Ta 0
R
(d) into (a)
2
 R 2
1 d d  b cb w& b R
qm
(
)

0
 d d
k
k (Ta 0  T )
Define

 b c b w& b R 2
k
 R 2
qm
 
k (Ta 0  T )
(d)
(e)
(f)
(g)
(f) and (g) into (e)
14
1 d
d
(
)      0
 d d
The boundary conditions become
(h)
d (0)
 0, or  (0)  finite
d
(i)
d (1)

 Bi [ (1)  1]
d
(j)
Bi is the Biot number
hR
Bi 
k
Homogeneous part of (h) is a Bessel differential equation.
The solution is

(k)
 ( )  C1 I 0 (   ) + C 2 K 0 (   ) 

15
Boundary conditions give
Bi [ 1 + ( /  )]
C1 
, C
2 0
 I1 (  ) + Bi I 0 (  )
(m) into (k)
T ( r )  Ta 0
 (r ) 

T  Ta 0
(5) Checking
(m)
Bi [ 1 + ( /  ) ]

I 0 (  r / R) 

 I 1 (  ) + Bi I 0 (  )
(n)
Dimensional check: Bi,and are dimensionless. The
arguments of the Bessel functions are dimensionless.
Limiting check: If no heat is removed (),arm reaches a
uniform temperature . All metabolic heat is transferred
to the blood. Conservation of energy for the blood:
16
   bcb w& b (To  Ta 0 )
qm
Solve for To
Set h  Bi  0

qm
To  Ta 0 +
 b cb w& b
(o)

qm
T ( r )  Ta 0 +
 b cb w& b
(p)
which agrees with (o)
(6) Comments
(i) Solution depends on 3 parameters: Bi, metabolic heat  ,
and blood perfusion parameter 
(ii) Setting r  0 and r  R in (n) gives center and
surface temperatures
17
(iii) The solution for zero metabolic heat production is
    0
obtained by setting qm
(iv) The solution for zero blood perfusion can not be
deduced from (n). Setting   0 in (n) gives .
Solution is obtained by setting   0 in (h) and then
solving for T:
T  T
1
1
2
(q)

+
1

(
r
/
R
)
2
 / k ) 2 Bi 4
( R qm


10.4.2 Chen-Holmes Equation
• First to show that equilibration occurs prior to reaching
the arterioles
• Accounts for blood directionality
• Accounts for vascular geometry
• The Pennes equation is modified to:
18
  kT +
w& b*  b cb (Ta*
T
 T )   b cb u  T +   k pT + qm   c
t
(10.4)
NOTE:
(1) w& b* = local perfusion rate
(2) Ta* = blood temperature upstream of the arterioles  Ta 0
(3) u  blood velocity vector, accounts for directionality
(4)  b cb u .T  energy convected by equilibrated blood.
Note similarity with convection term in moving fins
(eq.2.19) and with flow through porous media (eq. 5.6)
(5)   k pT  conduction due to temperature fluctuations in
equilibrated blood
19
(6) k p  “perfusion conductivity”, depends on blood velocity
and inclination relative to temperature gradient,
vessel radius and number density
Limitations
(1) Vessel diameter  300  m
Le
 0.6
(2)
L
(3) Requires detailed knowledge of the vascular network
and blood perfusion
10.4.3 Three-Temperature Model for Peripheral
Tissue
Rigorous Approach
• Accounts for vasculature and blood flow directionality
20
• Assign three temperature variables:
c
s
&
(1) Arterial temperature Ta
wc
p
Ts
(2)Venous temperature Tv
c
l
(3) Tissue temperature T
i
t
l
• Identify three layers:
(1) Deep layer: thermally
t
T
significant counterd l
x
current artery-vein pairs
a
v
(2) Intermediate layer:
0
Ta
Tv
porous media
(3) Cutaneous layer: thin,
F 1
independently supplied by counter-current artery-vein
vessels called cutaneous plexus
• Regulates surface heat flux
u
n
e
21
• Consists of two regions:
(i)
(ii)
Thin layer near skin with negligible blood flow
Uniformly blood perfused layer (Pennes model)
Formulation
Seven equation:
3 for the deep layer
2 for the intermediate layer
2 for the cutaneous layer
• Model is complex
• Simplified form for the deep layer is presented in the next
section
• Attention is focused on the cutaneous layer:
(i) Region 1, blood perfused. For 1-D steady state:
22
d 2T1  b cb w& cb
+
(10.5)
(Tc 0  T1 )  0
2
k
dx
T1  temperature variable in the lower layer
Tc 0  temperature of blood supplying the cutaneous pelxus
w& cb  cutaneous layer blood perfusion rate
x  coordinate normal to skin surface
(ii) Region 2, pure conduction , for 1-D steady state:
d 2T2
0
2
dx
(10.6)
10.4.3 Weinbaum-Jiji Simplified Bioheat
Equation for Peripheral Tissue
The 3 eqs. for Ta , Tv and T are replaced by one equation
23
•Effect of vasculature and heat exchange between artery,
vein, and tissue are retained
•Added simplification narrows applicability of result
Control Volume
 Contains artery-vein pairs
 Countercurrent flow, Ta  Tv
 Includes capillaries, arterioles a
and venules
Ta
(a) Assumptions
t
T
r
v
Tv
c
v
(1) Uniformly distributed blood
F 1
bleed-off leaving artery is
equal to that returning to vein
(2) Bleed-off blood leaves artery at Ta and enters the vein at Tv
24
(3) Artery and vein have the same radius
(4) Negligible axial conduction through vessels
(5) Equilibration length ratio Le / L  1
(6) Tissue temperature T is approximated by
T  (Ta + Tv ) / 2
(10.7)
(7) One-dimensional: blood vessels and temperature gradient
are in the same direction
(b) Formulation
Conservation of energy for tissue in control volume takes into
consideration:
(1) Conduction through tissue
(2) Energy exchange between vessels and tissue due to
capillary blood bleed-off from artery to vein
25
(3) Conduction between vessel pairs and tissue
Note: Conduction from artery to tissue not equal to
conduction from the tissue to the vein (incomplete
countercurrent exchange)
Conservation of energy for the artery, vein and tissue and
conservation of mass for the artery and vein give
T 
T

(10.8)
 c  ( k eff
) + qm
t  x
x
keff = effective conductivity, defined as

n
2 2

+
p

( 10.9)
k eff k 1
( b cb a u) 
2
 k s

a  vessel radius
n  number of vessel pairs crossing surface of control
volume per unit area
u  average blood velocity in countercurrent artery or vein
26
s  shape factor, defined as
s 
p
cosh( l / 2a )
(10.10)
l  center to center spacing between two parallel and
isothermal vessels
 NOTE
 k eff accounts for the effect of vascular geometry and blood
perfusion
 a,s , n and u depend on the vascular geometry
Conservation of mass gives u in terms of inlet velocity uo to
tissue layer and the vascular geometry. Eq. (10.9) becomes
k eff
 ( 2  b c b a o uo ) 2

 k 1 +
V ( )
2
kb


(10.11)
27
a o  vessel radius at inlet to tissue layer, x  0
V ( )  dimensionless vascular geometry function
(independent of blood flow)
  x / L  dimensionless distance
L  tissue layer thickness
uo  blood velocity at inlet to tissue layer, x  0
NOTE: ( 2  b cb ao uo / kb ) is independent of vascular geometry.
It represents the inlet Peclet number:
2  bcbao uo
Peo 
kb
Eq. (10.12) into eq. (10.11)
keff  k [1 + Peo2V ( )]
(10.12)
(10.13)
Notes on keff :
28
(1) For the 3-D case, orientation of vessel pairs relative
to the direction of local tissue temperature gradient gives
rise to a tensor conductivity
(2) The second term on the right hand side of eqs. (10.11) and
(10.13) represents the enhancement in tissue conductivity
due to blood perfusion
Cutaneous layer: Use eqs. (10.5) and (10.6)
d 2T1  b cb w& cb
+
(Tc 0  T1 )  0
2
k
dx
d 2T2
dx
2
0
(10.12)
(10.13)
Rewrite eq. (10.5) in terms of the Peclet number: Pe0
29
w& b 
p noao2 uo
(10.14)
L
no = number of arteries entering tissue layer per unit area
Eq. (10.12) into eq. (10.14)
p noao kb
w& b 
Peo
2 L bcb
Define R
L1w& bc
R
L w& b
(10.15)
(10.16)
R = total rate of blood to the cutaneous layer to the total rate
of blood to the tissue layer
L1 = is the thickness of the cutaneous layer
Eqs. (10.15) and (10.16) into (10.5)
30
d 2T1 p noao kb
+
RPe0 (Tc 0  T1 )  0
2
2kL1
dx
(10.17)
(c) Limitation and Applicability
 Results are compared with 3 - temperature model of
Section 10.4.3
 Accurate tissue temperature
prediction for:
(1) Vessel diameter < 200 μm
(2) Equilibration length ratio
Le / L  0.2
(3) Peripheral tissue thickness < 2mm
31
Example 10.2: Temperature Distribution in
Peripheral Tissue
Peripheral tissue
Skin surface at Ts
Blood supply temperature Ta 0
Neglect blood flow
through cutaneous layer
vascular geometry is
described by V ( )
V ( )  A + B + C 2
7  10
5
V ( )
keff 
k[1 + Peo2V ( )]

0
1
Fig. 10.7
5
5
5







A 6.32 10 , B
15.9 10 and C 10 10
(i) Use the Weinbaum-Jiji equation determine temperature
.
distribution
(ii) Express results in dimensionless form:
32
2  b c b a o uo
 L2
qm
T  Ts
, Pe 0 
,  
  x / L,  
k (Ta 0  Ts )
Ta 0  Ts
kb
(iii) Plot showing effect of blood flow &
metabolic heat
(1) Observations
s
Ts
k
q m
 Variation of k with distance is known
ke ( x )
 Tissue can be modeled as a single
x
layer with variable k eff
Ta0 0
• Metabolic heat is uniform
• Temperature increases as blood perfusion F 1
and/or metabolic heat are increased
i
(2) Origin and Coordinates. See Fig. 10.8
(3) Formulation
33
(i) Assumptions
(1) All assumptions leading to eqs. (10.8) and (10.9)
are applicable
(2) Steady state
(3) One-dimensional
(4) Tissue temperature at the base x = 0 is equal to Ta 0
(5) Skin is maintained at uniform temperature
(6) Negligible blood perfusion in the cutaneous layer.
(ii) Governing Equations. Obtained from eq. (10.8)
d
dT
  0
( k eff
) + qm
dx
dx
(a)
k eff  k [1 + Pe02 V ( )]
(b)
34
V ( )  A + B + C 2
(iii) Boundary Conditions
T (0)  Ta 0
T ( L)  Ts
(c)
(d)
(e)
(4) Solution
Define
 L2
qm
T  Ts
x
 , 
,  
k (Ta 0  Ts )
Ta 0  Ts
L
(f)
Substituting (b), (c) and (f) into (a)
d
d



2
2 d 
 1 + Pe0 ( A + B + C ) d  +   0


(g)
Boundary conditions
35
 ( 0)  1
 (1)  0
Integrating (g) once
1 +
Pe02 ( A +
(h)
(i)

d
 C1  
B  + C )
d
2
integrating again
 d
d
  C1

+ C2
2
2
2
2
1 + Pe0 ( A + B + C )
1 + Pe0 ( A + B + C )
(j)
integrals (j) are of the form



d
and
a + b + c 2
 d
 a + b + c 2
(k)
where
36
a  1 + APe02 , b  BPe02 ,
c  CPe02
(m)
Evaluate integrals, substitute into (j)
2
1 b + 2c
  C1

tan
d
d
 1
b
1 b + 2c
2
ln(a + b + c ) 
tan

c2
d
d
d  4ac  b 2

 + C 2
(n)
(o)
Boundary conditions (h) and (i) give the constants C 1 and C 2
37
 2
C1
d

1 b + 2c
 1 b + 2c 
tan

tan



d
d 
  1 a + b + c 2 b  1 b + 2c
 1 b + 2c  

tan
 tan
 ln



c 2
a+b+c
d
d
d 
(p)
where
 1 a+b+c
1   ln
+
c2
a
C1 
2 
1
tan
d 
b 
1 b
 1 b + 2c  
tan

tan

d 
d
d  
b
b + 2c 
 tan 1
d
d 
(q)
Note:
(1) a, b, c and d depend on Pe 0 . Listed in Table 10.1
(2) C 1 depends on both Pe o and  :
38
Pe 0  60 and   0.02 : C1  1.047
Pe0  180 and   0.6
:
Table 10.1
Pe o
C1  1.0176
60
(3) Table 10.2 lists enhancement in k
(4) Fig. 10.9 shows  ( )
180
a 1.2275
b -0.5724
c 0.36
d 1.44
3.0477
-5.1516
3.24
12.96
1.0

0
0.2
0.4
0.6
0.8
1.0
k eff / k
  0 .6
Pe o  60 Pe o  180
1.44
1.13
1.06
1.01
1.00
1.02
Table 10.2
3.05
2.15
1.51
1.12
1.02
1.14
Pe o  180
0.5

0

  0.02
Pe o  60
0.5
Fig. 10.9
1.0
39
(5) Checking
Dimensional check:  ,  , Pe0 and the arguments of
tan1
and ln are dimensionless
Boundary conditions check: Boundary conditions (h) and (i)
are satisfied
Qualitative check: Tissue temperature increases as blood
perfusion and metabolic heat are increased
(6) Comments
(i) Enhancement in k eff due to blood perfusion
(ii) Temperature distribution for Pe0  60 and   0.02 is
nearly linear. At Pe0  180 and   0.6 the
temperature is higher
40
(iii) The governing parameters are Pe0 and  . The two are
physiologically related
(iv) Neglecting blood perfusion in the cutaneous layer
during vigorous exercise is not reasonable
10.4. 5 The s-Vessel Tissue Cylinder Model
Model Motivation
• Shortcomings of the Pennes equation
• The Chen-Holmes equation and the Weinbaum-Jiji equation are
complex and require vascular geometry data
(a) Basic Vascular Unit
Vascular geometry of skeletal muscles has common features
• Main supply artery and vein, SAV
41
c
t
t
t
t
t
1 mm dia.
muscle
cylinder
s
P P
s
t
20  50 m dia.
t
c
P
0.5 mm
50  100 m dia.
s
s
s
100  300 m dia.
SAV
300  1000 m dia.
Fig. 10.10 Schematic of a representative vascular arrangement
• Primary pairs, P
• Secondary pairs, s
• Terminal arterioles and venules, t
• Capillary beds, c
42
NOTE: Blood flow in the SAV, P and s is countercurrent
Each countercurrent s pair is surrounded by a cylindrical
tissue which is approximately 1 mm
Diameter and typically 10-15 mm long
• The tissue cylinder is a repetitive unit consisting of arterioles,
venules and capillary beds
• This basic unit is found in most skeletal muscles
• A bioheat equation for the cylinder represents the governing
equation for the aggregate of all muscle cylinders
(b) Assumptions
(1) Uniformly distributed blood bleed-off leaving artery is
equal to that returning to vein of the s vessel pair
43
(2) Negligible axial conduction through vessels and cylinder
(3) Radii of the s vessels do not vary along cylinder
(4) Negligible temperature change between inlet to P vessels and
inlet to the tissue cylinder
(5) Temperature field in cylinder is based on conduction with a
heat-source pair representing the s vessels
(6) Outer surface of cylinder is at uniform temperature
(c) Formulation
• Capillaries, arterioles and venules are essentially in
local thermal equilibrium with the surrounding tissue
s vessels within the cylinder are thermally significant:
T  Ta  Tv
• Three temperature variables are needed:T, Ta
• Three governing equations are formulated
and Tv
44
Tlocal
L
L
R
vein
artery
x
tissue
(a) tissue cylinder
vein
lv
 l
tissue l
a
artery
(b) enlarged cross - section
Fig. 10.11
• Navier-Stokes equations of motion give the velocity field
in the s vessels (axially changing Poiseuille flow)
Boundary Conditions
(1) Continuity of temperature at the surfaces of the vessels
(2) Continuity of radial flux at the surfaces of the vessels
45
(3) Tissue temperature at cylinder radius R is assumed
uniform equal to Tlocal
T
(4) Symmetry at the mid-plane x  L gives
0
x
(5) Inlet artery bulk temperature at x  0 is specified as Tab0
(6) At x  L the flow in the s vessels vanishes and the artery,
vein and tissue are in thermal equilibrium at the local
tissue temperature Tlocal
(d) Solution
• The three eqs. for T, Ta andTv are solved analytically
• Solution gives Tvb0, the outlet bulk vein temperature at
x 0
Simplified Case
Assume:
46
(1) Artery and vein are equal in size
Tlocal
(2) Symmetrically positioned relative
to center of cylinder, i.e., l a  l v
R
tissue
Results
T  at x  0 is given by
Tab 0  Tvb0
A11

T 
 1+
+
Tab 0  T local
A12
vein
lv
 l
la
artery
simplified case
2
A11
2
A12
1  
la2  11 
A11   ln R(1  2 ) + 
4  
R  24 
1
(10.18)
(10.19)
47
1 R
2la2 cos  la4 
A12  ln 
1
+ 4
2
4  l
R
R 
(10.20)
(e) Modification of Pennes Perfusion Term
Eq. (10.18) gives
Tab 0  Tvb 0  T  (Tab 0  Tlocal )
(a)
Conservation of energy for blood at x  0 gives the total
energy q b delivered by blood to cylinder
qb   b cbp aa2 ua (Tab 0  Tvb 0 )
(10.21)
(a) into (10.21)
qb   b cbp aa2 ua T  (Tab 0  Tlocal )
Dividing by the volume of cylinder
48
qb
p aa2 ua



c

T
(Tab 0  Tlocal )
(10.22)
b b
2
2
pR L
pR L
Blood flow energy generation per unit tissue volume: qb
qb
qb 
(10.23)
2
pR L
Blood flow per unit volume w& b :
p aa2 ua
w& b 
p R2 L
(10.23) and 10.24) into (10.22)
qb   b cb w& b T  (Tab 0  Tlocal )
(10.25)
Since R  l , it follows that
Tlocal  T
(10.25) becomes
qb   b cb w& b T  (Tab 0  T )
(10.26)
(10.24)
49
qb in Pennes equation is given by
qb   b cb w& b (Ta 0  T )
(10.1)
Comparing (10.26) with (10.1):
(1) Artery supply temperature  body core temperature Ta 0
(2) A correction factor, T  , is added in (10.26)
Use (10.26) to replace the blood perfusion term in the
Pennes equation (10.3)
T
   c
  kT +  b cb w& b T  (Tab 0  T ) + qm
(10.27)
t
NOTE:
(1) This is the bioheat equation for the s-vessel cylinder
model
50
(2) T  is a correction coefficient defined in (10.18)
(a) It depends only on the vascular geometry of the tissue
cylinder
(b) It is independent of blood flow rate
(c) Its value for most muscle tissues ranges from 0.6 to 0.8
(d) This vascular structure parameter is much simpler
than that required by Chen-Holmes and WeinbaumJiji equations
(3) The model analytically determines the venous return
temperature
(4) Accounts for contribution of countercurrent heat
exchange in the thermally significant vessels.
(5) The artery temperatureTab 0 appearing in eq. (10.27) is
unknown
51
(a) It is approximated by the body core temperature in
the Pennes bioheat equation
(b) Its determination involves countercurrent heat
exchange in SAV vessels
(6) While equations (10.5) and (10.6) apply to the cutaneous
layer of peripheral tissue, eq. 10.23 applies to the region
below the cutaneous layer.
Example 10.3: Surface Heat Loss from Peripheral
Ts
cutaneos
Tissue
L1 w& cb
Peripheral tissue of thickness L
Cutaneous layer of thickness L1
Blood perfusion w& b
Primary vessel supply temperature Tab0
L w& b tissue
x
Tab 0
0
Tcb0
Fig. 10.12
52
Cutaneous plexus:Perfusion rate w& bc (uniformly distributed),
blood supply temperature Tcb0
Skin temperature Ts

Metabolic heat qm


Specified correction coefficient T
Use the s-vessel tissue cylinder model, determine surface
flux
(1) Observations
• Temperature distribution gives surface flux
• This is a two layer problem: tissue and cutaneous
(2) Origin and Coordinates. See Fig. 10.12
53
((3) Formulation
(i) Assumptions
(1) Apply all assumptions leading to (10.5) and
(10.27)
(2) Steady state
(2) One-dimensional
(3) Constant properties
(4) Uniform metabolic heat in tissue layer
(5) Negligible metabolic heat in cutaneous layer
(7) Uniform blood perfusion in cutaneous layer
(8) Tissue temperature at x = 0 is Tab 0
(9) Specified surface temperature
(ii) Governing Equations
Fourier’s law at surface:
54
q s   k
T1 ( L + L1 )
x
(a)
k  tissue conductivity
qs  surface heat flux
T1  temperature distribution in the cutaneous layer
Need 2 equations: one for tissue layer and one for cutaneous
Tissue layer temperature T: eq. (10.27):

d 2T  b cb w& b T *
qm
+
(Tab 0  T ) +
 0, 0  x  L
2
k
k
dx
(b)
Cutaneous layer temperature T1: eq. (10.5):
d 2T1  b cb w& cb
+
(Tcb0  T1 )  0 , L  x  L + L1
dx
k
(c)
(iii) Boundary Conditions
55
T (0)  Tab 0
T ( L)  T1 L)
(d)
(e)
dT ( L) dT1 ( L)

dx
dx
(f)
T1 ( L + L1 )  Ts
(g)
(4) Solution
Let
x
T  Tab 0
T1  Tab 0

,
,
L
Ts  Tab 0
Ts  Tab 0
(h)
(b) and (c) become
2



d 2  b cb w& b L2
q
L
*
b


T


 0, 0    1
2
k
k (Tab 0  Ts )
d
(i)
56
d 2  b cb w& cb L2 
Tab 0  Tcb 0 


 0, 1    1 + 0


2
k
Tab 0  Ts 
d

(j)
Dimensionless parameters:

 b cb w& b L2
k
, c 
 b cb w& cb L2
k
(k) into (i) and (j)
qbL2
Tab0  Tcbo
, 
,  
k (Tab0  Ts )
Tab0  Ts
(k)
d 2
*



T
    0,
2
d
d 2
  c +  c   0 ,
2
d
0 1
1    1 + 0
L1
0 
L
(m)
(n)
(o)
57
Boundary conditions
 (0)  0
(p)
 (1)   (1)
(q)
d (1) d (1)

dx
dx
(r)
 (1 +  0 )  1
(s)
Solutions to (m) and (n):

  A sinh T  + B cosh T  
 T *
  C sinh  c  + D cosh  c  + 
*
*
(t)
(u)
Boundary conditions (p)-(s) give constants
58

A

*
(1   ) cosh  c  1  cosh   T
+
+
+ C 1C 2 C 3
*
cosh  c (1 +  0 )
 T
sinh
  T *  C 2 C 3   T * cosh
T *
(v)

B
 T *
(w)


C  A T * cosh T * + C 1 C 2
(x)

1
D
 C 2 A T * cosh T *
cosh  c (1 +  0 )
(y)
C 1 , C 2 and C 3 are given by
C1 
(1   )  c sinh  c
sinh T 
*
cosh  c (1 +  0 )
T

*
59
C 2    c  cosh  c  sinh  c tanh  c (1 +  0 
1
C 3  sinh  c  cosh  c tanh  c (1 +  0 )
Surface heat flux:
q L
  c C cosh  c (1 +  0 ) + D sinh  c (1 +  0 )
k (Tab 0  Ts )
(z)
s
(5) Checking
Dimensional check: Parameters  ,  c ,  ,  ,  0 are dimensionless
Limiting check:
Special case: Ts  Tab 0  Tac 0 and q b  0, solutions (t), (u) and
(z) reduce to the expected results
T ( x )  T1 ( x )  Tab 0 and q s  0
60
(5) Comments
(i) Five governing parameters:  ,  c ,  ,  , and 0
(ii) Use solution (z) to examine the effect of cutaneous
blood perfusion on surface heat flux
(iii) Changing blood flow rate through the cutaneous layer
is a mechanism for regulating body temperature
(iv) The solution does not apply to the special case of
zero blood perfusion rate since and  c appear in
the differential equations as coefficients of the
variables and 
61