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
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CHAPTER 10
HEAT TRANSFER IN LIVING TISSUE
10.1Introduction
·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
10.2 Vascular Architecture and Blood Flow
2
10.3 Blood Temperature Variation
3
(a) Formulation
(1) Equilibration Site:
(2) Blood Perfusion:
(3) Vascular Architecture:
(4) Blood Temperature:
10.4 Mathematical Modeling of VesselsTissue
Heat Transfer
10.4.1 Pennes Bioheat Equation (1948)
Assumptions:
Arterioles, capillaries & venules
Neglects flow directionality. i.e.
isotropic blood flow
No influence
4
Let
¢
¢
¢
=
net rate of energy added by the blood per unit
q
b
volume of tissue
¢
¢
¢
=
rate of metabolic energy production per unit
q
m
volume of tissue
¢
¢
¢
¢
¢
¢
¢
¢
¢
=
=
+
&
(a)
E
q
dx
dy
dz
(
q
q
)
dx
dy
dz
g
b
m
Conservation of energy for the
element shown in Fig. 10.3:
Treat energy exchange due to blood perfusion as energy generation
5
(10.3)
6
Cartesian coordinates:
cylindrical coordinates:
spherical coordinates:
7
(10.3)
(1)
Equilibration Site:
Notes on eq. (10.3):
(b) Shortcomings of the Pennes equation
8
·
thermally significant
Occurs in the
pre

arteriole and
post

venule vessels (dia. 70

500
)
m
m
L
·
e
Thermally significant vessels
>
:
1
L
·
Equilibration length
=
: distance blood travels for
L
e
its temperatur
e to equilibrate with tissue
(2)
Blood Perfusion:
(3)
Vascular Architecture :
· Perfusion in not isotropic
· Directionality is important in energy interchange
9
(4)
Blood Temperature:
· Blood does not reach tissue at body core temperature
· Blood does not leave tissue at local temperature T
(c) Applicability
· Surprisingly successful, wide applications
· Reasonable agreement with some experiments
10
Model forearm as a cylinder
Blood perfusion rate
w
&
b
¢
¢
¢
Metabolic heat production
q
m
Convection at the surface
h
Heat transfer coefficient is
Ambient temperature is
T
¥
Use Pennes bioheat equation
to
determine the 1

D
temperature distribution
Example 10.1: Temperature Distribution
in the Forearm
(1)Observations
11
(2) Origin and Coordinates. See Fig. 10.4
(3) Formulation
(i) Assumptions
(1)Steady state
(2)Forearm is modeled as a constant radius cylinder
(3)Bone and tissue have the same uniform properties
(4)Uniform metabolic heat
(5)Uniform blood perfusion
(6)No variation in the angular direction
(7)Negligible axial conduction
12
(8) Skin layer is neglected
(9) Pennes bioheat equation is applicable
(ii) Governing Equations
Pennes equation (10.3) for 1D steady state radial heat transfer
(iii) Boundary Conditions:
(4) Solution
13
Rewrite (a) in dimensionless form. Define
(d) into (a)
Define
(f) and (g) into (e)
14
The boundary conditions become
Bi is the Biot number
Homogeneous part of (h) is a Bessel differential equation.
The solution is
15
Boundary conditions give
(m) into (k)
(5) Checking
Dimensional check:Bi,and are dimensionless. The arguments of the Bessel functions are dimensionless.
Dimensional check:
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:
Limiting check:
16
Solve for
which agrees with (o)
(6) Comments
17
10.4.2 ChenHolmes Equation
18
NOTE:
19
<
m
(1)
Vessel diameter
300
m
L
<
e
(2)
0
.
6
L
(3)
Requires detailed knowledge of the vascular network
and blood perfusion
Limitations
10.4.3 ThreeTemperature Model for Peripheral
Tissue
Rigorous Approach
20
(1) Arterial temperature
(2)Venous temperature
(3) Tissue temperature T
21
Formulation
22
2
r
&
d
T
c
w
+

=
1
b
b
cb
(10.5)
(
T
T
)
0
c
0
1
2
k
dx
=
temperature variable in the lower layer
T
1
=
temperature of blood supplying the cutaneous pelxus
T
c
0
&
=
cutane
ous layer blood perfusion rate
w
cb
=
coordinate normal to skin surface
x
The 3 eqs. for
and
are replaced by one equation
T
2
d
T
=
2
(10.6)
0
2
dx
T
,
T
a
v
(ii) Region 2, pure conduction , for 1D steady state:
10.4.3 WeinbaumJiji Simplified Bioheat
Equation for Peripheral Tissue
23
·
Contains artery

vein pairs
·
¹
Countercurrent flow,
T
T
a
v
·
Includes capillaries, arterioles
and venules
(1)
Uniformly distributed blood
bleed

off leaving artery is
equal to that returning to vein
T
T
(2)
Bleed

off blood leaves artery at
and enters the vein at
a
v
Control Volume
(a) Assumptions
24
(3) Artery and vein have the same radius
(4) Negligible axial conduction through vessels
<<
(5) Equilibration length ratio
L
/
L
1
e
(6) Tissue temperature
is approximated by
T
(7) Onedimensional: 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 bleedoff from artery to vein
25
é
ù
n
2
2
=
+
p
r
( 10.9)
k
k
1
(
c
a
u
)
ê
ú
eff
b
b
2
ë
û
s
k
=
vessel radius
a
=
number of vessel pairs crossing surface of control
n
volume per unit area
=
average blood velocity in countercurrent artery or vein
u
(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
26
·
NOTE
·
accounts for the effect of vascular geometry and blood
k
eff
perfusion
s
·
,
and
depend on the vascular geometry
a
n
u
,
Conservation of mass gives
in terms of inlet velocity
to
u
u
o
tissue layer
and the vascular geometry. Eq. (10.9) becomes
27
=
=
vessel radius at inlet to tissue layer,
x
0
a
o
x
=
dimensionless vascular geometry function
V
(
)
(independent of blood flow)
x
=
=
dimensionless distance
x
/
L
=
tiss
ue layer thickness
L
=
=
blood velocity at inlet to tissue layer,
u
x
0
o
r
is independent of vascular geometry.
(
2
c
a
u
/
k
)
NOTE:
b
b
o
o
b
Notes on
:
k
eff
It represents the inlet Peclet number:
Eq. (10.12) into eq. (10.11)
28
2
r
d
T
c
w
&
+

=
1
b
b
cb
(10.12)
(
T
T
)
0
c
0
1
2
k
dx
Cutaneous layer:
Use eqs. (10.5) and (10.6)
29
= number of arteries entering tissue layer per unit area
n
o
=
rate of blood to the cutaneous layer to the
rate
R
total
total
of blood to the tissue layer
= is the thickness of the cutaneous layer
L
1
Eq. (10.12) into eq. (10.14)
Define R
Eqs. (10.15) and (10.16) into (10.5)
30
·

temperature model of
Results are compared
with 3
Section 10.4.3
·
Accurate tissue temperature
prediction for:
(1)
Vessel diameter <
200
μm
(2)
Equilibration length ratio
<
L
/
L
0
.
2
e
(3)
Peripheral tissue thickness < 2mm
(c) Limitation and Applicability
31
Skin surface at
T

5
´
7
10
s
=
k
Blood supply temperature
eff
2
+
x
k
[
1
Pe
V
(
)]
x
o
V
(
)
T
a
0
0
x
1
Fig.
10.7
described by
x
V
(
)
2
x
=
+
x
+
x
V
(
)
A
B
C



5
5
5
=
´
=

´
=
´
A
6
.
32
10
,
B
15
.
9
10
and
C
10
10
Example 10.2: Temperature Distribution in Peripheral Tissue
Peripheral tissue
Neglect blood flow
through cutaneous layer
vascular geometry is
.
32
(1)
Observations
·
Variation of
k
with distance is known
·
Tissue can be modeled as a single
layer with variable
k
eff
(iii) Plot showing effect of blood flow &
metabolic heat
(2) Origin and Coordinates. See Fig. 10.8
(3) Formulation
33
x
(4)
T
issue temperature at the base
= 0 is equal to
(5) Skin is maintained at uniform temperature
(6) Negligible blood perfusion in the cutaneous layer.
(a)
(i) Assumptions
(1) All assumptions leading to eqs. (10.8) and (10.9) are applicable
(2)Steady state
(3)Onedimensional
(ii) Governing Equations. Obtained from eq. (10.8)
34
=
(d)
T
(
0
)
T
a
0
=
(e)
T
(
L
)
T
s
(iii) Boundary Conditions
(4) Solution
Define
Substituting (b), (c) and (f) into (a)
Boundary conditions
35
q
=
(h)
(
0
)
1
q
=
(i)
(
1
)
0
[
]
q
d
2
2
+
+
x
+
x
=

gx
1
Pe
(
A
B
C
)
C
0
1
x
d
x
x
x
d
d
ò
ò
q
=

g
+
C
C
1
2
2
2
2
2
+
+
x
+
x
+
+
x
+
x
1
Pe
(
A
B
C
)
1
Pe
(
A
B
C
)
0
0
(j)
x
x
x
d
d
ò
ò
and
(k)
2
2
+
x
+
x
+
x
+
x
a
b
c
a
b
c
Integrating (g) once
integrating again
integrals (j) are of the form
where
36
2
2
2
=
+
=
=
(m)
a
1
APe
,
b
BPe
,
c
CPe
0
0
0
+
x
2
b
2
c

1
q
=

C
tan
1
d
d
(n)
g
+
x
é
ù
1
b
b
2
c

2
1
+
x
+
x

+
ln
(
a
b
c
)
tan
C
ê
ú
2
ë
û
c
2
d
d
2
=

(o)
d
4
ac
b
Boundary conditions (h) and (i) give the
C
C
constants
and
1
2
Evaluate integrals, substitute into (j)
37
a
b
c
d
(1)
,
,
and
depend on
Listed in Table 10.1
Pe
.
0
(2)
depends on both
and
g
Pe
C
:
o
1
where
Note:
38
39
(i) Enhancement in
due to blood perfusion
k
eff
g
=
=
(ii) Temperature distribution for
and
is
0
.
02
Pe
60
0
g
=
=
nearly linear. At
and
the
0
.
6
Pe
180
0
temperature is h
igher
(5) Checking
Dimensional check:
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
40
g
(iii) The governing parameters are
and
. The two are
Pe
0
physiologically related
(iv) Neglecting blood perfusion in the cutaneous layer
during vigorous exercise is not reasonable
10.4. 5 The sVessel Tissue Cylinder Model
Model Motivation
(a) Basic Vascular Unit
Vascular geometry of skeletal muscles has common features
41
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 1015 mm long
(b) Assumptions
(1) Uniformly distributed blood bleedoff leaving artery is equal to that returning to vein of the s vessel pair
43
local thermal equilibrium with the surrounding tissue
,
and
T
T
T
a
v
(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
heatsource pair representing the s vessels
(6) Outer surface of cylinder is at uniform temperature
(c) Formulation
44
Boundary Conditions
(1)Continuity of temperature at the surfaces of the vessels
(2)Continuity of radial flux at the surfaces of the vessels
45
,
and
are solved analytically
T
T
T
a
v
, the outlet bulk vein temperature at
=
x
0
T
vb
0
(d) Solution
Simplified Case
Assume:
46
(1)
Artery and vein are equal in size
(2) Symmetrically positioned relative
=
to center of cylinder, i.e.,
l
l
a
v
Results
47
(e) Modification of Pennes Perfusion Term
Eq. (10.18) gives
(a) into (10.21)
Dividing by the volume of cylinder
48
Blood flow energy generation per unit tissue volume:
(10.23) and 10.24) into (10.22)
(10.25) becomes
49
(1)
Artery supply temperature
body core temperature
T
a
0
Use (10.26) to replace the blood perfusion term in the
Pennes equation (10.3)
NOTE:
(1) This is the bioheat equation for the svessel cylinder
model
50
(2) 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 ChenHolmes and Weinbaum
Jiji equations
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 Tissue
Peripheral tissue of thickness L
52
Cutaneous plexus:
Perfusion rate
(uniformly distributed),
w
&
bc
blood supply temperature
T
cb
0
Skin temperature
T
s
Metabolic heat
*
D
Specified correction coefficient
T
53
((3) Formulation
(i)Assumptions
(1) Apply all assumptions leading to (10.5) and
(10.27)
(2) Steady state
(2) Onedimensional
(3) Constant properties
(4) Uniform metabolic heatin tissue layer
(5) Negligible metabolic heat in cutaneous layer
(7) Uniform blood perfusion in cutaneous layer
(ii) Governing Equations
Fourier’s law at surface:
54
(a)
Need 2 equations: one for tissue layer and one for cutaneous
Tissue layer temperature T: eq. (10.27):
(iii) Boundary Conditions
55
(4) Solution
Let
(b) and (c) become
56
Dimensionless parameters:
(k) into (i) and (j)
57
Boundary conditions
Solutions to (m) and (n):
Boundary conditions (p)(s) give constants
58
59
Surface heat flux:
(5) Checking
Dimensional check:
Limiting check:
60
(5) Comments
61