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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


  • 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

  • Vessels

  • Artery/vein

  • Aorta/vena cava

  • Supply artery/vein

  • Primary vessels

  • Secondary vessels

  • Arterioles/venules

  • Capillaries

2


  • Blood leaves heart at

  • Blood mixing from various sources brings temperature to

10.3 Blood Temperature Variation

  • Equilibration with tissue: prior to arterioles and capillaries

  • Metabolic heat is removed from blood near skin

3


(a) Formulation

(1) Equilibration Site:

(2) Blood Perfusion:

(3) Vascular Architecture:

(4) Blood Temperature:

10.4 Mathematical Modeling of Vessels-Tissue

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):

  • This is known as the Pennes Bioheat equation

  • 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

  • Does not occur in the capillaries

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

  • 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

· 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


  • 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)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 1-D 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 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


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 Three-Temperature Model for Peripheral

Tissue

Rigorous Approach

  • Accounts for vasculature and blood flow directionality

20


(1) Arterial temperature

(2)Venous temperature

  • Assign three temperature variables:

(3) Tissue temperature T

  • Identify three layers:

  • Intermediate layer:

  • porous media

  • Cutaneous layer: thin,

  • independently supplied by counter-current artery-vein

  • vessels called cutaneous plexus

  • Regulates surface heat flux

21


  • Consists of two regions:

  • (i)Thin layer near skin with negligible blood flow

  • (ii)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


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 1-D steady state:

10.4.3 Weinbaum-Jiji 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

  • Effect of vasculature and heat exchange between artery,

  • vein, and tissue are retained

  • Added simplification narrows applicability of result

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) 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


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

  • 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)

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

  • Use the Weinbaum-Jiji equation determine temperature distribution

  • (ii) Express results in dimensionless form:

.

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

  • Metabolic heat is uniform

  • Temperature increases as blood perfusion

  • and/or metabolic heat are increased

(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)One-dimensional

(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 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


  • 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


  • Capillaries, arterioles and venules are essentially in

local thermal equilibrium with the surrounding tissue

  • Three temperature variables are needed:

,

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

heat-source pair representing the s vessels

(6) Outer surface of cylinder is at uniform temperature

(c) Formulation

  • Three governing equations are formulated

44


  • 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


  • The three eqs. for

,

and

are solved analytically

T

T

T

a

v

  • Solution gives

, 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 s-vessel 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 Chen-Holmes and Weinbaum-

Jiji equations

  • (3) The model analytically determines the venous return

    • temperature

  • (4) Accounts for contribution of countercurrent heat

    • exchange in the thermally significant vessels.

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

  • 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 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


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