Chapter 10: Heat Transfer in Living Tissue
10.1 Introduction
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Focal point: Temperature distribution in blood perfused tissue.
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Applications: cryosurgery, frost bite, hyperthermia, skin burns,….
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Thermal modeling: Must account for (1) blood perfusion, (2) the vascular architecture,
and (3) variation in thermal properties and blood flow rate.
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Objective: Formulating a heat equation for determining temperature distribution in blood
perfused tissue..
10.2 Vascular Architecture and Blood Flow
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Blood is supplied to tissues and organs through a network of arteries and capillaries.
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Blood is drained from tissue and organs through a network of veins and venules.
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Size of arteries and veins diminishes in the direction of blood flow through the arteries.
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Note vessel names and sizes illustrated in Fig. 10.1.
10.3 Blood Temperature Variation
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Blood leaves the heart through the aorta at Ta 0 .
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It flows through primary arteries, arterioles and capillaries where it either adds or
removes heat from tissue depending on local temperature T.
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At the capillaries thermal equilibration with tissue becomes complete.
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Blood returns to the heart through venules, primary veins and vena cava.
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Blood mixing due to venous confluence brings its temperature back to Ta 0 .
10.4 Mathematical Modeling of Vessels-Tissue Heat Transfer
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Modeling seeks to predict temperature distribution.
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Assumptions are made to simplify the complexity of the process.
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Pennes bioheat equation is one of earliest and simplest models (1948).
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Subsequent improved models are more complex.
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Five models will be presented.
10.4.1 Pennes Bioheat Equation
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(a) Formulation. Assumptions:
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(1) Equilibration Site: Arterioles. capillary beds venules.
(2) Blood Perfusion: Isotropic.
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(3) Vascular Architecture: Local vascular geometry plays no role.
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(4) Blood Temperature: Blood reaches the capillary beds at Ta 0 and equilibrates with
the local tissue at T .
Conservation energy for element
E& in + E& g − E& out = E&
•
(1.6)
Energy transport by blood is treated as energy generation qb′′′
qb′′′ = ρ b cb w& b (Ta 0 − T )
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Other terms in (1.6) follow Section 1.4. Equation (1.6) lead to the Pennes bioheat
equation
∇ ⋅ k∇T + ρ b cb w& b (Ta 0 − T ) + q m′′′ = ρ c
•
•
(10.1)
∂T
∂t
(10.3)
The first term in (10.3) takes on different forms depending on the coordinate system.
(b) Shortcomings
• (1) Equilibration Site: Thermal equilibration takes place in pre-arteriole and postvenule vessels and not in the capillaries.
•
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(2) Blood Perfusion: Not isentropic
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(3) Vascular Architecture: Not considered.
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(4) Blood Temperature. Pre-arteriole blood is not at core temperature Ta 0 and vein
return temperature is not equal to the local tissue temperature T.
(c) Applicability:
Vessel diameters > 500 μm.
Equilibration length to total length Le / L > 0.3.
10.4.2 Chen-Holmes Equation
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Blood equilibration with the tissue occurs prior to reaching the arterioles.
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Pennes perfusion term is modified taking into consideration blood flow directionality
and vascular geometry
∇ ⋅ k∇T + w& b* ρ b cb (Ta* − T ) − ρ b cb u ⋅ ∇T + ∇ ⋅ k p ∇T + q m = ρ c
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w& b* is the local blood perfusion.
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Ta* ≠ Ta 0 .
∂T
∂t
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The third term accounts for energy convected due to equilibrated blood.
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Directionality of blood flow is described by the vector u .
Assumptions and limitations:
(10.4)
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Mass transfer between vessels and tissue is neglected.
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Thermal properties are assumed the same as those of the solid tissue.
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Vessels diameter < 300 μm .
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Equilibration length ratio Le / L < 0.6.
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Requires detailed knowledge of the vascular network and blood perfusion.
10.4.3 Three-Temperature Model for Peripheral Tissue
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Three temperature variables are assigned: arterial Ta , venous Tv and tissue T .
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Three vascular layers were identified: deep layer, intermediate and cutaneous layer.
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Deep layer: Countercurrent artery-vein pairs that are thermally significant.
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Intermediate layer: Modeled as a porous media
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Cutaneous layer: Below skin. Independently supplied by countercurrent artery-vein
vessels called cutaneous plexus.
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Divided into two regions; (i) upper region with negligible blood effect and (ii) a
lower region governed by Pennes equation.
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This model is governed by seven equations: three in the deep layer, two in the
intermediate and two in the cutaneous.
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It accounts for the vascular geometry and blood flow directionality.
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Detailed vascular data.
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Require numerical integration
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Detailed formulation of all equations will not be presented. A simplified form will be
detailed in the next section.\
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The heat equations for the cutaneous layer for the one-dimensional case are:
Lower region:
d 2T1
dx
2
+
ρ b cb w& cb
Upper region:
k
(Tc 0 − T1 ) = 0
d 2T2
dx 2
=0
(10.5)
(10.6)
10.4.4 Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue
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The three deep tissue equations of the three-temperature model are combined into a
single equation.
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Conservation of energy accounts for:
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(1) Conduction through tissue, (2) conduction between vessel pairs and tissue, (3)
energy transport by blood bleed-off between vessels and tissue.
(a) Assumptions.
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(1) Bleed-off from artery is equal to flow to vein.
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(2) Bleed-off leaves artery at Ta and enters vein at the venous temperature Tv .
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(3) artery and vein have the same radius.
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(4) No axial conduction through vessels.
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(5) Equilibration length ratio Le / L << 1 .
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(6) Tissue temperature T is approximated by
T ≈ (Ta + Tv ) / 2
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(10.7)
(b) Formulation. Energy conservation for artery, vein and tissue for one-dimensional
case (blood vessels and temperature gradient are in the same direction) gives
ρc
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∂T
∂
=
(k eff ∂T ) + q m′′′
∂x
∂t ∂ x
(10.8)
k eff is the effective conductivity, defined as
n
⎡
⎤
k eff = k ⎢1 + 2 (π ρ b cb a 2 u ) 2 ⎥
⎣ k σ
⎦
(10.9)
σ = shape factor, for vessels at uniform surface temperatures with center to center
spacing l , it is
π
σ=
(10.10)
cosh(l / 2a )
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k eff represents the effects of (1) vascular geometry and (2) blood perfusion.
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These two effects are separated by rewriting (10.9). The variables a, σ , n and u
depend on the vascular geometry. Conservation of mass and vascular geometry relate
blood velocity u to inlet velocity u o . Thus
⎤
⎡ ( 2 ρ b cb a o u o ) 2
(10.11)
k eff = k ⎢1 +
V (ξ )⎥
2
kb
⎦⎥
⎣⎢
• V (ξ ) = dimensionless vascular geometry function, independent of blood flow
ξ = x / L = dimensionless distance
L = tissue layer thickness
u o = blood velocity at the inlet to the tissue layer at x = 0
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The coefficient (2 ρ b cb ao u o / k b ) is independent of the vascular geometry. It
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2 ρ b cb a o u o
.
kb
The second term in (10.11) represents the enhancement in conductivity.
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(10.12) into (10.11)
represents the inlet Peclet number Peo =
k eff = k [1 + Peo2 V (ξ )]
(10.13)
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The two regions of the cutaneous layer shown are governed by (10.5) and (10.6).
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Eq. (10.5)
is rewritten in terms of the Peclet number Pe0 . Blood perfusion
rate w& b in the tissue layer is given by
w& b =
π no a o2 u o
L
(10.14)
no = number of arteries entering the tissue layer per unit area.. (10.12) into (10.14)
w& b =
π no a o k b
Peo
2 Lρ b c b
(10.15)
Define R as
R=
L1 w& cb
L w& b
(10.16)
L1 = thickness of the cutaneous layer. (10.15) and (10.16) into (10.5)
d 2T1
dx
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2
+
π no a o k b
2kL1
RPe0 (Tc 0 − T1 ) = 0
(10.17)
(c) Limitation and Applicability.
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Eq. (10.8) gives accurate predictions for vessels smaller than 200 μm in dia. and
equilibration length ratio Le / L < 0.3 . Experiments on the rat spinotrapezius muscle
showed that (10.8) is valid for Le / L < 0.2 .
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Eq. (10.8) is based on vascular architecture of peripheral tissues less than 2 cm thick.
It does not apply to deeper layers and to skeletal muscles.
10.4. 5 The s-Vessel Tissue Cylinder Model
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Motivation: Address shortcomings of other models
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(a) The Basic Vascular Unit.
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Different types of skeletal muscles have significant common arrangements shown in
Fig. 10.10
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SAV = main supply artery and vein.
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P = primary pairs.
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s = secondary pairs.
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t = terminal arterioles and venules
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c = capillary beds.
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Vessel size decreases progressively from SAV to c.
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Blood flow in the SAV, P, t and s vessels is countercurrent.
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Each countercurrent s pair is surrounded by a cylindrical tissue.
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The tissue cylinder is a repetitive unit found in most skeletal muscles. It is used to
formulate a bioheat equation for aggregate of all cylinders comprising the muscle.
(b) Assumptions.
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(1) Bleed-off blood in vessels leaving the s artery is equal to that returning to the
vein and is uniformly distributed along each pair.
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(2) Negligible axial conduction.
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(3) Radii of the s vessels do not vary along tissue cylinder
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(4) Negligible changes in temperature between the inlet to the P vessels and the inlet
to the tissue cylinder.
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(5) Temperature field in the tissue cylinder is based on pure radial conduction with a
heat-source pair representing the s vessels.
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(6) The outer surface of the cylinder is at uniform temperature T local .
(c) Formulation.
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The s vessels are not in thermal equilibrium with tissue temperature.
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Three temperature variables are introduced: arterial Ta , venous Tv and tissue T .
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Three equations for the temperature field in the s vessels and tissue cylinder were
formulated.
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Solution to the Navier-Stokes equations gives the two-dimensional velocity field.
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Tab 0 = bulk temperature of the artery at x = 0.
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Tlocal = temperature of the artery, vein and tissue at x = L .
(d) Solution.
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Analytic solution to the three equations for the artery, vein and tissue temperatures
were obtained.
Solution gives Tvb 0 , the outlet bulk temperature of the vein at x = 0.
Results for the special case: equal size blood vessels symmetrically positioned
relative to the center of the cylinder, i.e., l a = l v ( Fig. 10.11). The dimensionless
artery-vein temperature difference, ΔT ∗ at x = 0 is given by
T −T
A
ΔT = ab 0 vb0 = 1 + 11 +
Tab 0 − T local
A12
∗
where
2
A11
2
A12
−1
(10.18)
l a2 ⎤ 11 ⎫⎪
1 ⎧⎪ ⎡
A11 = − ⎨ln ⎢ R(1 − 2 )⎥ + ⎬
4 ⎪⎩ ⎣⎢
R ⎦⎥ 24 ⎪⎭
A12 =
(10.19)
2 l 2 cos φ l a4 ⎤
1 ⎡R
+ 4⎥
ln ⎢
1− a 2
4 ⎢l
R
R ⎥⎦
⎣
(10.20)
φ , l a and l are defined in Fig. 11.
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(e) Modification of Pennes Perfusion Term. Conservation of energy to the blood
at x = 0 gives the total energy qb delivered by the blood to the tissue cylinder
qb = ρ b cb π a a2 u a (Tab0 − Tvb0 )
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(10.21)
Using (10.18), equation (10.21) becomes
qb = ρ b cb π a a2 u a ΔT ∗ (Tab0 − Tlocal )
Dividing by the volume of the cylinder
π a a2 u a
= ρ b cb
ΔT ∗ (Tab 0 − Tlocal )
2
2
πR L
πR L
qb
(10.22)
Volumetric energy generation due to blood flow qb′′′ is
qb
qb′′′ =
π R2L
Volumetric blood flow per unit tissue volume w& b
(10.23)
π a a2 u a
w& b =
π R2L
(10.24)
Eqs.(10.23) and (10.24) into (10.22) gives
qb′′′ = ρ b cb w& b ΔT ∗ (Tab0 − Tlocal )
(10.25)
Since R >> l , Tlocal ≈ T. Eq. (10.25) becomes
qb′′′ = ρ b cb w& b ΔT ∗ (Tab0 − T )
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(10.26)
Equation (10.26) replaces eq. (10.1) used in the formulation of the Pennes equation.
It differs from. (10.1) in two respects:
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(1) Artery supply temperature is not set equal to the body core temperature.
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(2) It includes the ΔT ∗ factor.
Using eq.(10.26) in the Pennes equation (10.3) gives
∇ ⋅ k∇T + ρ b cb w& b ΔT ∗ (Tab 0 − T ) + q m′′′ = ρ c
∂T
∂t
(10.27)
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Note the following:
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(1) The dimensionless factor ΔT ∗ is a correction coefficient. It it depends only
on the vascular geometry of the tissue cylinder and is independent of blood flow.
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(2) A closed-form solution is obtained for the determination of ΔT ∗ . Its value
for most muscle tissues ranges from 0.6 to 0.8.
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(3) The model determines the venous return temperature and accounts for
contribution of countercurrent heat exchange in the thermally significant vessels.
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(4) The artery temperature Tab0 in eq. (10.27) is unknown. It is approximated by
the body core temperature in the Pennes bioheat equation.
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Equations (10.5) and (10.6) apply to the cutaneous layer of peripheral tissue,
(10.23) applies to the region below the cutaneous layer.