BIOHEAT EQUATIONS
BIOHEAT EQUATIONS
Heat transfer in blood vessels and tissues
Mihir Sen
March 17, 2013
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BIOHEAT EQUATIONS
Outline
Bioheat transfer problem
Large blood vessels
Tissues and microvasculature
Pennes’s equation
Other models
Transient equations
Multiscale problem
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BIOHEAT EQUATIONS
Bioheat transfer applications
Body heat balance
Thermoregulation
Thermogeneration
Heat transfer in muscles and tissues
Skin burns
Surgical procedures
Ablative surgery
Cryosurgery
Therapeutic hyperthermia
Therapeutic hypothermia
Cryopreservation
Organs for transplant
Resuscitation medicine
Extracorporeal equipment
Measuring instruments
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BIOHEAT EQUATIONS
Properties
Thermodynamic properties (density, compressibility, specific
heat)
Transport properties (thermal conductivity)
Properties of frozen tissue
Temperature dependence of properties
Water and fat content dependence
Convective heat transfer coefficient
Rate of perfusion
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BIOHEAT EQUATIONS
Metabolism
Chemical reactions in living organisms to generate heat
Thermodynamically, metabolism maintains order by creating
disorder
Metabolic regulation
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BIOHEAT EQUATIONS
Allometry of metabolism
Savage et al. (2004): 626 mammalian species
BMR = basal metabolic rate [W], M = mass [g]
BMR ∼ M 3/4
Heart and respiratory rates, stride frequencies ∼ M −1/4
Life spans, times to first reproduction ∼ M 1/4
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BIOHEAT EQUATIONS
Summary of heat transfer mechanisms
Conduction (Fourier’s law)
q̇′′ = −k∇T
Advection
q̇′′ = ρcV (T − Tref )
Convection (Newton’s law of cooling)
q̇ ′′ = h (Tsurf − Tfluid )
Radiation (Stefan-Boltzmann’s law)
¢
¡ 4
4
q̇ ′′ = ǫσ Tsurf
− Tsurr
q̇ ′′ ≈
1
(Tsurf − Tsurr )
Rrad
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BIOHEAT EQUATIONS
Large blood vessels
z
Tw
′′
q̇w
Wall boundary conditions (inner surface)
Constant wall temperature, Tw = constant
′′ = constant
Constant wall heat flux, q̇w
Conjugate heat transfer
Continuity of temperature at wall
Continuity of heat flux at wall
Matching with external heat transfer
Bulk (mean, average or mixing cup) temperature
R
ρvcp T dA
Tm = A
ṁcp
Z R
2
=
v(r)T (r)r dr
vm R 2 0
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BIOHEAT EQUATIONS
Large blood vessels: energy equation
Integral approach
(
′′ P dz
q̇w
ṁc dTm =
h (Tw − Tm ) P dz
known wall heat flux
known wall temperature
Differential approach (cylindrical coordinates)
∂T
∂T
vθ ∂T
∂T
+ vr
+
+ vz
∂t
∂r
r ∂θ
∂z
"
#
1 ∂2T
k 1 ∂ ¡ ∂T ¢
∂2T
r
+ 2 2 +
=
+Φ
ρcp r ∂r ∂r
r ∂θ
∂z 2
Φ = viscous dissipation (conversion from mechanical to thermal
energy)
Nondimensional: Péclet number = V D/α
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BIOHEAT EQUATIONS
Large blood vessels: preliminaries
Thermally fully developed if
½
¾
∂ Tw (z) − T (r, z)
=0
∂z Tw (z) − Tm (z)
Entrance length Lthermal = Pr Lhydro ; for blood Pr = 10–25
Nusselt number Nu D = hD/k where q̇ ′′ = h(Tw − Tm )
Correlations Nu D = Nu D (Re, Pr )
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BIOHEAT EQUATIONS
Large blood vessels: laminar flow
Constant wall temperature Tw
Nu D = 3.66
Tw − Tm (z)
= e−z/Le
Tw − Tm (0)
ṁcp
Le =
(thermal equilibration length)
Ph
At the exit
Tw − Tm (L)
= e−L/Le
Tw − Tm (0)
Notice that Tm (L) = Tw for L ≫ Le .
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BIOHEAT EQUATIONS
Chen and Holmes (1980)
Vessels of diameter ≈ 175µm have anatomical length ≈
thermal equilibration length (these are called thermally
significant blood vessels)
Temperature in smaller vessels are quickly equilibrated and do
not contribute much to heat transfer.
Larger vessels are sparse and do not contribute much to heat
transfer.
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BIOHEAT EQUATIONS
Large blood vessels: laminar flow
′′
Constant wall heat flux q̇w
Nu D = 4.36
Tm (z) = Tm (0) +
′′ P
q̇w
z
ṁcp
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BIOHEAT EQUATIONS
Flow in porous media
http://www.medphys.ucl.ac.uk/research/mle/images.htm
Darcy’s law (1856)
−∇p =
µ
v
K
K = permeability [m2 ]
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BIOHEAT EQUATIONS
Modifications
Transient Darcy equation
−∇p =
µ
∂v
v + ρc ·
K
∂t
Darcy-Brinkman equation
−∇p =
µ
v − µeff ∇2 v
K
Darcy-Forchheimer equation
−∇p =
µ
cF ρ|v|v
v+ √
K
K
Darcy-Brinkman-Forchheimer equation
−∇p =
µ
cF ρ|v|v
v+ √
− µeff ∇2 v
K
K
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BIOHEAT EQUATIONS
Porous medium heat transfer: single-temperature model
∂T
′′′
+ (ρc)f v · ∇T = ∇ · (km ∇T ) + q̇m
(ρc)m
|{z}
{z
}
|
∂t
|
{z
}
| {z }
accumulation
advection
conduction
generation
(ρc)m = (1 − α)(ρc)s + α(ρc)f
km = (1 − α)ks + αkf
′′′ = (1 − α)q̇ ′′′ + αq̇ ′′′ heat generation per unit volume
q̇m
s
f
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BIOHEAT EQUATIONS
Porous medium heat transfer: two-temperature model
∂Ts
= (1 − α)∇ · (ks ∇Ts ) + h(Tf − Ts ) +(1 − α)q̇s′′′
∂t
| {z }
fluid to solid
µ
¶
∂Tf
(ρc)f α
+ v · ∇Tf = α∇ · (kf ∇Tf ) + h(Ts − Tf ) +αq̇f′′′
∂t
| {z }
(1 − α)(ρc)s
solid to fluid
α = porosity (fraction of fluid by volume)
′′′ = heat generation per unit volume
q̇s,f
s = solid
f = fluid
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BIOHEAT EQUATIONS
Circulation tree
Branching Rn+1 < Rn
Pin
Pout
p
Womersley number α = R ωρ/µ
α → 0 as R → 0, so flow is steady (not pulsating)
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BIOHEAT EQUATIONS
Branching in human cerebral cortex
Francis et al. (2009)
Typical arteriole (right) and venule (left), scale bar = 1 mm
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BIOHEAT EQUATIONS
Murray’s law (1926)
Applied to
Circulatory system
Respiratory system
Water transport system in plants (xylem)
Obtained from
Minimization of energy expenditure by an organism.
One parent branch of radius r with n daughter branches of radii ri
r3 =
n
X
ri3
i=1
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BIOHEAT EQUATIONS
Pennes’s bioheat equation (1948)
tissue
qx +
qx
artery
Ta T
∂qx
dx
∂x
∂T
′′′
= ∇ · (k∇T ) + q̇m
+ ωρb cb (Ta − T )
ρc
{z
}
|
∂t
perfusion
T = tissue temperature
′′′
= metabolic heat source rate [W/m3 ]
q̇m
ω = perfusion rate, volumetric flow rate of blood per volume of tissue
[s−1 ]
Ta = arterial blood temperature
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BIOHEAT EQUATIONS
Experiments
Wissler’s (1998) analysis of Pennes’s data (resting human forearm)
Experimental data and Pennes’s theoretical values
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BIOHEAT EQUATIONS
Whole-body heat transfer
Ferreira and Yanagihara (2009)
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BIOHEAT EQUATIONS
Heat transfer by conduction inside the body
∂T
′′′
= ∇(k · ∇T ) + q̇m
∂t
Heat transfer to outside by radiation and convection
ρc
′′
′′
q̇rad
+ q̇rad
= Crc (Tskin − Toutside )
Heat transfer to outside by evaporation (p = partial pressure)
′′
q̇evap
= Ce (pskin − poutside )
Heat transfer between blood and tissue (Pennes’s bioheat equation)
∂T
′′′
= ∇(k · ∇T ) + ωρb cb (Ta − T ) + q̇m
∂t
Thermoregulatory system
¡
¢
0
∆ωskin = K1 (Th − Th0 ) +K2 Tskin − Tskin
| {z }
ρc
hypothalamus
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BIOHEAT EQUATIONS
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BIOHEAT EQUATIONS
Berkeley Comfort Model (2001)
Huizenga et al. (2001)
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BIOHEAT EQUATIONS
Counterflow model for extremities
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BIOHEAT EQUATIONS
Thermal resistance
For electrical resistance
resistance =
voltage difference
electric current
Thermal resistance
R=
∆T
q̇
q̇ =
∆T
R
or
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BIOHEAT EQUATIONS
Thermal resistance representation of human body
Top: exposed skin
Second: clothed skin
Third: clothed skin with conductive contact
Fourth: bare skin
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BIOHEAT EQUATIONS
Steady state (left) and transient from 28 to 4.7◦ C (right)
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BIOHEAT EQUATIONS
Tumor detection
Thermography of skin (Agnelli et al., 2011)
Healthy tissue and tumor region in three-dimensional domain.
Temperature distribution (left), and temperature profile on the
skin surface (right).
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BIOHEAT EQUATIONS
Limitations of Pennes’s model
Assumption: leaving temperature = tissue temperature.
Ignores
Directional dependence of perfusion heat source.
Different diameters of blood vessels (µm to mm range).
Sharply varying material properties.
Heat generation by necrosis.
Vasculature geometry.
Transvascular transport of energy and mass.
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BIOHEAT EQUATIONS
Continuum models: Chen-Holmes
Single-temperature model
ρc
∂T
′′′
= ∇ · (k∇T ) + ωρb cb (Ta − T ) − ρb cb V · ∇T + ∇ · (kp ∇T ) + q̇m
∂t
Two-temperature model
hA
∂Ts
= (1 − α)∇ · (ks ∇Ts ) +
(Tf − Ts ) + (1 − α)q̇s′′′
∂t
V
∂Tf
hA
αρf cf
= α∇ · (kf ∇Tf ) +
(Ts − Tf ) + αq̇f′′′
∂t
V
(1 − α)ρs cs
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BIOHEAT EQUATIONS
Vasculature-based models
Weinbaum-Jiji-Lemons
dTa
= −qa
ds
dTv
ρb cb πrb2 V
= −qv
ds
∂T
= ∇ · (k∇T ) + ngρb cb (Ta − Tv )
ρc
∂t
d
− n πrb2 ρb cb V
(Ta − Tv ) + q̇m
ds
Simplified Weinbaum-Jiji
∂T
= ∇ · (keff ∇T ) + q̇m
ρc
∂t
(
¡
¢2 )
n ρb cb πrb2 V cos γ
keff = k 1 +
σ∆ k 2
ρb cb πrb2 V
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BIOHEAT EQUATIONS
Non-Fourier heat transfer
Conservation of energy
∂T
+ ∇ · q̇′′ = 0
∂t
Fourier =⇒ parabolic heat conduction
∂T
= ∇ · (k∇T )
q̇′′ = −k∇T =⇒ ρc
∂t
Cattaneo-Vernotte =⇒ hyperbolic heat conduction
ρc
∂ q̇′′
+ q̇′′ = −k∇T
∂t
Dual phase lag
τ
=⇒
τ ρc
∂2T
∂T
+ ρc
= ∇ · (k∇T )
2
∂t
∂t
∂2T
∂ q̇′′
+ q̇′′ = −k∇T − kτT
∂t
∂t∂x
µ
¶
2
∂ T
∂T
∂2T
τq ρc 2 + ρc
= ∇ · (k∇T ) + ∇ · kτT
∂t
∂t
∂t∂x
τq
=⇒
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BIOHEAT EQUATIONS
Experiments with processed meat
Mitra et al. (1995): Hyperbolic heat conduction
Antaki (2005): Dual phase lag
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