Acta of Bioengineering and Biomechanics
Vol. 5, No. 1, 2003
Numerical estimation of burn degree of skin tissue
using the sensitivity analysis methods
E. MAJCHRZAK, M. JASIŃSKI
Silesian University of Technology, Gliwice, Poland
The numerical analysis of thermal processes proceeding in the skin tissue due to external heat flux is
presented. Heat transfer in the skin tissue was assumed to be transient and one-dimensional.
Thermophysical parameters of successive skin layers are different, at the same time in subdomains of
dermis and subcutaneous region the internal heating resulting from blood perfusion and metabolism is
taken into account. The degree of the skin burn can be predicted on the basis of the so-called Henriques
integrals. The paper deals with the sensitivity analysis of these integrals with respect to the
thermophysical parameters. In numerical computations, the boundary element method has been used. In
the final part of the paper, the results of computations are presented.
Key words: burn of skin tissue, numerical estimation, thermophysical parameters
1. Introduction
The skin is treated as a multi-layer domain in which one can distinguish the
epidermis, dermis and subcutaneous region, with blood perfusion and metabolic heat
source in the latter two regions. The skin surface is subjected to an external heat flux
which is a function of time. Heat transfer in the skin tissue is assumed to be transient
and one-dimensional. In the paper, the mathematical model and also the numerical
algorithm based on the boundary-element method are presented. This model allows us
to predict the skin temperature and time that passes till the appearance of the
symptoms of the first-, second- and third-degree burns due to external heat flux effect.
Because thermophysical parameters of the skin vary widely from person to person, the
analysis of the sensitivity of temperature field and burn predictions to these variations
is also carried out. In the final part of the paper, the results of computations are shown.
E. MAJCHRZAK, M. JASIŃSKI
94
2. Governing equations
The skin is divided into three layers, i.e. epidermis Ω1 of thickness L1 [m], dermis
Ω2 of thickness L2 − L1 and subcutaneous region Ω3 of thickness L3 − L2.
Thermophysical parameters of these subdomains are equal to λe [W/(mK)] (thermal
conductivity) and ce [J/(m3 K)] (specific heat per unit of volume), e = 1, 2, 3.
The transient bioheat transfer in the domain of skin is described by the following
system of equations [6], [10], [11]:
x ∈ Ω e : ce
∂Te
∂ 2T
= λe 2e + ke (TB − Te ) + Qme ,
∂t
∂x
(1)
where Ge [(m3 of blood/s)/m3 of tissue)] is the blood perfusion rate, cB [J/(m3 K)] is the
volumetric specific heat of blood, TB is the arterial blood temperature and Qme [W/m3]
is the metabolic heat source. It should be stressed that for the epidermis subdomain
(e = 1) G1 = 0 and Qm1 = 0.
On the contact surfaces between subdomains considered the continuity conditions
in the following form
∂Te
∂T

= −λe +1 e +1 ,
− λ
x ∈ Γe, e +1 :  e ∂x
∂x

Te = Te +1
e = 1, 2
(2)
are given. Additionally
q = q0 ,

x ∈ Γ0 : 
q = −α (T1 − T ∞ ),
t ≤ t0 ,
t > t0 ,
(3)
where q1 = −λ1∂T1/∂x, q0 is the given boundary heat flux, t0 is the exposure time, α is
the heat transfer coefficient, T ∞ is the ambient temperature. For conventionally
assumed internal boundary limiting the system, no flux condition can be taken into
account. A quadratic initial temperature distribution between 32.5 °C at the surface
and 37 °C at the base of the subcutaneous region was used [11].
Thermal damage of skin begins when the temperature at the basal layer (the
interface between epidermis and dermis) rises above 44 °C (317 K). HENRIQUES [4]
found that the degree of skin damage could be predicted on the basis of the integrals
τ
 ∆E 
I b = Pb (Tb ) exp −
 dt
RT
t
(
)
b


0
∫
and
(4)
Numerical estimation of burn degree of skin tissue
95
τ

∆E 
I d = Pd (Td ) exp −
 dt ,
 RTd (t ) 
0
∫
(5)
where ∆E/R [K] is the ratio of activation energy to universal gas constant, Pb, Pd [1/s]
are the preexponential factors, while Tb [K] is the temperature of basal layer (the
surface between epidermis and dermis) and Td [K] is the temperature of dermal base
(the surface between dermis and subcutaneous region).
The first-degree burns are said to occur when the value of the burn integral (4) is
within the interval 0.53 < Ib ≤ 1, while the second-degree burns when Ib > 1 [4], [11].
The third-degree burns are said to occur when Id > 1. So in order to determine the
values of integrals Ib, Id the heating curves and next the cooling curves representing
the basal layer and dermal base must be known.
3. Sensitivity analysis
The sensitivity analysis of bioheat transfer has been carried out with respect to the
parameters λe, ce, Ge, Qme. These parameters we denote by ps, s = 1, 2, ..., 10, this
means p1 = λ1, p2 = λ2, p3 = λ3, p4 = c1, p5 = c2, p6 = c3, p7 = G2, p8 = G3, p9 = Qm2, p10 =
Qm3. If the direct method of sensitivity analysis is used [2], [3], [9], then we should
consider ten additional boundary-initial problems [8]





















where
 c ∂λe ∂ce  ∂Te
∂ 2U es
∂U es

−
− Ge cBU es +  e
= λe
2
∂ps ∂ps  ∂t
∂t
λ
∂
x
e

x ∈ Ωe :
∂G
∂Q
1 ∂λe
+
[Ge cBTe − (Ge cBTB + Qme )] + e cB (TB − Te ) + me ,
∂ps
∂ps
λe ∂ps
ce
1 ∂λ1

t ≤ t0 ,
Q1s = − λ ∂p q0 ,
1
s
x = 0: 
1 ∂λ1
Q1s = −αU1s −
q , t > t0 ,

λ1 ∂ps 1
U es = U e +1, s ,

x = Le :  1 ∂λe
1 ∂λe +1
qe + Qes =
qe +1 + Qe +1, s ,

∂
p
λ
λ
s
e +1 ∂ps
 e
x = L : Q3 s = 0,
t = 0:
U es = 0,
(6)
e = 1, 2,
E. MAJCHRZAK, M. JASIŃSKI
96
U es =
∂Te
∂ps
(7)
and
Qes = −λe
∂U es
.
∂x
(8)
For example, the sensitivity with respect to the thermal conductivity of epidermis
(p1 = λ1 ) is determined by the following boundary-initial problem



















∂U e1
∂ 2U e1
c ∂λe ∂Te
= λe
− Ge cBU e1 + e
2
∂t
λe ∂λ1 ∂t
∂x
x ∈ Ωe :
1 ∂λe
+
[Ge cBTe − (Ge cBTB + Qme )],
λe ∂λ1
ce
1

t ≤ t0 ,
Q11 = − λ q0 ,
1
x = 0: 
1
Q11 = −αU11 − q1 , t > t0 ,

λ1
U e1 = U e+1,1 ,

1 ∂λe+1
x = Le :  1 ∂λe
q + Qe1 =
qe+1 + Qe+1,1 ,
 λ ∂λ1 e
λ
e+1 ∂λ1
 e
(9)
x = L : Q31 = 0,
t = 0 : U e1 = 0,
where ∂λ1/∂λ1 = 1, ∂λ2/∂λ1 = 0, ∂λ3/∂λ1 = 0, while the sensitivity with respect to the
volumetric specific heat of dermis ( p5 = c2) is connected with the boundary initial
problem

x ∈ Ω e :


x = 0 :



 x = Le :


x = L :
t = 0 :
ce
∂U e5
∂ 2U e5
∂c ∂Te
,
= λe
− Ge cBU e5 − e
2
∂t
∂c2 ∂t
∂x
t ≤ t0 ,
Q15 = 0,

Q15 = −αU15 , t > t0 ,
U e5 = U e+1,5 ,

Qe5 = +Qe+1,5 , e = 1, 2,
Q35 = 0,
U e5 = 0,
where ∂c1/∂c2 = 0, ∂c2/∂c2 = 1, ∂c3/∂c2 = 0.
(10)
Numerical estimation of burn degree of skin tissue
97
It should be pointed out that the analysis of the temperature changes due to the
changes of ps requires the knowledge of primary solution, because in the mathematical
model of the sensitivity, the values resulting from this solution appear. Taking into
account forms (4), (5) of the functionals Ib , Id , the sensitivity of these integrals with
respect to the parameters ps is calculated using the formulas
τ
 ∆E
∂I b
∆E
exp −
= Pb
2
∂ps 0 RTb
 RTb
∫

 U bs d t

(11)
and
τ
 ∆E 
∂I d
∆E
exp −
= Pd
 U ds d t ,
2
∂ps 0 RTd
 RTd 
∫
(12)
where Tb = T1 (L1, t) = T2 (L1, t), Td = T2 (L2, t) = T3 (L2, t) (c.f. equations (2)) and Ubs =
U1s (L1, t) = U2s (L1, t), Uds = U2s (L2, t) = U3s (L2, t) (c.f. equations (6)).
The change of burn integrals connected with the change of parameter ps results
from the Taylor formula limited to the first-order sensitivity, this means
I b ( ps ± ∆ps ) = I b ( ps ) ±
∂I b
∆ps
∂ps
(13)
I d ( ps ± ∆ps ) = I d ( ps ) ±
∂I d
∆ps .
∂ps
(14)
and
4. Boundary element method
The primary and also the additional problems resulting from the sensitivity
analysis have been solved using the 1st scheme of the BEM for 1D transient heat
diffusion [1], [7]. So, the following equations for successive layers of skin are
considered
ce
∂Fe
∂ 2 Fe
= λe
+ Se ,
∂t
∂x 2
(15)
where Fe = Fe (x, t) denotes the temperature or functions resulting from the sensitivity
analysis, Se = Se (x, t) are the source functions. The functions Se take a form
• for the primary problem
E. MAJCHRZAK, M. JASIŃSKI
98
x ∈ Ω e : S e = Ge cB (TB − Te ) + Qme ,
(16)
• for the problems of the sensitivity with respect to ps
 c ∂λ ∂c  ∂T
1 ∂λe
S e = −Ge cBU es +  e e − e  e +
[Ge cBTe − (Ge cBTB + Qme )]
 λe ∂ps ∂ps  ∂t λe ∂ps
∂G
∂Q
+ e cB (TB − Te ) + me .
∂ps
∂ps
(17)
At first, we introduce the time grid
0 = t 0 < t1 < t 2 < ... < t f −1 < t f < ... < t F < ∞,
∆t = t f − t f −1 .
(18)
If the 1st scheme of the BEM is taken into account, then the boundary integral
equations (for successive layers of skin − e = 1, 2, 3) corresponding to the transition
t f–1 → t f are of the form [1], [7]
1
Fe (ξ , t ) + 
 ce
f
1
=
 ce
tf
∫
t f −1
x = Le

ξ , x, t , t ) J e ( x, t )dt 

x = Le −1
tf
∫
Fe* (
t f −1
f
x = Le
Le

*
f
J e (ξ , x, t , t ) Fe ( x, t )dt 
+
Fe* (ξ , x, t f , t f −1 ) Fe ( x, t f −1 ) dx

Le −1
x = Le −1
∫
1
+
ce
Le
∫
S e ( x, t
Le −1
f −1
tf
)
t
∫
Fe* (ξ , x, t f , t )dt dx ,
(19)
f −1
where Fe* are the fundamental solutions given by formula
Fe* (ξ , x, t f , t ) =
 ( x − ξ )2 
exp −
,
f
2 πae (t f − t )
 4ae (t − t ) 
1
(20)
where ξ is the point at which the concentrated heat source is applied and ae = λe /ce.
The heat fluxes resulting from the fundamental solutions are equal to
J e* (ξ , x, t f , t ) = −λe
 ( x − ξ )2 
λe ( x − ξ )
∂Fe* (ξ , x, t f , t )
=
exp
−
 , (21)
f
∂x
4 π [ae (t f − t )]3 / 2
 4ae (t − t ) 
while Je (x, t) = −λe ∂Fe/∂x. Assuming that for t ∈ [t f–1, t f ]: Fe(x, t) = Fe (x, t f ) and
Je (x, t) = Je (x, t f ) one obtains the following form of equations (19)
Fe (ξ , t f ) + g e (ξ , Le ) J e ( Le , t f ) − g e (ξ , Le −1 ) J e ( Le −1 , t f )
Numerical estimation of burn degree of skin tissue
= he (ξ , Le ) Fe ( Le , t f ) − he (ξ , Le −1 ) Fe ( Le −1 , t f ) + pe (ξ ) + ze (ξ ) ,
99
(22)
where
1
he (ξ , x) =
ce
tf
t
∫
J e* (ξ , x, t f , t ) d t =
f −1
sgn ( x − ξ )
erf
2
 | x −ξ | 
c

 2 ae ∆t 
(23)
and
1
g e (ξ , x) =
ce
=
tf
∫F
ξ , x, t f , t ) d t
*
e (
t f −1
 | x −ξ | 
 ( x − ξ )2  | x − ξ |
∆t
exp −
erf c 
,
−
2λe
πλe ce
 2 ae ∆t 
 4ae ∆t 
(24)
while
Le
Pe (ξ ) =
∫F
ξ , x, t f , t f −1 ) Fe ( x, t f −1 ) dx
*
e (
Le −1
=
1
2 πae ∆t
Le
∫
Le −1
 (x − ξ )2 
f −1
exp −
 Fe ( x, t ) dx,
 4 a e ∆t 
(25)
at the same time
Le
Z e (ξ ) =
∫
S e ( x, t f −1 ) g e (ξ , x) dx .
(26)
Le −1
For ξ → L+e −1 and ξ → L−e −1 for each domain considered one obtains the system of
equations
e
 g11

e
 g 21
e
  J e ( Le −1 , t f )  h11e
g12
=

f 
e
e  
g 22   J e ( Le , t )  h21
h12e   Fe ( Le −1 , t f )  Pe ( Le −1 )  Z e ( Le −1 )
+

+
 . (27)
f 
e  
h22   Fe ( Le , t )   Pe ( Le )   Z e ( Le ) 
E. MAJCHRZAK, M. JASIŃSKI
100
The final form of resolving system results from the continuity conditions for x = L1,
x = L2 and conditions given for x = 0 and x = L. So, for the primary problem and the
time t < t0 one has
1
 − h11

1
− h21

 0

 0

 0


 0
1
− h12
1
g12
0
0
1
− h22
1
h22
0
0
− h112
2
g11
− h122
2
g12
2
− h21
2
2
− h22
g 21
2
g 22
0
0
− h113
3
g11
0
0
3
− h21
3
g 21
1
q0 + P1 (0) + Z1 (0) 
0   T1 (0, t f )   − g11
 


0   Tb (t f )  − g 121q0 + P1 ( L 1) + Z1 ( L 1)
 


f





P2 ( L 1) + Z 2 ( L 1)
q (t )
0
=
 ,(28)
 b

P2 ( L 2 ) + Z 2 ( L 2 )
0   Td (t f )  
 





f
3  
P3 ( L 2 ) + Z 3 ( L 2 )
− h12   qd (t )  



f 
3  
P3 ( L) + Z 3 ( L)
− h22  T3 ( L, t ) 

at the same time
q1 (0, t f ) = −α [T1 (0, t f ) − T ∞ ] .
(29)
This system of equations for t > t0 is somewhat different (c.f. condition (3)) [5].
The solution of (28) determines the boundary temperature and heat fluxes at a time t f
for x = 0, L1, L2, L and next the temperature at the internal points can be found using
the following formula
ξ ∈ Ωe :
Te (ξ , t f ) = g e (ξ , Le −1 ) qe ( Le −1 , t f ) − g e (ξ , Le ) qe ( Le , t f )
+ he (ξ , Le )Te ( Le , t f ) − he (ξ , Le −1 )Te ( Le −1 , t f ) + Pe (ξ ) + Z e (ξ ) .
(30)
In the case of additional boundary initial problems (6) connected with the
sensitivity analysis, for transition t f–1 → t f one should solve the following system of
equations
1
1
 − h11
− h12

1
1
− h21
− h22

 0
− h112

2
 0
− h21

 0
0


0
 0
1
g12
0
0
1
h22
0
0
2
g11
− h122
2
g12
2
2
− h22
g 21
2
g 22
0
− h113
3
g11
0
3
− h21
3
g 21
0   U1s (0, t f ) 


0   U bs (t f ) 


0   Q1s (t f ) 


f


U (t ) 
0
  ds


f
3  
− h12   Q2 s (t ) 
f 
3  
− h22
 U 3 s ( L, t )
Numerical estimation of burn degree of skin tissue
101
1 1 ∂λ1


g11
q1 (0, t f ) + P1 (0) + Z1 (0)


λ1
∂ps


1 1 ∂λ1
f


g
q (0, t ) + P1 ( L 1) + Z1 ( L 1)


λ1 21 ∂ps 1

 
1 ∂λ2 1 ∂λ1 
2
f

 g11
q
t
P
L
Z
L
(
)
(
)
(
)
−
+
+
2
1
2
1

  λ2 ∂ps λ1 ∂ps  b
,
= 
 g 2 1 ∂λ2 − 1 ∂λ1  q (t f ) + P ( L ) + Z ( L ) 
2
2
2
2

 21  λ2 ∂ps λ1 ∂ps  b


 g 3  1 ∂λ3 − 1 ∂λ2  q (t f ) + P ( L ) + Z ( L )
3
2
3
2

 11  λ3 ∂ps λ2 ∂ps  d


 g 3  1 ∂λ3 − 1 ∂λ2  q (t f ) + P ( L) + Z ( L) 
3
3

 21  λ3 ∂ps λ2 ∂ps  d




(31)
 1 ∂λ2 1 ∂λ1 
f
−
Q2 s ( L1 , t f ) = Q1s ( L1 , t f ) − 
 qb (t )
∂
∂
λ
λ
p
p

2
s
 2 s
(32)
 1 ∂λ3 1 ∂λ2 
f
−
Q3s ( L 2 , t f ) = Q2 s ( L 2 , t f ) − 
 qd (t ) .
∂
∂
λ
λ
p
p
2
s 
 3 s

(33)
while
and
As previously, the resolving system for t > t0 is somewhat different [5]. The values
of sensitivity function Ues at the internal points are calculated on the basis of formula
ξ ∈ Ωe :
U es (ξ , t f ) = g e (ξ , Le −1 ) Qes ( Le −1 , t f ) − g e (ξ , Le ) Qes ( Le , t f )
+ he (ξ , Le ) U es ( Le , t f ) − he (ξ , Le −1 ) U es ( Le −1 , t f ) + Pe (ξ ) + Z e (ξ ) .
(34)
5. Examples of computations
In numerical computations, the following mean values of parameters have been
assumed [5], [11]: λ1 = 0.235 [W/(mK)], λ2 = 0.445 [W/(mK)], λ3 = 0.185 [W/(mK)],
c1 = 4.3068⋅10 6 [J/(m3 K)], c2 = 3.96⋅10 6 [J/(m3 K)], c3 = 2.674⋅10 6 [J/(m3 K)], cB =
3.9962⋅10 6 [J/(m3 K)], TB = 37 °C, G1 = 0, Ge = 0.00125 [(m3 of blood/s)/ m3 of tissue]
for e = 2, 3, Qm1 = 0, Qme = 245 [W/m3] for e = 2, 3 [11]. Preexponential factors: Pb =
1.43⋅1072 [1/s] for Tb ≥ 317 [K] and Pb = 0 for Tb < 317 [K], while Pd = 2.86⋅1069 [1/s]
for Td ≥ 317 [K] and Pd = 0 for Td < 317 [K]. The ratio of activation energy to universal
gas constant: ∆E/R = 55 000 [K]. The thicknesses of successive skin layers: 0.1, 2 and
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E. MAJCHRZAK, M. JASIŃSKI
10 [mm]. These layers have been divided into 10, 40 and 120 internal cells. Time step:
∆t = 0.05 [s].
Fig. 1. Temperature distribution (q0 = 9000 [W/m2], t0 = 15 [s])
Fig. 2. Distribution of the function λ2⋅∂T/∂λ2 (q0 = 9000 [W/m2], t0 = 15 [s])
Numerical estimation of burn degree of skin tissue
103
Fig. 3. Distribution of function c2⋅∂T/∂c2 (q0 = 9000 [W/m2], t0 = 15 [s])
In the first example, on the skin surface the heat flux q0 = 9000 [W/m2] is assumed,
the exposure time: t0 = 15 [s]. For t > t0 the Robin condition is accepted (α =
8 [W/m2K], T ∞ = 20 °C). In figure 1, the temperature distribution in the skin domain is
shown. Figures 2 and 3 illustrate the courses of the sensitivity functions λ2⋅∂T/∂λ2 and
c2 ⋅∂T/∂c2.
The time necessary for the appearance of the symptoms of the first- and second-degree
burn predicted for the mean values of parameters are equal to 11.8 [s] and 12.9 [s],
respectively (c.f. figures 4 and 5). The third-degree burn does not appear. The
sensitivity analysis of the burn integral Ib has been done with respect to all
thermophysical parameters and the changes of Ib connected with the following
changes of parameters: ∆ λ1 = 0.025 [W/(mK)], ∆ λ2 = 0.075, ∆ λ3 = 0.025, ∆ c1 =
1.32⋅104 [J/(m3 K)], ∆ c2 = 3.3⋅105, ∆ c3 = 3.86⋅105, ∆ Ge = 0.00125 [(m3 of blood/s)/m3 of
tissue],
∆ Qme
=
245 [W/m3 ], e = 2, 3 [5], [11] have been also calculated. It turned out that the changes
of thermal conductivity (figure 4) and volumetric specific heat (figure 5) of the dermis
subdomain are especially essential in the case considered. The diminution of these
parameters causes that the time necessary for the burn appearance decreases as well.
On the other hand, for the top value of λ2 the second-degree burn does not appear (c.f.
figure 4).
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E. MAJCHRZAK, M. JASIŃSKI
Fig. 4. Burn integral Ib and its sensitivity with respect to λ2 (q0 = 9000 [W/m2], t0 = 15 [s])
Fig. 5. Burn integral Ib and its sensitivity with respect to c2 (q0 = 9000 [W/m2], t0 = 15 [s])
Numerical estimation of burn degree of skin tissue
Fig. 6. Temperature profiles (q0 = 80000 [W/m2], t0 = 5 [s])
Fig. 7. Burn integral Id and its sensitivity with respect to λ2 (q0 = 80000 [W/m2], t0 = 5 [s])
105
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E. MAJCHRZAK, M. JASIŃSKI
Fig. 8. Burn integral Id and its sensitivity with respect to λ3 (q0 = 80000 [W/m2], t0 = 5 [s])
Fig. 9. Burn integral Id and its sensitivity with respect to c2 (q0 = 80000 [W/m2], t0 = 5 [s])
Numerical estimation of burn degree of skin tissue
107
Fig. 10. Burn integral Id and its sensitivity with respect to c3 (q0 = 80000 [W/m2], t0 = 5 [s])
In the second example, on the skin surface the heat flux q0 = 80000 [W/m2] is
assumed, the exposure time t0 = 5 [s]. In figure 6, the temperature distribution in the
skin domain is shown. The time until the appearance of the third-degree burn at mean
values of thermophysical parameters is equal to 15.75 [s]. In the case considered, the
changes of thermal conductivity and volumetric specific heat in the dermis and
subcutaneous region (c.f. figures 7, 8, 9 and 10) are especially essential. The changes
of epidermis parameters and also the changes of perfusion rate of blood and metabolic
heat source in the dermis and subcutaneous region have minimal influence on the
third-degree burn predictions.
6. Conclusions
The algorithm allows us to estimate the effects of variations in thermophysical
parameters on skin temperature and burn predictions. It turned out that blood
perfusion term and metabolic heat source in the Pennes bioheat transfer equation (1)
can be neglected in predicting the degree of burns of the skin tissue subjected to an
external heat flux.
Due to wide variations in the thicknesses of skin from person to person, the
sensitivity analysis with respect to the geometrical parameters of skin should be also
investigated.
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E. MAJCHRZAK, M. JASIŃSKI
Acknowledgement
The authors were supported by the State Committee for Scientific Research (KBN, Poland) through
the Grant No. 8 T11F 004 19.
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