1. Numerical study on the method of Active Dynamic Thermography on skin
burn wounds of irregular shape
Aditya Telang and Shong Leih-Lee*
Department of Power Mechanical Engineering
National Tsing-Hua University, Hsinchu 30013, Taiwan
* corresponding author
Email address: sllee@pme.nthu.edu.tw

2. Abstract:
Proper diagnostic assessment of burn wound depth is important to select the mode of
burn wound treatment. Active Dynamic Thermography (ADT), a diagnostic method developed
by Renkielska et al. [10], aimed to differentiate between skin burn wound areas which would
heal naturally within three weeks and areas which would not and would require surgery. The
main limitations of this method were due to the necessity of using proper thermal models of
living tissues [6]. Since then, new thermal models of human skin have been developed,
especially the three dimensional vascular model by Lee et al [4]. The purpose of this study is to
simulate skin burns of arbitrary shape using the thermal model developed by Lee et al [4], to
simulate the process of ADT on these burn wounds and finally to identify the skin areas which
require surgery using the cooling stage time constant as the parameter for identification of burn
grade. The identified skin areas which require surgery are then compared to the original irregular
wound to ascertain the accuracy of this method.

3. Nomenclature:
A Area
𝐵𝑖 Biot Number
𝑐 Dimensionless radius of secondary vessel
𝑐 𝑝 Specific Heat [𝐽 𝑘𝑔−1
𝐾−1
]
𝐸𝑣 Heat loss through evaporation [𝑊𝑚−2
]
𝑕 Heat transfer coefficient [𝑊𝑚−2
𝐾−1
]
𝑘 Thermal conductivity [𝑊𝑚−1
𝐾−1
]
𝐿 Thickness of Dermis [𝑚]
𝑄 Heat source term [𝑊𝑚−3
]
𝑄 𝑕 Heat flux at z=0 [𝑊𝑚−2
]
𝑄 𝑚 Heat generation by metabolism [𝑊𝑚−3
]
𝑞 𝑚 Dimensionless heat generation by metabolism
𝑇 Temperature [𝐾]
𝑡 Time [𝑠𝑒𝑐]
𝑋, 𝑌, 𝑍 Cartesian co-ordinates
𝑥, 𝑦, 𝑧 Dimensionless Cartesian co-ordinates

4. Greek symbols:
𝛼 Decaying factor
𝛼 𝑑 Thermal diffusivity [𝑚2
𝑠−1
]
𝛽 Dimensionless thickness of hypodermis
𝛾 Dimensionless heat capacity
𝛿 Dimensionless thickness of epidermis
𝜁 Dimensionless intensity of thermal radiation
Ω Pseudo-Biot number
𝜅 Dimensionless thermal conductivity
𝜃 Dimensionless temperature
𝜌 Density [𝑘𝑔 𝑚−3
]
𝜏 Dimensionless time
Δ𝑇 Reference temperature difference
Δ𝑡 Time step
Δ𝑥 Element length in X-direction
Δ𝑦 Element length in Y-direction
Δ𝑧 Element length in Z-direction

5. Subscripts:
𝑐 Core
𝑑 Dermis
𝑒𝑑 Epidermis
𝑕 Heat from external heat source (a halogen lamp)
𝑕𝑑 Hypodermis

6. Introduction:
The first quantitative description of heat transfer in human tissue taking blood flow into
consideration was attempted by Pennes [7]. In his model for the human forearm, he assumed the
forearm to be perfectly cylindrical. The two source terms in his bioheat equation were the heat
produced by tissue metabolism and the heat transferred from blood to tissue. The heat produced
by tissue metabolism, the volumetric glow rate of blood per unit volume of tissue per second and
the thermal conductivity of tissue was assumed to be spatially uniform. The source term
𝜔 𝑏 𝜌𝑏 𝑐 𝑝 𝑏
1 − 𝑘 (𝑇𝑎 − 𝑇) was used for heat transfer from blood to tissue, where 𝑇𝑎 and 𝑇
denote arterial blood and tissue temperatures and 𝜌𝑏 , (𝑐 𝑝) 𝑏 and 𝜔 𝑏 denote density, specific heat
and perfusion of blood respectively. The equilibrium factor 𝑘 was assigned the value 𝑘 = 0 by
assuming thermal equilibrium between venous blood and tissue. This was the well known
Pennes’ equation.
The use of the average blood perfusion by Pennes eliminated the necessity to model the
true vascular geometry and greatly simplified the form of the bioheat equation. Pennes
equation’s validity was questioned in many applications [15]. A drawback of the average
perfusion assumption was the significant errors it produced and hence it could not be used for all
tissues. This led to many amendments in the blood perfusion term by Weinbaum and Jiji[16],
Baish et al.[1], Wissler [14] and further corrections by Weinbaum et al.[17].
Ng et al. [5] developed a model of human skin where the steady state temperature during
burns was simulated using the boundary element method (BEM). The skin was modeled as three
layered – the epidermis, dermis and subcutaneous fat. The temperature distribution of the skin
was successfully simulated during burn injury.

7. A formulation for a bioheat equation with a superposition technique for normal and
burned skin was developed by Lee et. al [4].
Active Dynamic Thermography:
Renkielska et al [9] first proposed a method of static infrared thermal imaging (Static
Thermography, ST). However, this method was extremely sensitive to external conditions. For
more accurate burn wound depth assessment, Renkielska et al.[10] developed a noninvasive
diagnostic method termed as active dynamic thermography(ADT) for relatively large burn areas.
The higher accuracy of ADT when compared to ST was proved using animal experiments [8].
The essence of this method is that skin is optically heated resulting in a surface temperature rise
of 2.5°C and is allowed to cool. The cooling stage is modeled as exponentially decreasing with a
constant time constant. The time constant was calculated for each burn wound while the wound
depth was determined by the method of histological analysis [11]. It was found that when the
time constant was above a certain threshold, the wound would heal within 3 weeks otherwise it
would remain unhealed.
𝑇 𝑡 = 𝑇0 + ΔT exp −
t
τ
Renkielska had assumed that time constant does not change temporally during the
cooling stage. It was argued by Lee et al.[4] that two wounds with similar depth had significantly
different time constants and two time constants of essentially the same value corresponded to
burn wounds with totally different depths. This suggested that the time constant as suggested by
Renkielska et al. [10] was not an efficient parameter for assessment of burn wound grades. The
constant time constant assumption is not made in this study.

8. Model of Human Skin:
Governing Equation:
Figure 1 illustrates a schematic vascular system of the skin [4]. The blood vessels
(including arteries and veins) beneath the muscle are known as the primary vessels. The blood
circulates between the primary vessels and the cutaneous vessels by separate riser vessels. The
blood enters the secondary artery at the bottom of the dermis with a temperature 𝑇𝑎0 which is
approximately the same as that in the primary artery. Next, the blood rises to the top of the
dermis by the terminal artery and then flows to the terminal vein through a capillary bed (not
shown in Figure 1). Subsequently, the blood descends to the secondary vein by the terminal vein
at a temperature 𝑇𝑣0 which is slightly less than 𝑇𝑎0 because of heat loss to tissue. There is no
blood vessel in both the epidermis and the hypodermis.
Figure 2 illustrates a model for the cross section of skin. L is the thickness of the
epidermis. The thickness of the dermis and the hypodermis are represented as a scaling of L.
The non-dimensionalized governing equation for skin was described in Lee et. al [4] as
follows:
𝛾
𝜕𝜃
𝜕𝜏
=
𝜕
𝜕𝑥
𝜅
𝜕𝜃
𝜕𝑥
+
𝜕
𝜕𝑦
𝜅
𝜕𝜃
𝜕𝑦
+
𝜕
𝜕𝑧
𝜅
𝜕𝜃
𝜕𝑧
+ 𝑞 𝑚
Where, the following transformations have been used for non-dimensionalization:
𝜏 =
𝛼 𝑑 𝑡
𝐿2 ; 𝜃 =
𝑇−𝑇∞
Δ𝑇
; 𝑞 𝑚 =
𝑄 𝑚 𝐿2
𝑘 𝑑 Δ𝑇
; 𝑥 =
𝑋
𝐿
; 𝑦 =
𝑌
𝐿
; 𝑧 =
𝑍
𝐿
Where 𝛼 𝑑 and 𝑘 𝑑 are the thermal conductivity and thermal diffusivity respectively of the
dermis and 𝑄 𝑚 is the heat generation by metabolism. The dermis and epidermis belong to the

9. domain 0 ≤ 𝑧 < 1 and 1 ≤ 𝑧 < 1 + 𝛿, respectively. The dimensionless thermal conductivity and
specific heat are step functions with values unity in the dermis and with values 𝛾 =
𝜌𝑐 𝑝 𝑒𝑑
𝜌 𝑐 𝑝 𝑑
and
𝜅 =
𝑘
𝑘 𝑑
in the epidermis. The reference temperature was defined as Δ𝑇 = 𝑇𝑎0 − 𝑇∞ where 𝑇∞ is
the ambient temperature.
Boundary Conditions:
The boundary conditions in dimensionless form as described in Lee et. al [4] are:
At the cell walls,
𝜕𝜃 0,𝑦,𝑧,𝜏
𝜕𝑥
= 0 ;
𝜕𝜃 𝑥0,𝑦,𝑧,𝜏
𝜕𝑥
= 0 ;
𝜕𝜃 𝑥,0,𝑧,𝜏
𝜕𝑦
= 0 ;
𝜕𝜃 𝑥,𝑦0,𝑧,𝜏
𝜕𝑦
= 0
At the surface,
𝜕𝜃 𝑥, 𝑦, 𝑧 𝑒𝑑 , 𝜏
𝜕𝑧
= 𝐵𝑖𝜃 𝑥, 𝑦, 𝑧 𝑒𝑑 , 𝜏 + 𝑒 𝑣 − 𝜁(𝜏)
Where 𝐵𝑖 is the biot number defined as =
𝑕∞ 𝐿
𝑘 𝑒𝑑
, 𝐸𝑣 is the sweat evaporation rate and 𝑒 𝑣 =
𝐿𝐸 𝑣
𝑘 𝑒𝑑 Δ𝑇
.
Where 𝜁 𝜏 =
𝜁0, 0 ≤ 𝜏 ≤ 𝜏 𝑕
0, 𝜏 > 𝜏 𝑕
Lee et al. [5] used the superposition for the dimensionless temperature profile as
𝜃 𝑥, 𝑦, 𝑧, 𝜏 = 𝜙 𝑥, 𝑦, 𝑧, 𝜏 + 𝜓(𝑧)
Where, 𝜓 𝑧 is the temperature profile for
With the value of 𝜓(𝑧) as

10. 𝜓 𝑧 =
𝑞 𝑚
2
1 − 𝑧2
+ 𝑞 1 − 𝑧 + 𝛼 +
1
𝐵𝑖
𝑞+𝑞 𝑚
𝜅 𝑒𝑑
−
𝑒 𝑣
𝐵𝑖
+ 𝑐 for 0 ≤ 𝑧 ≤ 1
𝜓 𝑧 = 1 − 𝑧 + 𝛼 +
1
𝐵𝑖
𝑞+𝑞 𝑚
𝜅 𝑒𝑑
−
𝑒 𝑣
𝐵𝑖
+ 𝑐 for 1 ≤ 𝑧 ≤ 𝑧 𝑒𝑑
Where
𝑞
𝜅 𝑒𝑑
=
𝐵𝑖 𝑓 𝐵𝑖 + 𝑒 𝑣
1 + 𝐵𝑖 0.8𝜅 𝑒𝑑 + 𝛼
𝑓 𝐵𝑖 = 0.9797 − 0.2651𝐵𝑖 + 0.1411𝐵𝑖2
− 0.03627𝐵𝑖3
𝑐 𝐵𝑖 = 0.01812 − 0.03995𝐵𝑖 + 0.08439𝐵𝑖2
− 0.08799𝐵𝑖3
+ 0.03359𝐵𝑖4
Substituting in the bioheat equation, the governing differential equation for the burned
component of the temperature profile becomes
𝛾
𝜕𝜙
𝜕𝜏
=
𝜕
𝜕𝑥
𝜅
𝜕𝜙
𝜕𝑥
+
𝜕
𝜕𝑦
𝜅
𝜕𝜙
𝜕𝑦
+
𝜕
𝜕𝑧
𝜅
𝜕𝜙
𝜕𝑧
+ 𝑞 𝑚
With boundary conditions
𝜕𝜙 (0,𝑦,𝑧,𝜏)
𝜕𝑥
= 0 ;
𝜕𝜙 (𝑥0,𝑦,𝑧,𝜏)
𝜕𝑥
= 0 ;
𝜕𝜙 (𝑥,0,𝑧,𝜏)
𝜕𝑦
= 0 ;
𝜕𝜙 (𝑥,𝑦0,𝑧,𝜏)
𝜕𝑦
= 0
𝜕𝜙
𝜕𝑧
+ 𝐵𝑖𝜙 + 𝑒 𝑣 − ζ τ = 0
For normal skin
𝜕𝜙
𝜕𝑧
− Ωϕ = −Ωθc at 𝑥 𝑎 ≤ 𝑥 ≤ 𝑥 𝑏 ; 𝑦𝑎 ≤ 𝑦 ≤ 𝑦𝑏
For burned skin

11. 𝜕𝜙
𝜕𝑧
= −𝑞 at x < 𝑥 𝑎 ; 𝑥 > 𝑥 𝑏 ; 𝑦 < 𝑦𝑎 ; 𝑦 > 𝑦𝑏
And the initial condition
𝜙 𝑥, 𝑦, 𝑧, 0 = 𝑓0(𝑥, 𝑦, 𝑧)
Initially, 𝑓0(𝑥, 𝑦, 𝑧) was assigned the value zero everywhere in space. Then, the
simulation of the heating process was carried out with this value of the initial condition. The
final temperature profile at the end of heating was the value of 𝑓0(𝑥, 𝑦, 𝑧) for the cooling phase.
Numerical method:
A uniform fine grid system was employed in the vicinity of the terminal vessels
(−0.05 ≤ 𝑥 ≤ 0.05 and −0.05 ≤ 𝑦 ≤ 0.05),
𝑥𝑖 = −0.05 + 𝑖 − 1 Δ𝑥 ; 𝑦𝑗 = −0.05 + 𝑗 − 1 Δ𝑦
Where Δ𝑥 = Δ𝑦 = 0.001, i=1,2,3,…,201 and j= 1,2,3,…,201
Thermo physical properties of skin tissue and blood from the literature[5, 13] were
employed in the present computation:
𝑘 𝑑 = 0.37 ; 𝑘 𝑒𝑑 = 0.21 ; 𝑘 𝑕𝑑 = 0.16 ; 𝜌 𝑑 = 1200 ; 𝜌𝑒𝑑 = 1200 ; 𝜌𝑏 = 1100
𝑐 𝑝 𝑑
= 3200 ; 𝑐 𝑝 𝑒𝑑
= 3580 ; 𝑐 𝑝 𝑏
= 3300
Where the units of thermal conductivity, density ad specific heat are 𝑊𝑚−1
𝐾−1
, 𝑘𝑔𝑚−3
and 𝐽𝑘𝑔−1
𝐾−1
respectively. The evaporation rate is assumed to be 𝐸𝑣 = 10 𝑊𝑚−2
, while the
metabolism heat is negligible in skin (𝑄 𝑚 = 0). Under normal conditions, the blood perfusion
rate is 𝜔 𝑏 = 0.024 𝑚3
𝑠−1
𝑚3
.

12. The solution has been calculated by Lee et. al [4] for normal unburned skin. The
superposition method uses the idea that the temperature profile can be divided into two parts –
normal skin temperature profile and change induced by burns. The unsteady state solution of the
bioheat equation is solved in the thermal simulation of skin with the conventional central
difference scheme and the SIS solver [3].
Hypothetical Burn Shape:
Three types of hypothetical burn shapes are analyzed in this study – a square, an obtuse
scalene triangle and a composite of the maps of India and Taiwan to represent a general
unconnected arbitrary shape.
The arbitrary shapes were obtained using digital images of the maps of India and Taiwan.
The images were processed using MATLAB’s Image Processing Toolbox. ADT was simulated
through a FORTRAN 77 program in which the image shapes were used as hypothetical burn
wounds of arbitrary shape. The essence of this study was to be able to recreate these burn area
maps using the proposed method.
The ADT simulation resulted in 2-D maps of the time constant 𝛼 which changed as a
function of time. The process of cooling of well-hydrated and barely perfused tissue is faster than
in the unburned skin. The values of alpha for thermally damaged skin are shorter than those for
normal skin [9]. The inflection points of these alpha maps would give an outline of the burn
wound. Hence, the gradient of the time constant was calculated numerically for the entire map.
For internal grid points, the central difference method was used. At the boundary the value of
gradient alpha was assumed to be zero. This is justified by the condition in the simulation that
the total area chosen for simulation of ADT was large enough such that the analysis does not

13. suffer from boundary effects. Conversely, the boundary, unaffected by the burn wound, would
have constant alpha and thus zero gradient.
As the inflection point lies on the boundary between burned and unburned skin, a plot of
∇𝛼 has elevated burned areas, the boundary points of burned and unburned skin forming a
ridgeline over this elevated ‘mountain range’. This ridgeline was accentuated using the constant
variance image enhancement procedure suggested by Harris [2] . This procedure is, in essence, a
deblurring algorithm. The first step was to calculate the Gaussian weighted average of each point
taking into consideration the 8 neighboring points. The weight matrix of the Gaussian filter was
approximated using the method developed by Sami et al. [12]. The weights were assigned
according to the Gaussian filter matrix which is as follows:
𝑊 =
1
16
1 2 1
2 4 2
1 2 1
𝐿𝐴 = 𝐵1 ∗ 𝑊
Here, LA represents the local average value and 𝐵1 represents the input image. The
averaging was done at all points except the boundary points. At the boundary points, the average
was taken to be the value at that point itself. This is justified by the assumption that the wound is
far away from the boundary such that it is unaffected by boundary effects. Conversely, the
boundary is unaffected by the simulated burn wound and hence the value of alpha the same as
that of normal unburned skin which is constant near the boundary. Next, the local variance(LV)
and local standard deviation(LSD) is calculated in a similar fashion using the neighboring points.
𝐿𝑉 = 𝐵1 − 𝐵1 ∗ 𝑊 2
∗ 𝑊

14. 𝐿𝑆𝐷 = 𝐵1 − 𝐵1 ∗ 𝑊 2
∗ 𝑊
1
2
The variance and standard deviation at the boundary is zero. This local standard deviation
is used as a gain control. The difference between unfiltered and filtered maps was calculated and
normalized using the local standard deviation. This image enhancement procedure accentuates
the ridgeline.
𝐶𝑉𝐸 =
𝐵1 − 𝐵1 ∗ 𝑊
𝐵1 − 𝐵1 ∗ 𝑊 2 ∗ 𝑊
1
2
The maxima points along the ridgeline were then calculated. These points were sorted
using a minimum distance walker program. Thus the edge of the burn wound was outlined. Both
the original burn wound shape and the shape outline obtained by this method can be compared in
figures 8, 10 and 11(d).
Numerical Results and Discussion:
Three burn wound shapes are analyzed in this study – a square, an obtuse scalene triangle and a
composite of the maps of India and Taiwan which represents a completely arbitrary shape.
A square shaped wound was chosen first as it is the simplest shape in 2-D Cartesian co-
ordinate system. The square shaped hypothetical burn wound was defined at the center of the
skin model. The distance of the wound from the boundary of the skin model was large to avoid
boundary effects. In this first shape, plots of temperature, 𝛼, ∇𝛼 and the final calculated shape
are all shown. At steady state the temperature profile of the square wound is shown in Fig 3. The
heating stage of ADT was then simulated on the square wound. Fig 4(a) shows the temperature
profile after the heating stage. In the cooling stage, fig 4(b) shows the temperature profile of the

15. surface at different times in the plane y = 0.075 m. The y-coordinate is so chosen as it represents
the center of the wound along the y-axis. Fig 5(a) shows the plot of time constant 𝛼 at time t=0
just after heating. Fig 5(b) shows the variation of 𝛼 with time at y = 0.075 m. Fig 6(a) and 6(b)
shows the plot of ∇𝛼 on the skin surface at times t = 15.194 sec and at t = 29.952 sec
respectively. It is noted that the value of ∇𝛼 drops near sharp corners of the wound area. Fig 7
shows the variation of ∇𝛼 along y = 0.075 m at different times. Fig 8 shows the final calculated
wound area superposed on the original burn wound area.
The second burn shape considered was the triangular shape. An obtuse scalene triangle
was chosen to understand the effects of obtuse as well as sharp acute angles on the time constant.
Fig 9 is a 3-D plot of ∇𝛼 on the surface of the skin in the beginning of the cooling stage. Fig 10
shows the final calculated triangular wound area superposed on the original wound area. It can
be noted that the value of ∇𝛼 drops more sharply where the angle of the wound shape is acute.
Finally, an irregular shape was chosen as a practical choice for comparison to actual
wounds. The composite map served as an good sample of a shape of high irregularity. Fig 11(a)
shows the original arbitrary wound shape. Fig 11(b) shows the plot of ∇𝛼 vs the skin surface. Fig
11(c) shows the calculated burn wound area. Fig 11(d) shows the calculated area superposed on
the original wound area.
It is observed that this method successfully identifies burn wound shapes with reasonable
accuracy. The biggest advantage of this method is that it the original wound shapes were in the
form of digital images. Hence, a burn wound can be diagnosed by using only digital images of
the wounds and no other equipment is necessary.

16. References:
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diagnostic technique method for skin burn wounds. Burns. (Submitted for publication).
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18. FIGURE CAPTIONS
Fig. 1. A schematic vascular system of skin
Fig. 2. A cross-section of the three-dimensional burn model
Fig. 3. A contour plot of the temperature profile for a square shaped wound at steady
state before heating begins (time = -30.42 sec)
Fig. 4. (a) A contour plot of the temperature profile for a square shaped wound at time
t=0 just after heating at the beginning of the cooling stage
(b) Skin surface temperature on the plane 𝑦 = 0.075𝑚 at different time stages
Fig. 5. (a) Alpha contour plot at time 𝑡 = 0 𝑠𝑒𝑐 after heating
(b) Plot of 𝛼 vs x in the plane 𝑦 = 0.075𝑚 at different times
Fig. 6. Skin surface 3-D contour maps of ∇α for square shaped wound at
(a) 𝑡 = 15.194 𝑠𝑒𝑐 (b) 𝑡 = 29.952 𝑠𝑒𝑐
Fig. 7. Plot of ∇𝛼 vs x in the plane 𝑦 = 0.075𝑚 at different times
Fig. 8. The calculated wound area and actual wound area for the square shaped wound
Fig. 9. Skin surface 3-D contour map of ∇𝛼 for triangular wound at 𝑡 = 0.04 𝑠𝑒𝑐
Fig. 10. The calculated wound area and actual wound area for the triangular wound

19. Fig. 11. (a) Original wound of arbitrary shape
(b) Skin surface 3-D contour map of ∇𝛼 for wound of arbitrary shape
(c) The calculated wound area
(d) The calculated wound area and original wound area for wound of arbitrary
shape

20. FIGURES
Figure 1

21. Figure 2

22. Figure 3

23. Figure 4(a)

24. Figure 4(b)

25. Figure 5(a)

26. Figure 5(b)

27. Figure 6(a)

28. Figure 6(b)

29. Figure 7

30. Figure 8

31. Figure 9

32. Figure 10

33. Figure 11(a)

34. Figure 11(b)

35. Figure 11(c)

36. Figure 11(d)