skin burn henrique ecuations.pdf

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Burns 28 (2002) 713717

Effects of thermal properties and geometrical dimensionson skin burn injuries

S.C. Jiang, N. Ma, H.J. Li, X.X. Zhang

Department of Thermal Engineering, Tsinghua University, Beijing 100084, PR China

Accepted 20 May 2002

Abstract

A one-dimensionalmulti-layermodel is presentedto characterise theskinburnprocess resultingfrom theapplicationof a high temperature

heat source to a skin surface. Transient temperatures were numerically calculated using a nite difference method to solve the Pennesbioheat equation. A damage function denoting the extent of burn injury was then calculated using the Arrhenius assumptions. The modelwas used to predict the effects of thermal physical propertiesand geometrical dimensions on the transient temperature and damage functiondistributions. The results show that the epidermis and dermis thicknesses signicantly affect the temperature and burn injury distributions,while variations of the initial temperatures and the blood perfusion have little effect. 2002 Elsevier Science Ltd and ISBI. All rights reserved.

Keywords: Burn; Skin; Multi-layer; Thickness; Bioheat

1. Introduction

Accurate analysis of the skin burn process facilitates the

development of animal models. Burns result from a temper-ature elevation in tissue over a threshold value for a period of time. Both the temperature and the exposure duration greatlyaffect the extent of the burn. Henriques and Moritz [1] rstdiscussed the relationship between the temperature and thetime required to produce a specic degree of thermal injury.Their results showed that a burn injury of standard thresh-old severity can be produced by progressively decreasingtemperatures as the thermal insult period is logarithmicallyincreased.

Many investigators have discussed burn injuries for dif-ferent situations. Diller and Hayes [2] presented a nite el-ement model of burn injuries in blood-perfused skin. Torviand Dale [3] developed a nite element model of skin for aash re exposure and discussed the sensitivity of burn pre-dictions to variations in thermal physical properties. Ng andChua [4] presented a mesh-independent model to predictskin burn injuries using the nite element method. Ng andChua [5] compared one- and two-dimensional programmesfor predicting skin burns and showed that the temperaturedistributions predicted by the one- and two-dimensional pro-grammes were similar. Recently, Diller [6,7] proposed a

Corresponding author. Fax: + 86-10-62773412. E-mail address: [email protected] (X.X. Zhang).

one-dimensional model to calculate the transient tempera-ture and injury distributions in skin using the nite differ-ence method.

The model developed here presents the transient temper-ature and damage function distributions variations for thevariations of the initial temperature, blood perfusions andskin layer thicknesses. The results show that the epidermisand dermis thicknesses signicantly affect the temperatureand burn injury distributions.

2. Mathematical model

A one-dimensional multi-layer tissue model is shown inFig. 1. The skin is divided into three layers, the epidermis,the dermis, and the subcutaneous region. The region from theinner surface of the subcutaneous region to the core is calledthe inner region in this paper. The typical thermal physicalproperties [3] used in this paper are listed in Table 1 . Forblood, b = 1060kgm 3 and C b = 3770J kg 1 K 1 .

The analysis assumes that no interfacial resistance existsbetween the heating source and the skin surface. Therefore,the surface temperature remains constant during the expo-sure. The tissues are assumed to be homogeneous withineach layer. Blood perfusion, thermal conductivity, and heatcapacity are assumed to be constant in each layer despitethe temperature elevation. Metabolic heat generation is as-sumed to be zero relative to the other heat uxes involved.

0305-4179/02/$22.00 2002 Elsevier Science Ltd and ISBI. All rights reserved.PII : S0305- 4179(02 )00104 -3


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714 S.C. Jiang et al./ Burns 28 (2002) 713717

Nomenclature

C specic heat (J kg 1 K 1) H distance from skin surface to body corek thermal conductivity (W m 1 K 1)P pre-exponential factor (s 1)

R universal gas constant (8.314 J kmol 1

K 1

)t time (s)T temperature ( C or K) x space co-ordinate (m)

E activation energy (J kmol 1)

Greek letters density (kg m 3)b blood perfusion (m 3 s 1 m 3 tissue) damage function

Subscripts and superscriptsa ambientb bloodc coree eastf nali initial; node numberk time stepw west; wall

Fig. 1. Schematic diagram of the multi-layer tissues.

Table 1Thermal physical properties used in the skin

Thickness (m) Specic heat(Jkg 1 K 1)

Blood perfusion rate(m3 s 1 m 3 tissue)

Thermal conductivity(W m 1 K 1)

Density (kgm 3)

Epidermis 80 10 6 3590 0 0.24 1200Dermis 0.002 3300 0.00125 0.45 1200Subcutaneous 0.01 2500 0.00125 0.19 1000Inner tissue 0.03 4000 0.00125 0.5 1000

The most common model for simulating the thermal be-haviour of tissue is the Pennes bioheat transfer equation[8]. The one-dimensional form of Pennes bioheat equationwithout metabolic heat generation can be written as:

CT

t =

xk

T

x+ b bC b(T b T ) (1)

The burn process is modeled using the following initial andboundary conditions:

T(x, 0) = T i (x) for t = 0 (2)

T (0, t ) = T w for x = 0 (3)

T(H, t ) = T c for x = H (4)

where the initial temperature distributions, T i ( x), could beobtained by solving Eq. (1) with the following boundaryconditions:

kT i

x= h a(T i T a) for x = 0 (5)

T i = T c x = H (6)

Thermal damage begins when the tissue temperature risesabove 44 C. Theburn damageis predictedby calculating thedamage function, , at each point basing on the assumptionsof Henriques and Moritz [9] as follows:

(x) = t

0P exp

E

RT (x,t)dt (7)


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S.C. Jiang et al. / Burns 28 (2002) 713717 715

Threshold burn injuries occur if = 0.53 and third-degreeburn injuries occur if = 10 000 [2].

3. Numerical method

A one-dimensional nite difference method was used tosolve the bioheat Eq. (1). Non-uniform space meshes anduniform time steps were used in the model. The space stepsand time steps are small enough to ensure that the tran-sient temperatures were mesh-independent. An implicit timeschemewasused to discretizate thedifferential bioheat equa-tion as follows:

CT i k T i k 1

t x = ke

T ki + 1 T k

i

x kw

T ki T k

i 1x

+ b bC b(T b T k

i ) x (8)

where k was the effective thermal conductivity calculatedusing the harmonic mean. TheTridiagonalMatrix Algorithm[10] is used to solve the discretized equations. Then thedamage function at nodes with the temperatures higher thanthe threshold was calculated using:

(i) =kf

k= ki

P exp E

RT i kt (9)

where k i indicated the time step when the node temperaturewas above the threshold and k f indicated the nal time step.

4. Results

The time step independent temperature distribution waschosen by calculating for t = 0.01, 0.001 , and 0.0001 s.The results showed that t = 0.001 s was small enough toobtain time step independent temperature and damage func-tion distributions. The numerical results also showed that thetemperature variation decreased with depth. Therefore, thegrid space was increased from the epidermis layer towardsthe inner tissue.

The temperature eld in the tissue after 15 s of heatingwith a surface temperature of 90 C was shown in Fig. 2.Thetemperatures of theepidermisanddermis tissuechangedsignicantly, while the deeper tissue temperatures remainedalmost the same. Fig. 3 showed the damage function ( =0.53, = 10 000) distributions with time. Most of thetissues near the surface suffered severe injury immediatelyafter exposure, while in the deeper tissues, serious damageoccurred after a relatively longer time period. The presentresults agreed well with those of Ng and Chua [5] f or theconditions in their paper.

In another series of calculations, the initial temperaturedistribution was assumed to be uniform (34 C) to evaluatethe effect of the initial temperature distribution. The temper-ature distribution after 10 s and the damage function were

Fig. 2. Tissue temperature variations for a surface temperature of 90 C.

Fig. 3. Threshold ( = 0.53) and third-degree ( = 10 000) burndistributions for a surface temperature of 90 C.

Fig. 4. Temperature distributions 10 s after exposure for uniform andnon-uniform initial temperature distributions.

Fig. 5. Burn damage burn injuries for uniform and non-uniform initialtemperature distributions.


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716 S.C. Jiang et al./ Burns 28 (2002) 713717

Fig. 6. Temperature distributions 10s after exposure for different epidermisthickness and a dermis thickness of 2.0 mm.

compared with the previous results in Figs. 4 and 5. Thetemperature distributions within about 5 mm of the surfacewere almost the same with differences of several degreesdeeper in the tissue where the damage was less extensive.Fig. 5 showed that the damage burn injuries for both condi-

tions are almost the same. Therefore, the assumption of ini-tial uniform temperature distributions in the previous analy-ses [2,4,5] provided reasonable estimates of the burn injuriesresulting from short exposures.

The thicknesses of the epidermis and dermis differed fordifferent peoples and different locations. The transient tem-perature and burn injury distributions for different epidermisthickness and dermis thickness were shown in Figs. 69.They showed that the epidermis and dermis thickness bothsignicantly affected the transient temperature and burninjury distributions. Burn injuries greatly decreased with

Fig. 7. Threshold and third-degree burn injuries for different epidermisthickness and a dermis thickness of 2.0 mm.

Fig. 8. Temperature distributions 10s after exposure for different dermisthickness and an epidermis thickness of 0.08mm. The dermis thicknessis indicated by the square symbol.

Fig. 9. Threshold and third-degree burn injuries for different dermisthickness and an epidermis thickness of 0.08mm.

increasing epidermis and dermis thickness. Therefore, inestablishing an animal skin burn model, the animals shouldhave similar epidermis and dermis thickness to ensure therepeatability. In addition, the heat source should be appliedessentially the same location each time. An alternative

would be to apply different exposure times to each animalaccording to their specic epidermis and dermis thickness.

The effects of variations of blood perfusion in the der-mis layer and in deeper layers on the transient tempera-ture and burn injury distributions were also calculated. Theresults showed that the effect of blood perfusion can beignored in predicting the burn injuries for short durationexposures. The heat capacity of the tissues played an im-portant role in preventing the deeper tissues from thermaldamage.

5. Conclusion

A one-dimensional multi-layer nite difference modelwas developed in this paper to calculate transient temper-ature and burn injury distributions in living tissues. Themodel was used to analyse the effects of variations in ther-mal physical and geometrical properties on the transienttemperature and damage function distributions. The resultsshowed that the epidermis and the dermis thickness signif-icantly affected the transient temperature and burn injurydistributions, while the initial temperature distribution andthe blood perfusion had little effect. Therefore, animal skinburn studies should consider the effects of different der-mis and epidermis thicknesses, different heating locationsand different exposure times. The model accuracy was re-stricted by the accuracy of the thermal properties of thetissues during exposure and the accuracy of the Henriquesmodel.

References

[1] Henriques Jr FC, Moritz AR. Studies of thermal injury. I. Theconduction of heat to and through skin and the temperature attainedtherein. A theoretical and an experimental investigation. Am J Pathol1947;23:53149.


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[2] Diller KR, Hayes LJ. A nite element model of burn injury inblood-perfused skin. J Biomech Eng 1983;105:3007.

[3] Torvi DA, Dale JD. A nite element model of skin subjected to aash re. J Biomech Eng 1994;116:2505.

[4] Ng EYK, Chua LT. Mesh-independent prediction of skin burns injury.J Med Eng Technol 2000;24:25561.

[5] Ng EYK, Chua LT. Comparison of one- and two-dimensional pro-grammes for predicting the state of skin burns. Burns 2002;28:2734.

[6] Diller KR. Modeling thermal skin burns on a personal computer. JBurn Care Rehabil 1998;19:4209.

[7] Diller KR. Development and solution of nite difference equationsfor burn injury with spreadsheet software. J Burn Care Rehabil1999;20:2532.

[8] Pennes HH. Analysis of tissue and arterial blood temperatures inresting human forearm. J Appl Physiol 1948;1:93122.

[9] Henriques Jr FC. Studies of thermal injuries. V. The predictabilityand the signicance of thermally induced rate processes leading toirreversible epidermal injury. Arch Pathol 1947;43:489502.

[10] Tao WQ. Numerical heat transfer. 1st ed. Xian: Xian JiaotongUniversity, 1988 [in Chinese].

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