A heat transfer model of the human upper limbs

International Communications in Heat and Mass Transfer 39 (2012) 196–203

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A heat transfer model of the human upper limbs☆ M.S. Ferreira ⁎, J.I. Yanagihara Department of Mechanical Engineering, University of Sao Paulo, Av. Prof. Mello Moraes, 2231, 05508-900, Sao Paulo, SP, Brazil

a r t i c l e

i n f o

Available online 15 December 2011 Keywords: Bioheat transfer Thermoregulation Mathematical model

a b s t r a c t A steady state multi-segmented heat transfer model of the human upper limbs was developed. The main purpose was to evaluate the impact of blood flow through superficial veins and subcutaneous vascular structures in the palm of the hands over the heat transfer between the limbs and the environment. The distinguishing feature of the model is the inclusion of a detailed circulatory network composed of vessels with diameter larger than 1 mm. The model was validated by comparing its results from exposures to a hot, a neutral, and a cold environment to experimental data presented in the literature. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction In 1948, Pennes [1] presented a mathematical model of heat transfer in human tissue. In the resulting equation, called the bioheat equation, the effect of blood flow on heat transfer was modeled as a heat sink or source, the magnitude of which is proportional to the volumetric perfusion rate and difference between arterial and venous temperature [2]. Pennes assumed that thermal equilibrium occurs in the capillary beds, although Chen [3] later showed that it occurs before the blood enters the beds. In addition, a relevant phenomenon not taken into account in Pennes’ model is the countercurrent heat exchange between parallel arteries and veins with diameters larger than those of the capillaries (5–15 μm diameter). Some analytical models [2,4] were developed in order to describe temperature variations along countercurrent vessels from 50 to 1000 μm diameter. “Whereas the Pennes equation has gained widespread acceptance and has generally yielded results that agree with experimental observations, important questions about its validity remain unanswered” [5]. It is well known that the human body extremities play an important role in dissipating body heat. Hirata et al. [6] showed that the heat loss in the forearm is enhanced by the venous blood returning through the superficial veins originated from arteriovenous anastomoses (AVA) presented in the hands and fingers. These structures allow the enhancement of heat transfer from the body core to the extremities without an increase in vasodilation and with a smaller pressure loss. Recently, Grahn et al. [7] described the importance of heat transfer of subcutaneous vascular structures in the glabrous skin presented in the palm of the hands, soles of the feet, and regions of the face and ear pinnae. According to these researchers, blood flow

☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (M.S. Ferreira). 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.12.004

through these structures is extremely variable, ranging from near zero in cold stress to as much as 60% of the cardiac output during heat stress. Blood flow is regulated according to the opening degree of the AVAs. Also, these researchers proposed a way to effectively manipulate body core temperature. Application of sub-atmospheric pressure to the hand promotes the distention of the local vasculature and, consequently, an increase in blood flow with concomitant heat transfer enhancement. Using the appropriate thermal load, this technique can be used to restore to normothermia hypothermic postsurgical patients, cold-stressed healthy subjects [8], and heatstressed individuals. Heat transfer models of the human body have been used to evaluate thermal comfort or predict thermal response to exposure to different environments. In general, multi-segmented models represent the body by cylinders divided into annular concentric layers representing the tissues. The availability of MRI (Magnetic Resonance Imaging) data suggests the development of a more detailed geometric representation of the human body. However, the lack of physiological data, thermophysical properties of tissues and heat transfer correlations suggest the opposite. In addition, as presented before, the presence of AVAs in the hands and superficial veins in the upper limbs are important to thermoregulation. Consequently, instead of searching for a more detailed geometric representation, maybe the effort should be spent on the implementation of thermal relevant phenomena and structures. To what extent are these issues included in whole body or human body segments thermal models? Countercurrent heat exchange between arteries and veins is not considered in all multisegmented models found in the literature; this attribute is restricted to some of them [9–11]. In order to include countercurrent heat exchange, it is necessary to calculate global heat transfer coefficients between arteries and veins, which, in turn, require the calculation of two-dimensional steady state conduction heat transfer shape factors. Following a different path, some models include a more detailed description of the circulatory system [12–14], considering even pulsating blood flow, superficial veins, and arteriovenous anastomoses.

M.S. Ferreira, J.I. Yanagihara / International Communications in Heat and Mass Transfer 39 (2012) 196–203

Nomenclature A BF D Gz M Nu Q R T L c d h k q r u v x w

Plexus superficial area or segment superficial area [m 2] Blood flow rate [cm 3/h] Diameter [m] Graetz number Metabolic heat generation [W] Mean Nusselt number Blood flow rate [m 3/s] Cylinder radius [m] Temperature or tissue temperature [°C] Cylinder length [m] Specific heat [J/(kg K)] Distance between artery and vein [m] Combined heat transfer coefficient or convective heat transfer coefficient [W/(m 2 K)] Thermal conductivity [W/(m K)] Metabolic heat generation rate per unit volume [W/ m 3] or heat flux [W/m 2] Radial coordinate [m] Global heat transfer coefficient per unit volume [W/ (m 3 K)] Velocity [m/s] Axial coordinate measured from the left side of the cylinder [m] Perfusion rate [m 3/(m 3 s)]

Greek symbols α Shunt index ρ Density [kg/m 3] δ Palmar index

Subscripts D Vessel diameter air Air ar Arterial art Between arterial arch and tissue at Between artery and tissue b Back plexus bl Blood cr Core ev Evaporation (sweat plus diffusion) hd Hand op Operative p Palmar plexus sk Skin st Between superficial vein and tissue t Tissue vb Venous back plexus vbt Between venous back plexus and tissue ve Venous or vein vet Between venous plexus and tissue vp Venous palm plexus vpt Between venous palm plexus and tissue vs Superficial venous or vein vt Between vein and tissue x Axial coordinate

The aforementioned whole body models represent the hand with its fingers by one cylinder with the same features of other segments. Nevertheless, as presented before, hands and fingers have important

197

structures for heat transfer. It is worth stressing that our objective is not to present detailed information or criticize the models found in the literature; instead, the purpose is to access the impact of including the features described in the latter paragraphs in a model. In order to achieve this objective, a detailed heat transfer model of the human upper limbs was developed. Circular cylinders representing the arm, forearm, fingers, and a slab representing the hand compose the model. Except for the latter, the segments are subdivided into two concentric annular layers, the skin and core ones. Steady-state heat conduction in the radial and axial directions was considered. The distinguishing feature of the model is the inclusion of a detailed circulatory network composed of vessels with diameter larger than 1000 μm. The disposition and dimensions of these vessels are based on anatomical considerations. These vessels are treated as discrete structures with temperatures varying in the longitudinal direction. Superficial veins in the arm and forearm, arteriovenous anastomoses and subcutaneous venous plexus in the hand and fingers are structures included in the network. Countercurrent heat exchange between an artery and its venae comitantes is also taken into account. Small vessels were treated on a continuum basis. 2. Model description 2.1. Geometric model and circulatory system Seven circular cylinders representing the arm, forearm, and the five fingers compose the model. Each cylinder is divided into two annular concentric layers, the skin and core. A slab, also divided into skin and core, was used to model the hand. The macro-circulation arrangement proposed (Fig. 1) is based on anatomical considerations [15]. Only the main vessels with diameters larger than 1000 μm were considered. In the arm, the brachial artery with its venae comitantes, and the cephalic and basilic veins were considered. In the forearm, the radial and ulnar arteries with their venae comitantes, and four superficial veins were taken into account. The digital arteries and veins were considered to be representative of the arteries and veins in the fingers. In the hand, three venous

Fig. 1. Macro-circulation arrangement considered in the: (a) arm and forearm, (b) hand, and (c) finger.

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Fig. 2. Micro-circulation arrangement considered in the model.

(venous arch, back and palmar plexus) and one arterial plexus (arterial arch) were included. In order to simplify the figures, the number of superficial veins represented is smaller than the number considered in the model. See Table 3 for the correct quantities. Vessels with diameters smaller than 1000 μm form the micro-circulation, and are treated as part of a continuum. Blood leaves the main arteries perpendicularly, irrigates the tissue, and returns to the major veins (see Fig. 2). Mass balances are applied to the macro-circulation and afterwards to the micro-circulation. Eqs. (1)–(26) were used to calculate the blood flow rate (BF) inside each vessel in the model. Fig. 1 serves to illustrate these relationships. The relation between the blood flow in the micro and macro-circulation can be calculated by applying a mass balance to all arteries, deep veins and superficial veins in a segment. Assuming uniform perfusion in each layer, arterial blood flow decreases linearly in the segment axial direction. The opposite occurs in the deep veins. It was assumed that superficial veins in a segment have a uniform mass flow rate. Arteries: Brachial Artery BF in ¼ Arm BF þ Forearm BF þ Hand BF þ 5  Finger BF

ð1Þ

Brachial Artery BF out ¼ Forearm BF þ Hand BF þ 5  Finger BF

ð2Þ

Ulnar Artery BF in ¼ Radial Artery BF in ¼ Brachial Artery BF out  2

ð3Þ

Ulnar Artery BF out ¼ Radial Artery BF out ¼ ðHand BF þ 5  Finger BF Þ  2

ð4Þ

Arterial Arch in ¼ Hand BF þ 5  Finger BF

ð5Þ

Arterial Arch out ¼ 5  Finger BF

ð6Þ

Palmar Digital Artery BF in ¼ Finger BF  2

ð7Þ

Palmar Digital Artery BF out ¼ 0

ð8Þ

Veins: Brachial Vein BF in ¼ ðForearm BF þ Hand cr BF þ 5  Finger cr BF þ bÞ  2

Ulnar Vein BF out ¼ Radial Vein BF out ¼ ¼ ðForearm BF þ Hand cr BF þ 5  Finger cr BF þ bÞ  4

ð13Þ

Venous Arch BF out ¼ Hand BF þ 5  Finger cr BF þ b

ð14Þ

Cephalic Vein BF in ¼ Basilic Vein BF in ¼ ðHand sk BF þ 5  Finger sk BF Þ  α  2

ð15Þ

Cephalic Vein BF out ¼ Basilic Vein BF out ¼ Cephalic Vein BF in

ð16Þ

Superf icial Vein BF in ¼ ðHand sk BF þ Finger sk BF Þ  α  6

ð17Þ

Superf icial Vein BF out ¼ Superf icial Vein BF in

ð18Þ

Palmar Venous Plexus BF in ¼ 5  Finger sk BF  δ

ð19Þ

Palmar Venous Plexus BF out ¼ Palmar Venous Plexus BF in

ð20Þ

Back Venous Plexus BF in ¼ 5  Finger sk BF  ð1–δÞ

ð21Þ

Back Venous Plexus BF out ¼ Back Venous Plexus BF in

ð22Þ

Digital Vein BF in ¼ 0

ð23Þ

Digital Vein BF out ¼ Finger BF  4

ð24Þ

b ¼ Hand sk BF þ 5  Finger sk BF  ð1−α Þ

ð25Þ

g ¼ 5  Finger sk BF  ð1−α Þ

ð26Þ

Notes: the following property is valid, Arm BF = Arm sk BF + Arm cr BF. α is the shunt index (α = 0 means that all blood leaving the fingers and hand skin returns through the deep veins; alternatively, when α = 1, the blood from these regions returns through superficial veins). δ is an index that indicates the proportion of finger skin blood flow that returns through the palmar plexus (δ = 0 means that all blood leaving the finger skin returns through the back venous plexus of the hand; alternatively, when δ = 1, the blood from these regions returns through palmar plexus. 2.2. Heat transfer model

ð9Þ

Brachial Vein BF out ¼ ðArm BF þ Forearm BF þ Hand cr BF þ 5  Finger cr BF þ bÞ 2 ð10Þ Ulnar Vein BF in ¼ Radial Vein BF in ¼ ðHand cr BF þ 5  Finger cr BF þ bÞ  4

Venous Arch BF in ¼ 5  Finger cr BF þ g

ð11Þ ð12Þ

Applying a steady state energy balance around a cylindrical differential element of tissue, the following equation is derived: " #
 1∂ ∂T ∂2 T kr þ k 2 þ wbl ρbl cbl ½T ar −T  þ q þ uat ½T ar −T  r ∂r ∂r ∂x þ uvt ½T ve −T  þ ust ½T vs −T  ¼0

ð27Þ

where, respectively, the terms represent radial and axial heat conduction, heat exchange between tissue and micro-circulation (Pennes’ model), metabolic heat generation rate, heat transfer between an

M.S. Ferreira, J.I. Yanagihara / International Communications in Heat and Mass Transfer 39 (2012) 196–203

artery and the surrounding tissue, heat transfer between a vein and the tissue, and heat transfer between a superficial vein and the tissue. The arteries and veins in the macro-circulation were modeled as discrete structures that act as line heat sources or sinks, with axial varying intensity. The proposed model resembles another one [16], but differs from it because the medium vessels are placed in specific locations. Thereby, the global heat transfer coefficients, uat, uvt, and ust should be regarded as a function of radial coordinate (r). This procedure generates a temperature profile that is not derivable with respect to the radial direction in some locations. The global heat transfer coefficients (u) were calculated considering just a convective thermal resistance between the blood and the inner wall of the big vessels. The boundary conditions considered on the cylinder surface and in the cylinder center are, respectively: −ksk

  ∂T  r¼R ¼ h T−T op þ qev ∂r

 ∂T  ¼0 ∂r r¼0

ð28Þ

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in the above equations, the first terms mean the heat transferred between the venous plexus and the skin, the second terms are the heat transferred between the skin and the environment, the third terms are the heat transfer by blood perfusion, and the last terms are the tissue metabolic heat generation. (hA)vbt and (hA)vpt are the convective heat transfer coefficients in the vessels that form the plexus multiplied by the total superficial area of the back and palm plexus, respectively.   ρbl cbl Q ar;in T ar;in −T ar −ðhAÞart ðT ar −T cr Þ ¼ 0

ð36Þ

where the first term means the enthalpy entering the arterial arch and the second, heat transferred by convection between arterial arch and tissue. (hA)art is the convective heat transfer coefficient in the vessels multiplied by the total superficial area of the arterial arch.  ðhAÞvpt  ðhAÞvbt ðT vb −T cr Þ þ T vp −T cr þ ðhAÞvet ðT ve −T cr Þ þ ðhAÞart ðT ar −T cr Þ 2 2 þρbl cbl Q cr ðT ar −T cr Þ þ M cr ¼ 0

ð29Þ

ð37Þ

The differential equations describing the temperature profile inside arteries and veins can be obtained from the application of energy and mass balances. Applying an energy balance to infinitesimal control volumes inside an artery, a vein and a superficial vein, not taking into account axial heat conduction and considering constant properties, the following differential equations are obtained:

where the first, second, third, and fourth terms mean, respectively, the heat transferred between the core and the venous back plexus, the venous palm plexus, the venous arch, and the arterial arch. (hA)art is the convective heat transfer coefficient in the vessels multiplied by the total superficial area of the arterial arch. The fifth term is the perfusion one and the last is the metabolic heat generation of the core.

ρbl Q ar cbl

dT ar þ hat πDar ðT ar −T Þ ¼ 0 dx

ð30Þ

      ρbl cbl Q ve;in T ve;in −T ve þ ρbl cbl Q sk;p T sk;p −T ve þ ρbl cbl Q sk;b T sk;b −T ve þρbl cbl Q cr ðT cr −T ve Þ−ðhAÞvet ðT ve −T cr Þ ¼ 0

where hat is the convective heat transfer coefficient between blood and artery inner wall (used to calculate uat in Eq. (27)).

ð38Þ

dT ve D=2 þ hvt πDve ðT ve −T Þ þ 2πρbl cbl ∫0 wbl ðT ve −T Þrdr ¼ 0 ð31Þ dx

where the first, second, third, and fourth terms are the enthalpy entering the venous arch from the fingers, from the palm skin, from the back skin, and from the core, respectively. The last term is the heat transfer between venous arch and the core tissue.

ρbl Q ve cbl

where hvt is the convective coefficient between blood and vein inner wall (used to calculate uvt in Eq. (27)). ρbl Q vs cbl

dT vs þ hst πDvs ðT vs −T Þ ¼ 0 dx

ð32Þ

ð39Þ

where hst is the convective coefficient between blood and vein inner wall (used to calculate ust in Eq. (27)). Eqs. (30)–(32) are subjected to a prescribed temperature boundary condition at the vessel entrance. Convective heat transfer coefficients between blood and vessels were calculated using Eq. (33) valid for GzL b 1000 [17]: 1:58 logGzL

NuD ¼ 4 þ 0:155e

ð33Þ

Considering the hand, if steady-state energy balances are applied sequentially to control volumes around the hand back skin, palm skin, arterial arch, venous arch, back venous plexus, and palm venous plexus (see Fig. 1b), Eqs. (34)–(40) are obtained:  ðhAÞ h  i ðhAÞvbt  hd T vb −T sk;b − T sk;b −T op −qev;hd 2 2   M þ ρbl cbl Q sk;b T ar −T sk;b þ sk ¼ 0 2  ðhAÞ h  i ðhAÞvpt  hd T vp −T sk;p − T sk;p −T op −qev;hd 2 2  M þ ρbl cbl Q sk;p T ar −T sk;p þ sk ¼ 0 2

  ðhAÞ   ðhAÞ vbt vbt T vb −T sk;b − ðT vb −T cr Þ ρbl cbl Q vs;in ð1−δÞ T vs;in −T vb − 2 2 ¼0

ð34Þ

ð35Þ

  ðhAÞ   ðhAÞ   vbt vbt T vp −T sk;p − T vp −T cr ρbl cbl Q vs;in δ T vs;in −T vp − 2 2 ¼0 ð40Þ where the first terms in both equations are the enthalpies entering the venous back and palmar plexus from the fingers. The last terms are the heat transfer between the venous plexus and the skin and core. 2.3. Numerical solution The finite volume method was applied to the system of partial differential equations. An implicit scheme was considered. The resulting linear system was solved by the Gauss–Siedel procedure for each segment, in the following order: arm, forearm, and finger. Afterwards, Eqs. (34)–(40) representing the hand were solved directly. This sequence is repeated until convergence is achieved. The computer program was written in C++. A series of tests were performed in order to validate the numerical method and its computational implementation. No convergence problems were observed. In all simulations, a mesh of 30 × 30 volumes was employed in the arm, forearm, and finger.

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Table 1 Geometrical parameters.

Table 3 Vessels geometrical parameters.

Body segment

L/m

D/m

th/m

A/m2

V/m3

e/mm

Vessel

Number

L/m

D/mm

D*/mm

x/mm

Arm Forearm Hand Finger

0.31 0.28 0.10 0.07

0.085 0.067 0.085 0.021

– – 0.044 –

828 × 10− 4 589 × 10− 4 214 × 10− 4 46.2 × 10− 4

1759 × 10− 6 987 × 10− 6 374 × 10− 6 24.3 × 10− 6

4.5 4.5 – 2.0

Brachial artery Ulnar artery Radial artery Arterial palmar arch Proper palmar digital artery Brachial vein Radial vein Ulnar vein Venous arch Cephalic vein Basilic vein Superficial vein Venous palmar plexus Venous back plexus Palmar digital vein

1 1 1 10 2+ 2 2 2 10 1 1 6 10 5 4+

0.31 0.28 0.28 – 0.08 0.31 0.28 0.28 – 0.31 0.31 0.28 – – 0.08

4.0 2.6 2.6 1.6 1.0 2.9 2.0 2.0 1.6 3.3 3.3 1.8 1.0 1.8 1.0

3.5–4.4 1.8–4.3 1.9–3.1 1.4–1.8 0.5–1.3 2.9–3.8 1.6 2.3 – 1.9–3.8 1.9–5.1 1.8 – – 0.8–1.3

24 17 17 – 7.5 21 15 15 – 8 8 4 – – 8.5

where L is the cylinder length, D is the cylinder diameter (in the hand D means width), th is the segment thickness, A is the segment superficial area, V is the segment volume, and e is the skin layer thickness.

2.4. Physiological and physical parameters Geometrical parameters used in the model are listed in Table 1. The entries in this table were calculated from data presented in Ferreira [18] and Shitzer et al. [16]. The heat transfer parameters used are listed in Table 2. All heat transfer coefficients were taken from Dear et al. [19]. Exception is made to the finger convective heat transfer coefficient, which was calculated from a standard equation for a cylinder in cross flow [20]. Werner and Reents [21] measured the heat fluxes by evaporation in the arm and hand. Evaporative heat losses from the forearm and fingers were considered equal to that of the arm. Vessels diameters were based on a comprehensive literature review (Baptista-Silva et al. [22], Martin et al. [23], Planken et al. [24], He et al. [13], Shima et al. [25], Giannattasio et al. [26], Dobrosielski et al. [27], Fazan et al. [28], Braga Silva [29], Green et al. [30], Rotella et al. [31], Brodman et al. [32]. The diameters used and the range of variation are presented in Table 3. Vessels lengths were considered equal to the body segments they belong to. Distance between arteries and veins (d) were calculated using Eq. (47) (Shitzer et al. [16]. Distances x were estimated based on cross-section drawings from Drake et al. [15] and measurements from Ferreira [18]. pffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ 2 Dar Dve

ð41Þ

Blood flows for the upper limbs were taken from Salloum et al. [14] or Takemori et al. [12]. The values for the hand and finger were calculated based on the ratio between their volume and the total volume (hand plus fingers). The first blood flow data set is presented in Table 4 and the second one in Table 5. Using the data from Table 4 and the mass balance Equations, Eqs. (1)–(26), the blood flow in each vessel can be calculated (Table 6). Metabolic heat generation values used in the model were taken from Salloum et al. [14]. It was assumed that they also apply to cold and hot exposure. 3. Results and discussion Firstly, a steady state simulation was performed for each environment. In order to validate the model, the results were compared with experimental data from Werner and Reents [21]. Each condition was simulated twice, using each blood flow data set (Tables 4 and 5). The results are presented in Fig. 3. The comparison between the results Table 2 Heat transfer parameters. Cold (Top = 10 °C)

Neutral (Top = 30 °C)

Hot (Top = 40 °C)

qev/(W/m2)

qev/(W/m2)

Body segment

h/(W/ m2 K)

qev/(W/m2)

Arm Forearm Hand Finger

8.1 8.6 8.2 11.0

5 5 10 5

4 4 43 4

75 75 70 75

where h is the heat transfer coefficient (for vair b 0.1 m/s), convection plus radiation, Top is the operative temperature and, qev is the heat flux by evaporation (for vair b 0.2 m/s).

where L is the vessel length, D is the vessel diameter, D* refers to the minimum and maximum mean values found in the literature. For the arteries and deep veins, x is the distance from the vessel cross-section center to the central axis of the cylinder. For the superficial veins and plexuses, x is the distance from the surface of the segment. The values of x for the brachial, ulnar, and radial veins were calculated from the relationship used by Shitzer et al. [16]. + Note: for each finger.

Table 4 Minimum, basal, and maximum blood flow rates (first set, Salloum et al. [14] data). Body segment

Minimum sk BF/(cm3/h)

Basal sk BF/ (cm3/h)

Maximum sk BF/(cm3/h)

cr BF/ (cm3/h)

Arm Forearm Hand + fingers Hand Finger

0 0 627 473 30.7

910 508 1114 840 54.6

8319 5553 4454 3362 218

2667 1758 388 293 19.0

Minimum blood flow corresponds to the cold condition, basal blood flow to the neutral one, and maximum to the hot environment.

and the experimental data indicates that the model is capable of predicting skin temperatures under neutral and hot conditions, but fails to represent it under cold environment. A possible explanation is that a two-layer model (skin and core) was used. Core temperatures under small skin perfusion rates strongly influence skin temperatures because of the absence of a fat layer, which has low thermal conductivity. In addition, experimental data suggests that muscle blood flow is reduced as the forearm is cooled [34]. This phenomenon might also occur in the arm, hand, and fingers. The blood flow reduction results in smaller temperatures on the upper limbs than those observed in Fig. 3. In order to verify the potential of the model developed, a broader region subject to blood flow reduction under cold exposure was considered. The skin layer was extended beyond the thickness presented in Table 1 and a value of thermal conductivity (0.3 W m − 1 K − 1) between that of the skin (0.5 W m − 1 K − 1, [35]) and fat (0.2 W m − 1 K − 1, [35]) was used. The results, indicated by a triangle in Fig. 3, agree well with the experimental data for cold exposure. It could also be observed that the blood flow rates have a strong Table 5 Minimum, basal, and maximum blood flow rates (second set, Takemori et al. [12] data). Body segment

Minimum sk BF/(cm3/h)

Basal sk BF/ (cm3/h)

Maximum sk BF/(cm3/h)

cr BF/ (cm3/h)

Arm Forearm Hand + fingers Hand Finger

0 0 0 0 0

488 272 4699 3547 230

8319 5553 5723 4320 281

2941 1643 264 199 12.9

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Table 6 Minimum, basal, and maximum blood flow rates in each vessel (Takemori et al. [12] data). Vessel

Brachial artery Ulnar artery Radial artery Arterial palmar arch Proper palmar digital artery Brachial vein Radial vein Ulnar vein Venous arch Cephalic vein Basilic vein Superficial vein Venous palmar plexus Venous back plexus Palmar digital vein

Minimum BF/(cm3/h)

Neutral BF/ (cm3/h)

Maximum BF/ (cm3/h)

In

Out

In

Out

In

Out

4689 1,011 1011 26.4 6.45 1011 65.9 65.9 6.45 0 0 0 0 0 0

2022 132 132 6.45 0 2344 505 505 26.4 0 0 0 0 0 1.61

10,804 3613 3613 496 121 3378 1123 1123 110 235 235 78.3 11.5 11.5 0

7227 2480 2480 121 0 5167 1689 1689 449 235 235 78.3 11.5 11.5 30.4

24,286 6650 6650 598.9 146.9 3787 65.9 65.9 6.45 2863 2863 954.2 70.3 140.5 0

13,300 2994 2994 146.9 0 9280 1894 1895 36.7 2863 2863 954.2 70.3 140.5 36.7

influence over the temperatures of the finger, hand, and forearm and a small influence over the arm temperature. The variability in the results indicates the need for more experimental data on blood flow rates in the vessels with diameters larger than 1000 μm. Except for the finger, the model with data from Takemori et al. [12] yields better results. Therefore, their data were used thereafter.

Fig. 4. Axial temperature profiles in the upper limb under a cold environment - Takemori et al. [12] data. In the figure, BA= Brachial Artery, RA = Radial Artery, UA= Ulnar Artery, AA= Arterial Arch (in), DA= Digital Artery, BV= Brachial Vein, BaV = Basilic Vein, CV = Cephalic Vein, RV = Radial Vein, UV = Ulnar Vein, SV = Superficial Vein, VP = Venous Plexuses, and FSV = Finger Superficial Vein.

Under a neutral and hot environment, small temperature variations in the vessels of the arm and forearm were observed, but in the fingers, the axial temperature variation is relevant. Under a cold environment, major temperature variations are observed, even in the arm (Fig. 4). Almost a 2 °C variation is observed in the brachial

Fig. 3. Comparison between model skin temperatures and experimental data in the: (a) arm, (b) forearm, (c) hand, and (d) finger. *Simulation performed using Takemori et al. [12] data and an extended skin layer.

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artery and veins. In the forearm, the variation is greater, almost 8 °C in the ulnar and radial arteries, and almost 10 °C in the ulnar and radial veins. Under this environment, the counter-current heat transfer between arteries and veins is evident. It is interesting to compare this result to that considering no heat transfer between these big vessels and the tissue. The blood temperature in the axillary vein just after mixing the streams coming from the brachial veins and from the basilic and cephalic veins is a good index for comparison (see Fig. 1a). Considering counter-current heat exchange, the blood temperature in the axillary vein was 32.5 °C; without counter-current heat exchange, it was 32.3 °C. This 0.2 °C difference seems small but it represents a 5.8% increase in heat loss (from 41.7 to 44.1 W, if the two upper limbs are considered). It should be stated that the temperature variations under the cold environment are underestimated because a two-layer model was used. Observing blood temperature inside the big vessels, it can be concluded that for neutral and hot environment exposures it is not necessary to consider temperature variations along these vessels because they are small. The same is not valid for cold exposure. In whole body models (e.g. Wissler [9]), a common approach is to consider two blood reservoirs in each compartment. The temperature in each reservoir is considered to be homogenous and heat transfer is allowed to occur between the reservoirs, between them and the tissue. Global heat transfer coefficients between arterial and venous reservoirs and between them and the tissue are calculated using 2D conduction shape factors. The present results show that this kind of model will work well for neutral to hot environment exposure. Nevertheless, the present approach eliminates the need to calculate these global heat transfer coefficients. This seems to be an advantage because the shape factors are valid for 2D steady state situations but they are also used in transient simulations. The effect of superficial veins in heat transfer to the environment was also evaluated. For each environment, two simulations were performed. In the first one, all venous blood that comes from the hand and finger skin returns through the superficial veins (α = 1 in Eqs. (1)–(26). In the other, blood returns through deep veins (α = 0). Under a neutral environment, 3% increase of heat transfer from the upper limbs was observed when hand and fingers skin blood flow returns through the surface, as compared to the situation in which the blood returns through deep veins. A heat transfer enhancement of 3.3% was observed considering a higher activity level (Tin,axillary artery = 37.5 °C) and under a neutral environment. Under the extreme hot environment, no heat transfer increase was observed. This occurs because the environment temperature (40 °C) is higher than the temperature of the skin, and heat transfer to the environment is due solely to sweat evaporation. For the cold environment, the blood flow through the superficial veins was considered null. The heat transfer enhancement could be greater during transients, in cases in which the differences between the temperature of blood in the superficial veins and skin are higher, especially in the beginning of the transient. These simulations confirm the importance of superficial veins for thermoregulation, as previously observed [6]. Another point that should be addressed is that if one tries to implement this upper limbs model on a whole human body model, two more variable parameters should be defined, the shunt index (α) and δ. The shunt index indicates the proportion of blood flow leaving the hand and fingers that returns through superficial veins. The values used in the simulations were 1.0 for the hot environment and 0.1 for the neutral one. δ is an index that indicates the proportion of finger skin blood flow that returns through the palmar plexus. A value of 0.5 was used in all simulations. The model can be used to simulate the steady state heat loss in the upper limbs when the palm of the hand is in contact with a cold plate. Grahn et al. [33] described this heat extraction device. In this equipment, the hand is inside a confined space and the pressure can be varied. By lowering the pressure, the palm vessel plexus is mechanically distended and blood flow to the palm is increased, and consequently

Fig. 5. Heat loss to the environment from one upper limb, using the heat extraction device.

heat transfer with the surroundings is enhanced. Conditions used in the simulation were: environment temperature equal to 40 °C, axillary artery blood equal to 38 °C, plate superficial temperature equal to 20 °C, and epidermis thickness equal to 2.5 mm. The thermal resistance considered between the outer surface of the skin and the surface of the plate is 5 × 10 − 3 m 2 K/W. To simulate a pressure variation, the blood flow rate was varied from its base value in a hot environment (0.1 L/min) to a maximum value of 1.2 L/min. The results are presented in Fig. 5. The single point on the vertical axis in this figure indicates the heat loss without the use of the extraction device. A marked increase in heat loss can be observed. As pointed out by Grahn et al. [7], the structures previously described in the palm of the hands and fingers play an important role in thermoregulation. These researches demonstrate the ability to manipulate core temperature using the heat extraction/adding device. The simulations indicate a ninefold increase in heat loss while using this device under maximum blood flow to the palmar plexus [1.2 L/min]. These results agree with the estimative presented by the authors [7]. 4. Conclusions A steady state heat transfer model of the human upper limbs was successfully developed. Seven circular cylinders representing the arm, forearm, and the five fingers compose the model. Each cylinder is divided into two annular concentric layers, the skin and the core. A slab, also divided into skin and core, was used to model the hand. In the macro-circulation, main vessels with diameters larger than 1000 μm were represented. In the arm, the brachial artery with its venae comitantes, and the cephalic and basilic veins were considered. In the forearm, the radial and ulnar arteries with their venae comitantes, and four superficial veins were taken into account. The digital arteries and veins were considered to be representative of the arteries and veins in the fingers. In the hand, three venous plexuses (venous arch, back and palmar) and one arterial plexus (arterial arch) were included. Vessels with diameters smaller than 1000 μm form the micro-circulation, and are treated as part of a continuum. The model was validated by comparing its results from exposures to a hot, a neutral, and a cold environment to experimental data presented in the literature. Better prediction capabilities could be attained if more experimental data on blood flow through larger vessels were available. It was demonstrated that under neutral and hot environments, temperature variations along big vessels in the arm and forearm are very small, while the opposite was observed during cold exposure. Subsequently, the effect of superficial veins blood flow on heat loss to the environment was evaluated. It was concluded that a heat transfer enhancement was observed only under a neutral

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