INTRODUCTION

TOP ABSTRACT INTRODUCTION BODY CONSTRUCTION HEAT TRANSFER WITHIN THE... HEAT EXCHANGE WITH THE... NUMERICAL METHODS RESULTS REFERENCES APPENDIX

IN THE PAST THREE DECADES, several multisegmental mathematical models of human thermoregulation have been developed (e.g. Refs. 17, 34, 41) and have become valuable tools in the hands of their authors in contributing to a deeper understanding of regulatory processes. These models, however, have not yet gained widespread use. Reasons for this may include lack of confidence in their predictive abilities, limited range of applicability, and poor modeling of the heat exchange with the environment. This paper and the paper in preparation make a contribution to research efforts to formulate a more precise, flexible, and universal model of the human thermoregulatory system.

The main stimulus for the model is the need to predict the thermal response of human beings to the conditions that they experience in and around buildings. Other well-established models can predict ambient air and surface temperatures, air speeds, and humidity; therefore, these parameters are the boundary conditions that will act on the thermoregulatory model.

The research sought to draw together the most appropriate models describing individual parts of the whole thermoregulatory system. These were then combined within a mathematical framework to produce a robust, computationally fast, and accurate model that will be of practical value in the design of buildings and urban environments. The research thus seeks to deliver the products of fundamental physiological research to scientists working in other disciplines.

From the mathematical point of view, the human organism can be separated into two interacting systems of thermoregulation: the controlling active system and the controlled passive system. The active system is simulated by means of cybernetic models predicting regulatory responses, i.e., shivering, vasomotion, and sweating, as discussed elsewhere (16). The passive system, the subject of this paper, is modeled by simulating the physical human body and the heat-transfer phenomena occurring in it and at its surface. Comparisons between the model's predictions and known analytic solutions are made. Only the complete model, i.e., the passive system plus active system, can be compared with field measurements of human responses to environmental conditions and with the predictions of whole body models produced by others; such comparisons are, therefore, a subject of the upcoming paper.

The metabolic heat is produced within the body. This heat is distributed over body regions by blood circulation and is carried by conduction to the body surface, where, insulated by clothing, the heat is lost to the surroundings by convection, radiation, and evaporation (see Fig. 1). Additionally, some heat is lost to the surroundings by respiration. To simulate adequately all these heat-transport phenomena, the present model of the passive system accounts for the geometric and anatomic characteristics of the human body and considers the thermophysical and the basal physiological properties of tissue materials.

View larger version (43K): [in this window] [in a new window]   Fig. 1.   A schematic diagram of the passive system.

Whereas the model was inspired by the desire to evaluate environmental conditions in buildings, it also has applications in other areas, e.g., in the car industry, in textile industries, in the aerospace industry, in meteorology, in medicine, and in military applications. In these disciplines, the model can serve for research into human performance, thermal acceptability and temperature sensation, clinical and therapeutic treatments, safety limits, etc.

	BODY CONSTRUCTION

TOP ABSTRACT INTRODUCTION BODY CONSTRUCTION HEAT TRANSFER WITHIN THE... HEAT EXCHANGE WITH THE... NUMERICAL METHODS RESULTS REFERENCES APPENDIX

The passive system of the model was developed by using thermophysical and basal physiological properties of tissue materials, based on literature (10, 17, 40). The humanoid was formulated to represent an average human (see Table 1 for overall body data). The body was idealized as 15 spherical or cylindrical body elements: head, face, neck, shoulders, arms, hands, thorax, abdomen, legs, and feet. The use of lumped data (e.g., if body elements are divided into a core and a shell) is prone to errors during temperature transients and was thus avoided. In accordance with former studies (17), a division was therefore made whenever a significant change of body tissue properties occurred. As a result, the present multilayer model consists of annular concentric tissue layers and uses seven different tissue materials: brain, lung, bone, muscle, viscera, fat, and skin. Each of these is subdivided into one or more tissue nodes (see Table 2).

In accordance with the work of Weinbaum et al. (39), the skin was modeled as two layers with distinctly different physiological properties: the inner skin and the outer skin. The inner skin is ~1 mm thick and simulates the cutaneous plexus, a region in which metabolic heat is generated and blood is perfused (see Table 2). The outer layer of skin is of similar thickness; however, it contains neither any heat source nor any thermally significant blood vessels. This superficial cutaneous layer plays a role when evaporative heat losses through the skin are modeled. The layer contains sweat glands and simulates the vapor barrier for moisture diffusion through the skin.

Asymmetric removal of bodily heat is often experienced in practice. It may be due to inhomogenous ambient conditions, e.g., solar irradiation, cold surfaces, draught effects, nonuniform clothing, or to physiological effects on the skin, such as local differences in evaporative heat losses, etc. Except for the face and shoulders, each body element was, therefore, subdivided spatially into sectors. Most of the body elements were divided into three sectors: anterior, posterior, and inferior (see Table 2). Anterior and posterior segments permit the treatment of lateral environmental asymmetries. Inferior segments account for body sides, which are "hidden" by other body parts (e.g., section A-A' in Fig. 1) and thus have reduced radiant heat exchange with the environment.

The sectors were coupled thermally by introducing a core element around the cylinder axis (see Fig. 1, section A-A') or the midpoint of the sphere. In the model, except for the head, the radius of the core was defined to be identical to the radius of the innermost tissue material layer in each body element. In extremities, for instance, the radius of this domain is identical to the radius of the bone. However, the brain, which has an outer radius of r = 8.6 cm (Table 2), was given a core of 4-cm radius.

Glossary

Symbols , ,  

	Short-wave absorptivity
	Factor of the blood perfusion rate (W · K1 · m3)
	Difference
	Stefan-Boltzmann constant (W · m2 · K4) (=5.67051 × 108 W · m2 · K4)
	Long-wave emissivity
, ,	Time-independent coefficients arising from the numerical formulation of the bioheat equation
H2O	Heat of vaporization of water (= 2,256 × 103 J/kg3)
µbl	Proportionality factor (K1)
	Meachanical efficiency of the human body
	Density (kg/m3)
	View factor
	Linear operator
	Angle of a body element sector (°)
	Sum
	Geometry factor of an isothermal core domain of a body element
	Geometry factor of the bioheat equation
	Partial derivative
A	Area (m2)
a	Distribution or weighting coefficient
act	Activity level (met)
b	Regression coefficient
Bx	Arterial blood temperature parameter arising from hx
C	Heat loss by convection (W)
c	Heat capacitance (J · kg1 · K1)
Cc	Coefficient of the blood pool temperature column of the major coefficient matrix
CCX	Countercurrent heat exchange
CO	Cardiac output (l/m)
CplC	Coupling coefficient of the major coefficient matrix
E	Heat loss by evaporation (W)
fcl	Overall clothing surface area factor
f*cl	Local value of fcl
feff	Effective radiant area of the body
h	Surface heat transfer coefficient (W · m2 · K1)
hx	Countercurrent heat exchange coefficient (W/K)
H	Internal whole body workload (W)
icl	Overall moisture permeability index from the skin to the clothing surface
i*cl	Local value of icl
Icl	Overall intrinsic insulation from the skin to the clothing surface (m2 · K · W1)
I*cl	Local value of Icl (m2 · K · W1)
k	Conductivity (W · m1 · K1)
L	Lewis constant (La = 0.0165) (K/Pa)
Lc	Coefficient of the blood pool temperature line of the major coefficient matrix
m	Mass (kg)
M	Whole body metabolism (W)
MRT	Mean radiant temperature (°C)
N	Number of nodes
P	Water vapor pressure (Pa)
q	Heat flux density (W/m2)
qm	Metabolic rate (W/m3)
Q	Heat flux (W)
r	Radius (m)
Re	Moisture permeability resistance (m2 · Pa · W1)
rh	Relative humidity (%)
s	Short-wave irradiation (W/m2)
t	Time (s)
T	Temperature (°C)
T*	absolute temperature (K)
U*	Local clothing insulation coefficient from the skin to the ambient air (W · m2 · K1) (or) (W · m2 · Pa1)
v	Airspeed (m/s)
V	Volume (m3)
w	Perfusion rate (s1)
wt	Wetted area ratio

Subscripts and superscripts

	Last node next to an interface between two inhomogenous tissue shells
+	First node next to an interface between two inhomogenous tissue shells
0	Thermal neutrality
air	Air
acc	Accumulation
bas	Basal value
bd	Body
bl	Blood
bl,a	Arterial blood
bl,p	Central blood pool
bl,v	Venous blood
c	Convection
cl	Clothing
CN	Crank-Nicolson
Du	Dubois
e	Evaporation
eff	Effective
expl	Explicit
f	Fabric
frc	Forced
ifc	Interface
j	Running number variable
hy	Hypothalamus
impl	Implicit
m	Mean value
mix	Mixed
mus	Muscle
n	Total number
nat	Natural
o	Operative
osk	Outer skin layer
r	Node number
R	Long wave radiation
re	Rectal
rsp	Respiration
sat	Saturated
sf	Surface of body sector
sh	Shivering
sk	Skin
sR	Short-wave radiation
sr	Surrounding
sw	Sweating
t	Time step number
ty	Tympanic
w	Work
x	Countercurrent heat exchange

	HEAT TRANSFER WITHIN THE TISSUE

TOP ABSTRACT INTRODUCTION BODY CONSTRUCTION HEAT TRANSFER WITHIN THE... HEAT EXCHANGE WITH THE... NUMERICAL METHODS RESULTS REFERENCES APPENDIX

   The heat transport mechanisms occurring in the living tissue have been formulated by Pennes (28) in the so-called "bioheat equation." This differential equation describes the heat dissipation in a homogenous, infinite tissue volume

(1)

According to Eq. 1, from left to right, the radial heat flow from warmer to colder tissue regions [heat-conduction term, where k is the tissue conductivity (W · m¹ · K¹), T is tissue temperature (°C), r is radius (m),  is a geometry factor (dimensionless);  = 1 for polar coordinates and  = 2 for spherical coordinates (head)] is overlayed by metabolism q_m(W/m³) and blood perfusion [heat-convection term, where _bl is density of blood (kg/m³), w_bl is blood perfusion rate (s¹), c_bl is heat capacitance of blood (J · kg¹ · K¹), and T_bl,a is arterial blood temperature (°C)]. This combined effect is balanced by the storage of heat within the tissue mass [right-hand side of Eq. 1, where  is tissue density (kg/m³), c is tissue heat capacitance (J · kg¹ · K¹) and t is time (s)].

The bioheat equation was applied to all tissue nodes by using the appropriate material constants k, , and c, the basal heat-generation term q_m, and basal blood perfusion rate w_bl for each tissue layer (Table 2). It should be noted that the magnitudes of the physiological variables q_m and w_bl are affected by responses of the active system, as described elsewhere (16). The arterial blood temperature T_bl,a in Eq. 1 arises from the actual overall thermal state of the body and is obtained by simulating the human blood circulatory system.

Heat conduction. The heat-conduction term in Eq. 1 considers temperature variations in the radial direction. Angular heat flows were also neglected in the present model, even though the body elements were divided into sectors, and environmental asymmetries were considered. There were two reasons for this simplification. First, at the periphery (i.e., predominantly in the skin), although significant angular temperature gradients may occur due to asymmetric boundary conditions, the total amount of the heat conducted angularly from one skin sector to another is negligible, compared with the total amount of heat transferred radially through a sector band. This is because peripheral sectors have large shell areas, compared with the small interface areas between adjacent skin sectors. Second, ambient influences are insignificant in internal tissue layers where heat dissipation is dominated by metabolism and by blood circulation (28, 33). Therefore, inhomogeneities of q_m and w_bl between individual organs of the abdominal viscera (40), for instance, which can hardly be considered by symmetrical geometry models, govern local temperature profiles within the body and swamp the effect of environmental asymmetries.

Metabolism. In the model, the metabolic heat production of a finite tissue volume is treated as a sum of the basal value q_m,bas,0 and the additional heat q_m, which may be produced by local autonomic thermoregulation, while the subjects are exercising and/or shivering

(2)

(3)

(4)

(5)

(6)

(7)

1

Blood circulation. Blood circulation is of vital importance for the dissipation of heat within the human body. The dissipation effect was examined by simulations using the present model without blood circulation: the core temperature of the brain, which has a high metabolic rate (Table 2), exceeded 73°C when the passive system was exposed to steady neutral ambient temperatures of 30°C.

(8)

(9)

(10)

(11)

3

1

1

(12)

(13)

(14)

	HEAT EXCHANGE WITH THE ENVIRONMENT (BOUNDARY CONDITIONS)

TOP ABSTRACT INTRODUCTION BODY CONSTRUCTION HEAT TRANSFER WITHIN THE... HEAT EXCHANGE WITH THE... NUMERICAL METHODS RESULTS REFERENCES APPENDIX

At the body surface, heat is exchanged by convection with the ambient air q_c, by radiation with surrounding surfaces q_R, by irradiation from high-temperature sources q_sR, and by evaporation of moisture from the skin q_e. The rate of heat exchange varies over the body and is affected by the clothing ensemble worn. Hence, in the model, heat balances were established for each skin sector of each body element as boundary conditions to Eq. 1. In general, the net heat flux q_sk (W/m²) passing the surface of a peripheral sector is equivalent to the sum of these individual heat exchanges

(15)

The elements of this equation are considered in detail in following sections.

Convection. The convective heat exchange q_c between a skin sector of surface temperature T_sf and ambient air of temperature T_air was modeled by considering both natural and forced convection using combined convection coefficients h_c,mix

(16)

2

1

(17)

2

1

(18)

View larger version (29K): [in this window] [in a new window]   Fig. 2.   Overall convective heat transfer coefficient (hc,m) for natural convection. Tsk,m, mean skin temperature; Tair, air temperature; Fanger, Ref. 14; Bach, Ref. 3; Birkebak, Ref. 5; Rapp, Ref. 29.

View larger version (28K): [in this window] [in a new window]   Fig. 3.   Overall convective heat transfer coefficient hc,m for forced convection. Colin et al., Ref. 12; Nishi, Ref. 26.

Radiation. The exchange of heat by long-wave radiation q_R is usually of similar importance in the heat balance of the human body as the heat exchange by convection.

2

1

(19)

(20)

8

2

4

(21)

(22)

(23)

Clothing insulation. Because clothing plays an important role in the behavioral thermoregulation of human beings, efforts were made to model garments in some detail. Unfortunately, standard literature provides only overall clothing parameters, i.e., the intrinsic clothing insulation I_cl, the clothing area factor f_cl, and the moisture permeability index i_cl (for exact definitions see, e.g., Ref. 22). These do not reflect the real insulating properties of the garments but provide characteristics of a uniform (imaginary) clothing layer covering the entire body. Therefore, a simple computer model was developed that uses the present passive system and resimulates the experimental procedures for measuring the overall insulation characteristics of garments undertaken by McCullough et al. (22, 23). The program computes the required local insulation values I*_cl, f *_cl, and i*_cl of trousers, shirts, underwear, socks, etc., from measured uniform layer data I_cl, f_cl, and i_cl for these garments and from the evaporative resistance of the fabric R_e,f. These local values include the insulation effect provided by any layer of air trapped between the skin and the fabric. A database on clothing items was created, which was used to compose clothing ensembles in the present multisegmental model.

2

1

(24)

2

1

2

1

2

1

(25)

Evaporation. The present skin evaporation model ensures heat and mass transfer balances at each skin sector of each body element. The latent energy transport from a skin sector of area A_sk is given by

(26)

(27)

2

1

(28)

(29)

Respiratory heat losses. Most heat is lost through the body surface; however, heat exchange with the environment also occurs by respiration. The model for respiratory heat loss was obtained from the work of Fanger (14), taking into account the loss by evaporation and convection. The latent heat exchange is calculated according to Ref. 14 from the pulmonary ventilation as a function of the whole body metabolism [i.e.,  q_mdV (W)]; latent heat of vaporization of water; and the difference between humidity ratio of the expired and inspired air, which depends on the ambient air temperature T_air (°C) and the partial vapor pressure of the ambient air, P_air (Pa)

(30)

(31)

(32)

	NUMERICAL METHODS

TOP ABSTRACT INTRODUCTION BODY CONSTRUCTION HEAT TRANSFER WITHIN THE... HEAT EXCHANGE WITH THE... NUMERICAL METHODS RESULTS REFERENCES APPENDIX

In the model, each sector of each tissue shell was divided into nodes. These were unequally spaced in the radial direction, being closer together in the outer regions where temperature gradients are generally the steepest and where, therefore, the largest numerical errors could occur (17). The number of nodes per tissue shell N can be chosen arbitrarily in the model; it is only limited by the capacity of the computer memory. The current nodal configuration is shown in Table 2.

A finite-difference scheme was used to discretize Eq 1. The partial derivatives with radius were approximated by the central difference method, which provides improved (i.e., second-order) accuracy, compared with the first order of the forward or backward difference methods. The partial derivative for the temporal change was approximated by using a hybrid scheme, the Crank-Nicolson method, which also has a second order of accuracy. This method remains unconditionally stable, so the model can tolerate time steps of variable length. This means that the model can be more easily integrated with other dynamic simulation programs (e.g., of the building or vehicle environment).

The resulting discrete formulation of the bioheat equation was equally applicable to the cylindrical and to the spherical body parts. The passive system was organized as a structured system of time-independent "conduction" matrices, which collected all thermophysical tissue constants, and time-dependent "blood" matrices, the coefficients of which had to be computed for each time step of the program. Details are given in the APPENDIX. A quick solver was developed, which worked efficiently because it considered only the nonzero cells in the matrices.

	RESULTS

TOP ABSTRACT INTRODUCTION BODY CONSTRUCTION HEAT TRANSFER WITHIN THE... HEAT EXCHANGE WITH THE... NUMERICAL METHODS RESULTS REFERENCES APPENDIX

Because the passive system does not contain models for the thermoregulatory feedback mechanisms that exist in human beings, it was not possible to validate the model by comparisons with field measurements. It was, however, possible to verify the numerical accuracy of the passive-system model by comparing its predictions with existing analytic solutions for heat flow in homogeneous cylinders and spheres. Of most concern was the accuracy of the predicted temperatures, but the correct formulation of the boundary conditions was also examined. Tests were performed for a variety of thermal conditions in which the passive system was exposed to cold air. Because of producing steep temperature gradients in the tissues, these conditions reveal computational errors better than the warm-air or hot-air exposures (17).

Analytic solutions for conduction in cylinders and spheres as well as for conduction superimposed by internal heat generation (18) were employed to verify the model in steady-state conditions. These tests indicated that the model predicts steady-state tissue temperatures, which are so close to the known analytic results that the mean error is typically <0.02 K.

The performance of the model in transient conditions was of special interest. An analytic solution for transient heat conduction (without heat generation and fluid perfusion) in spheres derived by Gröber et al. (18) was compared with the predictions of the model for the head sphere made of brain tissue and setting q_m =  = 0. At t = 0, the sphere, which had an initially homogenous temperature of 37°C, was exposed to ambient temperatures of T_air = 0°C. Because there was no internal heat source, the sphere continuously cooled down. Good agreement with the solution of Gröber et al. was obtained for all temperatures in the sphere, as shown in Fig. 4 for three nodes (core, periphery, and a node in between). An error analysis of all nodal temperatures and times indicated a mean error of |T| = 0.19 K, with the largest deviations from analytic results arising in the node next to the core, |T| = 0.45 K (core: |T| = 0.15 K), and the smallest errors occurring at the periphery, |T| = 0.04 K. Simulations with a time step of 60 s did not significantly increase the accuracy for the given nodal configuration (see Table 2). The numerical scheme employed provided reasonable accuracies for both polar and spherical coordinates (mean error of |T| < 0.5 K), even for time steps (t) of 1 h.

View larger version (44K): [in this window] [in a new window]   Fig. 4.   Comparison of predicted head-sphere temperatures with exact, analytically derived (analyt.) values for conductive cooling using a time step of 5 min. Tinit, initial temperature; h, surface heat transfer coefficient; k, conductivity; , density; c, heat capacitance.

In another severe test of the model, the calculated temperature within a homogenous cylinder of muscle tissue was compared with an analytic solution of the bioheat equation quoted by Eberhart (13). The cylinder is initially in thermal equilibrium with the environment at T = T_air = 0°C. At t = 0, the cylinder is suddenly supplied with a constant rate of blood flow (w_bl = 0.001 s¹, T_bl,a = 37°C) and source of metabolic heat (q_m = 600 W/m³). The temperature of the cylinder therefore rises quickly before converging asymptotically to the final steady value. The predicted temperatures for three nodes in the cylinder, the core, the periphery, and a node between them, are compared with the analytically derived values in Fig. 5. The mean error for all temperatures at t = 5 min was |T| = 0.04 K. The reverse error behavior to the previous example was observed: larger deviations from the exact values occurred at the surface (|T| = 0.11 K) and the lower mean errors occurred in the core (|T| = 0.04 K).

View larger version (40K): [in this window] [in a new window]   Fig. 5.   Comparison of predicted muscle tissue temperatures in a leg-cylinder, with exact, analytically derived values after sudden supply of blood and metabolic heat. Prediction time step was 5 min. wbl, blood perfusion rate; qm, metabolic rate; Tbl,a, arterial blood temperature.

The effect of time-step length was investigated by using the previous test but with a higher blood perfusion rate (w_bl = 0.005 s¹), to ensure an even more rapid heating of the cylinder (reaching ~95% of the steady-state value after 10 min). The simulation was undertaken with a time step of 10 min, and the predictions were compared with the exact solution (Fig. 6). There was an initial "overshoot" of the predicted temperature in the outermost cylinder node, but the temperature oscillations were strongly damped, and the predicted values quickly approached the analytic solution. This demonstrates the stability of the model even for disproportionally large time steps.

View larger version (29K): [in this window] [in a new window]   Fig. 6.   Comparison of predicted muscle tissue temperatures in a leg-cylinder with exact, analytically derived values after instantaneous introduction of a high blood flow and metabolic heat. Prediction time step was 10 min.

Finally, the temperature distribution within, and at the surface of, the whole passive system was calculated for conditions of "thermal neutrality" of T_air = 30°C (4) (for a detailed specification see Table 3). The simulation of the nonregulated, reclining, and unclothed human body provided a mean skin temperature of T_sk,m = 34.4°C, and a head core temperature (hypothalamus) of T_hy = 37.0°C, values that agree with published data of T_sk,m = 34.4°C (14) and T_ty = 37.0°C (4) for reclining subjects exposed to thermoneutral conditions. Some further simulation results of this steady-state exposure are listed in Table 4.

Conclusions. The heat exchange with the environment plays a key role in the thermal state of the human body. Development of models for the passive system of human thermoregulation needs to reflect the importance and the complexity of these phenomena. In the present model, convective heat losses were simulated by accounting for both free and forced convection and their variation over the human body. Simulation of the radiant heat exchange with the surroundings considered spatial asymmetries of the environment and the (posture-dependent) degree of exposure of individual sites of the body elements. Respiration was modeled by considering both the convective and latent heat losses. Short-wave radiation was also included to enable the effects of solar irradiation and other high-temperature sources on the human body to be analyzed. A clothing model that accumulated nonuniform insulation properties was developed for use with the multisegmental model. In the skin evaporation model, the traditional skin wettedness concept was replaced by ensuring local heat and mass balances. These considered moisture diffusion through the skin, the evaporation of sweat, the evaporative resistance of garments, and the accumulation of the liquid sweat on the skin.