A Multi-Segmented Human Bioheat Model for Asymmetric High Temperature Environments

In workplaces such as steel, power grids, and construction, firefighters and other workers often encounter non-uniform high-temperature environments, which significantly increase the risk of local heat stress and local heat discomfort for the workers. In this paper, a multi-segment human bioheat model is developed to predict the human thermal response in asymmetric high-temperature environments by considering the sensitivity of the modeling to angular changes in skin temperature and the effects of high temperatures on human thermoregulatory and physiological responses simultaneously. The extended model for asymmetric high-temperature environments is validated with the current model results and experimental data. The result shows that the extended model predicts the human skin temperature more accurately. Under non-uniform high-temperature conditions, the local skin temperature predictions are highly consistent with the experimental data, with a maximum difference of 2 °C. In summary, the proposed model can accurately predict the temperature of the human core and skin layers. It has the potential to estimate human physiological and thermoregulatory responses under uniform and non-uniform high-temperature environments, providing technical support for local heat stress and local thermal discomfort protection.


Introduction
Non-uniform high-temperature environments are frequently encountered in the workplace of firefighters and other personnel working in the steel, power grid, and construction industry. The risk of local heat stress and local thermal discomfort for workers increases significantly due to exposure to unilateral fire or unilateral intense heat. Without appropriate protection measures, it would lead to irreparable local burn injuries or even lifethreatening injuries. Knowledge of human physiological and thermoregulatory responses to non-uniform high-temperature environments is important for assessing physiological performance, comfort, and safety. As a result, developing models of human thermophysiological responses appropriate for extremely non-uniform high-temperature environments is critical.
Mathematical modeling of thermal physiological responses is one of the most effective and valuable tools for simulating human physiological and thermoregulatory responses [1]. Some cutting-edge academic organizations and groups conducted a series of related studies and achieved numerous results, including greatly preventing the heat stress response of the human body and improving the thermal comfort of the human body in high-temperature environments. Firstly, the human bioheat model applicable to uniform thermal environments was developed and widely used in the prediction of human physiological and

Model Development
In this section, we first develop a human bioheat model for non-uniform environments based on Salloum's work and then extend it to consider the effect of high temperature on the human physiological and thermoregulatory response [17,32,33]. In general, the extended human bioheat model simulates the thermal response through two interacting systems: the passive system and the controlled active system. The passive system includes the energy balance integral equation and the circulation system for modeling the heat transfer that occurs. To obtain the local skin temperature in different regions of each segment, this paper expands the number of skin nodes in each segment by direction division to improve the passive system. The active system is used to model the thermoregulatory response in the human body controlled by the hypothalamus. This paper improves the model by extending

Modeling Asymmetric Passive Systems
According to Salloum's model, in the passive system, the human body was subdivided into 15 cylindrical segments: head, chest, abdomen, upper arms, forearms, hands, thighs, calves, and feet. Each segment contains a core node, a vein node, an artery node, and a skin node. Heat transfer between different segments is accomplished by the blood flow between adjacent body segments, including arterial blood from the heart to various body segments and back to the heart with venous blood. The heat transfer process in the same segment includes conduction, convection, and perfusion between different nodes. Blood perfusion included from the arterial node to the core, and from the core node to the skin node. Convection occurs between the arterial and venous blood flow and the core layer, and conduction occurs between core layer nodes and adjacent skin nodes.
However, given the actual operating scenarios of firefighters and workers in the steel, outdoor power grid, and construction industries, they are often exposed to intense heat on one or more sides. Therefore, the skin node was decomposed into four equal sectors (anterior, exterior, posterior, and inferior) in the proposed model by referring to the model of Kubaha [34]. Thence, the asymmetric human model is developed, which includes 15 segments, each including a core node, a vein node, an artery node, and four skin nodes, for a total of 105 nodes, as shown in Figure 1. The detailed parameters of each segment are shown in Table 1.
modeling the heat transfer that occurs. To obtain the local skin temperature in different regions of each segment, this paper expands the number of skin nodes in each segment by direction division to improve the passive system. The active system is used to model the thermoregulatory response in the human body controlled by the hypothalamus. This paper improves the model by extending physiological responses, such as metabolic rate changes, cardiovascular system changes, and sweating rates, induced by high temperatures.

Modeling Asymmetric Passive Systems
According to Salloum's model, in the passive system, the human body was subdivided into 15 cylindrical segments: head, chest, abdomen, upper arms, forearms, hands, thighs, calves, and feet. Each segment contains a core node, a vein node, an artery node, and a skin node. Heat transfer between different segments is accomplished by the blood flow between adjacent body segments, including arterial blood from the heart to various body segments and back to the heart with venous blood. The heat transfer process in the same segment includes conduction, convection, and perfusion between different nodes. Blood perfusion included from the arterial node to the core, and from the core node to the skin node. Convection occurs between the arterial and venous blood flow and the core layer, and conduction occurs between core layer nodes and adjacent skin nodes.
However, given the actual operating scenarios of firefighters and workers in the steel, outdoor power grid, and construction industries, they are often exposed to intense heat on one or more sides. Therefore, the skin node was decomposed into four equal sectors (anterior, exterior, posterior, and inferior) in the proposed model by referring to the model of Kubaha [34]. Thence, the asymmetric human model is developed, which includes 15 segments, each including a core node, a vein node, an artery node, and four skin nodes, for a total of 105 nodes, as shown in Figure 1. The detailed parameters of each segment are shown in Table 1.    [11,13,17].  For each segment, the schematic diagram of heat transfer between the core, artery, vein, and four skin nodes is shown in Figure 2. The energy balance equations for the seven nodes after modification of Salloum's energy balance equation [13] are as follows. For clothed body segments, the simplified clothing model described by Yang et al. [18], which was proven to be reasonably accurate [18,22], was adopted in this study.

Modeling of Active System
The thermoregulatory control system consists of four activities: skin vasoconstriction, vasodilation, shivering, and sweating. For vasomotor, the same governing equations of Salloum et al. [13] were adopted in this model. In particular, the mean skin temperature In addition, the sweating and the shivering thermoregulatory function follows the JOS model of Tanabe [11] and Kobayashi [12], and will not be presented here. The distribution coefficients used in this paper are shown in Table 2. Tsk0 is the set point temperature of the corresponding segment, and the thermal reference core temperature of the head is set as 36.9 °C [11]. αsk is the skin sensitivity. αsw, and αsh are the distribution coefficients of sweating and shivering, respectively.  For the core node where t is the time. C cr and T cr are the heat capacitance and the temperature of the core node, respectively. M cr is the basal metabolic rate of the core. W is the mechanical work made in the core. Q per f usion,a−cr = . m cr c bl (T bl,a − T cr ) is the perfusion heat transfer of the core, which is associated with the blood perfusion in the core . m cr and the temperature of the core node T cr and artery node T bl,a . c bl is blood-specific heat, usually taking the value of 4000 J/kg.
are the convection heat transfer between the core, and the artery, and vein, respectively; h a and h v are heat convection coefficients, which can be calculated by referring to the human circulatory system model of Salloum et al. [13]. In addition, for the core layer of the chest, there is also respiratory heat loss RES = 0.0014M cr (34 − T air ) + 0.0173M cr (5.87 − P air ), which is generated by convection and evaporation [35].
K cr−sk,i (T sk,i − T cr ) represents the conduction heat transfer between skin node i and the corresponding core node. K cr−sk,i is the thermal conductance be- tween the skin node i and to the core node.
represents the blood perfusion volume of skin layer i to the core, which is associated with the skin perfusion blood flow rate . m sk,i and the temperature of the skin node i T sk,i and core node T cr . The effect of non-uniform environments on the blood perfusion rate of the skin node i is evaluated by the parameter δ i , that is .
For the artery and vein node where C bl,a and C bl,a are the heat capacitance of the artery and vein node, respectively.   Finally, the energy balance equation for the four skin nodes after modification of Salloum's energy balance equation is given by where C sk is the heat capacitance of the skin node i (i = 1-4). The first term on the right-hand side of the equation indicates the conduction heat exchange between each skin node in the same segment, where, K sk,i denotes the skin thermal conductivity, L n denotes the segment length, th sk,i denotes the thickness of the skin layer in the segment, r is the radius of the skin layer, and θ is the corresponding angular coordinate of the skin node. M sk,i is the metabolic rate of the skin node and is equal to the total metabolic rate of the segment divided by the number of the skin node. CON sk,i , EVA sk,i , and RAD sk,i are the heat exchanges between the skin node i and the external environment caused by convection, evaporation, and radiation, respectively. For the exposed segments of the skin, we have where RAD sk,i,1 = A sk,i h r (T sk,i − T mrt ) is the radiation heat transfer when there is no intense heat on one or more sides, and T mrt is the mean ambient radiation temperature. When there is intense heat on one side i, an additional RAD sk,i,2 = σ · ε · F v f · (T sk,i + 273.15) 4 − (T r + 273.15) 4 needs to be introduced to represent the radiative heat transfer between the strong radiation source and the skin node i of the corresponding segment. T r is the temperature of an intense heat source. F v f is the view factor, which can be calculated based on the human body location and the radiation source distribution. The case will be configured according to the actual conditions and calculated using the additivity of the view factor [10]. For clothed body segments, the simplified clothing model described by Yang et al. [18], which was proven to be reasonably accurate [18,22], was adopted in this study.

Modeling of Active System
The thermoregulatory control system consists of four activities: skin vasoconstriction, vasodilation, shivering, and sweating. For vasomotor, the same governing equations of Salloum et al. [13] were adopted in this model. In particular, the mean skin temperature T sk,mean and skin blood perfusion rate . m sk,per f usion of the corresponding segment are given by [26]. .
In addition, the sweating and the shivering thermoregulatory function follows the JOS model of Tanabe [11] and Kobayashi [12], and will not be presented here. The distribution coefficients used in this paper are shown in Table 2. T sk0 is the set point temperature of the corresponding segment, and the thermal reference core temperature of the head is set as 36.9 • C [11]. α sk is the skin sensitivity. α sw , and α sh are the distribution coefficients of sweating and shivering, respectively.

Extension to High-Temperature Environments
The thermal response of the human body is affected by parameters such as ambient temperature, humidity, and wind speed. The human physiological response will change as the temperature increases when other conditions remain unchanged. In the working environment of firefighters and others working in the steel, power grid, and construction industries, the temperature is usually higher than 40 • C. The external environment mainly transmits heat to the human body through convection and radiation. Therefore, to ensure that the core temperature of the human body is within the normal range, the thermoregulatory system of the human body responds differently compared with normal environments. It leads to several significant differences, such as the effects of high temperatures on metabolic rate, cardiovascular system, and the role of heat-induced dehydration on sweating and skin vasodilation. Therefore, the extended non-uniform high-temperature environment model should consider the above specific thermal response of high temperature to the human body.

Effects of High Temperatures on Metabolic Rate
In high-temperature environments, heat stress causes an increase in the intensity of the metabolism, which provides energy for thermoregulatory activities. According to Weng et al. [17], the metabolic rate increases by 10-13% for every 1 • C increase in body temperature, when the environment temperature is greater than 39 • C. This process should be considered in the extended model. The relationship between the proportional rising in human metabolic rate M and the increase in body temperature T cr can be expressed as [17] where M 0 is the metabolic rate at room temperature.

Effects of High Temperatures on Cardiovascular System
When the ambient temperature rises, the heart activity will increase to meet the needs of the human body for heat dissipation and oxygen supply. Wenzel et al. [36] indicated a slow increase in heart rate over time in a hot environment. The increase in heart rate ∆HR can be calculated by [33,36] where D = 100 × t 0 Sweat × dt/74.30(%) is the percentage of cumulative water lost by the body.

Role of Heat-Induced Dehydration on Sweating
According to Zhao et al. [33], the percentage of water loss to the total body weight decreases sweating sensitivity and increases sweating threshold temperature to reduce sweating. Therefore, in the modified model, we considered the role of calculated water loss D in the sweating regulation by introducing the modified equation for sweating thermoregulatory function proposed by Montain et al. [37].
where T head is the head core temperature, T head,set = 37 • C is the set-point temperature, and Wrms and Clds are integrated signals from skin thermoreceptors. That is to say, each 1% increase in cumulative dehydration resulted in a 0.06 • C increase in the hypothalamic thermoregulatory point threshold, while resulting in a 0.068 kg/(h· • C) decrease in the sweat sensitivity of sweating.

Results and Discussion
In this paper, MATLAB (Math Works, Inc.) is used to solve the multi-segment human bioheat model in a non-uniform environment. Firstly, the simulated core temperature is validated using the existing experimental data and the simulation results in a uniform high-temperature environment. In an asymmetric high-temperature environment, the improved model was validated using the skin temperature data of different body segments reported by Fanger [38] in a non-uniform high-temperature environment.

Comparison with Uniform High-Temperature Environments
The core and mean skin temperatures simulated by the extended model were validated by comparing them with published experimental data and simulation results by Yang et al. [18]. In the experiments, five healthy males (wearing only shorts) were exposed to four conditions: first for 30 min in a neutral environment (29 • C; Stage 1), followed by 30 min in a hot environment (45 • C; Stage 2), and then back to the neutral environment (29 • C; Stage 3) for 30 min, finally in the hot environment (45 • C; Stage 4) for 30 min. During the experiments, the core and skin temperatures of each subject were measured. Figure 3 shows the comparison of (a) the core temperature and (b) the mean skin temperature between the experimental data and simulated results of Yang et al. [18] and the simulated results calculated with the current model. The relative errors between the predicted results of the two models and the experimental data are shown in Table 3. It is clear that the simulation results are in good agreement with the experimental data, and the maximum relative error is only ±1.62%. The core temperature predictions were within ±0.48% of relative error to the experimental data, which is comparable to the agreement shown by the model of Yang et al. [18]. The temperature prediction of the mean skin temperature was very close to the experimental data, especially under hightemperature conditions (stage 2 and stage 4), with a relative error of ±0.67%, which was significantly better than the model of Yang et al. [18].    This may be because the current model takes into account the effect of high temperature on metabolic rate as well as heart rate, which results in higher human metabolic rate and skin blood flow rate in the extended model in a hot environment than in the Yang et al. model. In addition, the role of heat-induced dehydration on sweating is introduced in the current model and well represents the sweating effect of the human body in a hot environment. Therefore, the validation results indicate that the extended model can accurately predict the core temperature and skin temperature in uniform high-temperature environments.

Comparison with Asymmetric High-Temperature Environments
A set of asymmetric radiation experiments was reported by Fanger et al. [38] in 1985.
In the experiment, 16 college students (8 males and 8 females, standard KSU uniforms) were at a distance of 0.5 m from the warming wall. During the first hour of the experiment, the warming wall was not heated (23 • C). Subsequently, the warm wall temperature was adjusted every half hour (32.6 • C, 42 • C, 51.6 • C, 61.1 • C, and 70.1 • C), while the ambient air temperature was down (21.9 • C, 20.7 • C, 19.3 • C, 17.9 • C, and 16.7 • C) to keep the operating temperature stable.
Concerning the core temperature (a) and mean skin temperature (b), the comparisons of the experimental data [38] with the simulation results of the current model are shown in Figure 4. As a result, the model-predicted core temperature and mean skin temperature differed most from the experimental data by 0.45 • C and 0.96 • C, respectively. Therefore, the modified model can accurately predict the core and average skin temperature throughout the process. Figure 5 shows the comparison of the left arm (exterior) skin temperature (a), right arm (exterior) skin temperature (b), and right thigh (anterior) skin temperature (c) between the experimental data [38] and the simulated results calculated with the current model. The calculated temperatures of the three body segments are in good agreement with the experimental data, and the maximum difference is within 2 • C. The results show that the extended model can accurately predict the human core temperature and skin temperature in asymmetric high-temperature environments.
Additionally, the unilateral skin temperature of the concerned body segment is more accurate compared to its average value from Figure 5. The reason is that the spatial temperature distribution is asymmetric, and the ambient radiation temperature on the side close to the heat source is higher, resulting in a higher risk of local heat stress and local heat discomfort in the corresponding human body segment. Conversely, on the side away from the heat source, the ambient radiant temperature is lower, resulting in a lower risk of localized heat stress on the corresponding body part. This means that previous models, which treat each part of the body as a uniform column, are not suitable for non-uniform high-temperature environments. Therefore, it is necessary to consider the sensitivity of modeling to changes in skin temperature angles when predicting the risk of local heat stress in workers in a non-uniform high-temperature environment.

Conclusions
In this paper, a multi-segment human bioheat model was first extended to the asymmetric environment by considering the sensitivity of the modeling to angular changes in skin temperature. The human body was subdivided into 15 cylindrical segments, each containing a core node, a vein node, an artery node, and four skin nodes, for a total of 105 nodes. Then, the effects of high temperatures on metabolic rate, the cardiovascular system, and the role of heat-induced dehydration on sweating and skin vasodilation were considered to further extend this model for human thermal response prediction in asymmetric high-temperature environments.
The comparison of predicted results from previous models and experimental data indicate that the extended model can accurately simulate core, average skin, and local skin temperatures under both uniform and non-uniform high-temperature conditions. The analysis of the results revealed that the extended model could predict the human skin temperature more accurately in uniform high-temperature environments. Under non-uniform high-temperature conditions, the local skin temperature predictions agreed better with the experimental data, with the maximum difference within 2 °C. Therefore, the model can be used as an effective tool to estimate human physiological and thermoregulatory responses under uniform and non-uniform high-temperature environments.
In conclusion, this paper established a thermophysiological response model for workers in a non-uniform high-temperature environment. The model can accurately calculate the heat transfer process between the non-uniform heat source and the human based on the actual scene between them. It can not only be applied to fire scenes, steel grid work environments, and architectural scenes, but also to life scenes with local heat sources, such as bathroom bathing and indoor heating. The application of the modified model is extended by taking into account specific human thermal responses and physiological equations in extremely high-temperature environments.  Figure 5. Comparison of the left arm (exterior) skin temperature (a), right arm (exterior) skin temperature, (b) and right thigh (anterior) skin temperature (c) between the experimental data [38] and the simulated results calculated with the current model.

Conclusions
In this paper, a multi-segment human bioheat model was first extended to the asymmetric environment by considering the sensitivity of the modeling to angular changes in skin temperature. The human body was subdivided into 15 cylindrical segments, each containing a core node, a vein node, an artery node, and four skin nodes, for a total of 105 nodes. Then, the effects of high temperatures on metabolic rate, the cardiovascular system, and the role of heat-induced dehydration on sweating and skin vasodilation were considered to further extend this model for human thermal response prediction in asymmetric high-temperature environments.
The comparison of predicted results from previous models and experimental data indicate that the extended model can accurately simulate core, average skin, and local skin temperatures under both uniform and non-uniform high-temperature conditions. The analysis of the results revealed that the extended model could predict the human skin temperature more accurately in uniform high-temperature environments. Under nonuniform high-temperature conditions, the local skin temperature predictions agreed better with the experimental data, with the maximum difference within 2 • C. Therefore, the model can be used as an effective tool to estimate human physiological and thermoregulatory responses under uniform and non-uniform high-temperature environments.
In conclusion, this paper established a thermophysiological response model for workers in a non-uniform high-temperature environment. The model can accurately calculate the heat transfer process between the non-uniform heat source and the human based on the actual scene between them. It can not only be applied to fire scenes, steel grid work environments, and architectural scenes, but also to life scenes with local heat sources, such as bathroom bathing and indoor heating. The application of the modified model is extended by taking into account specific human thermal responses and physiological equations in extremely high-temperature environments. The output parameter of the model is the local skin temperature, which supports the calculation of local thermal stress and local thermal discomfort. In particular, it can predict parameters such as localized thermal burns, protecting operators at high temperatures. It will also provide the improvement suggestions of protective clothing to avoid irreparable burns or life-threatening damage.
Future work will verify the modified model through experiments, adding a refined high-temperature protective clothing model, especially a cooling system. Subscript a artery a,adj adjacent element of artery blood air ambient air bl blood bl,a artery blood bl,a,adj adjacent element of artery blood bl,v vein blood bl,v,adj adjacent element of vein blood cr core cr-a exchange between core and artery cr-sk exchange between core and skin cr-v exchange between core and vein i skin node sector index (i = 1-4) mrt mean radiant temperature max maximum min minimum n serial number of different nodes of each body part perfusion, a-cr blood perfusion from artery to core perfusion, cr-v blood perfusion from core to vein perfusion, cr-sk blood perfusion between core and skin sh shiver sk skin sw sweat v vein v,adj adjacent element of vein blood Greek symbols α distribution coefficient θ the corresponding angular coordinate of the skin node