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Mathematical model to estimate the increase in firefighters’ core temperature during firefighting activity with a portable calorimeter

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We developed a mathematical model to estimate the increase in firefighters’ core body temperature from energy expenditure (EE) measured by accelerometry to prevent heat illness during firefighting. Wearing firefighter personal protective equipment, seven male subjects aged 23–42 years underwent a graded walking test on a treadmill while esophageal temperature (Tes) and skin temperature were measured with thermocouples and EE was measured with a tri-axial accelerometer. To estimate the increase in Tes from EE, we proposed a mathematical model composed of the heat capacity of active muscles (C1, kcal·°C−1), the heat capacity of the sum of resting muscles and skin (C2), the resistance to heat flux from C1 to C2 (R1, °C·min·kcal−1), and the resistance from C2 to the skin surface (R2). We determined the parameters while minimizing the differences between the estimated and measured changes in Tes profiles during graded walking. We found that C1 and C2 in individuals were highly correlated with their body weight (kg) and body surface area (m2), respectively, whereas R1 and R2 were similar across subjects. When the profiles of measured Tes (y) and estimated Tes (x) were pooled in all subjects, they were almost identical and were described by a regression equation without an intercept, y = 0.96x (r = 0.96, P < 0.0001), with a mean difference of − 0.01 ± 0.12 °C (mean ± SD) ranging from − 0.18 to 1.56 °C of the increase in Tes by Bland-Altman analysis. Thus, the model can be used for firefighters to prevent heat illness during firefighting.

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Tes :

Esophageal temperature

Tskin :

Skin temperature


Energy expenditure

Ta :

Atmospheric temperature


Relative humidity


Sweat rate

Tsuit :

Temperature of inner suits

VO2 :

Oxygen consumption rate

VM (G):

Vector magnitude


Respiratory quotient

C1 :

Heat capacity of the active muscles to produce heat

C2 :

Heat capacity of the sum of resting muscle, fat and skin tissue to absorb heat

R1 :

Resistance to heat flux from C1 to C2 via circulating blood

R2 :

Resistance to heat flux from C2 to the skin surface through skin tissue

T1 :

Change in muscle temperature

T2 :

Change in blood and inner skin tissue temperature

T3 :

Mean skin temperature

F0 :

Heat production rate

F1 :

Heat flux from the active muscle to the skin via circulating blood

F2 :

Heat flux from the inner skin tissue to the skin surface


Personal protective equipment


A self-contained breathing apparatus


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We thank the students of Shinshu University for participating in this study as subjects.


This study was funded by a grant from the Japan Society for the Promotion of Science (15H01830) and by Teijin Co. Ltd.

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Authors and Affiliations



T.K., R.Y., and H.N. initiated and designed the project. T.K., Y.O., H.H., Y.K., K.U., and K.M. performed the experiments. T.K., Y.O., K.U., and K.M. analyzed the data. T.K., Y.O. and H.N. wrote the manuscript. S.M. and H.N. critically reviewed the manuscript. H.N directed the research.

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Correspondence to Hiroshi Nose.

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The authors declare that they have no conflicts of interest.

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H. Nose retired from the Department of Sports Medical Sciences, Shinshu University Graduate School of Medicine on March 31, 2018, and he is a specially appointed professor of the Shinshu University School of Medicine.

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Supplemental Fig. 1

The front (left panel) and back (right panel) images of a subject wearing the firefighter personal protective equipment (PPE), including a firefighter suit with inner and outer suits, gloves, boots, helmets and a self-contained breathing apparatus (SCBA). (PPTX 517 kb)

Supplemental Fig. 2

The relationship between VO2 measured by respirometry (ΔmVO2) and VO2 estimated by accelerometry (ΔeVO2) during a graded treadmill walking test. The values were pooled from 7 subjects. The solid line indicates a regression line of all values. They were highly correlated (R2= 0.956, P < 0.0001) with a regression equation without intercept; y = 0.858x. (PPTX 87.5 KB)

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Kimura, T., Ogawa, Y., Hayashi, H. et al. Mathematical model to estimate the increase in firefighters’ core temperature during firefighting activity with a portable calorimeter. Int J Biometeorol 64, 755–764 (2020).

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