## 2 Why Are Mathematical Models of Value?
The heat loss by transepidermal water loss (E) can be formulated as:  
\[\mathrm{E}=\mathrm{h_{e}}\;\omega\;\mathrm{(P_{SH20}-P_{AH20})}\;\mathrm{F_{ pcl}} \tag{2}\]  
where \(\mathrm{h_{e}}\) is the evaporative heat loss coefficient, \(\omega\) is the fraction of skin area that is completely wet, \(\mathrm{P_{SH20}-P_{AH20}}\) is the water vapor partial pressure difference between the skin (\(\mathrm{P_{SH20}}\)) and the air, (\(\mathrm{P_{AH20}}\)), and \(\mathrm{F_{pcl}}\) is the reduction factor for water vapor transfer due to the diaper (together with a bonnet and/or a transparent plastic bag, if present). To take account of the fact that extremely-low-birth-weight neonates have a very thin skin (enhancing transepidermal water loss), \(\omega\) varied from 0.412 on the first day of life to 0.08 after 10 days of life [13]. Evaporative heat loss (due to breathing) can be expressed as:  
\[\mathrm{Eresp}=\mathrm{V_{E}}\;\delta\;\mathrm{(M_{e}-M_{i})} \tag{3}\]  
where \(\mathrm{V_{E}}\) is the pulmonary ventilation rate, \(\delta\) is the latent heat of water vaporization, and \(\mathrm{M_{e}-M_{i}}\) is the difference in water content between expired air (Me) and inspired air (Mi). When a neonate is receiving ventilatory support or oxygen support, \(\mathrm{M_{i}}\) is the water content of the supplied air or oxygen. The heat transfer from the skin surface to the moving air in the incubator (C) is quantified as:  
\[\mathrm{C}=\mathrm{h_{c}}\!\left(\mathrm{T_{a}-\overline{\mathrm{T}}_{ \mathrm{sk}}}\right)\!\mathrm{F_{cl}} \tag{4}\]  
where \(\mathrm{h_{c}}\) is the convective heat transfer coefficient, and \(\mathrm{T_{a}}\) is the incubator air temperature. Natural convection (also termed \"free convection\") is driven by the temperature difference between the air and  
Figure 1: A simplified block diagram of the heat exchanges between the neonate’s skin and the environment: K, C, R and E are respectively the conductive, convective, radiant and evaporative heat transfers. the skin. Forced convection occurs at air speeds of more than 0.2 m/s. Mixed-mode convection occurs between these two limits. The convective heat loss through the respiratory tract is defined as:  
\[\mathrm{Cresp}=\mathrm{V_{E}\,Cp\,(T_{e}-T_{i})} \tag{5}\]  
where Cp is the heat capacity of the air, and \(T_{e}-T\mathrm{i}\) is the temperature difference between expired air and inspired air. In a closed incubator, \(T_{i}=T_{a}\). When a neonate is receiving ventilatory support or oxygen support, \(T_{i}\) is the temperature of the supplied air or oxygen. The heat conducted from the skin to the mattress (K) is calculated using [PERSON]'s law:  
\[K=h_{K}\big{(}T_{m}-\overline{T}_{sk}\big{)} \tag{6}\]  
where \(h_{K}\) is the conductive heat transfer coefficient, and \(T_{m}\) is the temperature of the mattress surface. Lastly, the conventional heat balance equation for heat stored in the body (S) can be written as:  
\[S=M\pm R-E-E\mathrm{resp}\pm C\pm\mathrm{Cresp}\pm K \tag{7}\]  
Metabolic heat production was assessed as a function of post-natal age, as recommended by [PERSON] et al. for very low birth weight neonates [14]. When the heat storage is nil, the body reaches a thermal equilibrium with its surroundings, and the neonate's thermal environment is optimal. An example of the set of software variables used to calculate the thermal environment in a closed incubator is shown in Figure 2:  
The basic software settings (input parameters) were provided by the nursing staff. Output variables were calculated from the difference between the metabolism (Mreq) required to achieve thermal equilibrium and the calculated metabolism (M) (S = 0). The infant's degree of maturity was reflected by the body weight and the gestational and postnatal ages. To validate a model's predictive ability, the sum of the heat exchanges is sometimes compared with the metabolic heat production measured by indirect respiratory calorimetry.