## 2 Why Are Mathematical Models of Value? Interest in using mathematical models of body heat exchanges is growing steadily, despite the fact that the complex equations describing these exchanges have to be simplified for successful implementation. This growth in interest is illustrated by the development of computer programs for determining the optimal thermal environment for low-birth-weight neonates in a closed incubator. The development of this type of quantitative model is also driven by the need to better understand how a neonate responds to extreme body overheating or cooling; for obvious ethical reasons, these situations cannot be studied directly. In general, management of an incubator's thermal environment is based on the measurement of physiological variables. Environmental control is mainly focused on air temperature and thus does not take account of other relevant environmental parameters. The difference between peripheral skin temperatures (at the hand or foot) and central (abdominal) skin temperatures can also be used to assess heat stress because it disappears when the neonate is overheating. Hypothermia and hyperthermia can also be detected by measuring the internal (core) temperature, which is assumed to reflect changes in body heat storage. Although the rectal temperature remains the gold standard for measuring excessive body warming, the placement of a temperature probe in the rectum is not recommended in routine clinical practice because it is associated with a risk of rectal perforation especially in very small neonates [10]. However, few other measurement sites and monitoring devices have been studied. The tympanic temperature is thought to reflect the core body temperature, given its proximity to the hypothalamus (the coordinating centre for body temperature control) and its vascular supply. However, the ear drum can also be perforated by a temperature probe. The axillary temperature is a safe option but its accuracy is questionable; measurements can be influenced by the peripheral regional vasoconstriction that occurs in some cases of malignant hyperthermia (due to high levels of circulating catecholamines). As with the rectal temperature, measurement of the oesophageal temperature is not recommended in routine clinical practice. Moreover, these local temperatures do not correspond to the \"core body temperature\"--the erroneous use of the latter term in certain publications. [PERSON] et al. [11] reported that in both humans and rabbits, thermoregulatory adjustments involve opposite regional temperature changes. [PERSON] et al. [12] showed that in a thermoneutral environment, the rectal temperature was lower than the oesophageal and aural temperatures. Temperatures close to that of rectum can be found in the deep central organs but not elsewhere, and so it is better to express the risk of thermal stress in terms of body heat storage. Hence, computer models are based on the maintenance of a balance between metabolic heat generation in the body (M) and heat exchanges (by conduction, convection, evaporation and radiation) between the skin and the environment (Figure 1). The radiant heat flux (R) between the neonate's skin and its surroundings is described by Stephan-Boltzmann's law:  
\[\mathrm{R}=\mathrm{h_{r}}\!\left[\left(\overline{\mathrm{T}}_{\mathrm{sk}}+2 73\right)^{4}-\left(\overline{\mathrm{T}}_{\mathrm{r}}+273\right)^{4}\right] \mathrm{F_{cl}} \tag{1}\]  
where \(\mathrm{h_{r}}\) is the radiant heat transfer coefficient, \(\overline{\mathrm{T}}_{\mathrm{r}}\) is the mean radiant temperature of the surrounding radiant surfaces, \(\overline{\mathrm{T}}_{\mathrm{sk}}\) is the mean skin temperature, and \(\mathrm{F_{cl}}\) is the reduction factor due to due to the diaper (together with a bonnet and/or a transparent plastic bag, if present). For premature infant clothing insulation consisted of a diaper in combination with a bonnet covering the scalp and / or a transparent plastic bag covering the torso and the limbs.