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In this paper, we combine polynomial functions, Generalized Estimating Equations, and bootstrap-based model selection to test for signatures of linear or nonlinear relationships between body surface temperature and ambient temperature in endotherms. Linearity or nonlinearity is associated with the absence or presence of cutaneous vasodilation and vasoconstriction, respectively. We obtained experimental data on body surface temperature variation from a mammalian model organism as a function of ambient temperature using infrared thermal imaging. The statistical framework of model estimation and selection successfully detected linear and nonlinear relationships between body surface temperature and ambient temperature for different body regions of the model organism. These results demonstrate that our statistical approach is instrumental to assess the complexity of thermoregulation in endotherms.

Endotherms (mammals and birds) regulate body temperature primarily by balancing metabolic heat production with heat exchanged with the ambient [

temperature ( T s ) and ambient temperature ( T a ), Δ T = T s − T a , varies linearly or nonlinearly with ambient temperature [

Specimens of Gracilinanus microtarsus were live-trapped in savanna-like habitat at the Reserva Biológica de Mogi-Guaçu (RBMG) located in the district of Martinho Prado, Mogi-Guaçu, São Paulo (22˚15S; 47˚08W). Field work was carried out from October 2011 to April 2012. Trapping was done on three consecutive nights. A single 8 ´ 8 trapping grid with 64 trapping stations located 10 m from each other was used to capture individual G. microtarsus. A single Sherman live trap (7.5 cm ´ 9.0 cm ´ 23.5 cm; H. B. Sherman Traps, Inc., Tallahassee, Florida) was set on trees at each trapping station about 1.75 m above ground and baited with banana and peanut butter.

Individuals of G. microtarsus were housed in individual cages in an animal room maintained at approximately 23˚C with a 12h/12h light/dark cycle. Gracile mouse opossums were provided with ad libitum water and an amount of food (dry cat and dog food and mango) designed to keep weight gain (

Changes in radiative heat exchange at the surface of different body regions of G. microtarsus were monitored by video thermography as ambient temperature was varied from 8˚C to 38˚C. On the day of an experiment, each individual animal was transferred from its home-cage to an experimental container (7.5 cm ´ 9.0 cm ´ 23.5 cm). This container was framed with wire mesh (1 ´ 10 cm), which allowed the monitoring of body temperature surface without any considerable interference. The animal container was then placed inside a temperature-controlled chamber (FANEM Ltd., São Paulo) and the thermal camera (FLIR SC640, FLIR Systems, Inc.) was positioned under the animal cage, allowing us to track body surface temperature while manipulating ambient temperature. First, the individual was allowed one hour to habituate to the experimental conditions at 23˚C. We set the temperature change protocol for a 1˚C stepwise increment or decrement from the initial temperature (23˚C) up to 38˚C or down to 8˚C, respectively. This range of T a covers the range of T a the gracile mouse opossum is commonly exposed in its natural habitat (

4 days, in which animals were returned to their home-cages and allowed to feed and drink. Whether individual animals were first submitted to the incrementing and decrementing temperature protocols was a random decision.

Thermal images were recorded at 10 frames∙s^{−1} using an IR camera streamed to a computer where data acquisition was managed by the software Thermacam 2.9 (FLIR Systems, Inc). Thermal imaging cameras measure the amount of near infrared radiation (typically wavelengths: 8 - 12 nm) emitted by a surface and then convert this measured radiation to a radiative temperature reading according to the Stefan-Boltzmann equation. This equation states that the energy emitted, R (W∙m^{−2}), is proportional to the fourth power of its absolute temperature, T, in Kelvin degrees,

R = ϵ σ T 4 (1)

where, ϵ is the emissivity of the surface and σ is the Stefan-Boltzmann constant (5.67 ´ 10^{-8} W∙m^{−2}∙K^{−4}). We assumed that the body surface of G. microtarsus has an homogenous emissivity of 0.96, which is typical of organic materials [

We analyzed surface temperature ( T s ) separately for different body regions including ears (split into distal and proximal regions), feet, tail, chest, back (evaluated when the animal turned on its back) and ventral areas. Initially, we chose one frame for each T a from the last two minutes of the exposure period to any given T a . Thus, at the time used for data collection, animals had already been exposed to that particular T a for at least 10 min. For each chosen frame, the regions of interest were digitally drawn to obtain the average surface temperature of each body part. Typically, we used the same frame to analyze the T s from all body parts. In cases in which the animal positioning did not allow for the analysis of all the body regions on the same frame, we used the temporally closest frame to analyze the missing body region.

The relationship between body surface and ambient temperature can be one of two possible types depending on whether there is no vasomotor adjustment or whether there is vasomotor adjustment [

follows. If the rate of change is constant, we must have d t s d t a = c . Integrating both sides, we have ∫ d T s d T a d t a = ∫ c d T a , then f ( T s ) = c T a + d . Therefore, the function that describes the relationship between T s and T a is linear. If the rate of change is not constant, we must have d 2 T s d T a 2 = c . Integrating both sides we have ∫ d 2 T s d t a 2 d T a = ∫ c d T a , yielding d T s d T a = c T a + d . Integrating both sides again we have ∫ d T s d T a d T a = ∫ ( c T a + d ) d T a , yielding f ( T s ) = c T a 2 + d T a + e . Therefore, the

function describing the relationship between body surface temperature ( T s ) and ambient temperature ( T a ) is nonlinear, quadratic in T a . Continued differentiation and integration leads to higher order polynomials in T a . Polynomial functions thus capture the linearity and nonlinearity intrinsic to the relationship between surface temperature and ambient temperature associated with the mechanism of vasomotor adjustment. Therefore, we modeled the relationship between ( T s ) and ( T a ) with a family of polynomial functions.

A response variable that is repeatedly measured on the same subject at different time points is the key feature of longitudinal data sets. In this setting, the correlation between observations from a given individual must be accounted for. Otherwise, downstream analyses may be affected by a number of factors, including false conclusions due to underestimated variance terms. Generalized estimating equation (GEE) models are an extension of generalized linear models devised to analyze data, which arise commonly in applied sciences [

We defined Δ T as T s − T a , where T s is the surface temperature and T a is

the ambient temperature. Note that the d T s d T a introduced in the previous section

is proportional to Δ T , representing the rate of change of T s as T a changes. Our main goal was to describe the relationship between Δ T and T a for each body region. The relationship between Δ T and T a is linear when the body region is not capable of vasomotor adjustment and nonlinear when the body region is capable of vasomotor adjustment. Here, we considered as nonlinear the polynomial functions of degree 2 and 3. We used a two-step strategy to select models to describe variation in Δ T as a function of ambient temperature ( T a ) for each body region. In the first step, we selected a candidate model by minimizing the quasilikelihood under the Independence model Criterion (QIC) statistic over a grid comprised of correlation structures and model complexities (represented by increasing polynomial degrees). The QIC statistic is analogous to the familiar AIC (Akaike’s Information Criterion) statistic used for comparing models fitted with likelihood-based methods [

Biophysical modeling of the heat exchange process allowed us to estimate the amount of heat exchanged (Q) with the ambient by each body region of the gracile mouse opossum. We represented the gracile mouse opossum by a simple geometric model: the ventral body region as a semi-cylinder, the back as a horizontal plate, the chest as a horizontal plate, the tail as a horizontal cylinder, the feet as a flat plate, the fingers as horizontal cylinders (data for fingers were added to that of the feet), and the ears as vertical plates. We measured the characteristic dimension, d i , and surface area, A i , of each i-th body region from thermal images of the gracile mouse opossum using the shape tools of software ImageJ (

We estimated heat exchange (Q) in Watts for each body surface ( Q ears , Q feet , Q tail , Q chest , Q dorsal , Q ventral ), where Q ears is the sum of the estimated heat exchange by the proximal and distal regions of the ear (multiplied by two to account for the two ears); Q feet is the sum of the estimated heat exchange by the lower and upper soles (multiplied by four to account for the four feet), plus the estimated heat exchange by the fingers (multiplied by 20 to account for the 20 fingers); Q tail is the estimated heat exchange by the tail; Q chest is the estimated heat exchange by the chest; Q dorsal is the estimated heat exchange by the dorsal region; and Q ventral is the estimated heat exchange by the ventral region. We assumed that the gracile mouse opossum was in thermal equilibrium with the ambient. We defined Q total for each gracile mouse opossum by adding the Q values for all body surfaces as,

Q total = Q ears + Q feet + Q tail + Q chest + Q dorsal + Q ventral . (2)

Percent heat exchange by each i-th body region ( Q i ) was calculated as Q i Q total × 100 . The Q i for each body region was estimated using the following equation [

Q i = Q r + Q c , (3)

where Q r is the radiative heat exchange and Q c is the free convective heat exchange, such that,

Q r = ϵ σ A ( T s 4 − T a 4 ) , (4)

and

Q c = h c A ( T s − T a ) , (5)

where ϵ is the emissivity for biological tissues, assumed to be 0.96 according to Monteith and Unsworth [^{−8}); A is the surface area (m^{2}) of the i-th body region; T s is the surface temperature of the i-th body region; T a is ambient temperature (K˚); and h c is the heat transfer coefficient given by

h c = N u k d , (6)

where k is the thermal conductivity of the air at a particular T a (W∙m^{−1}∙˚K). The relationship between k and T a was estimated by Tattersall et al. [

k = 0.00241 + 7.5907 e − 6 × T a . (7)

N u is the dimensionless Nusselt number that is a measure of the ratio of buoyant to viscous forces. In free convection, the Nussel number is a function of the dimensionless Grashof (Gr) and Prandtl (Pr) numbers, written as: N u = f ( G r , P r ) . In forced convection, the Nussel number is a function of the dimensionless Reynolds (Gr) and Prandtl (Pr) numbers: N u = f ( R e , P r ) . Grashof, Prandtl, and Reynolds numbers are given by,

G r = a g d 3 ( T s − T a ) v 2 , (8)

R e = V d v , (9)

P r = c p μ k , (10)

where a is the coefficient of thermal expansion of air, g is the acceleration of gravity, d is the characteristic dimension of the body region (

Body region | Shape | d (m) | Surface area (m^{2}) | Gr | Re | Nu |
---|---|---|---|---|---|---|

Ear (proximal) | plate | 11.37 × 10 − 3 | 291.39 × 10 − 6 | −5 - 20 | 0.024 - 0.027 | 0.595 R e 0.5 |

Ear (distal) | plate | 11.37 × 10 − 3 | 291.39 × 10 − 6 | −8 - 7 | 0.024 - 0.027 | 0.595 R e 0.5 |

Feet | plate | 13.18 × 10 − 3 | 297.15 × 10 − 6 | −6 - 7 | 0.028 - 0.031 | 0.595 R e 0.5 |

Tail | cylinder | 5.37 × 10 − 3 | 2585.02 × 10 − 6 | −0.3 - 0.4 | 0.011 - 0.013 | 0. 891 R e 0.33 |

Dorsal | plate | 91.22 × 10 − 3 | 7050.43 × 10 − 6 | −489 - 7 ´ 10^{3} | 0.195 - 0.215 | 0.595 R e 0.5 |

Chest | plate | 24.15 × 10 − 3 | 415.95 × 10 − 6 | −20 - 261 | 0.052 - 0.057 | 0.595 R e 0.5 |

Ventral | semi-cylinder | 91.22 × 10 − 3 | 1361.45 × 10 − 6 | −856 - 1.6 ´ 10^{4} | 0.195 - 0.215 | 0.595 R e 0.5 |

convection considering a wind velocity of 0.1 ms^{−1} according to Tattersall et al. [

N u = c R e n , (11)

where c and n are constants related to shape.

Data in table are the shape and characteristic dimensions used for heat exchange calculations, calculated d and surface area, and relationship between Grashof (Gr), Reynolds (Re), and Nusselt (Nu) numbers [

Assuming that each gracile mouse opossum was in thermal balance during the experimental procedure we should expect that the relationship between Q total and T a would follow the relationship between metabolic heat production and T a . Based on the rates for oxygen uptake determined by Cooper et al. [

To estimate the role of each body region for heat exchange we defined the variable Q i as the percent heat exchange of each i-th body region. Q i was

calculated as Q i Q total × 100 . The relationship between Q i and T a was described

by linear regression models. We followed the same framework of Generalized Estimating Equations (GEE) as the estimation method for the model to account for correlated response.

Our primary objective is to search for signatures of linearity or non-linearity in the relationship between the differential Δ T = T s − T a and ambient temperature ( T a ). Linearity or nonlinearity between Δ T and T a arise as a consequence of the absence or presence of cutaneous vasodilation and vasoconstriction, respectively, which are major physiological mechanisms of vasomotor control, modulating heat exchange with the ambient [

Within the range of T a from 8˚C to 38˚C, surface temperatures ( T s ) for all body regions of the gracile mouse opossum increased with T a at increasing rates depending on the body region (

Candidate model | Selected model | Correlation matrix | r | Quantile 2.5% | Quantile 97.5% | Vasomotor adjustment |
---|---|---|---|---|---|---|

Ear (proximal) | ||||||

b 1 + b 2 T a + b 3 T a 2 + b 4 T a 3 b 1 + b 2 T a + b 3 T a 2 | * | autoregressive | 0.31 | 3.138 | 31.2 | yes |

Ear (distal) | ||||||

b 1 + b 2 T a + b 3 T a 2 + b 4 T a 3 b 1 + b 2 T a + b 3 T a 2 | * | autoregressive | 0.60 | −0.938 | 14.5 | yes |

Feet | ||||||

b 1 + b 2 T a + b 3 T a 2 + b 4 T a 3 b 1 + b 2 T a + b 3 T a 2 | * | independence | −0.994 | 27.6 | yes | |

Tail | ||||||

b 1 + b 2 T a + b 3 T a 2 b 1 + b 2 T a | * | autoregressive | 0.54 | 1.993 | 26 | yes |

Ventral | ||||||

b 1 + b 2 T a b 1 | * | autoregressive | 0.43 | 1087.259 | 1228.6 | no |

Dorsal | ||||||

b 1 + b 2 T a b 1 | * | autoregressive | 0.40 | 721.043 | 825.3 | no |

Chest | ||||||

b 1 + b 2 T a b 1 | * | autoregressive | 0.81 | 755.442 | 1044.27 | no |

*indicates which candidate model was selected. If 0 ∈ q 2.5 % , q 97.5 % , then we do not have evidence to reject the hypothesis that there are no gains in using a more complex model.

The comparison between the rates of heat production estimated from Cooper et al. [

Heat exchanged by each body region varied across the range of ambient temperature as indicated by the significant interactions between Q i and T a (

Coefficient | Estimate | SE | p-value |
---|---|---|---|

Ear (proximal) | −15.742 | 2.977 | 1.2 e − 07 *** |

Ear (distal) | −20.878 | 2.112 | < 2 e − 16 *** |

Feet | −30.396 | 3.097 | < 2 e − 16 *** |

Tail | −45.177 | 3.548 | < 2 e − 16 *** |

Ventral | 29.658 | 3.223 | < 0.001 *** |

Dorsal | 23.526 | 1.998 | < 2 e − 16 *** |

Chest | −22.635 | 2.451 | < 2 e − 16 *** |

T_{a}: Ear (proximal) | −0.810 | 0.153 | 0.597 |

T_{a}: Ear (distal) | −0.080 | 0.108 | 0.461 |

T_{a}: Feet | 0.763 | 0.164 | 3.1 e − 06 *** |

T_{a}: Tail | 1.359 | 0.176 | 1.4 e − 14 *** |

T_{a}: Ventral | −0.384 | 0.158 | 0.015* |

T_{a}: Chest | 0.039 | 0.114 | 0.731 |

*coefficients statistically significant at p-value £ 0.05; ***coefficients statistically significant at p-value £ 0.001.

Heat exchange in the surface of an endotherm is a process that can be modeled by the equation Q total = ϵ σ A ( T s 4 − T a 4 ) + h c A ( T s − T a ) , as described in classic textbooks and review articles of animal physiology and biophysical ecology [

Different body regions of the gracile mouse opossum exhibited distinct heat exchange properties in response to variation in ambient temperature. As a consequence, the relative contribution of each body region to total heat balance also changed with ambient temperature. Whereas furred body regions such as the chest, dorsal, and ventral region, exhibited a linear relationship between Δ T and T a , furless body regions such as the ears, feet, and tail exhibited a nonlinear relationship between Δ T and T a . The linear relationship between Δ T and T a for the furred regions of G. microtarsus indicates that mechanisms other than vasomotor adjustment dominated the dynamics of heat exchange at these regions. Indeed, this is readily observed by the decay of Δ T as the ambient temperature approaches T b (around 30.6˚C and 34.7˚C, [

For the range of temperatures tested, temperature equalization (i.e., Δ T = 0 ) for the furred body regions occurred only at the highest temperatures tested (differently from what happened with the furless body regions). At these temperatures (>35˚C), ambient temperature most likely had already surpassed internal body temperature. Therefore, we suspect that, in this case, fur insulation of the animals reduced heat gain from the environment at greater rates, shifting the point of temperature equalization to a temperature a few degrees higher than their normal body temperature. As at these temperatures energy expensive heat loss mechanisms are likely to intervene, preventing extra heat gain from the ambient might be highly relevant for the total heat balance and energetics of G. microtarsus. Finally, we should acknowledge the possibility that the atainement of equalization at higher than expected temperatures (normal T b ) could be attributable to hyperthermia. We did not monitored internal body temperature of our animals during the experiments. Nevertheless, the results from temperature equalization obtained for the furless regions, particularly the ear, suggest that the animals did not experience hyperthermia. Thus, we believe that the linear relationship between Δ T and T a for the furred regions of G. microtarsus reveals not only the insulative properties of the fur in preventing excessive heat loss at low temperatures, but also the less commonly acknowledged role of preventing excessive heat gain at higher temperatures.

Furless body regions such as the ears, feet, and tail exhibited a nonlinear relationship between Δ T and T a . However, there were differences among different furless regions. The relationship found for the ears was quite interesting. For T a higher than 35˚C, Δ T tended to zero and remained unchanged. Our interpretation of this result is that around 35˚C T a equalized with body temperature [

The differences in the heat exchange properties observed among the different body regions of G. microtarsus had a direct impact on the contribution of each body region to the total heat balance of the gracile mouse opossum. Furred regions were the largest in surface area and collectively responded for more than half of the total heat dissipated, especially at temperatures below 20˚C - 24˚C. However, as these regions lacked the capacity for vasomotor adjustment, as temperatures were elevated above that limit, the contribution of the furless regions of the feet and tail to the total heat balance of the animals gained prominence. Indeed, from almost negligible at low temperatures, the contribution of the feet and tail was elevated to more than half of the total heat exchanged at the highest temperature tested. This effect can be directly attributed to the vasomotor response exhibited by these body regions. Also, it is conceivable that some evaporative cooling mechanism concurrently improved heat dissipation through tail and feet at temperatures above the vasomotor threshold. At this point, it remains uncertain which evaporative heat dissipation mechanisms are employed by G. microtarsus.

Whereas examining heterothermy across different body regions can be straightforward with IRT, addressing the question of how T s can inform about T b regulation and the existence of thermal windows lacks a sound statistical approach. Here, we have shown that combining the use of polynomial functions and estimation by GEE can be a robust and informative strategy for addressing questions on the dynamics of either T s or surface heat exchange across animal body regions. Polynomial functions capture the linearity and nonlinearity in the relationship between T s and T a , making the task of detecting regions capable or not of vasomotor adjustments objective and unbiased. In addition, the GEE estimation method used with generalized linear models incorporates structures typical of IRT data, such as correlated responses, within-individual correlations, allows for missing data, time-varying covariates, irregularly-timed or mistimed measurements [

Tail and feet comprise less than 25% of G. microtarsus body surface although their contribution to total heat exchange at temperatures above 20˚C - 24˚C was considerable. A similar response is found in rabbits and elephant ears [

Our estimates of total heat exchange for the gracile mouse opossum were consistently lower than the rates of heat production estimated by Cooper et al. [

In conclusion, the process of heat exchange across an endotherm surface as a function of ambient temperature is complex and depends on as many dimensions of the organism and its surrounding ambient. Assessing this process under simplified conditions within the unifying principles of thermoregulation unveils and clarifies the rich diversity of mechanisms underlying the process. Under this approach, one is able to better describe how the dynamics of heat exchange mechanisms vary between body regions, allowing endotherms to thermoregulate across ambient temperatures. This approach is based on detailed understanding of the relative importance of each body region for the entire budget across the range of ambient temperature.

We are very grateful to Marcus A. M. de Aguiar, Eduardo G. Martins, Luiza C. Duarte, and Mathias M. Pires for carefully reading the first drafts and improving the clarity of the manuscript. João Del Giudice Neto and Marcos Mecca Pinto from Mogi-Guaçu Biological Reserve for logistical support. Research supported by Fundação de Amparo à Pesquisa do Estado de São Paulo, Brazil (FAPESP) [Denis O. V. de Andrade - 2007/05080-1 and 2013/04190-9; Sérgio F. dos Reis - 2005/51353-4]; Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil (CNPq) [Barbara Henning - 140773/2013-4; Denis O. V. de Andrade - 302045/2012-0 and 306811/2015-4; Sérgio F. dos Reis - 303544/2011-2]; and CAPES [Barbara Henning - 99999.004965/2014-00].

Henning, B., Carvalho, B.S., Boldrini, J.L., dos Reis, S.F. and Andrade, D.O.V. (2018) Statistical Estimation of Surface Heat Control and Exchange in Endotherms. Open Journal of Statistics, 8, 220-239. https://doi.org/10.4236/ojs.2018.81013