Analysis of Energy Expenditure

(The slides used in this presentation can be viewed by clicking on the hyperlinks.)

Lipids in biological samples, including grains and meats, have been determined using near-infrared methods for a number of years. The near-infrared spectrum of oleic acid shows the distinctive spectral regions that correspond to lipids. The peak structures around 1200 nm, 1700-1800 nm, 2100-2200 nm, and 2300-2400 nm, and the lack of peaks elsewhere in the spectrum are powerful indicators of the presence of lipids in a sample. The shifting of peaks in these regions to lower energies (longer wavelengths) generally indicates cis-to-trans isomerization or saturation of the lipid carbon-carbon double bond. Fatty acids like oleic acid have broad regions of the spectrum with relatively low absorbance, enhancing tissue penetration of near-IR light with these wavelengths.

To compare the performance of near-IR spectrometric imaging with a camera to direct measurements made with a conventional diffuse reflectance spectrometer, an experiment was performed with samples containing known amounts of different lipid mixtures. This picture in visible-light of four different rabbit chows and two optical reflectance standards (one black and one white, in the center of the image) was also imaged in the near-infrared using conventional tungsten blackbody light bulbs. The four different chow formulations were high cholesterol (1%), low fat, high saturated fat, and control. The purpose of the experiment was to determine whether camera imaging at six wavelengths was as effective at differentiating among the chows as scanning individual pellets in a diffuse reflectance spectrometer.

Multidimensional distances in standard deviations (SDs) were measured between samples of chow with the near-IR camera at six different near-IR wavelengths. In the table, distances less than 3 SDs denote a match between the samples, while distances greater than 3 SDs denote samples with statistically different spectra. The near-IR camera was able to identify perfectly all of the calibration and validation chow samples.

Similar distances in SDs between the chow samples were measured using a conventional near-IR diffuse reflectance spectrometer. The conventional spectrometer also achieved perfect identification of the different chow samples, in many cases with larger distances in SDs than the camera. This increased discrimination ability is not unusual because the conventional spectrometer is able to scan 701 wavelengths on a single chow pellet, with more even and controlled illumination of all pellet surfaces. However, in the cross validation analysis one chow pellet failed cross validation in the high cholesterol chow group. The failure to cross validate was probably due as much to the small size of the calibration and validation pellet groups (n_c=5, n_v=5) as to the increased sensitivity of the conventional spectrometer.

Determination of Heat from Temperature Measurements. Given heat capacity and mass, heat content can be determined from temperature measurements. Surface temperature can be determined from infrared photon (blackbody) emission. Infrared photon emission can be detected using infrared cameras based on InSb and PtSi CCD technology. In this manner, the energy output of an subject like a rat can be determined.

Two models are being used by our research group to predict core temperatures in metabolism studies from the surface temperatures of different body parts. The models are based on empirical data measurements or on computer simulations aided by empirical measurements.

Empirical Model

The regression of dose of a prototype drug against the temperature of various body zones, like the tail, dorsal region, and core temperature, yields an equation that can be used to predict total energy output. From similar dose measurements on new compounds, and images of skin temperatures from infrared (IR) photon emission, core temperatures can be predicted accurately. This type of empirical modeling has been demonstrated in the metabolism literature in rats and in human subjects.

Simulation Model

The simulation model is a semi-empirical model that uses the laws of heat transfer, bulk temperature and metabolic measurements, and a three-dimensional model of the body to calculate spatially resolved temperatures from the input data (including image data).

This IR emission image of an alert, freely moving rat shows areas with the highest surface temperatures in shades of red, and proceeds through the spectrum to portray decreasing temperatures in shades of yellow, green, blue, and violet. Areas with temperatures off the calibration scale on the high end are colored in white, while areas with temperatures off the calibration scale on the low end are colored in black.

Tissue temperature control is established by a balance between energy production and energy dissipation.

Heat conduction between materials is governed by an equation based on the temperature, thermal conductivity, area, and distance of conduction.

The variables and constants in equations governing heat transfer are given in this Table of Constants.

Heat flow in joules can also be calculated by an equation based on the specific heat, mass, and temperature change.

Heat convection can be modeled as heat conduction into a moving material followed by heat conduction out of the moving material and into another substance. This is similar in concept to the three-layer insulation problem that often appears in textbooks.

Heat radiation is governed by an equation known as Stefan's Law, which gives power output in J/sec given the surface emissivity, area, temperature, and temperature of the surrounding environment. Most of the heat lost by an subject like a rat can be calculated from heat conduction, convection, and radiation, However, some heat is carried off by vaporization of water. This amount of energy loss can be calculated using the latent heat of vaporization and mass of water lost.

Other measurements can be used in the total energy balance calculation to estimate of energy expenditure in chemical reactions. Oxygen consumption (by atmospheric Raman spectrometry), oxy- and deoxyhemoglobin (by short-wavelength near-IR spectrometry of electronic transitions), carbon dioxide and water (by infrared spectrometry) can also be made on the subject or the surrounding atmosphere to assist in the determination of overall energy balance. Such measurements typically require that the subject be housed in a metabolic calorimetry chamber.

The metabolic calorimetry chamber is a hybrid metabolic cage and Dewar, constructed from reflective aluminum and polystyrene foam panels. The chamber has multiple ports that can be instrumented and used for near-IR and visible-light cameras, air inlets and outlets, urine collection, food and water, temperature sensors, lighting, etc. The camera on the left of this image is a Mitsubishi PtSi CCD IR camera, while the camera on the right is a Cincinnati Electronics InSb camera converted to near-IR use. The top ports in the chamber are open, while the ports on the sides are still screwed on.

The semi-empirical computer model for heat transfer partitions the total energy information provided by temperature sensors, oxygen consumption, CO₂ and water measurements into body zones using the guidance of temperature images provided by IR photon emission. Conjugate-gradient or other optimization of the semi-empirical model parameters to the observed temperature data allow the model parameters to be adjusted until an adequate estimate of energy expenditure is obtained.

The objective of the semi-empirical modeling is to allow reasonable estimates of spatially resolved energy expenditure in freely moving subjects to be made. The accuracy of the semi-empirical modeling is dependent upon the coarseness of the three-dimensional grid(determined by the volume element, or voxel, size) used in the simulation. Finer gridspermit more accurate simulation of energy transfer, in part because the dx terms in heat conduction imply the use of an infinitesimally small volume element.

In a 3-D cubic grid, each cell has 26 nearest-neighbors. Changes in the temperature value in a given cell propagate through the other cells according to the conduction, convection, radiation, and vaporization rules established for the individual cells. The propagation of changes through the grid with each time tick follows a wave-front equation and integrated wave-front equation that determines how many cells are affected at each time "tick."

The semi-empirical computer model indexes the variables in each cell with values that indicate how the cells are permitted to transfer energy by conduction, convection, radiation, and in some cases, vaporization.

Because computer memory is contiguous and linear, functions are derived to map the locations of the 26 nearest-neighbors in 3-D to addresses in a 1-D array, creating a multi-linked list data structure. The pointers in the links to the temperature values in each cell are also used as pointers to the corresponding conduction, convection, radiation, and vaporization parameters used by the semi-empirical model.

Assessment of the semi-empirical model fit to the observed temperature-image data occurs through calculation of the difference image between the model and observed data, and minimization of the sum of the squared residuals. The parameters in the are adjusted semi-empirical model are adjusted to maximize the goodness-of-fit using an optimization method such as conjugate-gradient or simplex optimization. Monte Carlo ensemble forecasting is used to gauge the robustness of the solution by randomly varying the initial parameters slightly 30 times and running the simulation repeatedly to see whether it always converges to the same solution. (If 30 trials all reach the same solution, one can be much more confident of the results than one could be if 30 trials converged on 20 or 30 different solutions.)

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