How did Lofgreen and Garrett do the math?

Lofgreen and Garrett introduced a new system for predicting growing and finishing beef cattle energy requirements and feed values using net energy concepts. Based on data from comparative slaughter experiments they mathematically derived the California Net Energy System. Scaling values to body weight to the ¾ power, they summarized metabolizable energy intake (ME), energy retained (energy balance [EB]), and heat production (HP) data. They regressed the logarithm of HP on ME and extended the line to zero intake, and estimated fasting HP at 0.077 Mcal/kg^0.75, similar to previous estimates. They found no significant difference in fasting HP between steers and heifers. Above maintenance, however, a logarithmic fit of EB on ME does not allow for increased EB once ME is greater than 340 kcal/kg^0.75, or about three times maintenance intake. So based on their previous work, they used a linear fit so that partial efficiency of gain above maintenance was constant for a given feed. They show that with increasing roughage level efficiency of gain (slope) decreases, consistent with increasing efficiency of gain and maintenance with greater metabolizable energy of the feed. Making the system useful required that gain in body weight be related to EB. They settled on a parabolic equation, with significant differences between steers and heifers. Lofgreen and Garrett also used data from a number of experiments to relate ME and EB to estimate the ME required for maintenance (ME = HP) and then related the amount of feed that provided that amount of ME to the metabolizable energy content of the feed (MEc), resulting in a logarithmic equation. Then they related that amount of feed to the net energy for gain calculated as the slope of the EB line when regressed against feed intake. Combining the two equations, they estimate the net energy for maintenance and gain per unit feed (Mcal/kg dry matter) as a function of MEc: 0.4258 × 1.663^MEc and 2.544-5.670 × 0.6012^MEc, respectively. Finally, they show how to calculate net energy for maintenance and gain from experiments where two levels of a ration are fed and EB measured, where one level is fed and a metabolism trial is conducted, or when just a metabolism trial is conducted-but results are not consistent between designs.