Objective: The properties of Chinese materia medica are believed to be the summarization of the effects of biological performance on the various body states. Systemic discussion of chemical-factor elements, body-condition elements, biological-performance elements and their interrelationships is needed for research into the properties of Chinese materia medica. Following the practical characteristics of Chinese medicine, the three-element mathematical model was formed by introducing some mathematical concepts and methods and was used to study the cold or hot property of Chinese medicine, and to investigate the difference in biological performances of the two properties.
Methods: By using the concept of different functionality of Chinese medicine on abnormal states and the idea of interaction in mathematics, the effects of chemical-factor elements and body-condition elements were normalized to the amount of biological performance which was represented by some important indicators. The three-element mathematical model was formed with scatter plots through four steps, including effect separation, intensity calculation, frequency statistics and relevance analysis. A comparison pharmacology experiment of administration of hot property medicines, Fuzi (Radix Aconiti Lateralis Preparata) and Rougui (Cortex Cinnamomi), and cold property medicines, Huangbai (Cortex Phellodendri) and Zhizi (Fructus Gardeniae) on normal and glucocorticoid-induced yang-deficiency and yin-deficiency states was designed. The results were analyzed by the mathematical model. The scatter plots were the main output of model analysis. The expression of cold property and hot property was able to be quantified by frequency distribution of biological indexes of administrations on yang-deficiency and yin-deficiency states in the “efficacy zone” and “toxicity zone” of the plots and by the relevance analysis.
Results: The ratios of biological indicator frequency in the “efficacy zone” of administrations on yang-deficiency state and yin-deficiency state were 7∶3 for Fuzi, 3∶3 for Rougui, 4∶4 for Huangbai and 1∶5 for Zhizi. The sums of the biological indicator frequency in the “toxicity zone” of administration on the two states were 4 for Fuzi, 0 for Rougui, 2 for Huangbai and 4 for Zhizi. The relevance analysis showed that the order from Fuzi, Rougui, Huangbai to Zhizi was proportional to the change from “be true of yang-deficiency state” to “be true of yin-deficiency state”. The extent of the hot property decreased while that of the cold property increased in the order of Fuzi, Rougui, Huangbai and Zhizi. The stronger the efficacy of above medicines is, the more obvious the toxicity displayed.
Conclusion: The three-element mathematical model employed in this study is effectively capable of explaining the different biological expressions between hot property medicines and cold property medicines. This suggests that it may provide a mathematical tool and theoretical basis for the modern interpretation of cold property and hot property of Chinese medicine, and provide new ideas for further studing into the essence of Chinese medicine property theory.

Key words: cold (TCD), hot (TCD), four natures (TCD), models,theoretical, biological performance