Further development of mathematical description for combined toxicity: A case study of lead–fluoride combination

• Rats were exposed during 6 weeks to repeated i.p. injections of lead acetate, sodium fluoride or both in doses 0.05 LD50.
• The development of intoxication was estimated quantitatively with 54 functional, biochemical and morphometric indices.
• For mathematical description of effects a response surface regression model containing linear and cross terms was used.
• 10 types of combined action were found thus confirming the result of the similar study on Pb and Cd.
• Special attention is given to indices on which combined toxics produce opposite effects.

In this article, we check and develop further some postulates of the theory and mathematical modeling of combined toxic effect that we proposed earlier [1]. To this end, we have analyzed the results of an experiment on rats exposed during 6 weeks to repeated intraperitoneal injections of lead acetate, sodium fluoride or both. The development of intoxication was estimated quantitatively with 54 functional, biochemical and morphometric indices. For mathematical description of the effect that lead and fluorine doses produced alone or in combination, we used a response surface regression model containing linear and cross terms (hyperbolic paraboloid). It is shown that the combination of lead and fluoride features the same 10 types of combined effect that we found previously for the lead and cadmium combination. Special attention is given to indices on which lead and fluorine produce an opposite effect.