Hansch Approach
In 1964, after analyzing a large number of data sets, Hansch and Dunn formulated a more generalized approach to SARs. The critical breakthrough was the usage of a linear combination of different physicochemical parameters to delineate biological activity.78
log 1/C — alog P + pa + ■■• + constant [21]
C is the molar concentration that evokes a certain biological response. The utility of this model which was developed by using multiple linear regression (MLR) analysis has been demonstrated in all types of biological systems that run the gamut from interactions of chemicals with isolated receptors to toxicological interactions in whole animals.78,79
During this period, we continued to observe instances where the linear model was inadequate and not adept at explaining the biological data. At this juncture, we analyzed a number of datasets and proposed a model which stressed the nonlinear relationship of hydrophobicity (as expressed by log P) with biological activity.80
log 1 /C — alog P + b(log P)2 + ? + constant [22]
This parabolic model confirmed our earliest observations that drug transport/distribution was indeed a two-step process. We likened the movement of drug molecules in a biological system to a random walk process from the site of administration to the target receptor.81
Our first attempt to justify mathematically the parabolic relationship between drug effectiveness and lipophilic character was carried out in 1969 with the help of Bentley, a distinguished statistician in the Mathematics Department at Pomona College. Once again luck, in the form of Bentley, played a critical role in helping us derive the now classical mathematical model of the nonlinear dependence of biological activity on log P for nonequilibrium conditions using a single lipid barrier between two aqueous compartments.82'83 In this model, both the more polar compounds and more hydrophobic compounds are limited in their abilities to reach the target receptor. Only a compound with an intermediate hydrophobicity or optimum log P will arrive at the receptor and wield its biological effects. The concept of an optimum log P or cutoff was not novel. A few years earlier, Ferguson had determined that the chemical potential of isonarcotic substances, acting via mostly physical mechanisms, were the same up to a critical cutoff point.84,85 In later years, other models were developed to more adequately reflect the complex interactions involving the partitioning of drugs in the random walk process. These models will be briefly addressed later.
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