Abraham equation

Multiparametric models of solvent effects are the IMF and the Kamlet-Taft-Abraham equations. 

Our group elected to use Abraham s LFER model as our base equation becanse it is representative of the dermal QSPR approaches presently available (Abraham and Martins, 2004). Preliminary analyses applying 16 different LFER equations reviewed by Geinoz et al. (2004) to the entire PSFT (288 treatment combinations) and IPPSF data set (32 treatment combinations) demonstrated a superior fit of our data set to the Abraham equation compared to most other models reviewed. It must be stressed that the purpose of this research was not to identify the optimal LFER for predicting dermal permeation or to validate that this model is predictive of dermal absorption. Rather, we selected this model because it best described the available data and is widely accepted by the scientific community. 

The hoping rate (also treated as attempt frequency to hoping) from localized site i to j with energy Ei and Ej can be written by the Miller-Abraham equation... 

Integral equations have been developed for inhomogeneous fluids. One such integral equation is that of Henderson, Abraham and Barker (HAB) [88] who assumed the OZ equation for a mixture and regarded the surface as a giant particle. For planar geometry they obtained... 

The group of Abraham investigated many other solvent systems and derived equations to predict logP from structure for these solvent systems, which are also commercially available [94]. 

H-bonding is an important, but not the sole, interatomic interaction. Thus, total energy is usually calculated as the sum of steric, electrostatic, H-bonding and other components of interatomic interactions. A similar situation holds with QSAR studies of any property (activity) where H-bond parameters are used in combination with other descriptors. For example, five molecular descriptors are applied in the solvation equation of Kamlet-Taft-Abraham excess of molecular refraction (Rj), which models dispersion force interactions arising from the polarizability of n- and n-electrons the solute polarity/polarizability (ir ) due to solute-solvent interactions between bond dipoles and induced dipoles overall or summation H-bond acidity (2a ) overall or summation H-bond basicity (2(3 ) and McGowan volume (VJ [53] ... 

Abraham s equation (Eq. 4) is the most common tool to factorize log (called simply log P in Eq. 4) ... 

As the solute descriptors (E, S, A, B and V) represent the solute influence on various solute-solvent phase interachons, the regression coefficients e, s, a, h and V correspond to the complementary effect of the solvent phases on these interactions. As an example, consider the product aA in Eq. (4). Since A is the H-bond acidity of the solute, a is the H-bond basicity of the system. In other words, the intermolecular forces discussed in Sections 12.1.1.2 and 12.1.1.3 are present in all Abraham s log P factorization equations, with the exception of those interactions involving ions. This is the reason why Abraham s equahons are valid for neutral species only. 

We have not yet introduced the influence of the presence of point charges on the lipophilicity of a chemical. Nevertheless, Sections 12.1.1.2 and 12.1.1.3 do warn that the lipophilic behavior of an ionized molecule might be very different from that of its parent neutral compound. Indeed, in order to investigate the balance of forces governing the lipophiUcity of ionized species we must do without Abraham s equations, since they do not exist when ions are considered. Recently, Abraham et al. also demonstrated what had long been perceived intuitively - descriptors for ions are not the same as those for nonelectrolytes [12]. 

Du, C. M., Valko, K., Bevan, C., Reynolds, D., Abraham, M. H. Rapid gradient RP-HPLC method for lipophilicity determination a solvation equation based comparison with isocratic methods. Anal. Chem. 1998, 70, 4228-4234. 

The HINT method [49-51] refers to the idea of Abraham and Leo [52] that hydro-phobic fragment constants, reduced to atomic values, could be used to evaluate interactions between small and large molecules. The interaction energy term by scores atom-atom interactions ij) within or between the molecules using the following equation ... 

Kamlet, M. J., Doherty, R. M., Carr, P., Abraham, M. H., Marcus, Y Taft, R. W. Linear solvation energy relationships. 46. An improved equation for correlation and prediction of octanol-water partition coefficients of organic non-electrolytes (including strong hydrogen bond donor solutes)./. Phys. Chem. 1988, 92, 5244-5255. 

Abraham, M. H., Chadha, H. S., Applications of a solvation equation to drug transport properties, in Lipophilicity in Drug Action and Toxicology. Pliska, V., Testa, B.,... 

The coefficients of descriptors A, AB and S are not statistically significant, but the terms are left in the equation for comparison with Abraham and Le s [8] water-solubility equation, modified here to conform to the mathematical conventions used in this chapter ... 

Clearly, Eq. (5) and Eq. (6) are not exactly the same. In Abraham and Le s equation [8], the H-bond and volume terms have much larger coefficients than the corresponding ones in Eq. (5). This helps to explain why Eq. (6) did not predict the solubilities of the 58 compounds well. 

A second, more detailed approach was then investigated. From Scheme 1 it is clear that in order to calculate the solubility product, KB, it is necessary to know both the free energy of solution, AG9sol(A+,X ) and the lattice energy, Uc. Abraham s recent developments [9] of the Born Equation [10] make it possible to calculate AG9sol to within 10 kj mol1 from the ionic radii a+, a ), but it is not possible to calculate Uc... 

Kamlet M. J., Abboud J.-L., Abraham M. H. and Taft R. W. (1983) Linear Solvation Energy Relationships. 23. A Comprehensive Collection of the Solvatochromic Parameters, 7i, a, and ft, and Some Methods for Simplifying the Generalized Solvatochromic Equation, J. Org. Chem. 48, 2877-2887. 

One of the most comprehensive hydrogen-bonding scales is due to Abraham and his coworkers50, who have derived the general solvation equation 651... 

Recently68,69 Abraham and coworkers have applied equation 6 to the correlation of several physico-chemical and biological phenomena in binary systems. These include solvent-water partition coefficients70,71, HPLC capacity factors53,72 and the distribution... 

Kamlet MJ, Abboud JLM, Abraham MH, Taft RW (1983) Linear solvation energy relationships. 23. A comprehensive collection of the solvatochromic parameters, jr, a, and /3, and some methods for simplifying the generalized solvatochromic equation. J Org Chem 48 2877-2887. 

Abraham et al. [2] published a number of papers in which they analyzed Young s data set using MLR and gave a general solvation equation in which various solvent-solute interactions were described by solute descriptors and equation coefficients (Eq. 16)... 

Abraham s solute descriptors to yield a predictive regression equation. Further, the solute dipolarity/polarizability, hydrogen bond acidity, and hydrogen bond basicity were found to favor blood and solute size favor brain. 

For modeling the BBB penetration, authors used Abraham s data set of 57 compounds as the training set. The test set consisted of 13 compounds, 7 of which were taken from Abraham s data set and 6 from the data set of Lombardo and workers. In addition to the lipoaffinity descriptor, the other descriptors used by them include molecular weight and TPSA. Two models were developed one based on stepwise MLR and the other one based on ANN. To test the performance of different descriptors, they first carried out a simple LR of the 55 training set compounds (two outliers were removed) using TPSA as the only descriptor (Eq. 41). The equation was comparable to Clark s model (Eq. 33). 

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