**7. Abraham solvation parameter model: prediction of blood-to-tissue and gas-to-tissue partition coefficients**

Air-to-blood partitioning is a major determinant governing the uptake of chemical vapors into the blood and their subsequent elimination from blood to exhaled air. Air partitioning processes are becoming increasing more important in the pharmaceutical industry given the large numbers of drugs and vaccines that are now administered by inhalation aerosols and nasal delivery devices. Inhalation drug delivery is appealing given the large surface area for drug absorption, the high blood flow to and from the lung, and the absence of first pass metabolism that is characteristic of the lung. Inhalation drug delivery results in both a rapid clearance action and a rapid onset of therapeutic action, and a reduction in the number of undesired side effects. Eixarch and coworkers (2010) proposed the development of a pulmonary biopharmaceutical classification system (pBCS) that would classify drugs according to their ability to reside in the lung or to be transferred to the bloodstream. The classification scheme would need to consider factors associated with the lung's biology (metabolism, efflux transporters, clearance) and with the drug formulation/physicochemical properties (solubility, lipophilicity, protein binding, particle size, aerosol physics). Blood-totissue partitionings govern the distribution throughout the rest of the body once the drug has entered the bloodstream.

Abraham model correlations have been developed to describe the air-to-tissue and blood-totissue partition coefficients of drugs and volatile organic compounds (VOCs). The derived mathematical equations include:

*Muscle (*Abraham *et al.,* 2006c):

$$\begin{aligned} \text{log}\text{K}\_{\text{muscle/air}}\text{(in vitro)} &= -1.039 + 0.207\text{E} + 0.723\text{S} + 3.242\text{A} + 2.469\text{B} + 0.463\text{L} \\ \text{(N} &= 114, \text{ R}^2 = 0.944, \text{ SD} = 0.267, \text{ F} = 363) \end{aligned} \tag{26}$$

Prediction of Partition Coefficients and Permeability of Drug Molecules in Biological

**8. Abraham solvation parameter model: prediction of water-to-skin and blood-to-skin partition coefficients and skin permeability coefficients** 

Human skin is an important permeation barrier that controls the entry of chemicals into the body. The barrier properties of skin depend primarily on the outer skin cells, which are called the stratum corneum. The stratum corneum consists of multiple non-living layers of densely packed keratin-filled cells embedded in a lipid-rich extracellular matrix containing a mixture of ceramides, fatty acids, cholesterol and triglycerides (Monteiro-Riviere *et al*., 2001). The multiple layers are 7 – 16 micrometers in total thickness in most regions of the human body; however, in the palms of the hands and soles of the feet a much total layer thickness

N 196. R 0.938, SD 0.324, RMSE 0.319, F 572.8

N 127, R 0.91, SD 0.29, RMSE 0.286, F 242

N 155, R 0.34, RSME 0.332, F 474

logK human 1.18 0.39 0.97 3.80 2.69 0.41

logK rat 0.75 0.56 1.06 3.64 2.41 0.29

logK human or rat 1.069 0.456 1.083 3.738 2.580 0.376

reported by Abraham and coworkers (2005). For any fully characterized system/process (those with calculated values for the equation coefficients) further values of SP (see Eqns. 1 and 2) can be estimated for solutes with known values for the solute descriptors. Solute descriptors can be obtained by regression analysis of measured drug solubilities in organic solvents and measured water-to-solvent and organic solvent-to-organic solvent partition

**ESABL**

**ESABL**

**ESABL**

(37)

(38)

(39)

models

2

2

2

blood/air

blood/air

blood/air

coefficients as discussed above.

Systems with Abraham Model Solute Descriptors Derived from Measured Solubilities and… 119

needed as an indicator descriptor for carboxylic acids (**Ic** = 1 for carboxylic acids, **Ic** = 0 for noncarboxylic acid solutes) for the *in vivo* correlations involving drug molecules. The *in vivo* data sets included partition coefficient data for drug molecules such as nalidixic acid and valproic acid. No carboxylic acid solutes were contained in the *in vitro* data sets. The poor R2 statistics noted in several of the blood-to-tissue correlations are due, at least in part, to the small spread in the log P values and the increased experimental uncertainties as noted below. Each derived correlation was validated by training set and test set analyses. Based on the validation computations the derived correlations are expected to predict the log Ktissue/air and log Ptissue/blood values of additional compounds to within about 0.2 to 0.3 log units. As an informational note, the experimental data sets for the *in vitro* Abraham model correlations were determined using the equilibrium vial method. The gas-to-tissue partition coefficient of the VOC was calculated from the measured vapor phase composition in the headspace above the given tissue. The measured *in vitro* gas-to-tissue partition coefficients were converted to the corresponding blood-to-tissue values, Ptissue/blood values, through Eqn. 16. The Ptissue/blood include the experimental uncertainty in both the Ktissue/air and Pblood/air values. Should the *in vitro* experimental air-to-blood partitioning data not be available for the conversion, one can estimate the needed Pblood/air values from the three correlation

$$\begin{aligned} \text{log P}\_{\text{muscle/blood}} \left( \text{in vitro} \right) &= -0.185 \text{ - } 0.209 \text{E} - 0.593 \text{S} - 0.081 \text{A} - 0.168 \text{B} + 0.741 \text{V} \\ \left( \text{N} = 110 \text{, R}^2 = 0.537 \text{, SD} = 0.207 \text{, F} = 24 \right) \end{aligned} \tag{27}$$

$$\begin{aligned} \text{log P}\_{\text{muucle}/\text{blood}} \text{(in vivo)} \ 0.082 & - 0.059 \text{E} + 0.010 \text{S} - 0.248 \text{A} + 0.028 \text{B} + 0.110 \text{V} - 1.022 \text{lc} \\ \left( \text{N} = 60, \text{R}^2 = 0.745, \text{SD} = 0.253, \text{F} = 25.9 \right) \end{aligned} \tag{28}$$

*Fat* (Abraham and Ibrahim, 2006):

$$\begin{aligned} \text{logK}\_{\text{fat/air}} \text{(in vitro)} &= -0.052 + 0.051 \text{E} + 0.728 \text{S} + 1.783 \text{A} + 0.332 \text{B} + 0.743 \text{L} \\ \text{(N} &= 129 \text{, R}^2 = 0.958, \text{SD} = 0.194, \text{F} = 562.8 \text{)} \end{aligned} \tag{29}$$

$$\begin{aligned} \text{log P}\_{\text{fat/blood}} \text{(in vitro)} &= 0.474 + 0.016 \text{E} - 0.005 \text{S} - 1.577 \text{A} - 2.246 \text{B} + 1.560 \text{V} \\ \text{(N} &= 126, \text{ R}^2 = 0.847, \text{ SD} = 0.304, \text{F} = 132.7) \end{aligned} \tag{30}$$

$$\begin{aligned} \text{log P}\_{\text{fat/blood}} \text{(in vivo)} &= 0.077 + 0.249 \text{E} - 0.215 \text{ S} - 0.902 \text{A} - 1.523 \text{B} + 1.234 \text{V} - 1.013 \text{Ic} \\ \text{(N} &= 50, \text{R}^2 = 0.811, \text{SD} = 0.33 \text{F} = 30.7) \end{aligned} \tag{31}$$

*Liver* (Abraham *et al*., 2007a):

$$\begin{aligned} \text{log}\text{K}\_{\text{liver/air}}\text{(in vitro)} &= -0.943 + 0.836\mathbf{S} + 2.836\mathbf{A} + 2.081\mathbf{B} + 0.561\mathbf{L} \\ \text{(N} &= 124, \text{ R}^2 = 0.927, \text{ SD} = 0.256, \text{F} = 376.8) \end{aligned} \tag{32}$$

$$\begin{aligned} \text{log P}\_{\text{liver/blood}} \left( \text{in vitro} \right) &= -0.095 \text{ -- } 0.366 \text{S} - 0.357 \text{A} - 0.180 \text{B} + 0.790 \text{V} \\ \left( \text{N} = 125, \text{R}^2 = 0.583, \text{SD} = 0.228, \text{F} = 41.9 \right) \end{aligned} \tag{33}$$

$$\begin{aligned} \text{log P}\_{\text{liver/blood}} \left( \text{in } vivo \right) &= 0.292 \ - 0.296 \mathbf{S} - 0.334 \mathbf{A} + 0.181 \mathbf{B} + 0.337 \mathbf{V} - 0.597 \mathbf{I} \mathbf{c} \\ \left( \mathbf{N} = 85, \ R^2 = 0.522, \ \text{SD} = 0.420, \ \text{F} = 17.3 \right) \end{aligned} \tag{34}$$

*Lung* (Abraham *et al*., 2008a):

$$\begin{aligned} \text{log}\text{K}\_{\text{lung/air}}\text{(in vitro)} &= -1.250 + 0.639\text{E} + 1.038\text{S} + 3.661\text{A} + 3.041\text{B} + 0.420\text{L} \\ \text{(N}=44, \text{R}^2 = 0.968, \text{SD} = 0.250, \text{F} = 231.8) \end{aligned} \tag{35}$$

$$\begin{aligned} \text{R } \log \text{P}\_{\text{lung/blood}} \left( \dot{m} \,\, \text{vitro} \right) &= -0.143 \, - \, 0.383 \mathbf{B} + 0.308 \mathbf{V} \\ \left( \text{N} = 43, \, \text{R}^2 = 0.264, \, \text{SD} = 0.190, \, \text{F} = 7.2 \right) \end{aligned} \tag{36}$$

Correlations obtained by regression analysis of experimental drug partition coefficient data are denoted as "*in vivo*", and correlations pertaining to volatile organic compound partitioning are indicated as "*in vitro*". Human and rat partition coefficient data were combined into data set used in the regression analyses. The independent variable **Ic** was

log P 0.185 – 0.209 – 0.593 – 0.081 – 0.168 0.741

log P 0.082 – 0.059 0.010 – 0.248 0.028 B 0.110 – 1.022

*in vivo*

logK 0.052 0.051 0.728 1.783 0.332 0.743

log P 0.474 0.016 – 0.005 – 1.577 – 2.246 1.560

log P 0.077 0.249 – 0.215 – 0.902 – 1.523 1.234 – 1.013

logK 0.943 0.836 2.836 2.081 0.561

log P 0.095 – 0.366 – 0.357 – 0.180 0.730

log P 0.292 – 0.296 – 0.334 0.181 0.337 – 0.597

logK 1.250 0.639 1.038 3.661 3.041 0.420

log P 0.143 – 0.383 0.308

Correlations obtained by regression analysis of experimental drug partition coefficient data are denoted as "*in vivo*", and correlations pertaining to volatile organic compound partitioning are indicated as "*in vitro*". Human and rat partition coefficient data were combined into data set used in the regression analyses. The independent variable **Ic** was

**B V**

*in vitro*

*in vitro*

*in vitro*

*in vivo*

N 124, R 0.927, SD 0.256, F 376.8

*in vivo*

N 125, R 0.583, SD 0.228, F 41.9

*in vitro*

*in vitro*

*in vitro*

**ESABV**

**ESA V Ic**

**ESA BL**

**ESABV**

**E S A B V Ic**

**SABL**

**SABV**

**S A B V Ic**

**ESAB L**

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

(35)

(36)

2

liver/air

liver/blood

2

2

2

2

2

2

2

2

N 85, R 0.522, SD 0.420, F 17.3

 N 43, R 0.264, SD 0.190, F 7.2 *in vitro* 

N 44, R 0.968, SD 0.250, F 231.8

N 50, R 0.811, SD 0.33 F 30.7

2

muscle/blood

muscle/blood

*Fat* (Abraham and Ibrahim, 2006):

fat/air

fat/blood

fat/blood

*Liver* (Abraham *et al*., 2007a):

liver/blood

*Lung* (Abraham *et al*., 2008a):

lung/blood

lung/air

N 110, R 0.537, SD 0.207, F 24

N 126, R 0.847, SD 0.304, F 132.7

N 129, R 0.958, SD 0.194, F 562.8

N 60, R 0.745, SD 0.253, F 25.9

needed as an indicator descriptor for carboxylic acids (**Ic** = 1 for carboxylic acids, **Ic** = 0 for noncarboxylic acid solutes) for the *in vivo* correlations involving drug molecules. The *in vivo* data sets included partition coefficient data for drug molecules such as nalidixic acid and valproic acid. No carboxylic acid solutes were contained in the *in vitro* data sets. The poor R2 statistics noted in several of the blood-to-tissue correlations are due, at least in part, to the small spread in the log P values and the increased experimental uncertainties as noted below. Each derived correlation was validated by training set and test set analyses. Based on the validation computations the derived correlations are expected to predict the log Ktissue/air and log Ptissue/blood values of additional compounds to within about 0.2 to 0.3 log units. As an informational note, the experimental data sets for the *in vitro* Abraham model correlations were determined using the equilibrium vial method. The gas-to-tissue partition coefficient of the VOC was calculated from the measured vapor phase composition in the headspace above the given tissue. The measured *in vitro* gas-to-tissue partition coefficients were converted to the corresponding blood-to-tissue values, Ptissue/blood values, through Eqn. 16. The Ptissue/blood include the experimental uncertainty in both the Ktissue/air and Pblood/air values. Should the *in vitro* experimental air-to-blood partitioning data not be available for the conversion, one can estimate the needed Pblood/air values from the three correlation models

$$\begin{aligned} \text{logK}\_{\text{blood/air}} \text{(human)} &= -1.18 + 0.39\text{E} + 0.97\text{S} + 3.80\text{A} + 2.69\text{B} + 0.41\text{L} \\ \text{(N} &= 155, \text{R}^2 = 0.34, \text{RMSE} = 0.332, \text{F} = 474) \end{aligned} \tag{37}$$

$$\log \text{K}\_{\text{blood}, \text{air}} \left( \text{rat} \right) = -0.75 + 0.56 \text{E} + 1.06 \text{S} + 3.64 \text{A} + 2.41 \text{B} + 0.29 \text{L} \tag{38}$$

$$\left( \text{N} = 127, \text{R}^2 = 0.91, \text{SD} = 0.29, \text{RMSE} = 0.286, \text{F} = 242 \right)$$

$$\begin{aligned} \text{logK}\_{\text{blood/air}} \text{(human or rat)} &= -1.069 + 0.456 \text{E} + 1.083 \text{S} + 3.738 \text{A} + 2.580 \text{B} + 0.376 \text{L} \\ \text{(N} &= 196. R} = 0.938, \text{SD} = 0.324, \text{RMSE} = 0.319, \text{F} = 572.8 \text{)} \end{aligned} \tag{39}$$

reported by Abraham and coworkers (2005). For any fully characterized system/process (those with calculated values for the equation coefficients) further values of SP (see Eqns. 1 and 2) can be estimated for solutes with known values for the solute descriptors. Solute descriptors can be obtained by regression analysis of measured drug solubilities in organic solvents and measured water-to-solvent and organic solvent-to-organic solvent partition coefficients as discussed above.

### **8. Abraham solvation parameter model: prediction of water-to-skin and blood-to-skin partition coefficients and skin permeability coefficients**

Human skin is an important permeation barrier that controls the entry of chemicals into the body. The barrier properties of skin depend primarily on the outer skin cells, which are called the stratum corneum. The stratum corneum consists of multiple non-living layers of densely packed keratin-filled cells embedded in a lipid-rich extracellular matrix containing a mixture of ceramides, fatty acids, cholesterol and triglycerides (Monteiro-Riviere *et al*., 2001). The multiple layers are 7 – 16 micrometers in total thickness in most regions of the human body; however, in the palms of the hands and soles of the feet a much total layer thickness

Prediction of Partition Coefficients and Permeability of Drug Molecules in Biological

correlation coefficient decreases to R2 = 0.608.

stratum corneum per unit time (J) is given by

flux JSS and the donor concentration, Cdonor.

Systems with Abraham Model Solute Descriptors Derived from Measured Solubilities and… 121

The e · **E** and s · **S** terms were not statistically significant and were eliminated from the final derived correlation model. The poor R2 statistics for Eqn. 41 is due, at least in part, to the small spread in the values of log Pskin, from log Pskin = -0.82 to log Pskin = 1.61, for a range of only 2.43 log units. Carboxylic acids were found to be systematically retained in blood or plasma more than calculated. An indicator descriptor, **Iacid**, was needed to describe the log Pskin data of solutes containing a carboxylic acid functional group. The **Iacid** descriptor equals unity for carboxylic acid solutes, and takes the value of **Iacid** = 0 for all other compounds. The second indicator descriptor in Eqn. 41 was needed to combine the rat skin (**Irabbit** = 0) and rabbit skin (**Irabbit**) partitioning data into a single correlation model. The 0.059 **Irabbit** term amounts to a 0.059 log unit offset, which is likely less than the experimental uncertainty in the measured log Pskin data. If the 0.059 **Irabbit** term is omitted, the squared

Theoretical models of passive diffusion are based on Fick's law of diffusion and the conversation of particle numbers. Fick's law of diffusion states that a chemical diffuses from a region of higher concentration to a region of lower concentration with a magnitude that is directly proportional to the chemical's concentration gradient. When applied to transstratum corneum diffusion, the amount of chemical passing through a unit area of the

*K DC p sc* , *<sup>J</sup> <sup>h</sup>*

where Kp,sc is the chemical's solvent-to-stratum corneum partition coefficient, D represents the chemical's diffusivity in the stratum corneum lipid matrix and h is the apparent skin thickness (*i.e*., the diffusion pathlength). Under the assumption of constant donor concentration and sink conditions (zero receptor phase concentration) Eqn. 42 simplifies to

*p*,*sc donor*

*K DC*

The permeability coefficient, kp, is the coefficient of proportionality between the steady-state

Skin permeability experiments are generally performed *in vitro* using a Franz diffusion cell (shown in Figure 8). A freshly excised skin sample is mounted on the receptor compartment of the Franz cell with the stratum corneum facing upwards into the donor compartment and the dermis facing the receptor compartment. The latter compartment is filled with the receptor solution (often a phosphate saline solution buffered at pH of 7.4), and maintained at a constant temperature of 37 oC with a water jacketed cell under constant stirring. The donor compartment is filled with the vehicle solution containing the dissolved chemical of interest. At appropriate time intervals, aliquots of the receptor medium are withdrawn for analysis, and immediately replaced with an equal volume of fresh medium. Alternative diffusion cell designs and mathematical procedures for calculating the drug's diffusivity and permeability coefficient from the experimental permeation results are described in greater detail elsewhere (Friend, 1992; Hathout *et al*., 2010). For *in vitro* skin penetration studies, the skin retention of a drug can be assessed by the use of radiolabeled drugs (usually carbon-14 or tritium labeled). Skin samples should be exposed to the drug for no more than a maximum of 24 hours because of deterioration of skin integrity with time.

*SS*

(42)

*<sup>J</sup> <sup>h</sup>* (43)

of 400 – 600 micrometers is found (Holbrook and Odland, 1974) For a chemical to be absorbed into the body after dermal exposure, it must first dissolve in the stratum corneum and then diffuse through the remaining epidermis sub-layers and into the dermis layer, from where it will eventually enter the blood stream. Passive diffusion is the mechanism by which chemicals move through the stratum corneum. Passage through the remaining sublayers of the skin is more rapid.

Penetration of a compound into the skin is controlled by the compound's chemical structure and physicochemical properties. Lipophilicity and hydrogen-bonding character play a major role in a compound's skin absorption profile. In general, substances possessing the greater lipophilicity are more readily absorbed by the skin than compounds with lesser lipophilicity. Dermal absorption generally increases with increasing water-to-octanol partition coefficient from log POtOH/water = -1 to log POtOH/water = 3.5. Highly lipophilic compounds (those with log POtOH/water > 5) pass easily through the stratum corneum, but are generally too water-insoluble to pass through the remaining epidermis sub-layers to enter the blood stream. There has been increasing experimental evidence that ionized species can contribute to transdermal absorption (Netzlaff *et al*., 2006; Abraham and Martins, 2004; Michaels *et al*., 1975). When the penetrating compound can exist in both ionized and unionized forms, it is the unionized form that penetrates faster through the lipid regions. Some contribution of the ionized form to the overall permeability, however, is expected. The solubilizing vehicle and formulation ingredients can alter the skin penetration of a compound by affecting the barrier properties of the skin by a range of mechanisms including hydration, delipidization, fluidization and desmosome disruption in the stratum corneum, or by changing the partitioning of the compound into the stratum corneum.

Skin partitioning is important in the pharmaceutical industry as many medications are applied topically to the skin in ointments, in creams, in lotions and gels, and in skin patches. Once applied, the medication often needs to find its way into the blood system for delivery to the desired target site. Abraham and Martins (2004) developed a mathematical correlation between the water-to-skin partition coefficient, Ksc, and the Abraham solute descriptors

$$\begin{aligned} \text{Log K}\_{\text{sc}} &= 0.341 + 0.341 \text{E} - 0.206 \text{S} - 0.024 \text{A} - 2.178 \text{B} + 1.850 \text{V} \\ \text{(N} &= 45, \text{SD} = 0.216, \text{R}^2 = 0.926, \text{F} = 97) \end{aligned} \tag{40}$$

based on an experimental database containing 45 solutes, including several linear alcohols (*e.g*. methanol through 1-decanol) and several fairly large steroidal molecules (*e.g*. testosterone, progesterone, hydrocortisone, corticosterone, and aldosterone) and steroid esters (*e.g*. hydrocortisone-21 acetate, hydrocortisone-21 pentanoate, cortisone-21 acetate, cortisone-21 octanoate). Careful examination of Eqn. 40 reveals that the water-to-skin partition coefficient increases with increasing solute size, and decreasing with increasing solute polarity and solute hydrogen-bonding character.

Abraham and Ibrahim (2007) compiled experimental data on the distribution coefficients of drugs from blood or plasma to rat skin and rabbit skin. The authors analyzed the experimental log Pskin data in accordance with Eqn. 1 of the Abraham model

$$\begin{aligned} \text{log P}\_{\text{skin}} &= -0.253 - 0.189 \text{A} - 0.620 \text{B} + 0.713 \text{V} - 0.683 \text{I}\_{\text{acid}} + 0.059 \text{I}\_{\text{rabbit}}\\ \text{\(N = 59, SD = 0.26, R}^2 &= 0.733, \text{F = 29\)} \end{aligned} \tag{41}$$

of 400 – 600 micrometers is found (Holbrook and Odland, 1974) For a chemical to be absorbed into the body after dermal exposure, it must first dissolve in the stratum corneum and then diffuse through the remaining epidermis sub-layers and into the dermis layer, from where it will eventually enter the blood stream. Passive diffusion is the mechanism by which chemicals move through the stratum corneum. Passage through the remaining sub-

Penetration of a compound into the skin is controlled by the compound's chemical structure and physicochemical properties. Lipophilicity and hydrogen-bonding character play a major role in a compound's skin absorption profile. In general, substances possessing the greater lipophilicity are more readily absorbed by the skin than compounds with lesser lipophilicity. Dermal absorption generally increases with increasing water-to-octanol partition coefficient from log POtOH/water = -1 to log POtOH/water = 3.5. Highly lipophilic compounds (those with log POtOH/water > 5) pass easily through the stratum corneum, but are generally too water-insoluble to pass through the remaining epidermis sub-layers to enter the blood stream. There has been increasing experimental evidence that ionized species can contribute to transdermal absorption (Netzlaff *et al*., 2006; Abraham and Martins, 2004; Michaels *et al*., 1975). When the penetrating compound can exist in both ionized and unionized forms, it is the unionized form that penetrates faster through the lipid regions. Some contribution of the ionized form to the overall permeability, however, is expected. The solubilizing vehicle and formulation ingredients can alter the skin penetration of a compound by affecting the barrier properties of the skin by a range of mechanisms including hydration, delipidization, fluidization and desmosome disruption in the stratum corneum, or by changing the partitioning of the compound into the stratum corneum. Skin partitioning is important in the pharmaceutical industry as many medications are applied topically to the skin in ointments, in creams, in lotions and gels, and in skin patches. Once applied, the medication often needs to find its way into the blood system for delivery to the desired target site. Abraham and Martins (2004) developed a mathematical correlation between the water-to-skin partition coefficient, Ksc, and the Abraham solute descriptors

N 45, SD 0.216, R 0.926, F 97

solute polarity and solute hydrogen-bonding character.

N 59, SD 0.26, R 0.733, F 29

2

2

experimental log Pskin data in accordance with Eqn. 1 of the Abraham model

Log K 0.341 0.341 – 0.206 – 0.024 – 2.178 1.850

based on an experimental database containing 45 solutes, including several linear alcohols (*e.g*. methanol through 1-decanol) and several fairly large steroidal molecules (*e.g*. testosterone, progesterone, hydrocortisone, corticosterone, and aldosterone) and steroid esters (*e.g*. hydrocortisone-21 acetate, hydrocortisone-21 pentanoate, cortisone-21 acetate, cortisone-21 octanoate). Careful examination of Eqn. 40 reveals that the water-to-skin partition coefficient increases with increasing solute size, and decreasing with increasing

Abraham and Ibrahim (2007) compiled experimental data on the distribution coefficients of drugs from blood or plasma to rat skin and rabbit skin. The authors analyzed the

log P 0.253 – 0.189 – 0.620 0.713 – 0.683 0.059

**ABVI I acid rabbit** (41)

**ESABV**

(40)

layers of the skin is more rapid.

sc

skin

The e · **E** and s · **S** terms were not statistically significant and were eliminated from the final derived correlation model. The poor R2 statistics for Eqn. 41 is due, at least in part, to the small spread in the values of log Pskin, from log Pskin = -0.82 to log Pskin = 1.61, for a range of only 2.43 log units. Carboxylic acids were found to be systematically retained in blood or plasma more than calculated. An indicator descriptor, **Iacid**, was needed to describe the log Pskin data of solutes containing a carboxylic acid functional group. The **Iacid** descriptor equals unity for carboxylic acid solutes, and takes the value of **Iacid** = 0 for all other compounds. The second indicator descriptor in Eqn. 41 was needed to combine the rat skin (**Irabbit** = 0) and rabbit skin (**Irabbit**) partitioning data into a single correlation model. The 0.059 **Irabbit** term amounts to a 0.059 log unit offset, which is likely less than the experimental uncertainty in the measured log Pskin data. If the 0.059 **Irabbit** term is omitted, the squared correlation coefficient decreases to R2 = 0.608.

Theoretical models of passive diffusion are based on Fick's law of diffusion and the conversation of particle numbers. Fick's law of diffusion states that a chemical diffuses from a region of higher concentration to a region of lower concentration with a magnitude that is directly proportional to the chemical's concentration gradient. When applied to transstratum corneum diffusion, the amount of chemical passing through a unit area of the stratum corneum per unit time (J) is given by

$$J = -\frac{K\_{p,sc}D\Delta C}{\hbar} \tag{42}$$

where Kp,sc is the chemical's solvent-to-stratum corneum partition coefficient, D represents the chemical's diffusivity in the stratum corneum lipid matrix and h is the apparent skin thickness (*i.e*., the diffusion pathlength). Under the assumption of constant donor concentration and sink conditions (zero receptor phase concentration) Eqn. 42 simplifies to

$$J\_{SS} = \frac{K\_{p,sc} D C\_{droor}}{h} \tag{43}$$

The permeability coefficient, kp, is the coefficient of proportionality between the steady-state flux JSS and the donor concentration, Cdonor.

Skin permeability experiments are generally performed *in vitro* using a Franz diffusion cell (shown in Figure 8). A freshly excised skin sample is mounted on the receptor compartment of the Franz cell with the stratum corneum facing upwards into the donor compartment and the dermis facing the receptor compartment. The latter compartment is filled with the receptor solution (often a phosphate saline solution buffered at pH of 7.4), and maintained at a constant temperature of 37 oC with a water jacketed cell under constant stirring. The donor compartment is filled with the vehicle solution containing the dissolved chemical of interest. At appropriate time intervals, aliquots of the receptor medium are withdrawn for analysis, and immediately replaced with an equal volume of fresh medium. Alternative diffusion cell designs and mathematical procedures for calculating the drug's diffusivity and permeability coefficient from the experimental permeation results are described in greater detail elsewhere (Friend, 1992; Hathout *et al*., 2010). For *in vitro* skin penetration studies, the skin retention of a drug can be assessed by the use of radiolabeled drugs (usually carbon-14 or tritium labeled). Skin samples should be exposed to the drug for no more than a maximum of 24 hours because of deterioration of skin integrity with time.

Prediction of Partition Coefficients and Permeability of Drug Molecules in Biological

contributions, both in the sense of lowering log kp.

solubility and partition coefficient data.

**9. Conclusion** 

**10. References** 

246.

22 (2),73-83.

Systems with Abraham Model Solute Descriptors Derived from Measured Solubilities and… 123

where, kp, kp,ionic, and kp,neutral represent the overall permeation coefficient, that due to the ionic species, and that due to the neutral species; fionic and fneutral denote the fraction of ionic and neutral species at a given pH. For ionizable acids the skin permeability coefficient of the neutral molecule, kp,neutral, was so much larger than the skin permeability coefficient of the ionic form, kp,ionic, that the experimental unadjusted values of kp was adjusted to give kp,neutral from the fraction of the neutral form present under the experimental conditions of pH. For ionizable bases the ratio of kp,neutral to kp,ionic was assumed to be 17.5, and this value was used to obtain kp,neutral values from experimental unadjusted values of kp. If the experimental pH is near to the basic pKa, such an adjustment will be very close to the adjustment that assumes negligible permeation of ionizable species. But as the difference in (pH − pKa) becomes larger, the adjustment will be smaller than that of negligible permeation of ionic species. To account for the temperature differences, the authors adjusted experimental log kp values by 0.20 units from 32 °C to 37 °C, and by 0.48 units from 25 °C to 37 °C. The main factors that influence log kp are hydrogen bond basicity (b · B term) that decreases log kp, and solute volume (v · **V** term) that increases log kp. Solute dipolarity/polarizability (s · S term) and hydrogen bond acidity (a · **A** term) make minor

The Abraham solvation parameter model provides an in silico method for estimating ADMET properties of potential drug molecules in the early stages of drug discovery. To date mathematical expressions have been reported for predicting water-to-organic solvent partition coefficients and solubilities in more than 70 organic solvents, air-to-tissue and blood-to-tissue partition coefficients for 5 human and rat tissues, water-to-human skin and blood-to-rat/rabbit skin partitions, human skin permeability coefficients, and rat (Zhao *et al*., 2003) and human (Zhao *et al*., 2002) intestinal absorption. Expressions are also available for estimating Draize rabbit eye test scores for pure liquids and eye irritation thresholds in humans (Abraham *et al*., 2003), odor detection thresholds and nasal pungency of volatile organic compounds (VOCs) (Abraham *et al*., 2007b), and the minimum alveolar concentration (MAC) for inhalation anesthetics in rats (Abraham *et al.*, 2008b). The number of derived Abraham model correlations is expected in future years as more experimental data becomes available. Predictive applications require as input parameters the numerical values of the drug candidate's solute descriptors, which are easily calculable from measured

Abraham, M. H. & McGowan, J. C. (1987) The use of characteristic volumes to measure

Abraham, M. H. (1993a) Scales of solute hydrogen-bonding: their construction and

Abraham, M. H. (1993b) Application of solvation equations to chemical and biochemical

Abraham, M. H.; Hassanisadi, M.; Jalali-Heravi, M.; Ghafourian, T.; Cain, W. S. & Cometto-

processes. *Pure and Applied Chemistry* 65 (12), 2503-2512.

cavity terms in reversed phase liquid chromatography. *Chromatographia* 23 (4) 243–

application to physicochemical and biochemical processes. *Chemical Society Reviews*

Muniz, J. E. (2003) Draize rabbit eye test compatibility with eye irritation thresholds

Fig. 8. Franz diffusion cell used to measure skin permeability coefficients

The parallel artificial membrane permeability assay (PAMPA) has been suggested as a high throughput screening method for rapid determination of passive transport permeability in connection with gastrointestinal (GI) absorption (Sugano *et al*., 2002), blood-brain barrier penetration (Mensch *et al*., 2010 and skin permeation (Ottaviani *et al*., 2006). In the PAMPA method a 96-well filter plate coated with a liquid membrane is used to separate the donor and receptor compartments. Artificial membrane selection depends on the transport property to be determined. Ottaviani *et al*. (2006) found a reasonably accurate mathematical correlation between human skin permeability coefficient, kp, and the effective permeability coefficient, k*eff*, for a set of 31 compounds

$$\begin{aligned} \log \mathbf{k}\_{\mathbf{p}} &= 1.34 \log \mathbf{k}\_{\rm eff} + 0.28\\ \mathbf{(N = 31, SD = 0.42, R ^2 = 0.81, F = 31)} \end{aligned} \tag{44}$$

tested through an artificial membrane consisting of 70 % silicone and 30 % isopropyl myristate. The authors further noted that presence of isopropyl myristate as only a hydrogen-bond acceptor group in the artificial membrane was in accord with previous results demonstrating that stratum corneum lipids were better hydrogen-bond acceptors than hydrogen-bond donors.

Abraham and Martins (2004) reported an Abraham model correlation for human skin permeability coefficients from aqueous solution, kp,

$$\begin{aligned} \text{Log k}\_p(\text{cm/s}) &= -5.426 \text{ -- } 0.106 \text{E} - 0.473 \text{E} - 0.473 \text{A} - 3.000 \text{B} + 2.296 \text{V} \\ \text{(N} &= 119, \text{SD} = 0.461, \text{R}^2 = 0.832, \text{F} = 112) \end{aligned} \tag{45}$$

based on a database containing 119 experimental values at a common temperature of 37 oC. The authors adjusted the experimental data for ionization by assuming that the measured permeability coefficient was a simple addition of terms in Eqn. 46

$$\mathbf{k}\_{\mathbf{p}} = \mathbf{f}\_{\text{neutral}} \, \mathbf{k}\_{\mathbf{p}, \text{neutral}} + \mathbf{f}\_{\text{ionic}} \mathbf{k}\_{\mathbf{p}, \text{ionic}} \tag{46}$$

where, kp, kp,ionic, and kp,neutral represent the overall permeation coefficient, that due to the ionic species, and that due to the neutral species; fionic and fneutral denote the fraction of ionic and neutral species at a given pH. For ionizable acids the skin permeability coefficient of the neutral molecule, kp,neutral, was so much larger than the skin permeability coefficient of the ionic form, kp,ionic, that the experimental unadjusted values of kp was adjusted to give kp,neutral from the fraction of the neutral form present under the experimental conditions of pH. For ionizable bases the ratio of kp,neutral to kp,ionic was assumed to be 17.5, and this value was used to obtain kp,neutral values from experimental unadjusted values of kp. If the experimental pH is near to the basic pKa, such an adjustment will be very close to the adjustment that assumes negligible permeation of ionizable species. But as the difference in (pH − pKa) becomes larger, the adjustment will be smaller than that of negligible permeation of ionic species. To account for the temperature differences, the authors adjusted experimental log kp values by 0.20 units from 32 °C to 37 °C, and by 0.48 units from 25 °C to 37 °C. The main factors that influence log kp are hydrogen bond basicity (b · B term) that decreases log kp, and solute volume (v · **V** term) that increases log kp. Solute dipolarity/polarizability (s · S term) and hydrogen bond acidity (a · **A** term) make minor contributions, both in the sense of lowering log kp.
