**5. Calculation of Abraham solute descriptors from measured solubility and partition coefficient data**

The application of Eqn. 1 and Eqn. 2 requires a knowledge of the descriptors (or properties) of the solutes: **E**, **S**, **A**, **B**, **V** and **L**. The descriptors **E** and **V** are quite easily obtained. **V** can be calculated from atom and bond contributions as outlined previously (Abraham and McGowan, 1987). The atom contributions are in Table 6; note that they are in cm3 mol -1. The

Prediction of Partition Coefficients and Permeability of Drug Molecules in Biological

Pb 43.44 Bi 42.19

Cl 20.95 A 1.90 Br 26.21 Kr 2.46 I 34.53 Xe 3.29 Rn 3.84 Table 6. Atom contributions to the McGowan volume, in cm3 mol-1

Table 7. Solubility ratios for trimethoprim, as log (ratio)

Systems with Abraham Model Solute Descriptors Derived from Measured Solubilities and… 111

Exactly the same procedure is adopted if actual partition coefficients are experimentally available, rather than solubilities. The relevant equations are now those in Table 3 and Table 4. Of course if both solubilities and actual partition coefficients have both been experimentally determined, a combination of equations from Tables 1 and 2 and from Tables 3 and 4 can be used. Even though partition coefficients refer to partition into wet solvents, descriptors obtained from partition coefficients using equations in Table 3 and Table 4 can still be used to predict solubility ratios and solubilities in dry solvents for all the solvents listed in Table 1.

> C 16.35 N 14.39 O 12.43 Si 26.83 P 24.87 S 22.91 Ge 31.02 As 29.42 Se 27.81 Sn 39.35 Sb 37.74 Te 36.14

> H 8.71 He 6.76 B 18.32 F 10.48 Ne 8.51 Hg 34.00

> > Water-to-solvent calc obs Methanol 1.35 1.48 Ethanol 0.98 0.94 Propanol 0.75 0.80 Butanol 0.53 0.68 2-Propanol 0.61 0.51 2-Butanol 0.65 0.62 Tetrahydrofuran 1.18 1.02 Propanone 0.90 0.94 Gas to water 14.48 14.49

> > Gas-to-solvent calc obs Methanol 15.81 15.97 Ethanol 15.48 15.43 Propanol 15.23 15.29 Butanol 15.02 15.18 2-Propanol 15.14 15.01 2-Butanol 15.18 15.11 Tetrahydrofuran 15.70 15.51 Propanone 15.34 15.43 Gas to water 14.53 14.49

bond contribution is 6.56 cm3 mol -1 for each bond, no matter whether single, double, or triple, to be subtracted. For complicated molecules it is time consuming to count the number of bonds, Bn, but this can be calculated from the algorithm given by Abraham (1993a)

$$\mathbf{R}\mathbf{n} = \mathbf{N}\mathbf{t} \cdot \mathbf{1} + \mathbf{R} \tag{13}$$

where Nt is the total number of atoms in the molecule and R is the number of rings. Once **V** is available, **E** can be obtained from the compound refractive index at 20oC. If the compound is not liquid at room temperature or if the refractive index is not known the latter can be calculated using the freeware software of Advanced Chemistry Development (ACD). An Excel spreadsheet for the calculation of **V** and **E** from refractive index is available from the authors. Since **E** is almost an additive property, it can also be obtained by the summation of fragments, either by hand, or through a commercial software program (ADME Boxes, 2010). There remain the descriptors **S**, **A**, **B**, and **L** to be determined.

Partition coefficients and/or solubilities can be used to obtain all the four remaining descriptors (Abraham *et al*., 2004). Suppose there are available solubilities for a given compound in water and a number of solvents. Then solubility ratios, log (CA,organic/CA,water), can be obtained as shown in Eqn. 5 and Eqn. 6. If three solubility ratios are available for three solvent systems shown in Table 1, we have three equations and three unknowns (**S**, **A**, and **B**) so that the latter can be determined. Of more practical use is a situation where several solubility ratios are known. Then if we have, say, six solubility ratios and three equations, the three unknowns can be obtained as the descriptors that give the best fit to the six equations. The Solver add-on program to Excel can be set up to carry out such a calculation automatically. However, it is possible to increase the number of equations by the stratagem of converting the water-to-solvent solubility ratios into gas to solvent solubility ratios, CA,organic/CA,gas

$$\mathbf{C}\_{\text{A,organic}}/\mathbf{C}\_{\text{A,water}}\text{ \* }\mathbf{C}\_{\text{A,water}}/\mathbf{C}\_{\text{A,gas}} = \mathbf{ }\mathbf{ }\mathbf{C}\_{\text{A,organic}}/\mathbf{C}\_{\text{A,gas}}\tag{14}$$

The ratio CA,water/CA,gas is the gas-to-water partition coefficient, usually denoted as Kw. A further set of equations is available for gas-to-solvent solubility ratios, Table 2. Thus six water-to-solvent solubility ratios can be converted into six gas-to-solvent solubility ratios, leading to a set of 12 equations. If logKw is not known, it can be used as another parameter to be determined. This increases the number of unknowns from four (**S, A, B, L**) to five (**S, A, B, L**, logKw) but the number of equations is increased from six to twelve. In addition, two equations are available for gas to solvent partitions themselves, see the last entries in Tables 1 and 2, making for the present case no fewer than fourteen equations.

As an example, we use data on solubilities of trimethoprim in eight solvents (Li *et al*., 2008) converted from mol fraction to mol dm-3. The solubility in water was not given, but is known to be 2.09\* 10-3 in mol dm-3 (Howard and Meylan, 1997). The eight observed solubility ratios, CA,organic/CA,water, are in Table 7, as log (ratio). We took log Kw as another parameter to be determined, leading to no less than 18 equations: the eight original equations from solubilities in the eight solvents that led to CA,organic/CA,water, the corresponding eight equations for CA,organic/CA,gas, and two equations for CA,water/CA,gas (*ie* Kw). With **E** fixed at 1.892 and **V** fixed at 2.1813, the best fit values of the descriptors were **S** = 2.52, **A** = 0.44, **B** = 1.69, **L** = 11.81 and log Kw = 14.49; these yielded the calculated log (ratios) in Table 7. For all 18 values, the Average Error = -0.002, the Absolute Average Error = 0.092, the RMSE = 0.107, and the SD = 0.110 log unit. Not only do the original solubilities allow the derivation of descriptors for trimethoprim, but the latter, in turn, allow the prediction of solubility ratios and hence actual solubilities in all the solvents listed in Table 1.

bond contribution is 6.56 cm3 mol -1 for each bond, no matter whether single, double, or triple, to be subtracted. For complicated molecules it is time consuming to count the number of bonds, Bn, but this can be calculated from the algorithm given by Abraham (1993a)

Once **V** is available, **E** can be obtained from the compound refractive index at 20oC. If the compound is not liquid at room temperature or if the refractive index is not known the latter can be calculated using the freeware software of Advanced Chemistry Development (ACD). An Excel spreadsheet for the calculation of **V** and **E** from refractive index is available from the authors. Since **E** is almost an additive property, it can also be obtained by the summation of fragments, either by hand, or through a commercial software program (ADME Boxes,

Partition coefficients and/or solubilities can be used to obtain all the four remaining descriptors (Abraham *et al*., 2004). Suppose there are available solubilities for a given compound in water and a number of solvents. Then solubility ratios, log (CA,organic/CA,water), can be obtained as shown in Eqn. 5 and Eqn. 6. If three solubility ratios are available for three solvent systems shown in Table 1, we have three equations and three unknowns (**S**, **A**, and **B**) so that the latter can be determined. Of more practical use is a situation where several solubility ratios are known. Then if we have, say, six solubility ratios and three equations, the three unknowns can be obtained as the descriptors that give the best fit to the six equations. The Solver add-on program to Excel can be set up to carry out such a calculation automatically. However, it is possible to increase the number of equations by the stratagem of converting the water-to-solvent solubility ratios into gas to solvent solubility

 CA,organic/CA,water \* CA,water/CA,gas = CA,organic/CA,gas (14) The ratio CA,water/CA,gas is the gas-to-water partition coefficient, usually denoted as Kw. A further set of equations is available for gas-to-solvent solubility ratios, Table 2. Thus six water-to-solvent solubility ratios can be converted into six gas-to-solvent solubility ratios, leading to a set of 12 equations. If logKw is not known, it can be used as another parameter to be determined. This increases the number of unknowns from four (**S, A, B, L**) to five (**S, A, B, L**, logKw) but the number of equations is increased from six to twelve. In addition, two equations are available for gas to solvent partitions themselves, see the last entries in Tables

As an example, we use data on solubilities of trimethoprim in eight solvents (Li *et al*., 2008) converted from mol fraction to mol dm-3. The solubility in water was not given, but is known to be 2.09\* 10-3 in mol dm-3 (Howard and Meylan, 1997). The eight observed solubility ratios, CA,organic/CA,water, are in Table 7, as log (ratio). We took log Kw as another parameter to be determined, leading to no less than 18 equations: the eight original equations from solubilities in the eight solvents that led to CA,organic/CA,water, the corresponding eight equations for CA,organic/CA,gas, and two equations for CA,water/CA,gas (*ie* Kw). With **E** fixed at 1.892 and **V** fixed at 2.1813, the best fit values of the descriptors were **S** = 2.52, **A** = 0.44, **B** = 1.69, **L** = 11.81 and log Kw = 14.49; these yielded the calculated log (ratios) in Table 7. For all 18 values, the Average Error = -0.002, the Absolute Average Error = 0.092, the RMSE = 0.107, and the SD = 0.110 log unit. Not only do the original solubilities allow the derivation of descriptors for trimethoprim, but the latter, in turn, allow the prediction of solubility ratios and hence actual solubilities in all the solvents listed in Table 1.

where Nt is the total number of atoms in the molecule and R is the number of rings.

2010). There remain the descriptors **S**, **A**, **B**, and **L** to be determined.

1 and 2, making for the present case no fewer than fourteen equations.

ratios, CA,organic/CA,gas

Bn = Nt -1 + R (13)

Exactly the same procedure is adopted if actual partition coefficients are experimentally available, rather than solubilities. The relevant equations are now those in Table 3 and Table 4. Of course if both solubilities and actual partition coefficients have both been experimentally determined, a combination of equations from Tables 1 and 2 and from Tables 3 and 4 can be used. Even though partition coefficients refer to partition into wet solvents, descriptors obtained from partition coefficients using equations in Table 3 and Table 4 can still be used to predict solubility ratios and solubilities in dry solvents for all the solvents listed in Table 1.


Table 6. Atom contributions to the McGowan volume, in cm3 mol-1


Table 7. Solubility ratios for trimethoprim, as log (ratio)

Prediction of Partition Coefficients and Permeability of Drug Molecules in Biological

other blood-to-tissue partition coefficients as well.

workplace and in the environment.

brain/air

2

/ //

*P P xP x*

and blood-to-brain (Pbrain/blood) partition coefficients for VOCs in humans and rats

N 81, R 0.923, SD 0.346, RMSE 0.333, F 179.0

the product of the measured air-to-tissue partition coefficient, Ptissue/air, times the measured blood-to-air partition coefficient, Pair/blood. The *in vitro* partition coefficient data are important and are used as required input parameters in pharmacokinetic models developed to determine the disposition of volatile organic compounds that individuals inhale in the

Abraham and coworkers (2006a) reported correlation models for the air-to-brain (Pbrain/air)

Log P 0.987 0.263 0.411 3.358 2.025 0.591

*in vitro*

*tissue blood tissue air air blood*

Systems with Abraham Model Solute Descriptors Derived from Measured Solubilities and… 113

The logarithm of the blood-to-brain concentration ratio, log BB, is a thermodynamic quantity defining the extent of blood penetration. The log BB is mathematically given by

log ( ) *solute brain*

the ratio of the solute concentration in brain tissue divided by the solute's concentration in blood (or serum or plasma) at steady-state conditions. The blood/brain distribution ratio can be experimentally determined by intravenous administration of a single injection of 14Cradioactive isotope labeled test substance in rats. The animal is sacrificed at a specified time endpoint after equilibrium is achieved. The brain and blood are immediately harvested, and the concentration in each biological sample is quantified from the measured radioactivity. Isotopic labeling provides a convenient means to distinguish the injected test substance from all other chemicals that might be present in the body. Radioactive counting methods do not distinguish between the radioactive isotope in the injected test substance and any degradation products that might have been formed before the animal was sacrificed. The distribution experiments are usually carried out over a long time scale, possibly hours, and concentrations in blood and brain obtained as a function of time. The ratio, as Eq. 15, will change with time and only if it reaches a constant value can the ratio be taken as an equilibrium value. This is very time consuming indeed, as only one measurement can be made with each rat. Despite these shortcomings, radioactive labeling is one of the more popular methods for not only determining the blood-to-brain distribution coefficient, but

Blood-to-brain and blood-to-tissue partition coefficients have also been measured for volatile organic compounds using the *in vitro* vial method (see Figure 6). A known amount of animal sample is placed in a glass vial of known volume. The vial is then sealed and a minute known quantity of the volatile organic compound (VOC) is introduced by syringe through the rubber septum. After equilibration a sample of the headspace vapor phase is withdrawn from the glass vial for gas chromatographic analysis. The gas-to-tissue partition coefficient is computed from mass balance considerations as the total amount of solute added, the concentration of the vapor phase, the headspace volume and amount of tissue sample are all known. The blood-to-tissue partition coefficient, Ptissue/blood, is calculated as

, ,

*C C*

, , ( )( ) *solute tissue solute air*

**ESABL**

(17)

*solute air solute blood*

*C C* (16)

*BB*

*C*

, ,

*<sup>C</sup>* (15)

*solute blood*

Although we have set out the determination of descriptors from experimental measurements, it is still very helpful to use the ACD software (ADME Boxes, 2010) to calculate the descriptors at the same time. Occasionally there may be erroneous solubility measurements, or solubilities may be affected through solvate formation, and the calculated descriptors afford a useful check on the obtained descriptors from experiment measurements.
