**6. Abraham solvation parameter model: prediction of blood-to-brain and blood-to-iissue partition coefficient**

Successful drug development requires efficient delivery of the drug to the target site. The drug must cross various cellular barriers by passive and/or transporter-mediated uptake. Drug delivery to the brain is particularly challenging as there are two physiologically barriers – the blood-brain barrier (BBB) and the blood-cerebrospinal fluid barrier (BCSFB) – separating the brain from its blood supply controlling the transport of chemical compounds. The BBB is a continuous layer of microvessel endothelial cells, connected by highlydeveloped tight junctions, which effectively restrict paracellular transport of molecules irrespective of their molecular size. Tight junctions provide significant transendothelial electrical resistance to the brain microvessel endothelial cells and serves to further impede the penetration of the BBB. The electrical resistance between the endothelial cells is on the order of 1500 – 2000 Ω/cm2, as compared to and electrical resistance of 3.33 Ω/cm2 found in other body tissues (Alam *et al*., 2010). Under normal conditions the BBB acts as a barrier to toxic agents and safeguards the integrity of the brain. A compound may circumvent the BBB and gain access to the brain by the nose-to-brain route. The compound is transported to the brain via an olfactory pathway following absorption across the nasal mucosa.

Alternatively, compounds may permeate from the blood into the cerebrospinal fluid and permeate into the brain interstitial fluid. The BCSFB separates the blood from the cerebrospinal fluid (CSF) that runs in the subarachnoid space surrounding the brain. The BCSFB is located at the choroid plexus, and it is composed of epithelial cells held together at their apices by tight junctions, which limit paracellular flux. Hence compounds penetrate the barrier transcellularly. The CSF-facing surface of the epithelial cells, which secrete CSF into the ventricles, is increased by the presence of microvilli. The capillaries in the choroid plexus allow free movement of molecules via fenestractions and intracellular gaps. Transport across the BCSFB is not an accurate measure of transport across the BBB as the two barriers are anatomically different. However, as Begley *et al.* (2000) point out, for many compounds there is a permanently maintained concentration gradient between brain interstitial fluid and the CSF.

The transport of compounds into the brain can take place through 'passive' transport or 'active' transport. Nearly all the calculational models for transport into the brain deal with passive transport, although it is now known that many compounds are prevented from crossing the BBB through efflux mechanisms especially involving P-glycoprotein. The use of wildtype mice and knockout mice (the latter deficient in Pgp) has shown conclusively that for a number of drugs the brain to plasma distribution is much lower for the wildtype mice than for knockout mice. We will focus on passive transport, but it must be appreciated that any analysis might well include compounds that are actually subject to active transport and will appear as outliers in the analyses.

The steady-state distribution of a compound between the blood (or plasma) and brain, and the rate of permeation of a compound from blood (or from an aqueous saline solution) through the blood brain barrier, are two quantitative measures of drug uptake in the brain.

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

**6. Abraham solvation parameter model: prediction of blood-to-brain and** 

brain via an olfactory pathway following absorption across the nasal mucosa.

Alternatively, compounds may permeate from the blood into the cerebrospinal fluid and permeate into the brain interstitial fluid. The BCSFB separates the blood from the cerebrospinal fluid (CSF) that runs in the subarachnoid space surrounding the brain. The BCSFB is located at the choroid plexus, and it is composed of epithelial cells held together at their apices by tight junctions, which limit paracellular flux. Hence compounds penetrate the barrier transcellularly. The CSF-facing surface of the epithelial cells, which secrete CSF into the ventricles, is increased by the presence of microvilli. The capillaries in the choroid plexus allow free movement of molecules via fenestractions and intracellular gaps. Transport across the BCSFB is not an accurate measure of transport across the BBB as the two barriers are anatomically different. However, as Begley *et al.* (2000) point out, for many compounds there is a permanently maintained concentration gradient between brain interstitial fluid and the CSF. The transport of compounds into the brain can take place through 'passive' transport or 'active' transport. Nearly all the calculational models for transport into the brain deal with passive transport, although it is now known that many compounds are prevented from crossing the BBB through efflux mechanisms especially involving P-glycoprotein. The use of wildtype mice and knockout mice (the latter deficient in Pgp) has shown conclusively that for a number of drugs the brain to plasma distribution is much lower for the wildtype mice than for knockout mice. We will focus on passive transport, but it must be appreciated that any analysis might well include compounds that are actually subject to active transport and

The steady-state distribution of a compound between the blood (or plasma) and brain, and the rate of permeation of a compound from blood (or from an aqueous saline solution) through the blood brain barrier, are two quantitative measures of drug uptake in the brain.

Successful drug development requires efficient delivery of the drug to the target site. The drug must cross various cellular barriers by passive and/or transporter-mediated uptake. Drug delivery to the brain is particularly challenging as there are two physiologically barriers – the blood-brain barrier (BBB) and the blood-cerebrospinal fluid barrier (BCSFB) – separating the brain from its blood supply controlling the transport of chemical compounds. The BBB is a continuous layer of microvessel endothelial cells, connected by highlydeveloped tight junctions, which effectively restrict paracellular transport of molecules irrespective of their molecular size. Tight junctions provide significant transendothelial electrical resistance to the brain microvessel endothelial cells and serves to further impede the penetration of the BBB. The electrical resistance between the endothelial cells is on the order of 1500 – 2000 Ω/cm2, as compared to and electrical resistance of 3.33 Ω/cm2 found in other body tissues (Alam *et al*., 2010). Under normal conditions the BBB acts as a barrier to toxic agents and safeguards the integrity of the brain. A compound may circumvent the BBB and gain access to the brain by the nose-to-brain route. The compound is transported to the

check on the obtained descriptors from experiment measurements.

**blood-to-iissue partition coefficient** 

will appear as outliers in the analyses.

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

$$\text{Log } BB = \text{(}\frac{\text{C}\_{\text{solute}, bruin}}{\text{C}\_{\text{solute}, bluout}}\text{)}\tag{15}$$

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 other blood-to-tissue partition coefficients as well.

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

$$P\_{\text{tissue}/\text{blood}} = P\_{\text{tissue}/\text{air}} \text{x} \\ P\_{\text{air}/\text{blood}} = \left(\frac{\text{C}\_{\text{solute, tissue}}}{\text{C}\_{\text{solute,air}}}\right) \text{x} \left(\frac{\text{C}\_{\text{solute,air}}}{\text{C}\_{\text{solute,blood}}}\right) \tag{16}$$

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 workplace and in the environment.

Abraham and coworkers (2006a) reported correlation models for the air-to-brain (Pbrain/air) and blood-to-brain (Pbrain/blood) partition coefficients for VOCs in humans and rats

$$\begin{aligned} \text{Log P}\_{\text{train/air}} \text{(in vitro)} &= -0.987 + 0.263 \mathbf{E} + 0.411 \mathbf{S} + 3.358 \mathbf{A} + 2.025 \mathbf{B} + 0.591 \mathbf{L} \\ \text{(N = 81, R = 0.923, SD = 0.346, RMSE = 0.333, F = 179.0)} \end{aligned} \tag{17}$$

Prediction of Partition Coefficients and Permeability of Drug Molecules in Biological

 

N 233, R 0.75, SD 0.33, F 113

carboxylic acids, **Ic** = 0 for noncarboxylic acid solutes).

brain/blood 2

initial rate of unidirectional transfer

proteins, Eqn. 21 is modified as follows

or plasma.

Pbrain/blood data

Systems with Abraham Model Solute Descriptors Derived from Measured Solubilities and… 115

brain partition coefficients of VOC because it refers specifically to blood rather than to blood

A follow-up study (Abraham *et al*., 2006b) considered the partitioning behavior of drugs and drug candidates (measured by *in vivo* experimental methods), as well as the VOC *in vitro* partition coefficient data discussed above. The Abraham model correlation for the *in vivo* log

log BB log P 0.547 0.221 – 0.604 – 0.641 – 0.681 0.635 – 1.216

differs from the correlation equation for the VOCs (see Eqn. 18). In particular, the c-coefficients differ appreciably 0.547 (SD = 0.078) as against -0.024 (SD=0.069), which suggests that there is a systematic difference between the *in vivo* and *in vitro* distributions. The authors went on to show that the difference resulted in part because the two sets of compounds (drugs versus VOCs) inhabit different areas in chemical space. The *in vivo* drug compounds had much larger solute descriptors, and included compounds having a carboxylic acid functional group. The independent variable **Ic** was needed as an indicator descriptor for carboxylic acids (**Ic** = 1 for

The blood-to-brain partition coefficient provides valuable information regarding a compound's ability to penetrate the blood-brain barrier. Cruciani *et al*. (2000) noted that compounds having log BB values greater than 0.0 (concentration in the brain exceeds concentration in the blood) should cross the barrier, whereas compounds having log BB less than -0.3 tended not to cross the barrier. Li and coworkers (2005) used a slightly different classification scheme (see Figure 7) of dividing compounds into BBB-penetrating (BBB+) or BBB-non-penetrating (BBB-) according to whether the log BB value was ≥ -1 or ≤ -1, respectively. Many times an actual numerical log BB is not needed in the decision making process, and in such cases, an indication of BBB+ or BBB- is often sufficient. Zhao *et al*. (2007) proposed a fairly simple decision tree for classifying drug candidates as BBB+ or BBB- based on their Abraham solute descriptors (See Figure 7). Solute acidity and solute basicity were the two most important properties governing BBB penetration, with solute excess molar refraction playing a much smaller role. The proposed classification scheme correctly

As noted above permeation of a compound from blood (or from an aqueous saline solution) through the blood brain barrier can be used to indicate drug uptake in the brain. The membrane permeability-surface area product, PS, is a kinetic parameter used in describing

where kin is the measured transfer constant and F is the perfusion fluid flow expressed in milliliters per second per gram. For solutes that bind rapidly and reversibly to plasma

assuming that the unbound and bound forms of the drug are in equilibrium in the fluid. In Eqn. 22, fu is the fraction of the unbound drug in the perfusion fluid. In a typical experiment,

/ (1 ) *PS F*

/ (1 ) *fu PS F*

*in kF e* (21)

*in kF e* (22)

**E S A B V Ic**

(20)

*in vivo*

predicted the BBB penetration of 90 % of the 1093 compounds considered.

 brain/blood 2 Log BB Log P 0.057 0.017 – 0.536 – 0.323 – 0.335 0.731 N 78, R 0.725, SD 0.203, RMSE 0.196, F 37.9 *in vitro* **ESABV** (18)

Fig. 6. Equilibrium vial technique depicting removal of the equilibrated headspace vapor above the animal/human tissue

In Eqns. 17 and 18, N is the number of data points in the regression analysis, R2 represents the squared correlation coefficient, SD denotes the standard deviation and RMSE corresponds to the root mean square error. Note that in a multiple linear regression equation, the denominator in the definition of SD is N – P – 1 and in the definition of RMSE it is N – P, where P is the number of independent variables in the equation. The derived correlations provided a reasonably accurate mathematical description of the observed partition coefficient data as evidenced by the high squared correlation coefficients and reasonably small standard deviations. Both correlations were validated using training set and test set analyses. In comparing calculated biological data to observed values one must remember that the measured values do have larger experimental uncertainties. A reasonable estimated uncertainty for the measured log Pbrain/air would be about 0.2 log units based on independent values from different laboratories. Rat and human partition coefficient data for each given VOC were averaged (if both values were available), and the average values were combined into a single regression analysis. In a comparison of experimental human and rat partition coefficient data for 17 common compounds, the authors had shown that the two sets of data (human versus rat) differed by only 0.062 log units, which is likely less than the experimental uncertainty associated with the measured experimental values. For the compounds studied, human and rat partition coefficient data were identical for all practical purposes. The authors also showed that blood-to-brain and plasma-to-brain partition coefficients were sufficiently close and could be combined into a single Abraham model correlation

$$\log \text{P}\_{\text{brain}/(\text{blood}, \text{plasma})} = -0.028 + 0.003 \text{ E} - 0.485 \text{ S} - 0.117 \text{ A} - 0.408 \text{ B} + 0.703 \text{ V} \tag{19}$$

$$\left( \text{N} = 99, \text{R}^2 = 0.703, \text{SD} = 0.197, \text{RMSE} = 0.191, \text{F} = 44.1 \right)$$

Eqs. (18) and (19), are not substantially different, and the statistics are almost the same. It is a moot point as to whether further values of blood to brain partition coefficients should best be predicted through Eqn. 18 or 19. We recommend that Eqn. 18 be used to predict blood-to-

**ESABV**

(18)

(19)

Log BB Log P 0.057 0.017 – 0.536 – 0.323 – 0.335 0.731

*in vitro*

Fig. 6. Equilibrium vial technique depicting removal of the equilibrated headspace vapor

In Eqns. 17 and 18, N is the number of data points in the regression analysis, R2 represents the squared correlation coefficient, SD denotes the standard deviation and RMSE corresponds to the root mean square error. Note that in a multiple linear regression equation, the denominator in the definition of SD is N – P – 1 and in the definition of RMSE it is N – P, where P is the number of independent variables in the equation. The derived correlations provided a reasonably accurate mathematical description of the observed partition coefficient data as evidenced by the high squared correlation coefficients and reasonably small standard deviations. Both correlations were validated using training set and test set analyses. In comparing calculated biological data to observed values one must remember that the measured values do have larger experimental uncertainties. A reasonable estimated uncertainty for the measured log Pbrain/air would be about 0.2 log units based on independent values from different laboratories. Rat and human partition coefficient data for each given VOC were averaged (if both values were available), and the average values were combined into a single regression analysis. In a comparison of experimental human and rat partition coefficient data for 17 common compounds, the authors had shown that the two sets of data (human versus rat) differed by only 0.062 log units, which is likely less than the experimental uncertainty associated with the measured experimental values. For the compounds studied, human and rat partition coefficient data were identical for all practical purposes. The authors also showed that blood-to-brain and plasma-to-brain partition coefficients were sufficiently

brain/blood 2

above the animal/human tissue

brain/(blood,plasma)

2

close and could be combined into a single Abraham model correlation

N 99, R 0.703, SD 0.197, RMSE 0.191, F 44.1

log P 0.028 0.003 – 0.485 – 0.117 – 0.408 0.703

Eqs. (18) and (19), are not substantially different, and the statistics are almost the same. It is a moot point as to whether further values of blood to brain partition coefficients should best be predicted through Eqn. 18 or 19. We recommend that Eqn. 18 be used to predict blood-to-

**ESABV**

N 78, R 0.725, SD 0.203, RMSE 0.196, F 37.9

brain partition coefficients of VOC because it refers specifically to blood rather than to blood or plasma.

A follow-up study (Abraham *et al*., 2006b) considered the partitioning behavior of drugs and drug candidates (measured by *in vivo* experimental methods), as well as the VOC *in vitro* partition coefficient data discussed above. The Abraham model correlation for the *in vivo* log Pbrain/blood data

$$\begin{aligned} \text{log BB} &= \log \text{P}\_{\text{train}/\text{blood}} \left( \dot{m} \,\text{vino} \right) = 0.547 + 0.221 \text{E} - 0.604 \text{S} - 0.641 \text{A} - 0.681 \text{B} + 0.635 \text{V} - 1.216 \text{kC} \\ \text{(N} &= 233, \text{R}^2 = 0.75, \text{SD} = 0.33, \text{F} = 113) \end{aligned} \tag{20}$$

differs from the correlation equation for the VOCs (see Eqn. 18). In particular, the c-coefficients differ appreciably 0.547 (SD = 0.078) as against -0.024 (SD=0.069), which suggests that there is a systematic difference between the *in vivo* and *in vitro* distributions. The authors went on to show that the difference resulted in part because the two sets of compounds (drugs versus VOCs) inhabit different areas in chemical space. The *in vivo* drug compounds had much larger solute descriptors, and included compounds having a carboxylic acid functional group. The independent variable **Ic** was needed as an indicator descriptor for carboxylic acids (**Ic** = 1 for carboxylic acids, **Ic** = 0 for noncarboxylic acid solutes).

The blood-to-brain partition coefficient provides valuable information regarding a compound's ability to penetrate the blood-brain barrier. Cruciani *et al*. (2000) noted that compounds having log BB values greater than 0.0 (concentration in the brain exceeds concentration in the blood) should cross the barrier, whereas compounds having log BB less than -0.3 tended not to cross the barrier. Li and coworkers (2005) used a slightly different classification scheme (see Figure 7) of dividing compounds into BBB-penetrating (BBB+) or BBB-non-penetrating (BBB-) according to whether the log BB value was ≥ -1 or ≤ -1, respectively. Many times an actual numerical log BB is not needed in the decision making process, and in such cases, an indication of BBB+ or BBB- is often sufficient. Zhao *et al*. (2007) proposed a fairly simple decision tree for classifying drug candidates as BBB+ or BBB- based on their Abraham solute descriptors (See Figure 7). Solute acidity and solute basicity were the two most important properties governing BBB penetration, with solute excess molar refraction playing a much smaller role. The proposed classification scheme correctly predicted the BBB penetration of 90 % of the 1093 compounds considered.

As noted above permeation of a compound from blood (or from an aqueous saline solution) through the blood brain barrier can be used to indicate drug uptake in the brain. The membrane permeability-surface area product, PS, is a kinetic parameter used in describing initial rate of unidirectional transfer

$$k\_{in} = F\left(1 - e^{-PS/F}\right) \tag{21}$$

where kin is the measured transfer constant and F is the perfusion fluid flow expressed in milliliters per second per gram. For solutes that bind rapidly and reversibly to plasma proteins, Eqn. 21 is modified as follows

$$k\_{\rm int} = F\left(\mathbf{1} - e^{-f\mu \,\mathrm{PS}/F}\right) \tag{22}$$

assuming that the unbound and bound forms of the drug are in equilibrium in the fluid. In Eqn. 22, fu is the fraction of the unbound drug in the perfusion fluid. In a typical experiment,

Prediction of Partition Coefficients and Permeability of Drug Molecules in Biological

<sup>2</sup>

less than those for the neutral bases.

has entered the bloodstream.

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

muscle/air

2

(N 114, R 0.944, SD 0.267, F 363

**gas-to-tissue partition coefficients** 

N 88, R 0.810, SD 0.534, F 48.8

model

Systems with Abraham Model Solute Descriptors Derived from Measured Solubilities and… 117

are used together with the descriptors originally chosen for nonelectrolytes. This ensures that values of **S**, **A** and **B** for ions and ionic species are on the same scale as those for nonelectrolytes. Solute descriptors have been reported for many simple cations and anions, for carboxylates, for phenoxides, and for protonated amines and protonated pyridines. The j+ and j- equation coefficients have been determined (Abraham and Acree, 2010a,b,c,d) for several of the organic solvents listed in Table 1. Abraham (2011) recently reanalyzed the published log PS data in terms of Eqn. 24 to yield the following correlation

log PS 1.268 – 0.047 – 0.876 – 0.719 – 1.571 1.767 0.469 1.663

The 88 log PS values in Eqn. 25 were for compounds that existed in the saline perfusate entirely (or almost entirely) as neutral molecules or entirely (or almost entirely) as charged species, and which underwent perfusion by a passive process. Abraham showed that log PS values for carboxylate anions are about two log units less than those for the neutral carboxylic acids, and that log PS values for protonated base cations are about one log unit

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

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

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

logK 1.039 0.207 0.723 3.242 2.469 0.463

*in vitro*

**ESABL**

**ESABV J J**

(25)

(26)

the drug (dissolved in blood or in an aqueous saline solution) is perfused into the internal carotid artery and the rate of drug uptake is determined by a radioisotope assay method. The animals are sacrificed at various time intervals. The time scale needed to perform the perfusion study is very short – typically no more than a few minutes. Because of the small time scale, perfusion measurements are less subject to degradation effects than are log BB measurements, although the same difficulties over passive and active transport still exist.

Fig. 7. Decision tree for predicting whether drugs pass through the BBB based on their Abraham solute descriptors. BBB+ indicates BBB penetrating whereas BBB- denotes BBB non-penetration. The right-handside of any decision branch is no penetration (red box), and the left-handside is yes penetration (green box). (The right-hand side of any decision branch is yes, and the left-hand side is no.)

Abraham (2004) derived the following mathematical correlation

$$\begin{aligned} \text{log PS} &= -0.716 \text{ -- } 0.974 \text{S -- } 1.802 \text{A -- } 1.603 \text{B + } 1.893 \text{V} \\ \text{y} \left( \text{N} = 30 \text{, R}^2 = 0.868, \text{SD = 0.52, F = 42} \right) \end{aligned} \tag{23}$$

by regression analysis of the experimental log PS data for 30 neutral compounds from protein-free saline solution buffered at pH of 7.4. The contribution of the e · **E** term was not significant and was removed from Eqn. 23. The negative equation coefficients in Eqn. 23 indicate that an increase in compound polarity of any kind, that is dipolarity/polarizability, hydrogen-bonding acidity or hydrogen-bonding basicity, results in a decrease in the rate of permeation. Increased solute size (**V** solute descriptor), on the other hand, results in a greater permeation rate.

The Abraham model correlations that have been presented thus far pertain to neutral molecules. The basic model has been extended to include processes between condensed phases involving ions and ionic species

$$\mathbf{SP} = \mathbf{c} + \mathbf{e} \cdot \mathbf{E} + \mathbf{s} \cdot \mathbf{S} + \mathbf{a} \cdot \mathbf{A} + \mathbf{b} \cdot \mathbf{B} + \mathbf{v} \cdot \mathbf{V} + \mathbf{j}\_+ \cdot \mathbf{J}^+ + \mathbf{j}\_- \cdot \mathbf{J}^- \tag{24}$$

by adding one new term for cations and one new term for anions. **J**+ is used whenever a cation is the solute, **J**- whenever an anion is the solute, and neither is used whenever the solute is a nonelectrolyte. It is very important to note that the two new ionic descriptors

the drug (dissolved in blood or in an aqueous saline solution) is perfused into the internal carotid artery and the rate of drug uptake is determined by a radioisotope assay method. The animals are sacrificed at various time intervals. The time scale needed to perform the perfusion study is very short – typically no more than a few minutes. Because of the small time scale, perfusion measurements are less subject to degradation effects than are log BB measurements,

Fig. 7. Decision tree for predicting whether drugs pass through the BBB based on their Abraham solute descriptors. BBB+ indicates BBB penetrating whereas BBB- denotes BBB non-penetration. The right-handside of any decision branch is no penetration (red box), and the left-handside is yes penetration (green box). (The right-hand side of any decision branch

log PS 0.716 – 0.974 – 1.802 – 1.603 1.893

by regression analysis of the experimental log PS data for 30 neutral compounds from protein-free saline solution buffered at pH of 7.4. The contribution of the e · **E** term was not significant and was removed from Eqn. 23. The negative equation coefficients in Eqn. 23 indicate that an increase in compound polarity of any kind, that is dipolarity/polarizability, hydrogen-bonding acidity or hydrogen-bonding basicity, results in a decrease in the rate of permeation. Increased solute size (**V** solute descriptor), on the other hand, results in a

The Abraham model correlations that have been presented thus far pertain to neutral molecules. The basic model has been extended to include processes between condensed

by adding one new term for cations and one new term for anions. **J**+ is used whenever a cation is the solute, **J**- whenever an anion is the solute, and neither is used whenever the solute is a nonelectrolyte. It is very important to note that the two new ionic descriptors

**SABV**

SP c e · s · a · b · v · j · j · **ESA B VJ J** (24)

(23)

Abraham (2004) derived the following mathematical correlation

<sup>2</sup>

N 30, R 0.868, SD 0.52, F 42

is yes, and the left-hand side is no.)

greater permeation rate.

phases involving ions and ionic species

although the same difficulties over passive and active transport still exist.

are used together with the descriptors originally chosen for nonelectrolytes. This ensures that values of **S**, **A** and **B** for ions and ionic species are on the same scale as those for nonelectrolytes. Solute descriptors have been reported for many simple cations and anions, for carboxylates, for phenoxides, and for protonated amines and protonated pyridines. The j+ and j- equation coefficients have been determined (Abraham and Acree, 2010a,b,c,d) for several of the organic solvents listed in Table 1. Abraham (2011) recently reanalyzed the published log PS data in terms of Eqn. 24 to yield the following correlation model

$$\begin{aligned} \text{log PS} &= -1.268 - 0.047 \text{E} - 0.876 \text{S} - 0.719 \text{A} - 1.571 \text{B} + 1.767 \text{V} + 0.469 \text{J}^{+} + 1.663 \text{J}^{-} \\ \text{(N = 88, R = 0.810, SD = 0.534, F = 48.8)} \end{aligned} \tag{25}$$

The 88 log PS values in Eqn. 25 were for compounds that existed in the saline perfusate entirely (or almost entirely) as neutral molecules or entirely (or almost entirely) as charged species, and which underwent perfusion by a passive process. Abraham showed that log PS values for carboxylate anions are about two log units less than those for the neutral carboxylic acids, and that log PS values for protonated base cations are about one log unit less than those for the neutral bases.
