Prediction of the Blood-Brain Barrier Permeability Using RP-18 Thin Layer Chromatography

Abstract The Blood-Brain Barrier (BBB) permeability is an important factor governing a drug’s ability to act upon the Central Nervous System. The measure of the BBB permeability used throughout this study is the log BB (the blood/brain partitioning coefficient) measured in vivo or calculated. Useful yet simple models of the BBB permeability were developed by Stepwise Multiple Regression Analysis based on the chromatographic parameters Rf and Rf/PSA obtained by RP-18 TLC with acetonitrile - pH 7.4 phosphate buffered saline 70:30 (v/v) as mobile phase, combined with descriptors - the number of H-bond donors (HD), the number of H-bond acceptors (HA), energy of the highest occupied molecular orbital – (eH), energy of the lowest unoccupied molecular orbital (eL). The ability of the solutes to cross the BBB has been studied qualitatively using Discriminant Function Analysis. Almost all compounds with the known BB vivo parameter were correctly classified as CNS+/-. The classification functions based on Rf/PSA have been verified using an external group. The results of the chromatographic analysis proposed in this study (RP-18 TLC) are a source of valuable information on the BBB permeability of compounds available even on a very small scale.


Introduction
The blood-brain barrier (BBB) is the main physical and enzymatic barrier that separates brain tissues and blood, prevents the penetration of the central nervous system (CNS) by undesired substances (peripherally acting drugs or harmful compounds) and facilitates its penetration by neurotherapeutics [1][2][3]. The most common measures of the BBB permeability are log BB -the blood/brain partitioning coefficient at equilibrium conditions, i.e. the ratio of the drug concentration in the brain to that in the blood plasma measured at some defined time point and the less frequently used kinetic permeability log PS, where P is the observed permeability across BBB (cm/s) and S is the surface area of brain capillary endothelium (cm 2 /g) [3,4]. The evaluation of the compound's ability to cross the BBB is a vital step in assessing its therapeutic potential because many promising neurotherapeutics miss their goal due to their inability to enter the brain or, oppositely, peripherally acting drugs exhibit unwanted side effects due to their undesired brain penetration. The in vivo, in vitro and in silico methods used currently to study the drug brain penetration are discussed in a number of review papers [2,5,6]. High costs and low throughput of any in vivo experiments as well as the need to use lab animals are the reasons why several in vitro cellbased methods of the BBB permeation assessment have been developed [7]. Concurrently with the development of the in vitro cell-based techniques, some non-cell-based assays have been introduced [8]. As more data on the CNS activity of different compounds were gathered, several attempts were made to generate computational models linking the BBB permeability of compounds to their experimentally measured and/or theoretically calculated physicochemical features [4,5].
It has been known for several years that the lipophilicity of a molecule is an important factor influencing its BBB passage. The relationship between the compound's lipid solubility and its ability to enter the brain was observed as early as in the late nineteenth century [9]. Later studies have focused on correlations between the brain uptake and the lipophilicity of the solutes often expressed as the partition coefficient log P between water and certain water-immiscible solvents, of which 1-octanol is the most important example (log P oct ). It was observed that for a congeneric series of compounds of the hypnotic activity the most active ones are those with log P oct around 2 ("the rule of 2") [10]. Further research supported this hypothesis for a series of imidazoles injected intravenously into rats, linking the brain uptake expressed as log (C brain /C i.v. ) with the water-pH 7.4 buffer apparent partition coefficient log P' oct via the quadratic equation: log (C brain /C i.v. ) = -0.94 + 0.574 log P' oct -0.133 (log P' oct ) 2 (n = 14, R 2 = 0.897) with the maximum (as it can be easily calculated) at log P' oct = 2.16 [11], and for a series of C-11 labelled radiopharmaceuticals [12].
Later studies [13] did not confirm the general nature of the log BBlog P oct parabolic relationship. It was soon observed that the octanol-water based lipophilicity is not an ideal parameter to predict the BBB permeation for larger and more diverse groups of compounds. The application of other measures of lipophilicity was proposed of which log P in other solvent systems such as cyclohexane-water (log P cyc ), chloroform-water (log P chlor ) or Δlog P = log P oct log P cyc seemed promising but in spite of their relatively high predictive value they have not attracted broader attention because only few log P values for these systems are available.
Attention was then turned to the partitioning of solutes in other bi-phase systems such as chromatographic systems resembling the properties of biomembranes. Thin layer chromatography is a low-cost, rapid and easy-toperform separation technique whose main application in drug development process is at present the octanol/water lipophilicity determination [14][15][16]. The most important thin layer chromatographic parameter used to evaluate the lipophilicity is the R M value introduced by Bate-Smith and Westall: R M = log (1/R f -1) [17].
Reversed-phase thin layer chromatography is particularly useful for the determination of solutes' partition between water and the biomembrane because the partition between a non-polar sorbent and a purely aqueous mobile phase resembles that between water and the biophase. Because reversed-phase chromatography with the purely aqueous mobile phase is impractical due to very long retention times (or very low R f ), the R M 0 values are obtained by extrapolation of plots of R M versus the concentration of the organic modifier to zero concentration of the modifier (pure water/buffer). This extrapolation is usually conducted by using the linear Soczewinski-Wachmeister equation (called sometimes the "Soczewiński-Matysik" equation) [18]: R M = R M 0 -Sφ (φ is the volume fraction of the organic modifier in the mobile phase); some authors suggest that the relationship between R M and φ should be described with the quadratic equation R M = R M 0 + bφ + aφ 2 [19]. The slope S of the linear Soczewiński-Wachmeister equation is another useful measure of lipophilicity, related to the hydrophobic surface of a molecule [19]. On the basis of R M 0 an alternative lipophilicity parameter, C 0 , has been proposed. This parameter is calculated according to the equation: C 0 = R M 0 /S and represents the volume fraction of the organic solvent in the mobile phase for which the amount of the solute in the mobile phase is equal to that in the stationary phase [20].
The RP-thin layer chromatographic descriptors R M 0 and S have been used to evaluate the brain penetration of solutes by several authors [21][22][23][24][25][26]. The relationships between log BB and R M 0 obtained by these authors are, generally speaking, linear or quadratic, depending on the group of solutes; the relationships between log BB and S described in Ref. [23] are parabolic.
Although less frequently than by RP TLC, the retention data used as the blood-brain barrier permeability predictors may be collected by other planar chromatographic techniques [24,27]. In normal-phase TLC the relationship between φ (the volume fraction of the more polar component of the mobile phase) and R M can be expressed by the equation: R M = R M 0 -b log φ. NP-TLC R M 0 values can be correlated with log BB via the parabolic relationships [27]. Quite recently salting-out thin layer chromatography (SOTLC) -the separation technique involving the application of concentrated solutions of inorganic salts as mobile phases on silicagel, aluminum oxide or cellulose supports [28] -has been used to study the BBB permeability of solutes according to a novel procedure by Ciura et al. [29] utilizing the chromatographic parameter C 0 = R M 0 /m (derived from the equation R M = R M 0 + mC, where C is the salt molar concentration).
Although lipophilicity is a key factor governing the solutes' ability to cross the BBB and hence the chromatographic lipophilicity descriptors play a dominant role in several BBB permeation models, there are other molecular parameters (related mainly to the molecule's size and polarity) that influence the BBB passage [30]. According to some earlier research, Polar Surface Area (PSA) alone or combined with lipophilicity is an important predictor of the BBB permeability and two useful yet simple models of the blood-brain partitioning were proposed by Norinder (log BB = 0.139 + 0.152 log P -0.0148 PSA) [31] and Clark (log BB = 0.547 -0.016 PSA) [32].
Milosevic et al. [24] proposed the multiple regression BBB permeation models involving chromatographic descriptors (RPTLC R M 0 or NPTLC C 0 ) combined with topological polar surface area (TPSA) and according to Ciura [29] the SOTLC C 0 parameter should be combined with several other descriptors, including TPSA.
In our earlier research we created some useful models, capable of the quantitative prediction of the BBB permeability of solutes [33,34] involving the retardation factor R f (RP-18 TLC) determined for a single composition of the mobile phase (without the need to extrapolate as in the approaches based on R M 0 or C 0 ). The single chromatographic run approach, although used less frequently than the extrapolation technique, is recommended by some authors for lipophilicity determination [35,36] and the chromatographic data obtained via this approach proved useful in the BBB permeability estimations presented in Ref. [33,34]. The R f values were combined with some easily calculable molecular descriptors, selected by stepwise multiple regression analysis (MR) and traditionally linked to good brain penetration (log D, PSA, the number of hydrogen bond donors and acceptors HD and HA, dipole moment DM). The same RP-chromatographic descriptors (R f , R f / PSA) were used in our earlier research to study the solutes that cross the BBB (CNS+) and the compounds that do not exhibit good brain penetration (CNS-) [33,34].
Initially the efficiency of our methodology based on a RP 18 TLC model was confirmed for a group of CNSactive drugs [33]; then the applicability of our approach was acknowledged for some sunscreens and excipients in pharmaceutical and cosmetic preparations, whose CNS bioavailability should be limited [34]. In this paper we are proving a high predictive value and a universal applicability of our analytical model for a large group of cases of significant structural diversity and different sites of action.
In our current study we managed to demonstrate that the proposed analytical and mathematical models can be applied to any chemical compound of both CNS+ and CNS-activity. We confirmed the retardation factor R f and the chromatography-derived descriptor R f /PSA proposed in our previous paper [33] as universal predictors of good brain absorption and suggested adding them as novel parameters to the list of the currently recognized conditions of the BBB permeability [30].

Materials
154 drugs analyzed during these investigations ( Figure 1 [33,34]. The chromatographic data and physicochemical descriptors of these compounds were introduced into the analysis.

Thin layer chromatography
Thin layer chromatography was performed on 10x20 cm glass-backed TLC plates coated with an RP-18 F 254s layer from Merck, Germany (layer thickness 0.25 mm). Before use, the plates were pre-washed with methanoldichloromethane 1:1 (v/v) and dried overnight in ambient conditions. Solutions of compounds 1-154 in methanol (1 μg·μL -1 , spotting volume 1 μL), were spotted with the Hamilton microsyringe, 15 mm from the plate bottom edge, starting 10 mm from the plate edge, at 8 mm intervals. The chromatographic plates were developed in a vertical chromatographic chamber lined with filter paper and previously saturated with the mobile phase vapor for 20 min. The mobile phase consisted of acetonitrile -pH 7.4 phosphate buffered saline 70:30 (v/v). The development distance was 95 mm from the plate bottom edge. After development, the plates were dried at room temperature and examined under UV light (254 nm) and with the Desaga CD60 densitometer (Multiwavelength Scan, 200-300 nm at 20 nm intervals). All chromatograms were repeated in duplicate, and the mean R f values were used in further investigations.

Statistical analysis
154 compounds analyzed during these investigations were divided into two subsets: the training set (compounds with the known experimental BBB permeability BB vivo, values 1-46) and the training set (compounds without the known experimental BBB permeability BB vivo, values 47-154).

Stepwise Multiple Regression Analysis
The physicochemical parameters related to the compounds' BBB permeability were determined by the use of MR analysis. The stepwise multiple regression analysis and the correlation analysis were carried out using STATISTICA 10.0 [42]. The values of the BBB permeability (BB vivo), determined for 46 cases, and the calculated values of log BB (B1 and B2) for 154 analyzed compounds were used as dependent variables and other physicochemical molecular descriptors and chromatographic data as independent variables. The statistical significance (p-level) of a result as an estimated measure of the degree to which it represents the population was determined as p ≤0.05. The correlation matrix was used to correlate the biological activities with the various variables. If two independent variables showed a correlation greater than R 2 >0.4 one of them was removed.
Validation of the correlation models was carried out by the general internal cross-validation procedures: "leaveone-out" (LOO) and "leave-many-out" (LMO). In the LOO approach, one element is removed from the whole data set and used to verify the model generated with the remaining n-1 elements; the procedure is then repeated with another element. In the LMO method the data set is repeatedly divided into two subsets used for model generation and its verification, respectively. The predictive power of the developed models was evaluated using the following indicators: cross-validated squared correlation coefficient , predicted residual sum of squares (PRESS), standard deviation based on PRESS (S PRESS ), and standard deviation of error of prediction (SDEP). The LMO crossvalidation was applied by deleting 25% of the compounds in four cycles and predicting the BBB permeability of compounds deleted in each cycle from the corresponding equations derived from the reduced data set. Some criteria for the reliability prediction and robustness of the models are suggested in Ref. [43][44][45][46]: R 2 > 0. 6

Discriminant Function Analysis (DFA)
Investigations of the CNS activity of the drugs analyzed throughout this study were based on the discriminant function analysis (DFA) using the physicochemical and chromatographic data connected with the BBB permeability and selected by MLR analysis. All results were compared with the models obtained and tested in the previous investigations [33,34]. In this DFA 111 structurally different compounds were assigned to the CNS+ or CNSgroup of activity (defined according to Ref. [40], the remaining 43 compounds are not defined as CNS+/-). The classification functions were determined and validated for the compounds with the experimentally obtained BBB permeability (44 cases) and for compounds with calculated BBB permeability (B2) (111 cases). Discriminant function analysis is a multivariate technique that has two purposes: to separate cases from distinct populations; and to allocate new cases into previously defined populations [47]. The DFA was performed by STATISTICA 10.0 [41] software. In all subsequently performed analyses the stepwise method was applied. The model was formed by introducing subsequent variables that mostly contributed to group discrimination. After introducing sufficient grouping variables to the model (i.e. after obtaining the maximum probability of a priori classification) discriminant functions (roots) discriminating the activity groups were calculated. The maximum number of functions will be equal to the number of groups minus one, or the number of variables in the analysis, whichever is less. The quality of the discriminant function was evaluated by Wilks' lambda parameter, which is a multivariate analysis of variance statistics that tests the quality of group means for the variable(s) in the discriminant function [47]. The Wilks' lambda can assume values in the range of 0 (perfect discrimination) to 1 (no discrimination) and the statistical significance of roots (discriminant functions) used for interpretation was established on the basis of χ 2 tests of subsequent roots. Using statistically significant discriminant functions as the basis, canonical values were determined for the particular grouping variables. The scatter diagrams of the canonical values of the subsequent cases for the first two roots determined in the course of the analysis cannot be drawn to evaluate the discriminant power of the obtained models, because there are only two discriminated groups. The final phase of the qualitative analysis of the compounds was to determine the classification functions for each activity group. After calculation of the classification scores for a case, it is easy to decide how to classify it: in general we assign a case to a group for which it has the highest classification score. The tool used to determine how well the classification functions predict the group membership of cases is a classification matrix. The classification matrix shows the number of cases that were correctly classified (on the diagonal of the matrix) and those that were incorrectly classified.
The obtained discriminant models were evaluated by classification of 67 cases not included in the model (test set 47-113) with the known CNS+/-activity [40]. The values of the more important variables obtained with the DFA methodology were calculated for the test set. Than these values were introduced into the discriminant functions (12) and (13) obtained in validated DFA. We classify the case as belonging to the group for which it has the highest classification score. The new compounds were assigned to the CNS+ or CNSgroup of activity.
Ethical approval: The conducted research is not related to either human or animal use.

Stepwise multiple regression analysis
A number of computed or measured chromatographically (RP-18 TLC) BBB permeation descriptors have been presented in our previous papers [33,34,48]. The relationships between these descriptors and the observed (in vivo) BBB bioavailability were studied by MR and DF analysis for a group of compounds of diverse structures and successfully validated. The chromatographic data of the proposed analytical model (RP-18 TLC with the acetonitrile -pH 7.4 phosphate buffered saline 70:30 (v/v) mobile phase) -R f and R f /PSA were correlated with the parameters describing the ability of the compounds to penetrate the brain. The correlations are visible for the experimental BBB permeation as well as for the computed parameters B1, B2 and they are completed with the physicochemical parameters that are traditionally linked to the BBB permeation -log D, HD, HA, DM, log P, PSA, S a , M w [49]. The other chromatographic data (R M , R M /PSA, R f /M w , R M /M w ) were tested as the parameters describing the ability of the compounds to penetrate the brain, but only the R f and R f /PSA descriptors are significant and have a predictive capability. The theoretical BBB permeation parameters are given by the already mentioned equations: B1 (log BB = 0.139 + 0.152 log P -0.0148 PSA) [31] and B2 (log BB = 0.547 -0.016 PSA) [32]. It was demonstrated that the theoretical values B1, B2 describing the BBB permeability can be used to predict the BBB bioavailability of compounds even if they are not physically available [33]. A useful model of the blood-brain distribution was developed and validated. Models for discrimination between the CNS+ and CNScompounds were built on the basis of the chromatographic and physicochemical parameters (HD, HA, R f , R f /PSA, DM and B2) [33] and used successfully for a different group of compounds [34]. All compounds examined throughout these study were successfully assigned to one of two clusters: CNS+ or CNS-.
Such favorable results of analysis require confirmation. Our current research is aimed at proving the value of the previously proposed [33] analytical and mathematical models with the experiment involving a large (154 cases) and structurally diverse group of compounds.
The correlations of the BBB permeability factors (for the compounds with the experimentally obtained BBB permeability) with the chromatographic parameters (R f and R f /PSA) have been investigated (Equations (1)-(3)). The strongest relationship (R = 0.67, n = 46) can be found for B2 (Equation (3)), which is in line with the observation that simpler models are usually better [48]. All models contain the chromatographic parameters R f and R f /PSA. At the next stage of these investigations the stepwise analysis was carried out using all the physicochemical parameters collected for 46 compounds from the training set. After the analysis the following relationship (4) was obtained, containing statistically significant descriptors: On the basis of the stepwise analysis the parameter HA is introduced into the model. However, this parameter is too strongly correlated with another independent variable -B2 so it had to be removed from the equation. One of the descriptors used in the model was again B2; additionally, the ability of the compound to cross the BBB depends on the distribution coefficient (log D), the total dipole moment (DM) and the energy of the highest occupied molecular orbital (eH). The positive influence of log D (the parameter that takes into account the molecule's ionization level) on the compound's partition between the phases is clearly visible. The model explains 59% of the total variance. The possibility of replacing the difficult-to-obtain BB vivo parameter with the B2 value as the BBB penetration index was considered. Initially, the variables of the equation (4) given above were introduced, including HA and performing the regression analysis by the standard method: The result of the MR analysis for the B2 descriptor performed exclusively for the compounds with the known biological activity is very promising and reveals the importance of the same physicochemical parameters. The model explains 83% of the total variance. Next, the stepwise analysis by the forward selection method was performed. The result was very similar, although its statistical significance and the correlation coefficient R are higher. The resulting model explains 92% of the total variance in the studied group (46 compounds). The B2 parameter may be successfully used as a measure of the BBB permeability. It is determined by the same molecular descriptors so it may replace the difficult-tomeasure BB in vivo parameter which makes it possible to expand the group of studied compounds by including the molecules without the BBB permeability determined in vivo (in this case to the total of 154 cases). The MR analysis performed with the physicochemical data confirms this conclusion. The equation ( The computed parameter B2 is well defined by the chromatographic data which confirms the applicability of chromatography to predict the CNS bioavailability. The model (3) explains 44% of the total variance.  set 1-46) gives an equation that explains 31% of B2 variance (R = 0.56; n=154). The molecular descriptors and the chromatographic data as described above were used to generate the model given by Equation   The MR analysis based on the independent variables given by the B2 variability model presented earlier [33] has been repeated. The model obtained in this manner explained 95% of the variance for the group of 34 cases and involved the variables HA, HD, R f /PSA [33]. The resulting equation There are many parameters that enable the simplified prediction of the ability of drugs to enter the CNS. The molecular descriptors, describing the physicochemical properties of solutes and gathered in the "Rule of 5" have been selected after the analysis of thousands of compounds.
Complex patterns of drug -BBB interactions in vivo lead to discrepancies of the results of this rule application. Numerous sorptions and desorptions of a drug over its distribution path intensify these differences. Studying the structure and physicochemical properties arising from the compound structure is crucial but usually insufficient for the quantitative description of the distribution effect.
All the parameters proposed in the "Rule of 5" have the limited ranges of effectiveness for the BBB permeation (CNS+). These ranges are usually quite narrow and their extension or shifting is usually connected with other properties of a compound. There are very few mathematical models of biological activity based on a single independent value. Hence, the prediction of a drug brain absorption exclusively on the basis of its physicochemical properties (e.g. "Rule of 5") has certain limitations. Descriptors derived from these parameters are used. Linking two single parameters has given rise to the descriptors such as: log P -N+O, %PSA (PSA and S A ). Frequently used logarithmic derivatives (e.g. log M w ) facilitate a better fit to the complex behavior phenomena (with the tendency to reach a state of equilibrium) or to other logarithmic measures of activity (pA 2 , pD 2 , log BB etc.).
If compounds under investigation are available, their activity may be studied in specially designed analytical models, mimicking the conditions in vivo. The subtle differences in behavior must be multiplied to become traceable and expressed quantitatively. The application of a chromatographic method gives such possibilities both in studying the CNS bioavailability [29,33,34] and in other areas of medicinal chemistry [50]. The analytical models are used to study and compare the solutes of a great diversity of physicochemical properties. In the case of the chromatographic models the phase systems  are required that would make it possible to study all interesting cases without changing the conditions. Every chromatographic model imitates the complex conditions of in vivo interactions and must, in a simplified way, represent a specific phenomenon, not the universal features of the basic chromatographic separation. This is the reason why it is essential to introduce special features related to the studied biological activity. The phase affinity in the adsorption retention mechanism is of a hydro-lipophilic nature and the hydrophobic interactions are also important. The RP-18 phase exhibits the features of a pseudo-adsorptive support. Introducing a measure of the analyte polar surface area refines the effect of the chromatographic separation (R f ) with the properties strongly connected with the modeled phenomenon of the BBB permeation. It seems that taking into account the polarity of the molecules surface extends the possibility of studying the solutes of different structures. The variation of R f /PSA is larger than that of R f for the studied group of compounds.
In the experiment for 111 cases presented above (21 cases with the defined CNSparameter and 90 cases from the CNS+ group), the values of the R f /PSA parameter were ≥ 0.009 for CNS+ (14 errors with values 0.004-0.008) and < 0.009 for CNS-(3 errors with values 0.0092-0.0099); the values of R f for the same group were 0.10-0.90 for CNS+ and 0.50-0.90 for CNS-. These results clearly indicate the greater stability of the R f /PSA parameter towards the predicted BBB permeability.
The parameter B2 is also very stable in this experiment. In the group of 111 cases with the defined CNS+/-bioavailability the values of B2 for 90 CNS+ cases were ≥ -0.9 (6 errors were detected with the values -0.97 to -1.38). The values of the B2 parameter in the group of 21 CNScases were < -0.9 (3 errors were detected with the values 0.41 to -0.81). The analytes with the known permeability descriptor BBB vivo were within the same limits. Out of 6 CNScases 2 solutes were misclassified and in the group of 39 CNS+ compounds the number of incorrectly classified compounds was also 2.
The values of the BBB vivo parameter were > -0.9 for CNS+ (3 errors in the group of 30 cases) and < -0.9 for CNS-(6 cases without errors).

Discriminant Function Analysis
Drugs whose ability to cross the BBB has been measured experimentally are often described in the literature and databases [40,48] as CNS positive (CNS+) or CNS negative (CNS-). The qualitative parameter CNS+/CNSmay be used as a grouping variable in DFA to establish classification functions to predict the bioavailability of new drugs. 111 compounds from the training set of 154 cases were found in the database [40] and separated into two groups: CNS-(21 cases, code 0) and CNS+ (90 cases, code 1).
The stepwise DFA was carried out for 44 cases: 1-9, 11-31 and 33-46 with the measured BBB permeability BB vivo (for two solutes -acyclovir and tibolon no CNS+/classification has been found in Ref. [40]) on the basis of standardized data, using the chromatographic and physicochemical parameters selected in the course of the multiple regression analysis (8) The outcome of the last DFA was verified demonstrating the high classification power of the model (Wilks' lambda parameter = 0.394087, χ 2 = 37.71291, p-level = 0.000000).
The reliability of the model derived from DFA was determined by a cross-validation test based on the leaveone-out methodology. 44 cases (1-9, 11-31, 33-46) with the measured BBB permeability (BB vivo) were examined. The procedure is described above. The results obtained with the methodology are summarized in a cross-validation matrix ( Table 5). Using the results obtained via the cross-validation methodology we can confirm the reliability of the DFA model (the cross-validation error is 0%). The examination of the cross-validation matrix suggested that the classification probability was the same as the classification probability obtained a posteriori (97.73%) ( Table 5). The reliability of the analyses was proved. The models presented above not only described precisely the investigated groups of cases but have diagnostics value for new cases.
In order to validate the methods and confirm the discriminating value of the descriptors selected in the course of this study a group of 67 cases without the measured BBB vivo but with the BBB permeability known from the database [40] (47-113) was introduced.
The predictive values of the determined classification functions (equations: (12) and (13)) proposed for the classification of compounds as CNS+/were assessed. For this purpose, the variables calculated for the other compounds  with the known BBB permeability: CNS-(15 cases) and CNS+ (52 cases) were added to the raw variable file for the compounds 1-9, 11-31, 33-46 and then standardized together.
The classification functions: (12) and (13)  The effectiveness of the classification functions calculated in the previous paper (Equations 9 and 10 in Ref. [33]) and used successfully in Ref. [34] has been investigated for the cases 47-113. The results are presented in Table 6.  [33]) and presented below furnished 1 (out of 15) misclassified CNScase and 7 (out of 52) misclassified CNS+ cases. The total number of correctly classified cases reached 88.06%.
The investigation of cases with the known BB vivo parameter revealed 6 CNScompounds (2 classification errors) and 38 CNS+ cases with 5 classification errors, respectively. The results confirm the value of the chromatographic parameters R f /PSA and B2 in simple defining the ability of compounds to cross the BBB. The total number of the correctly classified cases was 84.09%.
Just like in the case of the training set (1-9, 11-31, 33-46) the stepwise DFA was carried out for the set of all compounds with defined CNS+/-(111 cases: 1-9, 11-31 and 33-113) involving the independent variables used in the previous analyses (Equations (14) and (15)). The classification matrix for the total of 111 cases is given in Table 7. The quality of DFA leading to the model (14)/(15) was verified by Wilks' lambda parameter 0.531980; χ 2 =67.21731; p-level = 0.000000 confirmed the model's high discriminating potential. Only 9 cases in the group of 111 compounds were incorrectly classified. Examination of the cross-validation matrix proved that the classification probability was higher than the classification probability obtained a posteriori (Table 8).

Conclusions
The results of our studies confirmed the values of the previously proposed [33] analytical and mathematical models for the experiment with a large and structurally diverse group of compounds. The presented results show that the proposed analytical and mathematical models are universal and can be applied to any chemical compound. The chromatographic data (R f and R f /PSA ) of the proposed analytical model (RP-18 TLC) are correlated with the parameters describing the ability of the compounds to penetrate the brain. The chromatographic parameters (R f and especially R f /PSA) obtained from the previous studies have a universal predictive capacity and can be therefore considered as one of the descriptors supplementing the "Rule of 5" parameter group [49].
The correlations are visible for the computed parameter B2 and they are completed with the physicochemical parameters that are traditionally linked to the BBB permeation -log D, HD, HA, DM ("Rule of 5", Ref. [49]). The results of the simple, inexpensive and very rapid chromatographic analysis proposed in this study may be applied to predict the BBB permeability of compounds isolated or synthetized on a very small scale.
The parameter B2, given by the equation B2 = log BB = 0.547 -0.016 PSA [32] has a predictive and discriminating value corroborated by the equations (12)- (15). The confirmed value of the computed BBB bioavailability indicator can be used to predict the BBB bioavailability of compounds without the experimental BBB permeability data [48]. The proposed chromatographic model of the BBB permeation based on RP-18 TLC and the computed B2 parameter can be used in future investigations of compounds of different chemical structures. The results of this and earlier [33,34] investigations confirm not only the assumed ability of compounds to cross the BBB but also the universal character and efficiency of all generated mathematical and analytical models to predict the BBB permeability of compounds available on a very small scale (sufficient for thin-layer chromatographic analysis) or even physically unavailable.
Taking into consideration the great importance of the calculated log BB descriptors -B1 and B2 revealed in the earlier analyses [33,34,48], a new element theoretically connected with the BBB permeation has been introduced that describes the molecular structure and is present in the B1 and B2 algorithms [31,32] (PSA value). A new descriptor R f /PSA has been created.
Introducing a measure of the analyte polar surface area refines the effect the chromatographic separation (R f ) with the properties strongly connected with the modeled phenomenon of the BBB permeation. It seems that taking into account the polarity of the molecules surface extends the possibility of studying the solutes of different structures. The variation of R f /PSA is larger than that of R f for the studied group of compounds. All the experiments performed so far [33,34] confirm the importance of the R f / PSA parameter as the value closely related to the CNS+/descriptor.
It was therefore concluded that the reversed-phase thin layer chromatographic data obtained on the RP-18 F 254s support and with acetonitrile-pH 7.4 phosphate buffered saline 70:30 (v/v) mobile phase may be used as the BBB permeability descriptors. The range of the R f / PSA parameter indicating the likely CNS bioavailability was, for the group of 34 cases studied earlier [33], ≥ 0.009 for CNS+ and < 0.009 for CNS-(without the errors). In an identical experiment for 111 cases presented above, the values of the R f /PSA parameter were ≥ 0.009 for CNS+ and < 0.009 for CNS-. These results clearly indicate the greater stability of the R f /PSA parameter towards the predicted BBB permeability.
The parameter B2 is also very stable in this experiment. In the group of 111 cases with the defined CNS+/-bioavailability the values of B2 for 90 CNS+ cases were ≥ -0.9. The values of the B2 parameter in the group of 21 CNScases were < -0. 9.
The values of the BBB vivo parameter were > -0.9 for CNS+ and < -0.9 for CNS-.