Combined Micellar Liquid Chromatography Technique and QSARs Modeling in Predicting the Blood–Brain Barrier Permeation of Heterocyclic Drug-like Compounds

The quantitative structure–activity relationship (QSAR) methodology was used to predict the blood–brain permeability (log BB) for 65 synthetic heterocyclic compounds tested as promising drug candidates. The compounds were characterized by different descriptors: lipophilicity, parachor, polarizability, molecular weight, number of hydrogen bond acceptors, number of rotatable bonds, and polar surface area. Lipophilic properties of the compounds were evaluated experimentally by micellar liquid chromatography (MLC). In the experiments, sodium dodecyl sulfate (SDS) as the effluent component and the ODS-2 column were used. Using multiple linear regression and leave-one-out cross-validation, we derived the statistically significant and highly predictive quantitative structure–activity relationship models. Thus, this study provides valuable information on the expected properties of the substances that can be used as a support tool in the design of new therapeutic agents.


Introduction
The development of new drugs with desired properties is a tedious, laborious, timeconsuming, and expensive process. Quantitative structure-activity relationship (QSAR) methods should be useful tools on this complicated path. The approach is based on the assumption that biological (pharmacokinetic) properties of structurally similar compounds can be quantitatively described by mathematical models. In addition, these models should predict with good probability the activity of structural analogs not yet synthesized. Although the establishment of QSAR models involves a number of steps and conditions, such as use of reliable and accurate input data, selection of relevant descriptors, and use of appropriate software and validation of the suggested model, the advantages of the QSAR methodology are not in doubt [1][2][3][4][5][6][7][8]. First of all, it reduces overhead costs, decreases the time of obtaining positive results, reduces animal testing, and respects the principles of Green Chemistry [5].
A crucial factor for a drug candidate is transport through the blood-brain barrier (BBB). Satisfactory transport through the BBB is an essential prerequisite for a potential drug to affect the central nervous system. However, to avoid side effects, the agents that act peripherally should not cross the BBB. In both cases, the permeability of the BBB must be known and should be evaluated at the earliest possible stage of testing [3][4][5][9][10][11][12].
where [M] is the total concentration of surfactant in the mobile phase minus the critical micellization concentration, cmc, K AM is the constant that describes solute-micelle binding, and k m is the solute retention parameter at zero micellar concentration, i.e., at surfactant monomer concentration equal to cmc. The K AM and k m parameters can be evaluated from the slope and intercept of experimental 1/k vs.
In our investigation, for the first time, the lipophilic properties of all the compounds (1-65) were experimentally determined by micellar liquid chromatography (MLC). The micellar lipophilicity parameters from MLC (as an experimental in vitro technique) and in silico data were combined with the aim of building the most satisfactory models for the prediction of the BBB permeation of 65 heterocyclic drug-like compounds. The linear quantitative structure-activity relationship models were produced using multiple linear regression on a database that consisted of different lipophilicity, polarity, electronic, and molecular size descriptors. The predictive ability of the developed models was validated by leave-one-out cross-validation (LOOcv).   belonging to the particular classes (I-VII).

Class
General Structure No R 1 R 2 I Int. J. Mol. Sci. 2022, 23, x FOR PEER REVIEW 4 of molecular size descriptors. The predictive ability of the developed models was validate by leave-one-out cross-validation (LOOcv). Table 1. Heterocyclic molecules  belonging to the particular classes (I-VII).    belonging to the particular classes (I-VII).

Class
III molecular size descriptors. The predictive ability of the developed models was validate by leave-one-out cross-validation (LOOcv).
IV molecular size descriptors. The predictive ability of the developed models was validate by leave-one-out cross-validation (LOOcv).

Results
Chromatographic retention parameters (k) were calculated according to th following relationship: where tR and tM are retention times for a given solute and an unretained compound respectively.
To describe the effect of surfactant concentration on solutes retention, we applie the Foley equation and four effluents with different SDS concentrations: 0.1, 0.105, 0.1 and 0.12 mol L −1 . The obtained results are presented in Figure 1

Results
Chromatographic retention parameters (k) were calculated according to th following relationship: where tR and tM are retention times for a given solute and an unretained compoun respectively.
To describe the effect of surfactant concentration on solutes retention, we applie the Foley equation and four effluents with different SDS concentrations: 0.1, 0.105, 0.1 and 0.12 mol L −1 . The obtained results are presented in Figure 1 as the 1/k vs. [M relationships (Equation (1)) for five chosen compounds. The parameters of Equation ( for all compounds tested, together with coefficients of determination R 2 , are presented Table 2.

Results
Chromatographic retention parameters (k) were calculated according to the following relationship: where t R and t M are retention times for a given solute and an unretained compound, respectively. To describe the effect of surfactant concentration on solutes retention, we applied the Foley equation and four effluents with different SDS concentrations: 0.1, 0.105, 0.11, and 0.12 mol L −1 . The obtained results are presented in Figure 1 as the 1/k vs. [M] relationships (Equation (1)) for five chosen compounds. The parameters of Equation (1) for all compounds tested, together with coefficients of determination R 2 , are presented in Table 2.
Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log P o/w ) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (  ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010).

Chromatographic Data
In our previous studies [75], all the compounds presented in Table 1 were analyzed using RP HPLC with three different stationary phases imitating biological partitioning: ODS (octadecylsilyl), IAM (artificial immobilized membrane), and Cholester (immobilized cholesterol). As lipophilicity descriptors, log k w parameters were used, describing solute retention in the system with 100% aqueous mobile phase, calculated by linear extrapolation. Presently, to mimic biodistribution, we have used the micellar chromatography technique with SDS as the mobile phase component and the Foley equation to describe solute retention. Very good linear relationships (R 2 > 0.9) were obtained for all tested compounds ( Figure 1, Table 2), confirming that the Foley equation correctly describes the retention of solutes in the tested chromatographic systems. Unfortunately, due to the strong retention of the tested compounds, the intercepts for all equations are negative. This is inconsistent with the physico-chemical interpretation of the regression coefficient of this equation: the intercept is equal to the reciprocal of k m , i.e., the retention factor in the system in which the concentration of unbound surfactant ([M]) in the effluent is equal to zero. This value may in no case be less than zero. For this reason, we decided to use log (k m /K AM ) values calculated from the slopes of Equation (1) as micellar lipophilicity descriptors of the compounds. The rationale is that both parameters (k m and K AM ) characterize lipophilic properties of solutes: their affinity to the stationary phase modified by the surfactant (k m ) and binding to the micelles (K AM ). Moreover, in our previous research [76] on the group of pesticides, we compared both micellar parameters (log k m and log K AM ), obtaining a very good rectilinear relationship with R 2 = 0.9724. To assess the correctness of our deductive reasoning, we examined the correlation between log (k m /K AM ) values and partition coefficients log P o/w obtained in silico from molecular structures of compounds, commonly accepted as lipophilicity descriptors. They were compared with analogous relationships for other chromatographic lipophilicities (log k w ) evaluated for ODS, IAM, and Cholester columns. The graphs presented in Figure 2 show the correct (direct proportion) relationships with moderate fit (R 2 > 0.6) but the best one for the micellar parameter (R 2 = 0.7980). The correlations between different chromatographic parameters considered as lipophilicity descriptors (log k w and log (k m /K AM ) are also moderate-R 2 in the range 0.6002-0.6990. The above relationships confirm that micellar parameters can be considered as lipophilicity descriptors.

In Silico Data
The compounds investigated have in silico log BB values in the range −0.293-0.712 (Table 3), and they penetrate the blood-brain barrier better (log BB > 0) or weaker (log BB < 0). Without more in-depth research, it is impossible to decide which may be CNS-active. It is clear that CNS activity requires BBB permeation, but some drugs that are not CNS-active may still pass through the BBB and show no activity because they do not interact with any CNS targets. Similarly, some drugs with an expected peripheral site of action may pass through the BBB, leading to undesirable side effects on the CNS. Molar weights of compounds range from 242.28 to 387.26 g mol −1 (Table 3) and meet the rule formulated by Lipiński and coworkers (The Rule of 5, Ro5) [77], i.e., MW ≤ 500 g mol −1 or one of the "Rules of Thumb" proposed by Clark [78] (MW ≤ 450 g mol −1 ) for brain permeation by drugs and clinical candidates. The numbers of hydrogen bond acceptors HBA ≤ 10, and the numbers of hydrogen bond donors HBD ≤ 5. HBDs also fulfill Ro5. The investigated molecules are bases: HBD = 0 for all compounds [75], with HBA ranging from five to eight (Table 2). Moreover, the polar surface areas, described by TPSA values, are in the range 48.27-57.50 Å 2 , which meets the next principle given by Clark [78]. For good brain permeation, the polar surface area of the compound should be below a certain limit. In the literature on the subject, there are two differing limits: 90 Å 2 suggested by van de Waterbeemd et al. [36] and a lower limit of 60-70 Å 2 proposed by Kelder et al. [33]. The test substances (except compounds from groups II and III) satisfy both limits. Table 3 also provides values of parachor ( Physico-chemical parameters characterizing the investigated compound logarithm of the partition coefficient (log Po/w) in the n-octanol/water sy numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bon molecular weight (MW), topological polar surface area (TPSA), polarizability parachor (Ƥ), are presented in Table 3. Also included are the log BB paramete values were evaluated in silico from molecular structures (ACD Percepta softw log BB* parameters from Table 3 were calculated in our previous studies [75], equation derived for 23 structurally similar compounds with known experimen values. The relationship between both, log BB and log BB*, parameters is linear good (R 2 = 0.9010). the regression coefficient of this equation: the intercept is equal to the reciprocal of km, i.e., the retention factor in the system in which the concentration of unbound surfactant ([M]) in the effluent is equal to zero. This value may in no case be less than zero. For this reason, we decided to use log (km/KAM) values calculated from the slopes of Equation (1) as micellar lipophilicity descriptors of the compounds. The rationale is that both parameters (km and KAM) characterize lipophilic properties of solutes: their affinity to the stationary phase modified by the surfactant (km) and binding to the micelles (KAM). Moreover, in our previous research [76] on the group of pesticides, we compared both micellar parameters (log km and log KAM), obtaining a very good rectilinear relationship with R 2 = 0.9724. To assess the correctness of our deductive reasoning, we examined the correlation between log (km/KAM) values and partition coefficients log Po/w obtained in silico from molecular structures of compounds, commonly accepted as lipophilicity descriptors. They were compared with analogous relationships for other chromatographic lipophilicities (log kw) evaluated for ODS, IAM, and Cholester columns. The graphs presented in Figure 2 show the correct (direct proportion) relationships with moderate fit (R 2 > 0.6) but the best one for the micellar parameter (R 2 = 0.7980). The correlations between different chromatographic parameters considered as lipophilicity descriptors (log kw and log (km/KAM) are also moderate-R 2 in the range 0.6002-0.6990. The above relationships confirm that micellar parameters can be considered as lipophilicity descriptors.

In Silico Data
The compounds investigated have in silico log BB values in the range −0.293-0.712 (Table 3), and they penetrate the blood-brain barrier better (log BB > 0) or weaker (log BB < 0). Without more in-depth research, it is impossible to decide which may be CNS-active. It is clear that CNS activity requires BBB permeation, but some drugs that are not CNS-active may still pass through the BBB and show no activity because they do not interact with any CNS targets. Similarly, some drugs with an expected peripheral site of action may pass through the BBB, leading to undesirable side effects on the CNS. Molar weights of compounds range from 242.28 to 387.26 g mol −1 (Table 3) and meet the rule formulated by Lipiński and coworkers (The Rule of 5, Ro5) [77], i.e., MW ≤ 500 g mol −1 or The relationships between log P o/w and log k w, ODS , log k w, IAM , log k w , Cholester [75], and log (k m/ K AM ) values.
In our procedure, parameters characterizing the lipophilic, structural, and electronic properties of molecules, i.e., micellar parameter (log k m /K AM ), MW, TPSA, HBA, α, and  , will be used as independent variables. To ensure the variables have a minimal impact on each other (to keep the principle of orthogonality), we checked for similarities among them. The results are presented in Figure 3. Here, we can see three groups of strongly correlated descriptors: (I) including TPSA and HBA (99.62%) characterizing the polar nature of the molecule and its ability to form hydrogen bonds; (II) consisting of polarizability, parachor, and molar weight (93.16%) related to the size of the molecule; and a single-element group (III) containing NRB describing molecule flexibility.

Establishment of Quantitative Structure-Activity Relationships
The establishment of QSAR models involves the use of reliable and accurate input data, selection of relevant descriptors, use of appropriate software, and validation of the suggested model [79,80] Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). Physico-chemical parameters charact logarithm of the partition coefficient (lo numbers of hydrogen bond donors (HBD), molecular weight (MW), topological polar parachor (Ƥ), are presented in Table 3. Als values were evaluated in silico from molec log BB* parameters from Table 3 were calc equation derived for 23 structurally similar values. The relationship between both, log good (R 2 = 0.9010).
, log BB, and log BB*) and in vitro (log k m /K AM ) data. The models were produced using the multiple linear regression (MLR) technique on a database that consisted of 65 recently discovered druglike compounds. The linear quantitative structure-activity relationships (QSARs) were presented for the modeling of log BB values. The developed models were validated by leave-one-out cross-validation (LOOcv).
van de Waterbeemd et al. [36] and a lower limit of 60-70 Å 2 proposed by Kelder et al. [33]. The test substances (except compounds from groups II and III) satisfy both limits. Table 3 also provides values of parachor (Ƥ) ranging from 497.81 to 763.55 m 3 mol −1 and polarizability (α) ranging from 27.55 to 41.70 Å 3 . In our procedure, parameters characterizing the lipophilic, structural, and electronic properties of molecules, i.e., micellar parameter (log km/KAM), MW, TPSA, HBA, α, and Ƥ, will be used as independent variables. To ensure the variables have a minimal impact on each other (to keep the principle of orthogonality), we checked for similarities among them. The results are presented in Figure 3. Here, we can see three groups of strongly correlated descriptors: (I) including TPSA and HBA (99.62%) characterizing the polar nature of the molecule and its ability to form hydrogen bonds; (II) consisting of polarizability, parachor, and molar weight (93.16%) related to the size of the molecule; and a single-element group (III) containing NRB describing molecule flexibility.

Establishment of Quantitative Structure-Activity Relationships
The establishment of QSAR models involves the use of reliable and accurate input data, selection of relevant descriptors, use of appropriate software, and validation of the suggested model [79,80]. Our models have involved descriptors characterizing solutes' lipophilicity (micellar log (km/KAM) values), polarity (TPSA, HBA), flexibility (NRB), and size (MW, α, Ƥ). We used in silico (HBA, NRB, MW, TPSA, α, Ƥ, log BB, and log BB*) and in vitro (log km/KAM) data. The models were produced using the multiple linear regression (MLR) technique on a database that consisted of 65 recently discovered drug-like compounds. The linear quantitative structure-activity relationships (QSARs) were Validation is a necessary step to establish the quality of a QSAR model [81][82][83]. In our investigations, traditional validation metrics were applied: the mean squared error (MSE), the coefficient of determination (R 2 ), the determination coefficient adjusted (R 2 adj ), and the determination coefficient predicted (R 2 pred ). R 2 adj is used to compare the goodness-of-fit for regression models that contain differing numbers of independent variables while R 2 pred determines how well a regression model makes predictions. These coefficients (R 2 , R 2 adj , R 2 pred ) have values between zero and one, and the closer to one, the more accurate the model. The MSE is used to assess the predictive ability and accuracy of the model, and models with small MSE values yield more highly reliable predictions. The derived models were compared and assessed by leave-one-out cross-validation (LOO), and the resulting determination coefficient (Q 2 LOO) and PRESS were calculated (Table 4). PRESS is a good estimate of the real prediction error of the model. It assesses a model's predictive ability and, in general, the smaller the PRESS value, the better the model's predictive ability [84]. The calculated global PRESS value must be lower than the sum of the squares of the response values of the total observations (SS). This proves that the developed models predict better than chance [83]. A reasonable QSAR model should have Q 2 LOO values greater than 0.6 or the ratio of PRESS/SS smaller than 0.4 [84]. QSAR models are only valid in the domain they were validated [85] so the determination of applicability domain (AD) is of great importance [86]. AD is a space of (physico-chemical) information, on which the model has been developed and for which it is applicable to make predictions for new compounds.
In the present work, we used the leverage approach (Williams plot) where the warning leverage, h*, was calculated according to: where n is the total number of samples in, and p is the number of descriptors involved in the correlation [87]. Table 4. Statistics of the established models M1-M12, M5*, M11*, and M12*: the coefficient of determination (R 2 , Q 2 ), the determination coefficient adjusted (R 2 adj ), the determination coefficient predicted (R 2 pred ), the predicted residual error sum of squares (PRESS), the variance inflation factor (VIF), the sum of squared differences from the mean (SS), the mean squared error (MSE), F-value, p-value; * the highest value. Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). We searched for relationships between solute property (SP), i.e., log BB, and its lipophilicity, molar size, and flexibility descriptors, that is: log SP = a 0 + a 1 lipophilicity + a 2 d I + a 3 d II + a 4 d III (4) where a 0 -a 4 are regression coefficients, and d I , d II , and d III are molecular descriptors from group I, II, and III, respectively. Accounting for possible combinations of independent variables (descriptors from groups I, II, and III) for both parameters (log BB and log BB*), twelve models denoted as M1-M6 (for log BB) and M7-M12 (for log BB*) have been obtained. Table 4 contains the R 2 , R 2 adj , R 2 pred , PRESS, and VIF (variance inflation factors) values calculated for these models, which became the basis for their preliminary evaluation leading to the selection of the most promising ones. Very high R 2 values (>>0.8) indicate that all M1-M12 equations are very good for modeling the data included (good feet of dataset). All R 2 pred values are >>0.6, indicating high predictive ability of the models. The decrease in the value of      7)). They explain both the direction and strength of the impact of a given descriptor on the calculated biological parameter. The correlations shown in segments A illustrate the relationships between the actual and predicted response, i.e., between log BB or log BB * values from Table 3, and these predicted by the QSAR models were developed (Equations (5)- (7)). The applicability domain (AD) was also evaluated and visualized as the Williams plots (segments C). The results proved that the obtained models are valid within the domain for which they were developed.

Model
The results (Figures 4B, 5B and 6B) indicate that the NRB parameter seems to be of negligible importance in the case of BBB permeation. Therefore, it was checked whether the omission of this descriptor would affect the evaluation of the derived models. The rationale is to capture the most important properties of compounds and build them into the QSAR model without overfitting the data. The results are presented as the following QSAR equations: The analysis of statistics of the above models, i.e., lower R 2 adj , R 2 pred , and PRESS values ( Table 4), indicates that the new models (omitting the NRB) are better for predicting substance permeation through the BBB. The new models were also cross-validated, and the results are presented graphically in Figures 7-9.        The results ( Figures 4B, 5B and 6B) indicate that the NRB parameter seems to negligible importance in the case of BBB permeation. Therefore, it was checked wh the omission of this descriptor would affect the evaluation of the derived models. rationale is to capture the most important properties of compounds and build them the QSAR model without overfitting the data. The results are presented as the follo QSAR equations: The analysis of statistics of the above models, i.e., lower R 2 adj, R 2 pred, and PR values ( Table 4), indicates that the new models (omitting the NRB) are bette predicting substance permeation through the BBB. The new models were cross-validated, and the results are presented graphically in Figures 7-9.

Lipophilicity
All derived models predict an increase in the BBB permeation with an increase of the substance lipophilicity. Thus, lipophilic compounds have a greater BBB permeability than hydrophilic. However, this influence is not indicated as dominant. Lipophilicity is undoubtedly an important parameter affecting BBB penetration, and it is the base parameter used in different QSARs modeling. Generally, transport of small molecules through membranes occurs via passive diffusion: a molecule dissolves in the phospholipid bilayer, diffuses across it, and then dissolves in the aqueous solution at the other side of the membrane. This process is closely related to the lipophilicity of the molecule: it cannot be too lipophilic, because it will not dissolve in the aqueous environment surrounding the bilayer on both sides; however, it also cannot be too hydrophilic, because it will not penetrate the lipid bilayer. Compounds investigated in this research are moderately lipophilic. Their lipophilicity, as assessed by log P o/w , ranges from 0.868 to 4.638, for which a positive effect on log BB is always expected.

HBA
Our results indicate HBA as the dominant factor affecting the log BB values. As the number of hydrogen bond acceptors in the molecule increases, its ability to permeate the BBB decreases. Solute polarity and the ability to form hydrogen bonds increase its solubility in the aqueous environment of the membrane, and highly polar molecules do not easily enter the hydrophobic environment of the BBB. In most QSARs models, the dominant descriptor of molecules' polarity is the polar surface area. Clark [78] and Kelder et al. [33] presented linear regression between log BB and PSA only, for a group of 45 drugs. In our models, this parameter was also used (M1-M3 and M9), but due to the statistical evaluation the models including HBA turned out to be more accurate. However, it should be noted that the TPSA values of the test compounds meet the requirements for active substances.

Lipophilicity
All derived models predict an increase in the BBB permeation with an increase o substance lipophilicity. Thus, lipophilic compounds have a greater BBB permeab than hydrophilic. However, this influence is not indicated as dominant. Lipophilic undoubtedly an important parameter affecting BBB penetration, and it is the parameter used in different QSARs modeling. Generally, transport of small mole through membranes occurs via passive diffusion: a molecule dissolves in phospholipid bilayer, diffuses across it, and then dissolves in the aqueous solution a other side of the membrane. This process is closely related to the lipophilicity o molecule: it cannot be too lipophilic, because it will not dissolve in the aqu environment surrounding the bilayer on both sides; however, it also cannot be hydrophilic, because it will not penetrate the lipid bilayer. Compounds investigate this research are moderately lipophilic. Their lipophilicity, as assessed by log Po/w, ra from 0.868 to 4.638, for which a positive effect on log BB is always expected.

HBA
Our results indicate HBA as the dominant factor affecting the log BB values. A number of hydrogen bond acceptors in the molecule increases, its ability to permeat BBB decreases. Solute polarity and the ability to form hydrogen bonds increas solubility in the aqueous environment of the membrane, and highly polar molecule not easily enter the hydrophobic environment of the BBB. In most QSARs models dominant descriptor of molecules' polarity is the polar surface area. Clark [78] Kelder et al. [33] presented linear regression between log BB and PSA only, for a grou 45 drugs. In our models, this parameter was also used (M1-M3 and M9), but due t statistical evaluation the models including HBA turned out to be more accu However, it should be noted that the TPSA values of the test compounds mee requirements for active substances. The highest TPSA value is 83.80 Å 2 , and the num of hydrogen bond acceptors HBA does not exceed 10.

Molecular Size
In this study, three descriptors for molecular size were proposed, i.e., molecular weight MW, polarizability α, and parachor Physico-chemical parameters characterizing the investigated com logarithm of the partition coefficient (log Po/w) in the n-octanol/w numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotata molecular weight (MW), topological polar surface area (TPSA), pola parachor (Ƥ), are presented in Table 3. Also included are the log BB p values were evaluated in silico from molecular structures (ACD Percep log BB* parameters from Table 3 were calculated in our previous stud equation derived for 23 structurally similar compounds with known ex values. The relationship between both, log BB and log BB*, parameters good (R 2 = 0.9010). Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010). Physico-chemical parameters characterizing the investigated compou logarithm of the partition coefficient (log Po/w) in the n-octanol/water numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable molecular weight (MW), topological polar surface area (TPSA), polarizab parachor (Ƥ), are presented in Table 3. Also included are the log BB param values were evaluated in silico from molecular structures (ACD Percepta s log BB* parameters from Table 3 were calculated in our previous studies [ equation derived for 23 structurally similar compounds with known experim values. The relationship between both, log BB and log BB*, parameters is lin good (R 2 = 0.9010). values, the permeation of the test substances through the BBB increases. This is in contrast to the results obtained by different researchers who noticed the negative effect of molecule size on compounds' permeation through biological membranes [88][89][90]. The positive effect of molecular size on the log BB (and log BB*) values observed herein could be explained by the partition mechanism of this process. Similarly, Platts et al. [91] obtained the positive effect of molecular size on the permeability through the skin. This relationship is a reflection of the correlation between the size of the molecule and its lipophilicity. It is especially important in the case of low molecular weights (MW < 400 mg mol −1 ), such as those in the presented research. Kouskoura et al. [12] noted that moderately increased molecular weight of the compound guarantees that its lipophilicity is sufficient to dissolve in the phospholipid bilayer and enter the BBB via passive diffusion. The same effect was observed in our previous research [92].

Flexibility
Research reveals that blood-brain barrier partitioning is governed not only by solute lipophilicity and polarity but also solute flexibility, solute-membrane flexibility, and solutemembrane binding. In general, ease in traversing the membrane depends on the flexibility of a compound. Limited flexibility can be considered as a merit, and higher flexibility can be proved to be a demerit [93]. The increase in BBB penetration of the solute with increasing solute flexibility was described by Iyer et al. [3]. Veber et al. [94] have found that increasing solute molecular flexibility (measured by the number of rotatable bonds) promotes a decrease in oral bioavailability. It suggests a parabolic relationship between log BB and molecular flexibility. That is, some "amount" of flexibility enhances log BB, but too much flexibility will diminish log BB. Our research ( Figures 4B, 5B and 6B) indicates a slight and ambiguous (positive for M12 and negative for M5 and M11 models) influence of the NRB value on the penetration of the tested substances through the blood-brain barrier. The numbers of rotatable bonds calculated for tested compounds are in the range from two to six, with average value equal to 3.2. Probably, in the case of the investigated molecules, these values are close to the maximum of the aforementioned parabolic relationship. Ultimately, NRB was considered a negligible factor and omitted in subsequent QSAR models.

Chromatographic Conditions
As the mobile phases buffered, SDS mixtures (0.10; 0.105, 0.11, and 0.12 mol L −1 ) with 7% (v/v) addition of isopropanol were used. The buffer was prepared from 0.01 mol L −1 solutions of disodium phosphate and citric acid, and the pH 7.4 value was fixed before mixing with an organic modifier. The flow rate was 1 mL min −1 . Solutes samples were dissolved in acetonitrile-c.a. 0.005 mg mL −1 . The compounds were detected under UV light at λ max 254 nm. All measurements were carried out at 25 • C. The dead time values were measured from non-retained compound (e.g., sodium chloride) peaks. All reported k values are the average of at least three independent measurements.

In Silico Calculations
Molecular weight (MW), topological polar surface area (TPSA), polarizability (α), parachor ( Physico-chemical parameters characterizing the investigated compounds, i.e., the logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, the numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (NRB), molecular weight (MW), topological polar surface area (TPSA), polarizability (α), and parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All the values were evaluated in silico from molecular structures (ACD Percepta software). The log BB* parameters from Table 3 were calculated in our previous studies [75], using the equation derived for 23 structurally similar compounds with known experimental log BB values. The relationship between both, log BB and log BB*, parameters is linear and very good (R 2 = 0.9010).

Conclusions
The linear quantitative structure-activity relationship models are presented for the modeling and prediction of the BBB permeation of heterocyclic drug-like molecules with promising activity. The models were produced using the multiple linear regression technique on a database that consisted of 65 recently discovered compounds. Among the different lipophilicity, polarity, electronic, and molecular size descriptors that were considered as inputs to the model, four variables were selected, i.e., micellar parameter characterizing the solutes lipophilicity log (k m /K AM ), the number of hydrogen bond acceptors HBA connected with polarity, and parachor or molecular weight (MW), describing the molecular size. The rationale was to combine in the model in vitro (micellar lipophilicity parameters log (k m /K AM )) and in silico (α, MW, Physico-chemical parameters characterizing the inv logarithm of the partition coefficient (log Po/w) in the numbers of hydrogen bond donors (HBD), acceptors (HBA molecular weight (MW), topological polar surface area ( parachor (Ƥ), are presented in Table 3. Also included are values were evaluated in silico from molecular structures log BB* parameters from Table 3 were calculated in our p equation derived for 23 structurally similar compounds w values. The relationship between both, log BB and log BB* good (R 2 = 0.9010).
) data. The accuracy of the proposed MLR models was illustrated using LOO cross-validation. The predictive ability of the developed models was found to be satisfactory and could be used for designing a similar set of heterocyclic compounds. Our research confirmed that solute polarity is one of the most important properties affecting the BBB permeation. The increase of HBA values decreases the log BB (or log BB*) values. In the QSARs models established in our studies, the number of HBAs was indicated as the dominant factor. The log BB (as well as log BB*) values increase with micellar (chromatographic) log (k m /K AM ) parameters. This means that more lipophilic drugs have a greater BBB permeability than less lipophilic. This is not the same as their CNS activity, because some compounds that are CNS-inactive may still pass through the BBB and show no activity because they do not interact with any CNS targets. Molecule size descriptors (MW, Physico-chemical parameters characterizing the investigated compounds, i.e., logarithm of the partition coefficient (log Po/w) in the n-octanol/water system, numbers of hydrogen bond donors (HBD), acceptors (HBA), and rotatable bonds (N molecular weight (MW), topological polar surface area (TPSA), polarizability (α), parachor (Ƥ), are presented in Table 3. Also included are the log BB parameters. All values were evaluated in silico from molecular structures (ACD Percepta software). log BB* parameters from Table 3 were calculated in our previous studies [75], using equation derived for 23 structurally similar compounds with known experimental log values. The relationship between both, log BB and log BB*, parameters is linear and v good (R 2 = 0.9010).