INTRODUCTION

Age-related macular degeneration, diabetic retinopathy, retinitis pigmentosa, optic nerve damage associated with glaucoma, and bacterial/fungal endophthalmitis are some of the vision impairing conditions afflicting the posterior segment of the eye (1). The exposure of affected target tissues to therapeutic quantities of drugs for appropriate lengths of time is an enormous challenge faced in the successful treatment of these conditions. Topical, systemic, and periocular routes of administrations are limited by their systemic side effects and solute permeability limitations (2). Intravitreal injections, providing more efficient drug delivery to the back of the eye compared to the above routes, are effective for a wide range of therapeutics including low molecular weight drugs as well as macromolecules. Beneficial effects have been shown in humans for a variety of drugs including low molecular weight drugs (3), oligonucleotides (Fomivirsen®) (4), and monoclonal antibodies or their fragments (Avastin® and Lucentis®) (5). The duration of effects of an intravitreally administered drug depends on the retention of the injected drug at the site of administration. The higher the intravitreal half-life of a drug injected in the vitreous of the eye, the greater is the anticipated duration of the pharmacological response. Longer half-life of a drug makes it amenable for less frequent dosing. However, it is time consuming and cost intensive to determine the intravitreal half-life of drugs in vivo. A scientific understanding of the relationship between physicochemical properties such as molecular weight (MW), lipophilicity, and solubility and drug elimination is essential for the development of therapeutic agents with desired intravitreal pharmacokinetic (PK) properties.

Quantitative structure–pharmacokinetic relationship (QSPKR) approaches attempt to derive relationships between the molecular properties of drugs and their PK properties. Once a validated model is developed, it might be useful in predicting PK parameters of a new compound or in selecting a molecule with desired PK properties from a library of compounds. Such an approach, serving as an initial screen for compounds, is expected to minimize the animal use. Recent years have witnessed a steep rise in the QSPKR based approaches to establish the relationship between the physicochemical/molecular properties and various PK parameters such as volume of distribution (6) and clearance (7). In addition, QSPKR had been used to predict the aqueous solubility (8) and plasma protein binding (9) of drug molecules.

In ocular drug delivery, QSPKR type of analysis was attempted to correlate physicochemical properties to drug permeation across cornea (10–14), conjunctiva (12,15,16), sclera (12), and retinal pigment epithelium (17). Maurice (18,19) was the first to study the QSPKR in vitreous humor. His group established the relationship that high molecular weight (MW) compounds have a high aqueous/vitreous ratio indicating the elimination of compounds by way of the anterior chamber. Also, there were few reports aimed at understanding spatial–temporal distribution of drug in the vitreous using mathematical modeling and computer simulations (20–25). These studies aimed at simulating the diffusion and distribution of the drug in the vitreous humor after intravitreal injection.

A systematic study was carried out by Miller (26) to correlate the physicochemical properties and drug elimination from vitreous following intravitreal injection. These authors had shown an excellent correlation between lipophilicity (Log D at pH 7.2) and vitreal elimination half-life following intravitreal injection in albino rabbits. However, the QSPKR was developed using a small set of structurally similar molecules (ciprofloxacin, fleroxacin, ofloxacin, and sparfloxacin). In addition, there were some investigational efforts to correlate drug elimination with MW (27,28). Dias and Mitra (27) had shown an inverse relationship between the Log MW and vitreous elimination rate constant for high MW hydrophilic compounds (FITC–dextrans). Maurice (28) reported computer-generated concentration contours for the diffusional loss of drugs from the human vitreous and demonstrated that the vitreal half-lives of compounds increased with increasing MW. Thus, some previous reports indicated the dependence of vitreal half-life on MW or lipophilicity. The earlier QSPKR studies for vitreal elimination half-life based on a small set of chemically similar molecules limits their application to predict half-lives of structurally different molecules. The objective of the present study was to develop a QSPKR based model with diverse unrelated compounds from the available literature reports and to understand the influence of some important physicochemical properties including MW, lipophilicity, and solubility on intravitreal elimination half-life.

METHODS

Data Collection for Intravitreal Half-Lives of Solutes

Literature search was conducted using the database PubMed (1966–May 2007) to identify prior literature reports for analysis. Search terms included ‘intravitreal’, ‘vitreal’, ‘pharmacokinetics’, ‘elimination’, ‘half-life’, and ‘clearance’. Reference lists from all retrieved articles were also reviewed and citations were retrieved if they referred to intravitreal pharmacokinetics. Since most of the literature reported intravitreal half-life of molecules in rabbits (few in human and other species), data pertaining to studies conducted exclusively in rabbits were considered for analysis. Data from both albino (more commonly assessed) strain as well as pigmented rabbit strain was collected. Although not likely complete, we believe we captured majority of the literature reports related to vitreal half-life in this study. From literature reports, besides half-life, other data concerning the drugs including the salt form administered, injected dose, injection volume, and dosage form (solution or suspension) was also collected.

In most of the earlier literature reports, investigators estimated the elimination half-life of the compounds using a few time points without any compartment modeling. The half-lives estimated using this noncompartmental approach was listed in the tables as terminal/elimination half-life. Half-lives estimated in literature reports using one compartment model was listed as half-life based on K 10 (the only elimination rate constant from one-compartment model). In case of two compartment models, any reported half-lives including distribution half-life or t 1/2,α, elimination half-life or t 1/2,β, and t 1/2,K10 or half-life for elimination from the central compartment were tabulated.

Pharmacokinetic Analysis of Some Literature Reported Concentration–Time Profiles

For literature reports with full concentration–time course data and limited detail on the relevant half-lives, pharmacokinetic analysis was carried out in this study using WinNonlin (version 1.5, Pharsight Inc., CA) to determine the PK parameters. The concentration–time data were fit to different PK models, and the best fit model was selected on the basis of the correlation coefficients and Akaike Information Criterion (AIC) values. Estimated half-life values for these drugs were compared with the literature reported values and used for regression analysis.

Collection of Molecular Descriptors

MW, Log P, Log D at pH 7 (hereafter mentioned only as Log D) and intrinsic solubility (IS) for the listed compounds were collected from SciFinder Scholar, which reports values calculated using Advanced Chemistry Development (ACD/Labs) Software V8.14 for Solaris (1994–2007 ACD/Labs). Solubility at pH 7.4 for all the molecules was estimated using ACD/I-Lab Web service (ACD/Aqueous solubility 8.02). For macromolecules with no reported molecular descriptors in SciFinder Scholar, wherever available, literature reported values for MW, lipophilicity, and solubility were collected. Logarithmic transformation was carried out for those variables showing broad range or high standard deviation (MW, intrinsic solubility, solubility at pH 7.4 and half-life).

Model Construction

Selection of Independent Variables

Since the dataset included half-lives from albino and pigmented rabbits, a dummy variable pigmentation factor (PF) denoting 0 for albino and 1 for pigmented was included in the analysis. Also a dummy variable salt factor (SF) was included denoting 0 for non-salt form of molecules and 1 for molecules administered in the salt form. To understand the effect of dose administered on the vitreal disposition of molecules, a new variable dose number (DN = dose injected/solubility at pH 7.4) was included as an independent variable in the analysis. For regression analysis, the variables collected were divided into two sets: the first set included Log MW, Log D, DN, PF and SF. The second set included Log MW, Log P, DN, PF and SF. Multicollinearity (condition that exists when independent variables are highly correlated with each other) was tested for each set by evaluating the relationship between the above variables using the correlation matrix (SPSS 11.5, SPSS Inc., IL). Two variables were considered collinear, if the correlation coefficient (R) between them was greater than 0.5.

Multiple Regression Analysis of Data for Model Development

Multiple linear regression analysis of the selected non-collinear independent variables was performed to determine statistically significant relationships between the variables and vitreal half-life using SPSS 11.5 (SPSS Inc., IL) statistical software. Half-life from both the strains was included for regression analysis only if the difference in half-lives were greater than 5 h. In order to determine the individual contributions of the predictor variables and to attain a best predicting model, backward stepwise regression was used to develop the model. In this model, all variables were fit first, and then the non-significant variables were removed one at a time until all the remaining variables in the model contribute significantly to the regression equation. At each step, the variable with the smallest contribution to the model (highest P value) was removed until the P value was less than 0.05. The above procedure was undertaken for (1) the entire dataset (all), (2) albino, (3) pigmented, (4) acids, (5) bases, (6) zwitterions, (7) neutrals, (8) suspensions, (9) macromolecules, (10) all without suspensions, (11) all without macromolecules, (12) salts, and (13) all without salts.

From the various models generated with each group, the best-fit model was selected based on the goodness-of-fit statistical parameters—high R 2, larger F values, and lowest P value. Only cases with valid values for all variables were included in the analysis. Since the data set included different half-lives, regression analysis was done in two different ways. First, regression analysis was done using t 1/2,β from two-compartment models, t 1/2,terminal/elimination from non-compartmental analyses, and t 1/2,K10 for one-compartment models. Second, wherever full time course data is available for drugs obeying two-compartment model, beta half-lives were replaced with their K 10 half-lives. The half-lives exhibiting good correlation with the independent variables based on the statistical parameters were selected for model development. Outliers were identified based on the residual values obtained from the regression analysis. Only those outliers whose removal improved the correlation coefficient significantly were removed and the data reanalyzed.

Models for Subclasses of Compounds/Formulations

To better understand the vitreal elimination for compounds/formulations of various types, dataset was classified in five different ways. In the first classification, molecules were divided into two sets, albino and pigmented based on the rabbit strain used. In the second classification, molecules were divided as acids, bases, zwitterions and neutrals based on their pKa values. Based on the dose administered and its solubility, the third classification consisted of formulations separated as solutions and suspensions. If the dose administered (in grams per liter) exceeded the solubility of the compound, it was classified as a suspension. The fourth classification consisted of macromolecules (MW > 1000) and all compounds without macromolecules (MW < 1000). The fifth classification consisted salts and non-salt forms of the drugs. Multiple linear regression analysis was performed for each subset in a particular classification at a time. The best predicting model and the influencing variables were identified.

Validation of the Models

The best-fit model obtained from each subset was cross-validated against other subsets (e.g., albino equation assessed for its predictability of pigmented group, etc.). Also the best-fit model obtained from the entire dataset was tested for its predictive ability on various subsets.

RESULTS

Data Collection

Literature reports on vitreous half-life in rabbit models were identified for a total of 68 compounds. Of these, 49 were assessed in New Zealand white or albino rabbits and 19 were assessed in pigmented rabbits (11 were assessed in both). Table I summarizes the physicochemical properties of the compounds and their respective vitreal half-lives in New Zealand white rabbits. Table II summarizes the properties of compounds in pigmented rabbits. Compounds in both the datasets covered a wide range of structurally different molecules, with vitreal half-life values ranging from 0.87 to 218 h. The data set included molecules as small as methanol (MW 32) to macromolecules like bevacizumab (MW 149,000). The lipophilicity of the molecules spanned the entire range, from highly hydrophilic (Log P = −3.64 and Log D = −9.2) to highly lipophilic (Log P = 3.44 and Log D = 2.75) and the solubility ranged from 0.0018 to 31,722 g/L.

Table I Physicochemical properties and intravitreal half-lives of various solutes in New Zealand white (albino) rabbits
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Table II Physicochemical properties and intravitreal half-lives of various solutes in pigmented rabbits
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Pharmacokinetic Analysis of Data from Literature Reports

For 14 drugs (eight in albino; six in pigmented) for which entire concentration–time profile was available, pharmacokinetic analysis was carried out. Most of the drugs exhibited two compartments pharmacokinetics except gentamicin and methicillin in the albino group and etanercept and GSSG in the pigmented group. The literature reported vitreal half-life values (t 1/2,terminal from noncompartmental analysis) were found close to the calculated K 10 half-life values for most of the drugs except erythromycin in albino rabbits (Table III). Erythromycin exhibited two compartment kinetics with a beta half-life of 4.44 h against literature reported 10 h. In case of pigmented rabbits, significant differences were observed between the reported and estimated half-life for foscarnet (77 vs 32.55 h) and vancomycin (62.34 vs 32.67 h). Multiple regression analysis was carried out using these estimated values.

Table III Vitreal half-lives estimated in this study based on some previous reports
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Classification of Entire Dataset

Based on the acidic and basic pKa values, the molecules in the data set were divided into acids [13], bases [15], zwitterions [18] and neutrals [9] as shown in Table IV. Table V summarizes the physicochemical properties of 11 molecules administered in the suspension form. Molecules with MW > 1,000 are summarized in Table VI. Out of these ten macromolecules, all physicochemical properties could be obtained for only six molecules including three different MW ranges of FITC–dextrans. Other macromolecules with missing data were not included in the multiple regression analysis.

Table IV Classification of the molecules in the data set as acids, bases, zwitterions, and neutrals
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Table V Physicochemical properties and intravitreal half-lives of various molecules administered as suspensions to albino or pigmented rabbits
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Table VI Physicochemical properties and intravitreal half-lives of macromolecules (molecular weight > 1,000)
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Multiple Linear Regression Analysis of Entire Data

Correlation matrix of all the predictor variables indicated that Log D and Log P are highly correlated (R = 0.782, Table VII) with each other. The R values of all other variables were less than 0.5. Initial multiple linear regression analysis of the entire dataset using K 10 half-lives (from both one and two compartments) as dependent variable yielded linear relationship with the independent variables (both Log P set and Log D set). On the other hand, models were not significant when terminal half-lives were used. Hence, further analysis was carried out by substituting K 10 half-lives from one or two compartments, wherever feasible, along with terminal/elimination half-life for the rest of the compounds as the dependent variable. The results of the regression analysis along with the best-fit equations obtained and statistical parameters are summarized in Table VIII for Log D set of variables and in Table IX for Log P set of variables. Also the scatter plots of the best-fit models are shown in Fig. 1 (Log D set of variables) and Fig. 2 (Log P set of variables).

Fig. 1
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Scatterplots of the best-fit models developed using Log D set of variables along with the best-fit equation for entire dataset (a), and various subsets including albino (b), pigmented (c), acids (d), bases (e), zwitterions (f), neutrals (g), suspensions (h), all without suspensions (i), macromolecules (j), all without macromolecules (k), salts (l), all without salts (m). The excluded outliers are shown in the circles.

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Fig. 2
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Scatterplots of the best-fit models developed using Log P set of variables along with the best-fit equation for entire dataset (a), and various subsets including albino (b), pigmented (c), acids (d), bases (e), zwitterions (f), suspensions (g), all without suspensions (h), all without macromolecules (i), salts (j), all without salts (k). The excluded outliers are shown in the circles. The best-fit equation obtained for macromolecules and neutral compounds is the same as mentioned in the Log D set of variables.

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Table VII Correlation matrix of the predictor variables
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Table VIII Summary of the models developed with Log MW, Log D, PF, SF and Dose/sol 7.4 from different groups with the best-fit equations and statistical parameters
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Table IX Summary of the models developed with Log MW, Log P, PF, SF and dose/sol 7.4 from different groups with the best-fit equations and statistical parameters
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The best-fit models obtained from regression analysis of the entire dataset using Log D set of variables (Table VIII, Eqs. 1, 1a) or Log P set of variables (Table IX, Eqs. 1, 1a) indicated that amphotericin B and TA (both low dose (0.3 mg) and medium dose (4 mg) in albino group) were outliers. Correlation values (R 2) increased from 0.5 to 0.7 with the omission of above data for each drug, justifying their exclusion from the dataset. Further, influence analysis indicated the significance of TA high dose (16 mg) for the model. Also, exclusion of this observation decreased the R 2 values of models and hence it was retained in the dataset. Both the best-fit equations (Fig. 1a and Fig. 2a) included Log MW, Log D or Log P and DN as the contributing variables for half-life.

Models for Subclasses of Data

For regression analysis of the first classification, initially the albino group was subjected to backward regression analysis. Similar to the entire dataset, exclusion of amphotericin B and TA (both low dose (0.3 mg) and medium dose (4 mg) in albino group) improved the model-fits with high correlation (R 2 = 0.820 in the Log D variables set and 0.738 in the Log P variables set; see equations 2 and 2a in Tables VIII and IX). In case of pigmented group, no statistically significant model was obtained with Log D variable set (Eq. 3, Table VIII). However, replacing Log D with Log P yielded a model with high correlation value (R 2 = 0.852, Fig. 2c) indicating the influence of Log P along with Log MW and DN in this group of molecules (Eq. 3, Table IX).

In the second classification, the best-fit equation obtained for acidic molecules indicated only Log D (Eq. 4, Table VIII; R 2 = 0.554) or Log P (Eq. 4, Table IX; R 2 = 0.435) as the influencing variable. However, both the equations had low correlation values. In case of bases, the best-fit equations obtained did not include Log MW (Eq. 5, Tables VIII and IX) and had high correlation values (R 2 = 0.930, Fig. 1k and R 2 = 0.88, Fig. 2j). In case of zwitterions, exclusion of amphotericin B improved the model and the model included Log D (Eq. 6a, Table VIII) or Log P along with SF (Eq. 6a, Table IX) in the best-fit equations. The best-fit models developed for neutral molecules had only Log MW and DN in their equations with high R 2 values (0.904, Fig. 1m; Eq. 7 in Tables VIII and IX).

In the third classification, the best-fit models obtained for suspensions included Log MW and DN (Eq. 8, Table VIII) or Log MW, Log P and DN (Eq. 9, Table IX) in the equation. The equation obtained with Log P had high correlation (R 2 = 0.844, Fig. 2d) and in both the models of suspensions, all the doses of TA were included. The best-fit equation obtained for the dataset without suspensions indicated Log MW and Log D (Eq. 9, Table VIII) or Log MW and Log P (Eq. 9, Table IX) as the contributing variables for vitreal half-life. However, the R 2 values for these models were less (0.555). Exclusion of amphotericin B from these models improved the best-fit significantly in both the cases (R 2 = 0.705, Fig. 1e; R 2 = 0.646, Fig. 2e).

The equation obtained with macromolecules had high correlation value (R 2 = 0.980, Eq. 10, Tables VIII and IX) and included Log MW, PF and SF as the contributing variables. However, this group included only six molecules including FITC–dextrans of three different MW (Fig. 1f). Exclusion of macromolecules resulted in models including all three variables Log MW, DN and Log D (Eq. 11, Table VIII) or Log MW, DN and Log P (Eq. 11, Table IX). Exclusion of amphotericin B and TA low and medium doses from this dataset improved the correlation values significantly (R 2 = 0.720, Fig. 1g; R 2 = 0.731, Fig. 2f).

The best-fit model obtained from molecules administered in the salt form included Log MW and Log D (R 2 = 0.911, Eq. 12, Table VIII) or only Log P (R 2 = 0.909, Eq. 12, Table IX) as the independent variable contributing to the vitreal half-life. However, the exclusion of these molecules from the entire dataset resulted in inclusion of Log MW, DN and Log D (R 2 = 0.480, Eq. 13, Table VIII) or Log MW, DN and Log P (R 2 = 0.516, Eq. 13, Table IX) in the model. Exclusion of amphotericin B and the low and medium doses of TA from this dataset improved the coefficient values in both the cases (R 2 = 0.710, Fig. 1i; R 2 = 0.691, Fig. 2h).

Cross-Predictive Ability of the Models Developed

The best-fit equations obtained from various models of different subsets were used for predicting the vitreal half-lives of the remaining molecules in the excluded set that was not used for the model development. The coefficient of determination between the predicted and actual half-lives (Q 2) of the different best-fit models is shown in Fig. 3. Poor correlation (Q 2) was observed in the presence of amphotericin B and triamcinolone acetonide in all the predicted subsets. However, the exclusion of both these molecules from all the predicted subsets increased the predictive ability of the models. In case of Log D set of variables, the model obtained without salts had good predictive ability on salts as indicated by its high correlation value (Q 2 = 0.8007, Fig. 3). The best-fit equations obtained with pigmented set (Q 2 = 0.6695), acids (Q 2 = 0.6649) and zwitterions (Q 2 = 0.6687) had reasonable predictive ability on their excluded sets. In the case of Log P set of variables, the model obtained without salts had good predictive ability on salts as indicated by its high correlation value (Q 2 = 0.9043). The next best-fit equation with high predictive ability was from the albino set (Q 2 = 0.8137) which predicted the pigmented set of molecules with good accuracy. The other best-fit equations with reasonable predictive ability in this group were from acids (Q 2 = 0.5926), pigmented set (Q 2 = 0.5736) and zwitterions (Q 2 = 0.5189).

Fig. 3
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Cross-predictive ability of the models developed from different subsets on rest of molecules in the excluded set that was not used for the model development.

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The predictive ability of the best-fit equations developed from the whole dataset using either Log D or Log P was tested on various subsets (Fig. 4). The best-fit equation developed using Log D had better predicting ability on the following subsets: albino (Q 2 = 0.8178), suspensions (Q 2 = 0.8765), all without suspensions (Q 2 = 0.7115), all without macromolecules (Q 2 = 0.7773), all without salts (Q 2 = 0.7587), acids (Q 2 = 0.4234) and bases (Q 2 = 0.8637). The best-fit equation developed using Log P had better predicting ability on pigmented set (Q 2 = 0.8146), macromolecules (Q 2 = 0.6770), salts (Q 2 = 0.9042), zwitterions (Q 2 = 0.7403) and neutral molecules (Q 2 = 0.9128).

Fig. 4
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Predictive ability of the best-fit equations obtained with entire dataset on various subsets. The best-fit equation obtained using Log D from the whole dataset without outliers (Log t 1/2 = −0.178 + 0.267 Log MW − 0.093 Log D + 0.003 dose/sol 7.4 + 0.153 PF) predicted the following subsets: albino, suspensions, all without suspensions, all without macromolecules, all without salts, acids, and bases with high correlation values. The best-fit equation obtained using Log P from the whole dataset without outliers (Log t 1/2 = −0.320 + 0.432 Log MW − 0.157 Log P + 0.003 dose/sol 7.4) predicted the following subsets: pigmented, macromolecules, salts, zwitterions, and neutral molecules with high correlation values and less residual errors.

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DISCUSSION

In the present study, QSPKR based models were developed to understand the important physicochemical properties that influence the vitreal disposition of molecules in rabbit eyes. Unlike the previous models developed from a small set of molecules with structural similarity, the present data set included structurally diverse compounds with a wide range of molecular weights, lipophilicity, and solubility. The data was analyzed as a whole as well as under select classifications based on eye pigmentation, acid/base properties of the drugs, solubility, and molecular weights. Log MW, Log P (partition coefficient), Log D (distribution coefficient at pH 7), dose number (DN = Dose injected/Solubility at pH 7.4), pigmentation factor (PF) and salt factor (SF) were selected as the independent variables for regression analysis. Multiple linear regression analysis was done using two sets of variables. In the first case Log MW, Log D, DN, PF and SF were used as the independent variables for model development. In the second case Log D was replaced by Log P for model development. Although both Log D (distribution coefficient) and Log P (partition coefficient) denote the lipophilicity of a molecule, Log D takes into account the ionization of compounds at different pH values. Hence regression analysis was performed using both these variables to better understand their influence on molecular properties. The selected independent variables were checked for multicollinearity (condition that exists when two or more independent variables are related to each other leading to unreliable estimation of regression coefficients) and the best-fit models were obtained for the entire dataset and different subsets. The best-fit models developed indicated that Log MW, Log D or Log P and dose number (DN = dose/solubility at pH 7.4) as the important noncollinear variables that best describe the vitreal disposition of molecules in various sets.

Multiple linear regression analysis of the selected non-collinear variables was found to be highly correlated with K 10 half-lives of both one and two compartments rather than the beta half-lives. In case of one compartment pharmacokinetic model, the elimination half-life of the drug from the injected compartment (vitreous) is represented only by the K 10 half-life. However, from a two compartment model (which includes a distribution compartment) alpha, beta and K 10 half-lives can be estimated. The alpha half-life represents the distribution phase and the beta half-life denotes the elimination phase. The K 10 half-life reflects the elimination of drug from the injected compartment (central compartment) and is derived from the relation (alpha × beta)/K21. In this study, inclusion of K 10 half-lives resulted in good correlations with the predictor variables.

Exclusion of amphotericin and triamcinolone acetonide (TA, low and medium doses) from the entire dataset and other subsets (except suspensions) resulted in models with high correlation value (R 2) and statistically significant coefficients for the independent variables. However, TA high dose (16 mg) was well fit in the models developed. In the two studies reporting the intravitreal clearance of amphotericin B (29,30), the time points included for half-life estimation were taken on days 1, 3, 6, 9, 12 or on days 1, 5, 10, and 15. Thus, insufficient time points were taken during the first 24 h, indicating that the half-life calculated with these points could be of the beta phase. Similar was the case with triamcinolone acetonide where time points were days 1, 2, 4, 7, 12, 20, and 30 in albino rabbits (TA low dose, 0.3 mg) (31), 1, 7, 14, 28, 42, 56, and 120 days (TA medium dose, 4 mg) in another study with albino rabbits (61), and days 1, 2, 3, 6, 13, 21, and 46 in pigmented rabbits (32). When sufficient time points were not included in pharmacokinetic studies, the results might under- or over-estimate some of the important pharmacokinetic parameters. Hence, these two drugs were excluded and stepwise regression analysis was performed without them to determine the variables that most influence the vitreal half-life.

The best-fit equations developed after excluding amphotericin B and TA (low and medium doses) from the entire dataset included all three important variables, namely, Log MW, Log P or Log D and DN. After exclusion of the outliers, the R 2 values increased significantly (>0.7), justifying their omission. The equation obtained with Log D set of variables included pigmentation factor as an additional variable when compared with Log P set of variables. These equations predicted the different subsets with good accuracy (Q 2 > 0.71) except acids (Q 2 = 0.4234) and macromolecules (Q 2 = 0.6770) as shown in Fig. 4. The predictive abilities of both these equations improved after excluding amphotericin B, and triamcinolone acetonide (low and medium doses) from the predicted datasets. Although both the equations developed with either Log D or Log P had similar predictive abilities (not much difference in the Q 2 values), the influence of Log P was more pronounced in case of pigmented set and zwitterions (Fig. 4). Both these subsets were well predicted using the best-fit equation employing Log P.

Log MW positively correlated with vitreal half-life of molecules. An increasing trend in the vitreal half-life was observed for molecules with increasing MW as seen in Fig. 5 except for the subsets acids (Fig. 5d), bases (Fig. 5e) and zwitterions (Fig. 5f). MW is an important physicochemical parameter that is known to influence the permeability of the molecule across different layers including sclera (33) and RPE (17). Similar results were reported earlier indicating an increased vitreal half-life for high MW compounds. Maurice (28) had shown that high MW compounds were eliminated from the vitreous slowly (high half-life) through the aqueous humor pathway. Also other investigators had shown the dependence of vitreal elimination on Log MW (27). However, the results obtained in this study indicate that Log MW alone does not influence the vitreal clearance of molecules tested. This was further confirmed by the poor predictive ability (Q 2 = 0.0977) of the previously reported equation involving only Log MW (Elimination rate constant K = 8.3767 − 1.5017 × Log MW (27)) when applied to the entire dataset of this study (excluding amphotericin B, TA and FITC–dextrans) and only to macromolecules including FITC–dextrans (Q 2 = 0.0886). The increased vitreal half-life (slow clearance) could also be attributed to the slow diffusion of high MW compounds in the vitreous. Exclusion of this variable from the best-fit equations resulted in poor R 2 values (Eqs. 3, 4, 6 in Table VIII; Eq. 4 in Table IX) or poor predictive ability (less Q 2 values as shown in Fig. 3).

Fig. 5
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Plot of vitreal half-life as a function of molecular weight for entire dataset (a), albino group (b), pigmented group (c), acids (d), bases (e), zwitterions (f), neutrals (g), suspensions (h), and macromolecules (i). Both vitreal half-life and molecular weight are shown in logarithmic scale.

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Log D and Log P correlated negatively with vitreal half-life, indicating shorter half-life for highly lipophilic molecules. This trend is clearly indicated in the scatter plots shown in Figs. 6 and 7. The decreasing trend is observed for the entire dataset and various subsets except neutral molecules (Figs. 6g and 7g), macromolecules (Figs. 6i and 7i), and suspensions (Fig. 7h). Intravitreally administered molecules are eliminated from the vitreous by either anterior (aqueous humor) or posterior (retina) route (28). Molecules eliminated by the anterior route diffuse into the anterior chamber and are excreted through the canal of Schlemm. Due to this long route and relatively less effective area of elimination pathway, molecules eliminated by this route have a long vitreal half-life and it is hypothesized as the major route of elimination for hydrophilic molecules. Retinal route is the major elimination pathway for lipophilic drugs. Because of the large surface area, tissue partitioning as well as active transport mechanisms, the vitreal half-life of molecules eliminated by this route is usually short. Thus, lipophilicity of a molecule determines its elimination pathway from the vitreous and thereby influences the vitreal half-life. Liu et al. (26) had previously reported a linear correlation between the lipophilicity (Log D at pH 7.2) and beta half-life using a small group (four) of structurally similar quinolones. However, the predictive ability of the equation (t 1/2,β = −1.8172 × Log D pH 7.2 + 2.1239) when applied to the dataset of the current study (excluding amphotericin B, triamcinolone acetonide, and the four quinolones used by the authors to develop the model) was low with a Q 2 of 0.1217. Also the best fit models obtained in this study involving only Log D (Eqs. 4 and 6 in Table VIII) or only Log P (Eqs. 4 and 12 in Table IX) exhibited poor predictive ability (less Q 2 values, Fig. 3). All these results clearly indicate that lipophilicity alone is not useful as a predictor of vitreal half-life of diverse compounds.

Fig. 6
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Plot of vitreal half-life as a function of Log P for entire dataset (a), albino group (b), pigmented group (c), acids (d), bases (e), zwitterions (f), neutrals (g), suspensions (h), and macromolecules (i). Vitreal half-life is shown in logarithmic scale.

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Fig. 7
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Plot of vitreal half-life as a function of Log D for entire dataset (a), albino group (b), pigmented group (c), acids (d), bases (e), zwitterions (f), neutrals (g), suspensions (h), and macromolecules (i). Vitreal half-life is shown in logarithmic scale.

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Dose number (DN = dose/solubility at pH 7.4) is an important variable in the models developed. DN positively correlated with vitreal half-life of molecules (Fig. 8). During the initial regression analysis of the dataset where dose was not included as a variable, the models developed had R 2 values in the range of 0.348 (suspensions) to 0.918 (macromolecules) and no best-fit model could be obtained for suspensions. Inclusion of DN, which accounts for the dose administered and the solubility of the molecule, improved the best-fit models and resulted in high R 2 values and good predictive ability. Addition of DN resulted in highly significant best-fit model for all suspensions (R 2 = 0.844, Fig. 2d) which could account for different doses of TA administered. When the dose injected exceeds the solubility of a molecule, a declining fraction of the injected dose gets absorbed and/or cleared, resulting in dose-dependent pharmacokinetics or half-life for suspensions.

Fig. 8
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Plot of vitreal half-life as a function of dose number (dose/solubility at pH 7.4) for entire dataset (a), albino group (b), pigmented group (c), acids (d), bases (e), zwitterions (f), neutrals (g), suspensions (h), and macromolecules (i). Both vitreal half-life and dose number are shown in logarithmic scale.

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The best-fit model generated with the data obtained from albino rabbits indicated that three parameters (Log MW, Log D or Log P and DN) were important in predicting the vitreal half-life of molecules (R 2 = 0.820, Fig. 1b; R 2 = 0.738, Fig. 2b). However, the best-fit equation involving Log P predicted the pigmented data with high accuracy (Q 2 = 0.8137, Fig. 3). Another interesting observation in this analysis was the role of Log P in pigmented groups. No significant model could be obtained in the pigmented group when Log D was used as the independent variable (Table VIII, Eq. 3, R 2 = 0.351). Also, the predictive ability of albino group equation developed using Log D was very poor on the pigmented set (Q 2 = 0.3271, Fig. 3). However, when Log P was used, a model with high correlation coefficient could be developed for the pigmented set (R 2 = 0.852, Fig. 2c). This was further supported by the high predictive ability of the albino set equation developed with Log P on the pigmented set (Q 2 = 0.8137, Fig. 3). These observations clearly indicate that Log P (not Log D) is an important variable that influences the pharmacokinetic properties of molecules in pigmented animals.

None of the models obtained from acids, bases, and zwitterions had Log MW in their best-fit equations (Tables VIII and IX). Lipophilicity turned out to be the important variable that influences the vitreal half-life of these molecules. These molecules get ionized at physiological pH at the injected site (vitreous) and hence changes occur in their solubility and lipophilicity (ionized molecules were more water soluble than the nonionized form). Hence the best-fit models developed from these ionizable molecules included either Log D or Log P in the best-fit equations. The best-fit model obtained for acids included only Log D (Fig. 1j) or Log P (Fig. 1i) in the equation and had poor R 2 and Q 2 values (Fig. 3). In case of bases, the molecules were best-fit using both Log D and DN or Log P (R 2 > 0.88). In case of zwitterions, Log P resulted in better-fit model (R 2 = 0.766, Fig. 2k) when compared with Log D. This could be a result of the difficulty in estimation of Log D values for zwitterions due to their multiple charges. In case of neutral molecules, the best-fit model was obtained with Log MW and DN. Since neutral molecules exhibit neutral charge at physiological pH, the vitreal half-life of these molecules was influenced by Log MW rather than lipophilicity. Although these models had high R 2 (>0.76 except acids), their predictive abilities were poor. All these data further strengthen the fact that all three predictor variables (Log MW, Log D or Log P and DN) are essential for predicting the vitreal disposition of molecules.

The advantage of including DN as a variable was obvious from the best-fit model obtained for suspensions. Without inclusion of DN, no best-fit models could be obtained for this group. The inclusion of DN along with Log MW and Log P improved the fit of this group (R 2 = 0.844, Eq. 8, Table IX) and also accounted for different doses of TA which was otherwise a clear outlier in other groups. Freely soluble compounds diffuse well in the vitreous and reach retina or aqueous humor and get eliminated based on their molecular properties. In case of suspensions where a high dose exceeding the solubility of molecule was injected, the dissolution of drug from the administered dosage form (suspension) becomes a crucial factor. This was further supported by a recent report by Amrite and Kompella (34), where inclusion of a dissolution–release step improved the model fits for suspensions after periocular injection. The model developed from the dataset excluding suspensions (Eq. 9, Tables VIII and IX) included only Log MW and Log D or Log P. However, the predictive ability of this equation which did not include DN was very poor (Q 2 = 0.0209, Fig. 3) indicating the importance of all three variables in predicting the vitreal half-life of molecules.

In case of macromolecules, the best-fit model obtained included only Log MW in the equation (Eq. 10, Tables VIII and IX). Only small molecules can freely diffuse in vitreous. Large MW molecules have decreased diffusivity and hence were slowly excreted from the vitreous. Hence in case of macromolecules, rather than lipophilicity and dose administered, the MW of the macromolecule influences the vitreal half-life. Though this model had the highest R 2 (0.980, Fig. 1f), its predictive ability on smaller molecules was less (Q 2 = 0.0155, Fig. 3). When macromolecules were excluded from the whole dataset, the best-fit equations obtained included all three variables (Eq. 11, Tables VIII and IX) showing the dependence of small molecules on lipophilicity and DN in addition to MW.

The best-fit model obtained with salts indicated Log MW and Log D or only Log P as the influential variables. Molecules in the salt form were formulated to overcome the solubility related issues during administration. Depending on the pKa values of the parent moiety, the salt form could have similar or altered partition/distribution coefficients after injection. Thus the best-fit models indicated the lipophilicity parameters as the important variable. The classification of salt forms in the present dataset was based on the identification of salt form by the source literature. However, in other literature reports, the exact salt form administered was not mentioned. Hence this classification should be interpreted with caution. Finally, exclusion of molecules administered as salts from the entire dataset included all three parameters and also had good predictive ability on the salt forms (Q 2 = 0.9043, Fig. 3) and on the experimental data.

Although drug concentration in the vitreous can be predicted based on drug physicochemical properties, drug levels in target macula region and drug persistence at the target are critical determinants of drug effects in posterior segment diseases such as AMD. However, with the present data it would be difficult to predict the drug distribution at such target sites. The data provided in the current study was an initial attempt to understand the influence of physicochemical properties on the vitreal half-life of molecules. In future, such an analysis can be potentially extended to target tissues such as the macula.

In conclusion, best-fit models were developed to predict the half-life of molecules in the vitreous after intravitreal injection. Unlike the previous reports, the present study indicates that no single parameter predicts the vitreal half-life of a diverse class of molecules. The best-fit models obtained clearly indicated Log MW, Log D or Log P and DN (dose/solubility at pH 7.4) as the critical physicochemical parameters that most influence the vitreal half-life of molecules. The best-fit model obtained in the present study from the entire dataset could be used to predict the vitreal half-life of diverse group of molecules. Half-lives for the subsets bases, suspensions, non-suspensions, non-macromolecules, non-salts, and albino rabbits could be best predicted using the general equation, Log t 1/2 = −0.178 + 0.267 Log MW − 0.093 Log D + 0.003 dose/solubility at pH 7.4 + 0.153 PF, with R 2 > 0.7. Half-lives for the subsets zwitterions, neutrals, macromolecules, salts, and pigmented rabbits could be best predicted using the general equation, Log t 1/2 = −0.320 + 0.432 Log MW − 0.157 Log P + 0.003 dose/solubility at pH 7.4, with R 2 > 0.7. Since a large number of molecules were used in arriving at the two general equations mentioned above, these are likely to be more universal and reliable in predicting vitreal half-lives compared to other equations developed with smaller datasets. However, the smaller datasets, when expanded in future, might provide more unique and simpler equations for various subsets. The models developed for subsets of compounds provide some interesting insights. For instance, while the best-fit models for individual groups of acids, bases and zwitterions involve Log D or Log P, the best-fit model for macromolecules includes Log MW and not Log D or Log P, suggesting the importance of MW over lipophilicity in this group. To the best of our knowledge, the general model developed in the current study with the entire dataset was the first of its kind, providing a universal single model that could be used to predict the vitreal half-life of structurally diverse molecules. The best-fit models developed and approach followed in the present study will benefit not only the discovery scientists but also the clinicians in selecting the proper drug or drug form for prolonged therapy.