Mathematical Models of Ocular Drug Delivery

Abstract

Drug delivery is an important factor for the success of ocular drug treatment. However, several physical, biochemical, and flow-related barriers limit drug exposure of anterior and posterior ocular target tissues during drug treatment via topical, subconjunctival, intravitreal, or systemic routes. Mathematical models encompass various barriers so that their joint influence on pharmacokinetics (PKs) can be simulated in an integrated fashion. The models are useful in predicting PKs and even pharmacodynamics (PDs) of administered drugs thereby fostering development of new drug molecules and drug delivery systems. Furthermore, the models are potentially useful in interspecies translation and probing of disease effects on PKs. In this review article, we introduce current modeling methods (noncompartmental analyses, compartmental and physiologically based PK models, and finite element models) in ocular PKs and related drug delivery. The roles of top-down models and bottom-up simulations are discussed. Furthermore, we present some future challenges, such as modeling of intra-tissue distribution, prediction of drug responses, quantitative systems pharmacology, and possibilities of artificial intelligence.

Proper drug delivery is an essential feature in successful eye treatment, because administration of medicinal products should result in adequate drug concentrations in the target tissues at safe dosing levels. Furthermore, therapeutic drug concentrations must be maintained long enough to enable reasonable dosing intervals. These features can be optimized with modification of the drug, dosing regimen, formulation, and the delivery system. The most important delivery routes of ocular drugs and the main barriers are summarized in Figure 1. 

Figure 1.

 

A schematic representation of the ocular anatomy, blood-aqueous barrier, blood-retinal barrier, and selected routes of ocular drug administration.

Topical administration of eye drops and ointments results in drug absorption to the eye via cornea and bulbar conjunctiva, but usually less than 5% of the instilled dose of a small molecule drug is actually delivered to the anterior chamber and much smaller levels to the posterior eye segment.^1,2 Biologics cannot be used as topical eye products, because these large molecules are not absorbed even to the anterior eye segment. Ocular drug absorption is limited by the rapid removal of the applied drug from the ocular surface and the tight tissue barriers of the cornea and conjuntiva on the ocular surface.¹ Therefore, topical drugs must be administered frequently (1–8 times daily) to maintain therapeutic drug concentrations in the target tissues. Longer duration of action, topical delivery of biologics, and improved drug bioavailability are unmet needs in anterior segment drug delivery with topical medications. Only the anterior eye segment can be treated with topical medications, because the drug concentrations in the posterior segment are too low for effective treatment. 

Posterior segment eye diseases are usually treated with intravitreal drug injections as solutions, suspensions, or implants (see Fig. 1) because intravitreally injected drugs can reach their targets in the retina and choroid at therapeutical levels.^1,3 Potent anti-VEGF proteins have half-lives of at least several days and they are injected usually at 1 to 2 month intervals, resulting in approximately 100-fold fluctuations of drug concentrations in the vitreous during treatment. Soluble small molecule drugs are rapidly eliminated from the vitreous: typical half-lives being 1.5 to 10 hours, and the longest reported half-life approximately 27 hours. Their retention can be extended to months or even years with implant technology.⁴ Longer duration of drug action, targeted delivery into the retinal cells, and less invasive modes of drug delivery (e.g. topical and systemic) are among the unmet drug delivery needs in the posterior segment treatment. 

Per oral drug tablet administration would be a convenient way to treat eye diseases, but it is only rarely applicable. Blood-ocular barriers (see Fig. 1) limit the access of systemic drugs to the eyes thereby necessitating use of high systemic doses that may be associated with adverse side effects. Targeted and selective drug distribution from the blood circulation to the ocular target tissues is essential for wider utilization of patient friendly per oral dosing in ocular treatment. 

Regardless of the route of drug administration, ocular pharmacokinetic (PK) studies are performed to quantitate drug concentrations in the eye. Due to obvious ethical reasons, drug concentrations cannot be determined in human tissues after drug administration to volunteers or patients. Because only sparse drug concentration data are available from human aqueous and vitreous humor,^5–7 ocular PKs depends on preclinical animal experiments that have been mostly performed with rabbits. Unfortunately, these invasive studies require euthanization of many animals. For example, a PK study of 6 time points and 6 eyes at each time point requires 36 animals, if the contralateral eye is used for estimation of systemic contribution of drug distribution. Therefore, 210 animals would be needed to compare 5 different drugs or formulations. Noninvasive fluorophotometry is useful in quantitation of fluorescent compounds in transparent ocular tissues (cornea, aqueous humor, and vitreous), but labeling may change PKs and lead to artifacts. 

The rabbit is the most widely used animal in ocular PK studies. Compared to the eyes of rodents, the rabbit's eye size is closer to the size of human eyes. The anterior segment size in rabbits is similar to the human eye, but the volume of vitreous in the rabbit eye is about 1.5 mL being smaller than the vitreous of the human eye (approximately 4 mL).⁸ Despite some differences, such as low blinking frequency and differences in the retinal anatomy, the rabbit is still considered to be a reasonable animal model for human translation. Monkey eyes have similarities with human eyes, but their use is limited by high price and low access. Mice and rats are widely used for drug response investigations in disease state models, but only limited PK rodent data are available. Inter-species scaling represents a major challenge in this field. 

The PK modeling is an important tool in the development of ocular drugs and delivery systems. The models integrate together many factors that act in concert, such as liquid flows, membrane permeability, partitioning to the tissues, and receptor binding. Therefore, the models describe and/or predict how drug modifications may affect ocular PKs. Furthermore, the models for different animal species and humans can be used to facilitate animal-to-human kinetic translation. The PK/pharmacodynamic (PK/PD) modeling extends from PKs to drug responses and provides predictions for drug action. With improving software and computational power the models have become more reliable and user-friendly tools for investigators. In this review, we present the state-of-the-art and future prospects of mathematical modeling for ocular drug delivery. 

Basics of Models and Parameters

Ocular PKs describe changes of drug concentrations in the ocular tissues with time, whereas PD informs about the time course of drug response intensity. Both PK and PD can be modeled with appropriate model structures and parameters, and the PD model shows the correlation between the drug concentrations and responses. Thus, PK/PD models build up relationships among dose, drug exposure, and responses over time. 

The models can be classified into two major categories: (1) empirical descriptive models that are used to derive parameter values from PK in vivo data using the top-down approach, and (2) mechanistic explanatory models that are used to predict concentration profiles in vivo based on mechanistic parameters and the bottom-up modeling approach. Various combinations can also be used (e.g. middle-out modeling).⁹ The classification of PK and PD models is schematically presented in Figure 2. Descriptive models are based on the top-down approach, whereas predictive models are using the bottom-up methods (see Fig. 2). Furthermore, the individual parameters can be determined experimentally (in vitro) to be used in predictive models (in silico; see Fig. 2). The parameters obtained from in silico, in vitro, and top-down in vivo experiments can be used as components of predictive models (Fig. 3). Predictive models are suitable for interspecies translation and augments clinical translation of drug candidates. Descriptive population-based models are important in defining the properties of clinical drug candidates. Overall, different model types should be used in concert for maximal scientific benefit (see Fig. 3). 

Figure 2.

 

Classification of model types. Individual factors (e.g. receptor affinity and membrane permeability) can be used as parameters in models. Clearance, volume of distribution and half-life are pharmacokinetic parameters that depend on binding and partitioning (affects volume and half-life) and permeability and active transport (affects clearance and half-life). Machine learning methods correlate parameter values with chemical structure, thereby providing predictions. Likewise, quantitative structure property relationship (QSPR) models generate such correlations. Docking and molecular dynamics and other molecular modeling tools are used for molecular level processes, such as receptor binding. Pharmacokinetic models integrate various factors even at the level of whole eye providing descriptive and predictive information.

Figure 3.

 

Schematic representation of modelling dynamics. Descriptive models of in vivo data provide pharmacokinetic parameters that can be further used as components in the bottom-up predictive models. Such models can utilize also individual parameters that are based on in vitro assays and in silico models. Modeling can be further extended to pharmacodynamics by adding response related parameters and linking them to pharmacokinetics in the effect compartment. Quantitative systems pharmacology aims to incorporate also complex biochemical systems biological pathways to the models.

PKs rely on parameter values that are defined experimentally (in vitro or in vivo) or predicted in silico (for parameters, see Table 1). The quality of individual parameters defines, together with the model structure, how well the model will function. 

Table 1.

 

Typical Parameters Needed in Building and Running PK and PD Models

 

Model Types

This chapter provides a brief introduction to the main model types that can be applied in ophthalmic drug research. Table 2 shows the main features of the models. 

Table 2.

 

Comparison of Compartmental, Population, and Physiologically Based Models

 

Compartmental Models

Compartmental analysis models are the most widely used classical top-down PK models (see Table 2). They are used to model the concentration versus time profiles by using theoretical interconnected compartments that are not directly related to anatomic organs or tissues. Each compartment is assumed to be well-stirred resulting in homogeneous drug concentration within each compartment. The model structure is determined by the route of administration and the PK data of the drug. The simplest case is a one-compartmental model^4,8 with one-phase decline in the semi-logarithm plot of the drug concentration-time profile after bolus injection (e.g. vitreous humor) exemplified in Figure 4. Models may also have two or three compartments if the drug concentrations in the compartments are not reaching pseudo steady state during the observation period.¹⁰ Using nonlinear regression analysis, the model is fitted to the data by successive iterations until the sum of the squared residuals between the observed concentrations and the predicted concentrations reaches the minimum (a process that is commonly named as curve fitting). The final mathematical equation representing the best fit describes the data with parameters, such as clearance, volume of distribution, and rate constants that are characteristic to the combination of the drug, delivery system, and physiological system. The obtained parameters can be further used in bottom-up simulations to predict concentration-time curves as a function of dose or infusion time. 

Figure 4.

 

Schematic representation of the one- and two-compartment models after bolus injection. The differential and final solved equations are shown, where A_o is the injected dose, CL is the clearance, and V is the volume of distribution in the compartment, 1 = central compartment, 2 = peripheral compartment, k_el = constant rate of elimination form the central compartment, k₁₂ = transfer rate constant from the central to the peripheral compartment, k₂₁ = transfer rate constant from the peripheral to the central compartment. C is the concentration in the central compartment at any time point, C₀ is the initial concentration (at time 0), and the sum of A and B in the two-compartmental model, α correspond to the slope of the initial-phase decline in semi-logarithm plot, and β to the terminal-phase, and both relate to the micro-constants (e.g. α + β = k₁₂ + k₂₁ + k_el) in the two-compartmental model.

Compartmental PK models can be combined with PDs by using the sigmoid maximum effect (E_max) model for graded drug effects. It is an empirical function for describing nonlinear concentration-effect relationships. It has the general form presented in Equation 1:  

\begin{eqnarray}E = \frac{{{E_{max}} \times {{\rm{C}}^\gamma }}}{{EC_{50}^\gamma + {{\rm{C}}^\gamma }}}\end{eqnarray}

(1)

where E_max is the maximum effect, C is the concentration of the drug, EC₅₀ is the concentration of the drug that yields half of the maximal effect, and γ is called the shape factor. Depending on the γ value, specific cases of the sigmoid E_max model are met, for example, γ = 1 corresponds to the simple E_max model, a very high value leads to sharp response (nearly quantal) versus concentration relationship, whereas a low γ value results in shallow linear relationship. More complicated models may be needed to describe complex concentration-effect relationships.¹¹ Moreover, the model will depend on the type of response (e.g. quantal or graded), the link between the measured concentration (e.g. in plasma and aqueous humor), and the effect at the action site in solid tissue (direct or indirect), and the response mechanism (immediate or delayed). Both compartmental PK and PK/PD models rely on the use of equations, which describe the rate of change of drug concentration in different tissues. 

NonCompartmental Analysis Model

The noncompartmental analysis (NCA) method does not rely on assumptions on compartments to describe concentration-time profiles. Instead, the NCA is based on calculation of the area under the concentration time curve (AUC) using the trapezoidal method. Based on AUC values, further PK parameters, such as clearance and bioavailability, can be calculated. The NCA is frequently applied for topical ophthalmic drugs, because concentration profiles in aqueous humor include only a few data points and they are difficult to fit by compartmental modeling.^2,12,13 

Population Pharmacokinetic/Pharmacodynamic Model

Population PK/PD modeling uses the compartmental model structure as previously described (see Fig. 4). However, population models simultaneously use data from all individuals in a test population, whereas traditional compartmental models are used to fit data from each individual or mean concentrations from the test group. Population PK and PK/PD are based on nonlinear mixed effects modeling that allows fitting a single model to the observed data from several subjects, while maintaining subject individuality. Thereby, it is possible to identify and account for sources of interindividual PK and/or PD variability; a very useful feature in clinical studies.¹⁴ 

Population models contain two components: (1) the structural model (i.e. compartmental model for PK data), and (2) the statistical or variance model (to describe the interindividual, interoccasion, and interstudy variability). Moreover, a covariate model may be included as third component (Fig. 5). The covariate model can be used to include a significant patient or subject property to explain variability in the concentration or response of the drug (e.g. weight, age, or genetics).¹⁴ Such models are powerful in dealing with sparse data, obtaining estimates of the population, and individual parameters that maximize the probability of properly describing the data. For ease of computing, the maximization of this likelihood is expressed as negative two times the logarithm of likelihood (−2* log-likelihood). 

Figure 5.

 

Schematic representation of the several components of a population model: structural, statistic, and covariate models²⁰ using the example of CL of one-compartment model after intravitreal injection (the same applies to V1; adapted from Ref. 20). The statistical model describes the variability in the observed data, η_indv (intersubject variability) accounts for the difference between an individual's parameter value and the population value, ε (residual or unexplained intrasubject variability) describes the difference between the observed data for the individual and the model prediction at each time point. The covariate model shows a property of the subject (weight) which explains part of the variability in the CL parameter. The statistical and covariate models add the influence of the various sources of variability for the estimation of the population predicted concentration (C_{pop pred}) and individual predictive concentrations (C_{indv. pred}).

Nonlinear mixed effects models have been sparsely used in preclinical and clinical studies of ophthalmology,^15–18 even though this technique would be useful in studies where individual drug concentrations are obtained from aqueous humor at different times post-dosing from a large number of patients who undergo cataract surgery. 

Physiologically-Based Pharmacokinetic Model

The physiologically-based pharmacokinetic (PBPK) modeling involves compartments that represent real physiological spaces (tissues or organs) that are interconnected by fluid flows. In the case of whole body PBPK, the organ compartments are interconnected by blood flows (arterial and venous) that are distributing the drug to the tissues (Fig. 6A). In the ocular PBPK model, the eye tissues, and fluid flows (ocular blood flow and aqueous humor) are connected and further linked to the systemic circulation (Fig. 6B). 

Figure 6.

 

Schematic examples of PBPK models. (A) In whole body PBPK models the drug is distributed to the tissues by blood flow (Q) and eliminated from tissues by clearances (CL; e.g. renal excretion and hepatic metabolism). Drug partitioning from the arterial blood flow to the tissues is determined by membrane permeability and tissue/blood distribution coefficients. (B) Scheme of a simple PBPK model for intravitreal injection of a protein drug. SA, surface area; P_RPE, permeability across retinal pigment epithelium.

The PBPK models are built using differential equations that are solved numerically. The equations are based on blood-flow rates in the tissues, tissue volumes, tissue/blood distribution coefficients of drugs, drug permeability, transport, and metabolism in the tissue. The parameters can be classified to organism related (e.g. blood-flow rate and transporter/enzyme expression), drug related (e.g. permeability coefficient, degree of ionization, and affinity to transporter protein), and formulation related (e.g. route of administration, dose, interval of administration, and drug release).^19,20 Drug and formulation parameters are similarly relevant in the eyes. Local drug delivery is affected by the flows of ocular fluids, such as tears and aqueous humor, but drug partitioning, permeation, and transport are based on the same factors as elsewhere. Overall, PBPK models require large number of parameters that can be estimated in vitro or predicted in silico, but the potential strengths of PBPK models are important: predictivity, inclusion of individual tissues, interspecies scaling, and inclusion of disease related factors. The complexity of the models depend on the intended application, but many researchers aim to build models that include only the essential tissues and parameters. Commercially available software, such as GastroPlus or SimCyp Simulator, show extensive inclusive networks of parameters. The ultimate goal of PBPK models is to predict concentration versus time profiles of drugs in different tissues, allowing extrapolation to different scenarios (e.g. effects of disease processes, drug interactions, and species). The simulated concentration-time profiles can be utilized as inputs for PD models, obtaining a PBPK/PD model. 

In the case of local administration, such as ophthalmic one, a semimechanistic model can be more adequate. Ophthalmic semimechanistic models (middle-out models)⁹ have been published related to intravitreal injections, based on a two-compartmental model (vitreal and aqueous humor compartments),²¹ or a three-compartmental model (adding a compartment for the retina).^22,23 

Finite Element Models

Computational fluid dynamics (CFDs) is a mathematical method that predicts and simulates physical fluid flows. The fluid flow is described by differential equations that are solved using numerical finite element methods. CFD software has been used to model a myriad of applications involving fluid flow (e.g. architecture, engineering, aerodynamics, and meteorology). In the same way, fluid flow, temperature gradients, and drug concentrations in the eyes can be simulated with CFD software tools. The development of an ocular CFD model begins with the in silico construction of a geometrically accurate model of an eye that is divided into a finite number of non-overlapping tiny compartments or mesh elements (Fig. 7). Solving the differential equations describing the fluid flow for each of these compartments allows the simulation of localized drug concentrations in the eyes. Thus, whereas compartmental models assume the homogeneous drug concentration in each tissue, CFD models can capture the realistic drug concentration gradients within ocular tissues. In addition to meshing, a CFD model sets physically meaningful material properties, boundary conditions, fluid flow, pressure, and temperature constraints for different ocular tissues.²⁴ Since the 1990s, CFD models have been used to simulate ocular drug concentrations,^25–32 but only in the last 10 years, the whole eye has been accurately modeled and validated with published in vivo data.^33,34 In the next paragraphs, a few examples of ocular CFD models are discussed. 

Figure 7.

 

Overview of the finite element model (FEM) illustrating a geometrically accurate eye model divided in 36,613 mesh elements, the equations governing the transport of macromolecules via diffusion and convection, and the calculated macromolecule concentrations in the vitreous (blue), retina, and aqueous humor (light blue) after intravitreal injection (purple). The transport of macromolecules in the vitreous and the retina is driven by diffusion, whereas convection dominates in the aqueous humor. Convection was set to zero in the retina. D is the diffusion coefficient (m² s⁻¹), k_B is the Boltzmann constant (1.381 × 10⁻²³ J K⁻¹), T is the absolute temperature (310.15 K), η is the dynamic viscosity (0.00069 kg m⁻¹ s⁻¹), r_H is the hydrodynamic radius of the molecule (m), P_app is the apparent permeability (m s⁻¹), h is the thickness of the layer (m), and K is the retina/vitreous partition coefficient (0.5). From Lamminsalo et al.³⁶

Quantitative Structure-Property Relationship Models

Quantitative structure-property/activity relationship (QSPR/QSAR) models are regression or classification models that can correlate structural features of molecules (e.g. physicochemical descriptors) with a property or activity (Fig. 8). For that, multivariate statistical methods are used that can be classified into linear methods, such as principal component analysis (PCA) or partial least squares (PLS), and nonlinear methods, such as random forest (RF) or support vector machine (SVM), among the classic supervised machine learning algorithms. 

Figure 8.

 

Basic scheme of QSPR modelling. The models find correlation between molecular descriptors and pharmacological property (e.g. receptor binding affinity, permeability, clearance, and metabolic reaction). The model can then be used to predict the property for new compounds based on structure, even before the compounds are synthesized. When the relationship between a set of descriptors and a given response is weak or highly nonlinear, it may be possible to create a useful classification model.³⁷

PCA is a statistical technique that can transform a given set of variables (e.g. physicochemical descriptors) into a smaller set of uncorrelated variables so that most of the original information is preserved. Therefore, it reduces the dimensionality of correlated data and can be used together with other statistical methods, such as PLS, to first select a subset of variables. PLS allows a linear regression between the property to predict and the final variables that best describe the property.^4,35–39 The regression model contains coefficients reflecting the influence of these variables on the property. The model is validated using different statistical tests and by determining cross-validation, internal and external test set validation, and Y-randomization test. These methods have been used to reliably predict corneal and conjunctival permeabilities based on two-dimensional descriptors and a relatively small compound data set.^35,38,39 

The QSPR model can be used to predict properties of compounds prior to their synthesis. However, for building (and validating) the model, a data set of drugs with a broad range of the investigated property and a wide chemical space is first required. Once built, it will be able to reliably predict the property of a new molecule that belongs to the same chemical space of the training set. On the other hand, the set cannot be too diverse; for example, including both proteins and small molecule drugs does not yield a reliable QSPR model. 

For more complex predictions, such as melanin binding, extremely randomized decision trees⁴⁰ have been used to build a model based on binding data from a large chemical library (more than 3000 compounds). The random forest is a combination of several machine learning algorithms that are used to build multiple decision trees randomly. Each decision tree is trained with a different data subset, and the predictions from all decision trees are averaged to produce the final prediction. Extremely randomized decision trees (in short, extreme trees) are also ensemble machine learning algorithms, but they utilize the entire original dataset and split the nodes in a random manner. The validated collections of decision trees are used to make predictions. 

Modeling for Ocular Drug Delivery Routes

Delivery

Ocular drug absorption, distribution, and elimination after topical eye drop instillation are complicated processes, as illustrated in Figure 9. Ocular absorption is limited by the loss processes on the ocular surface (drainage of eyedrop, systemic absorption through conjunctiva, and tear turnover) and tissue barriers (corneal and conjunctival epithelia).⁴¹ The loss processes lead to rapid decline in the precorneal drug concentration, thereby seriously limiting contact time with the ocular surface.⁴¹ Drainage of the eye drops and trans-conjunctival systemic drug absorption are the main factors that remove the drug, whereas normal tear turnover is a minor contributor.⁴¹ From the corneal and conjunctival epithelia, the drug distributes to the corneal and conjunctival layers and further to the aqueous humor and iris/ciliary body (from the cornea) and to the sclera and iris/ciliary body (from the conjunctiva). Distribution from the aqueous humor to the lens is minimal and the drug concentrations in the lenses are much lower than in the aqueous humor, iris, and ciliary body.⁴² Overall, drug bioavailability in the aqueous humor is usually less than 5% of the instilled dose, often < 1%,² much less than the systemic bioavailability from eye drops (usually > 50%).⁴³ Drug elimination from the anterior chamber takes place via the aqueous humor turnover and blood flow of the iris and ciliary body: all drugs are eliminated in the aqueous humor outflow, whereas permeable compounds are also able to cross the blood vessel endothelia in the iris and ciliary body to the blood circulation.¹ These processes are affected by the chemical features of drug molecules. In addition, formulation factors have influence on drug absorption to the cornea and conjunctiva. As all PK processes are ongoing simultaneously, modeling is an attractive approach to describe and predict drug concentration profiles in the eyes. The impact of multiple parameters of on ocular PKs is shown in Figure 9. 

Figure 9.

 

Pharmacokinetic processes involved in topical ocular drug administration. Drug loss by solution drainage and systemic absorption through the conjunctiva limit the contact time of the instilled drug to a few minutes on the ocular surface. Drug absorption takes place via the cornea and conjunctiva. The conjunctiva is about nine times more permeable than the cornea,³⁹ but the major part of conjunctivally absorbed drug is lost to the systemic blood circulation and not distributed to the inner eye.⁴⁹ Drug elimination from the anterior chamber is governed by aqueous humor outflow (approximately 3 µL/min) and blood flow of the iris-ciliary body (clearance even 30 µL/min). Intracameral injection data are needed to determine drug clearance from the anterior chamber and volume of drug distribution. Furthermore, AUC values of drug in the aqueous humor after intracameral and topical administration are used to determine topical ocular drug bioavailability. Unfortunately, ocular bioavailability (range of 0.1–5%) has been calculated only for less than 10 compounds.²

Rat and mouse eyes are small and dissection of small tissues is very difficult. Therefore, quantitative PK information from rat and mouse eyes is missing, even though these species are widely used in ocular drug response and toxicology studies. Features of the small rodent eyes may lead to misleading conclusions. In particular, topical delivery of drugs to the posterior eye segment is much easier in the rodents, but these results rarely translate to the bigger eyes of humans. Therefore, most PK studies on topical ocular drug application have been performed using rabbits. Dissection of rabbit eye tissues is feasible and many PK parameters are similar with the human eyes.⁸ Blinking frequency is the most remarkable difference between human and rabbits eyes, as humans blink much more frequently than rabbits.^44,45 Overall, drug distribution in rabbit and human eyes follows the same main patterns. For example, multiple dosing of brimonidine and timolol to human eyes resulted in 50 to 65 times higher concentrations in the aqueous humor than in the vitreous,⁴⁶ whereas this difference in rabbit eyes was 25 to 50 fold.⁴⁷ In humans, the mean timolol concentration in aqueous humor was 3.16 µM, whereas timolol levels in rabbits’ aqueous humor 4 hours after the last eye drop of 1 week dosing was 0.6 to 1.3 µM.⁴⁷ After a single dose, the peak concentration of timolol in aqueous humor was 1.97 µM.⁴⁸ Melanin binding may cause major differences in drug concentrations of rabbit's albino tissues and pigmented tissues in humans and rabbits.⁴⁵ 

Status of Modeling

Mathematical models for topical ophthalmic drugs were pioneered by the late David Maurice and the late Joseph Robinson. The work of Maurice was mostly based on noninvasive fluorophotometry and he has compiled ocular PKs in a comprehensive book chapter in 1984.⁴⁹ Research of Joseph Robinson was dependent on the use of ³H and ¹⁴C labeled drugs that were quantitated in dissected rabbit tissues.^51–53 Neither method is able to detect drug metabolism and fluorophotometry is limited to fluorescent compounds. These classic studies utilized the top-down approach, that is, the PKs was studied in vivo in rabbits and the rate constants and other PK parameters were solved with curve fitting of compartmental models.⁵⁴ Even though these studies yielded an overall framework of ocular PKs, they are not feasible for bottom-up simulations and predictions. 

As illustrated in Figure 3, in silico predicted parameters can be used in bottom-up models. For example, corneal and conjunctival permeability across isolated tissues from rabbits and pigs can be predicted with QSPR models with a few calculated molecular descriptors (e.g. logD and polar surface area) even before synthesis of compounds.^35,38,39 However, permeability across isolated tissues is measured at pseudo steady-state situation in diffusion cells, which differs from the transient exposure after eye drop instillation in vivo.⁵⁵ After eye drop application, the drug concentration in the tear fluid decreases rapidly and the absorption to the ocular surface tissues takes place in only a few minutes.^51,56,57 Kinetics in the rabbit’s tear fluid were studied in detail by Lee et al.⁵¹ who were able to quantitate and model the contributions of various factors, such as the roles of tear turnover, instilled volume dependent solution drainage, corneal absorption, and absorption across the conjunctiva and nictitating membrane to the blood stream. Therefore, the model should include both transient absorption into the corneal epithelium and distribution from the epithelium further, as discussed previously.⁵⁸ Some PK parameters cannot be predicted from chemical structures, but they can be determined in vivo in rabbit experiments. For example, the rate of drug elimination from tear fluid can be determined by sampling lacrimal fluid with 1 µL capillaries⁵¹ and the clearance from rabbit aqueous humor based on drug concentrations after intracameral injections.⁵⁹ Because these experiments cannot be performed with large numbers of compounds, predictive in vitro and in silico approaches are needed to yield generalizable data for accelerating ocular drug development. 

Several factors in ocular PKs of eye drops are still without proper PK submodels. First, instillation of an eye drop may induce lacrimation that affects drug elimination from the ocular surface, but the extent of induced lacrimation is impossible to predict and difficult to measure. Lacrimation is most probably affected by the drug and formulation features (osmotic pressure, pH, viscosity, and preservative).⁵¹ Second, contributions of corneal and conjunctival drug absorption into the eyes are difficult to model. Some drugs, like timolol, are absorbed primarily via the cornea.⁶⁰ On the other hand, brinzolamide concentration ratios in iris-ciliary body/aqueous humor after topical administration (approximately 8) were much higher than after intracameral injection (approximately 0.5) suggesting significant non-corneal absorption directly to the iris-ciliary body through the conjunctiva and sclera.¹² Third, the rate and extent of drug dissolution in vivo after topical application of suspensions is not known and in vitro dissolution methods may not predict the in vivo dissolution rate properly.¹³ Likewise, modeling of ointments,⁶¹ gels, bioadhesive formulations, and microemulsions has been rarely presented. Fourth, the impact of molecular features on drug retention in the corneal epithelium and stroma is not known⁶² even though QSPR models for corneal and conjunctival permeability in isolated tissues have been developed.^35,39 Fifth, the range of drug clearance values in the aqueous humor is about 10-fold,² but the data are not adequate for generating structure based predictions of clearance. 

PBPK models have been developed for topically applied ocular medications. The most notable is the Ocular Compartmental Absorption and Transit model of GastroPlus simulation package. Le Merdy et al.^61,63 has published the model and its parameters. The model was used mainly to simulate the impact of topical formulations on absorption (e.g. suspensions and ointments). Later, the PBPK model was built for levofloxacin, moxifloxacin, and gatifloxacin eye drop kinetics in the rabbit eye.⁶⁴ The models were subsequently used to simulate drug exposure after ocular solution administrations in humans. The physiological parameters were scaled to match human ocular physiology. Interspecies translation worked well as the simulated concentrations matched the observed concentrations in the human aqueous humor that was collected during cataract and keratoplasty surgery. Some parameters were solved using curve fitting, but the number of parameters in the complex models is so large that good fits are always obtained, even though the parameter values can be unrealistic.^61,64,65 It is obvious that PBPK modeling is a powerful approach, but reliable parameter values are needed for drugs in wide chemical space. Interestingly, PBPK modeling of timolol eye drops was recently successfully extended to intraocular pressure reducing responses of the drug.⁶⁶ 

Computational Fluid Dynamic Models

The CFD models have been developed for topical ocular drug delivery. The study by Missel and Sarangapani²⁴ has been the pioneer in this field. Their model simulated ocular concentrations of moxifloxacin, timolol, and pilocarpine after topical dosing. The drug flux through the cornea was calculated as flux = amplitude × k_per × c_tear, where the in vitro rabbit corneal permeability k_per and the drug concentrations in the tear film c_tear were experimentally measured. The amplitude is an empirically derived factor that accounts for non-productive losses of drug after administration. The ocular CFD model was able to simulate the corneal absorption of moxifloxacin and distribution within the eye, even at intra-tissue level (Fig. 10). 

Figure 10.

 

Simulated intraocular moxifloxacin concentrations at different times after eye drop instillation. Drug concentrations within ocular tissues are clearly discerned.^24,68

The simulated concentrations in the eye were also compared with measured drug levels (Fig. 11). The model gave an accurate prediction of the concentrations of the three drugs in the cornea and aqueous humor, but the levels in the posterior eye segment were not well simulated. For example, pilocarpine and timolol appear in the vitreous and lens quicker than explained by the model. A potential explanation for this discrepancy is that this model does not consider the non-corneal absorption via the conjunctiva and the sclera to the posterior segment. It is also challenging to estimate the contribution of intra-ocular blood flows of ciliary body and choroid; these factors are particularly difficult to study with experiments and simulation (i.e. elimination to systemic circulation versus re-distribution in the eye). 

Figure 11.

 

Concentration profiles for topically administered (A) pilocarpine nitrate (0.27%, 25 µL), (B) timolol maleate (0.65%, 25 µL), and (C) moxifloxacin hydrochloride (0.3%, 30 µL) in ocular tissues. Simulated concentrations are shown as lines, and corresponding experimental data as circles. The CFD model predicts well drug concentrations in the cornea, aqueous humor, and iris-ciliary body, but the predicted drug concentrations in the posterior segment are less accurate. Reproduced from Refs. 24, 68.

Intravitreal Injections

Intravitreal injections are extensively used in drug delivery to the posterior part of the eye, because these injections bypass many ocular barriers that are associated with topical and systemic drug administration.⁶⁷ Thus, higher retinal bioavailability is achieved for small molecule drugs (e.g. corticosteroids and antibiotics),^69,70 and macromolecules (anti-VEGF antibodies, Fab-fragments, and soluble receptors).⁷¹ After intravitreal injections, the compounds distribute in the vitreous and retina, and then eliminate posteriorly across the blood-retina barrier (BRB) and anteriorly via aqueous humor turnover.⁵⁰ Hydrophilic and/or large molecules with low BRB permeability are eliminated via the anterior chamber, as shown for albumin,⁷² FITC-dextran, and antibodies,^73,74 and small hydrophilic molecules like gentamycin and sucrose.^50,75 On the contrary, small lipophilic drugs with considerable BRB permeability are mainly eliminated posteriorly to the blood circulation.^4,49 In this case, vitreal elimination is much faster (half-lives of hours) than via anterior route (half-lives of days). 

Compartmental Models

Del Amo et al. (2015) analyzed all literature of intravitreal PK studies and curve fitted the data with compartmental models. In most cases, the one-compartment model was adequate, whereas the two-compartment model was needed when clear distribution and elimination phases were distinguished, presumably due to back-diffusion from the tissues. The curve fits resulted in about 50 values of volume of distribution and clearance. The narrow range of the volume of distribution values in rabbit eyes (1.2–2.2 mL) indicated only limited back diffusion from the surrounding tissues into the vitreous. In contrast, the overall range of clearance values spanned 118-fold range from macromolecules to lipophilic small compounds. 

Semi-Mechanistic Bottom-Up Models

Vitreal diffusion and retinal permeability are the main physical factors that affect elimination from the vitreous.^73,76 In the models, the vitreous is assumed to be a sphere, and the elimination mechanism is defined by two steps: diffusion in the vitreous after injection and elimination across BRB or anteriorly (Fig. 12). 

Figure 12.

 

Mechanistic model of intravitreal pharmacokinetics based on the vitreal diffusion and trans retinal permeability of compounds.^73,76 The lens does not allow trans-lenticular permeation. Anterior elimination takes place only through the limited space between the ciliary body, iris, and lens via the posterior chamber to the anterior chamber. Drugs are eliminated from the aqueous humor by outflow and permeation to the circulation of the anterior uvea.

If BRB permeability is low (hydrophilic or macromolecules), the anterior chamber is the main elimination route, whereas compounds with retinal permeability are also eliminated posteriorly (see Figs. 12, 13) (Supplementary Material 1). If the vitreal diffusion time and BRB permeability are known, the half-life of elimination from the vitreous can be estimated. The diffusion time (T_diff) from the center of the vitreal cavity to the retinal surface is calculated with Equation 2, where r_diff is the radius of the vitreal cavity, and D is the diffusion coefficient (assumed to be equal with the diffusivity in water).⁷⁷ The elimination rate constant (k_e) from the vitreal cavity without membrane control of blood-ocular barriers is derived from the diffusion time (Equation 3).  

\begin{eqnarray}{T_{{diff}}} = \frac{{r_{{diff}}^2}}{{6D}}\end{eqnarray}

(2)

 

\begin{eqnarray}{k_e} = \frac{1}{{{T_{{diff}}}}}\end{eqnarray}

(3)

 

Figure 13.

 

Comparison of calculated vitreal clearance of fluorescein, sucrose, and ranibizumab with the experimental data of various molecules with wide size range. The original references are found in a previous publication.⁴ Clearance of the three model molecules were calculated from the vitreal elimination constants as CL = V/k_v, where V is the volume of drug distribution (1.5 mL).

The available area of posterior (Equation 4) and anterior elimination (Equation 5) can be considered mathematically. In the equations, k_v is the vitreal elimination rate constant, S_ant is the surface area ratio of anterior chamber – vitreous interface compared to the whole vitreal outer surface area (rabbits = 0.23 and humans = 0.15), and S_ret is the respective available area ratio for posterior eliminations across the retina (rabbits = 0.68 and humans = 0.80).^21,76 

\begin{eqnarray} {\rm{k}}_{{\rm{v}}} = \left( {\rm{S}}_{{\rm{ant}}} + {\rm{S}}_{{\rm{ret}}} \right) \times {\rm{k}}_{{\rm{e}}} \end{eqnarray}

(4)

 

\begin{eqnarray}{{\rm{k}}_{\rm{v}}} = {{\rm{S}}_{{\rm{ant}}}} \times {{\rm{k}}_{\rm{e}}}\end{eqnarray}

(5)

 

In rabbits, the elimination rate constant of fluorescein can be calculated as 0.33 hours⁻¹ (corresponds to the half-life = 2.1 hours) and similarly half-life of sucrose would be approximately 10 hours. The experimental vitreal half-life values for fluorescein and sucrose in rabbits are 2.5 and 15 hours, respectively, whereas the calculated values in humans are 4.3 hours for fluorescein and 33 hours for sucrose. Fluorescein was assumed to utilize both anterior and posterior routes (Equation 4) and sucrose only the anterior route (Equation 5). In the case of a macromolecule (ranibizumab) diffusivity is slower (D = 1.34 × 10⁻⁶ cm²s⁻¹)⁷³ and elimination takes place only anteriorly. These assumptions lead to a vitreal half-life of 2.2. days in rabbits, slightly shorter than the reported experimental half-lives (2.88 and 3.17 days).^78,79 In humans, the estimated half-life would be approximately 7.2 days, close to the experimental value of 7.19 days.⁸⁰ Figure 13 presents the experimental clearance values of various small and large molecules in rabbit vitreous. The new calculated clearance for fluorescein, sucrose, and ranibizumab are in line with the experimental data. Excel calculator in Supplementary Material 2 is available for simulations of drug concentrations in the vitreous at different injected doses, clearances and volumes of distribution. 

Previously, del Amo et al.⁴ developed predictive QSPR models based on vitreal clearance of 40 small molecules. The models showed correlation between polarity related molecular descriptors (e.g. logD_7.4 and the polar surface area) and vitreal clearance, assuming BRB permeability as the critical parameter (as CL = permeability times membrane surface area). Unfortunately, Equations 4 and 5 do not consider the differences in the BRB permeability. More sophisticated models have been developed by Hutton-Smith et al. (2017) considering RPE permeability, but there is still need for more complete models that take into account geometric factors, vitreal diffusion, and BRB permeation. 

PK/PD and PBPK Models

In their pioneering work, Hutton-Smith et al.⁷³ modeled ranibizumab PKs and VEGF suppression in the posterior eye segment. The relationships between vitreal diffusion and half-life, target affinity, and duration of anti-VEGF action were probed with the model. A more detailed PBPK model, including both ocular and systemic PBPK, was later published.⁸¹ A more detailed PK/PD model was further developed including the retina compartment.²³ 

The PBPK model for monoclonal antibodies, including both ocular and systemic PBPK, have been published.⁸¹ The PBPK model for monoclonal antibodies is rich in parameter values of various tissues and including even some intracellular factors (e.g. endocytosis, FcRN receptor binding, and lysosomal degradation). The model predicted experimental concentrations of intravitreally injected monoclonal antibodies in the vitreous, retina, aqueous humor, and plasma.⁸¹ Remarkably, the authors constructed similar detailed PBPK models for the eyes of monkeys and humans.⁸² 

Anti-VEGF effect was measured in human aqueous humor samples after injections of intravitreal ranibizumab.⁸³ It was assumed that VEGF, ranibizumab, and their complexes are eliminated from the eye via anterior route. Interestingly, VEGF was suppressed in the aqueous humor for about 1 month, and, thereafter, the levels rose consistently. Furthermore, VEGF suppression levels of ranibizumab and aflibercept were later successfully correlated with drug responses in neovascular age-related macular degeneration (AMD).⁸⁴ Later, more detailed model, including retinal barriers inner limiting membrane and RPE, showed that VEGF suppression in the retina may have shorter duration than in the aqueous humor.²³ The authors also propose that modeling may give insights to the interpatient variability in antineovascular responses of anti-VEGF drugs. 

Computational Fluid Dynamic Model

Missel et al. has also pioneered CFD models of intravitreal drug injections.⁸⁵ His model was based on magnetic resonance imaging, and it was validated against experimental data. Later, we enhanced the anterior elimination part of Missel's model by introducing a circulatory flow pattern of aqueous humor in the anterior chamber.³⁴ The flow pattern improved mixing in the anterior chamber, thereby providing more accurate simulations of the elimination of macromolecules after intravitreal injection. As can be seen from Figure 14, some aqueous humor exits the anterior chamber via a pressure outlet (trabecular meshwork), whereas the rest of the fluid starts a new circle that mixes the drug in the aqueous humor. 

Figure 14.

 

The aqueous humor flow is enhanced with a temperature gradient, heat transfer, and gravity. This generates a circulatory flow in the anterior chamber, improves drug mixing in the aqueous humor, and leads to faster and more realistic drug elimination. From Lamminsalo et al.³⁴

This model was used to simulate the PK of an intravitreally injected antibody. The model was capable of simulating concentration gradients inside the vitreous and concentration versus time curves in the vitreous, retina, and aqueous humor (Fig. 15). The model gave an accurate prediction of the antibody concentrations in the vitreous and the retina, whereas concentrations in the aqueous humor were underestimated. The simulated concentrations in the retina and vitreous were not dependent on ocular pressure, whereas antibody concentrations in the aqueous humor were pressure sensitive, resulting in most accurate predictions at an intraocular pressure of 10.1 Torr. The model was also used to estimate the diffusion coefficient of the antibody in the retina and retinal pigment epithelium-choroid as well as the relative contributions of anterior (76%) and posterior (23%) elimination pathways. 

Figure 15.

 

(A) Antibody concentration profiles predicted by Lamminsalo's CFD model³⁵ at 0, 0.3, 3, and 30 days after intravitreal administration. An apparent initial distribution volume of 400 µL and intraocular pressure of 10.1 Torr were used. At day 30, the highest antibody concentration is about 200 times less than at day 0. (B) The simulated concentration profiles in the vitreous, retina, and aqueous humor are shown as lines, and corresponding experimental data as circles.

Bottom-Up Modeling of Intravitreal Drug Delivery Systems

Nano- and microparticles have been studied for prolonged retinal drug delivery as intravitreal injections.^86–89 Figure 16 illustrates the intravitreal kinetics of nano- and microparticles. After injection, particles distribute in the vitreous by convection and diffusion.^77,90 During their distribution, particles gradually release the loaded drug that will diffuse in the vitreous and distribute to the retina based on the principles of soluble drug kinetics. Liposomes (and probably other nanoparticles) are eliminated from the rabbit vitreous anteriorly.⁹¹ Elimination may be accelerated by endotoxin contamination (Supplementary Material 1). 

Figure 16.

 

Schematic illustrating the pharmacokinetics of intravitreal nano- and microparticles. These formulations typically release drug in the vitreous, while being eliminated at the same time.

The inner limited membrane (ILM) at the vitreo-retinal interface limits material transfer from the vitreous into the retina (Fig. 16). The ILM is permeable even for macromolecules,^22,79,92 but retinal distribution of particles depends on their properties and the species. For example, liposomes with hydrodynamic diameters above 100 nm failed to pass the ILM in an ex vivo bovine eye model, but pegylated liposomes of 50 nm permeated to the retina.⁹³ In such cases, the particles may release some cargo in the vitreous and some in the retina. The RPE is a tighter barrier that allows only minimal permeation of biologics or particles from retina to blood circulation.⁹⁴ 

Mathematical simulations can give insights on how the interplay of retention and drug release from intravitreal particles will affect concentrations of the released drug in the vitreous (Fig. 17).⁹¹ The simulations illustrate the importance of extended particle retention for prolonged drug effects: if the particles are eliminated faster than the drug is released, most of the drug is eliminated from the eye before being released. In that case, retinal bioavailability of the released drug and duration of its effects are suboptimal. Therefore, synchronizing drug release and vitreal retention of particles is of utmost importance in posterior segment drug delivery. Degradation rate of the particles may further complicate the kinetics, because it is likely that the decreasing size of the particles during the degradation process may accelerate their elimination from the vitreous. However, the clear relationship between the particle size and elimination rate constant is missing, because chemical particle features and their interactions with the vitreal components affect particle retention in the vitreous. 

Figure 17.

 

Simulated kinetics of released drug in the vitreous after intravitreal injection of particles with elimination half-lives of 1 week (A) and 4 weeks (B). Three different first-order release rates (release half-lives 1, 4, and 8 weeks) were simulated and the elimination half-life of the free drug was 3 hours. The volume of drug distribution was considered to be the anatomic volume of the vitreous humor.

Intravitreal implants are used in the clinics for prolonged drug action.⁹⁵ They release corticosteroids, like fluocinolone acetonide and dexamethasone, for months or even years.^96–100 Figure 18 shows schematically the kinetics of an intravitreal drug-releasing implant. 

Figure 18.

 

Compartmental model of an intravitreal implant. The implant releases the drug with zero-order or first-order kinetics in the vitreous. The released corticosteroids are eliminated via the posterior and anterior routes.

Modeling of Drug Binding to the Vitreous

In general, vitreal binding of small drug molecules is not significant, does not correlate with their binding to plasma proteins, and the binding is expected to affect drug elimination from the vitreous only modestly.¹⁰¹ In principle, binding to the vitreal components, like collagen, hyaluronic acid, and proteoglycans, is expected to retard drug elimination and prolong ocular retention. 

Previously, we constructed a PK model for a compound based on its fast association with the vitreal binding partner (Fig. 19).¹⁰² Like previously, the diffusion of a soluble drug in the vitreous was assumed to be similar with its diffusion in water at + 37°C.^73,103 After intravitreal injection, distribution and vitreal binding of the drug to a single binding site takes place instantaneously, generating an equilibrium between the bound and the unbound drug. Only the unbound drug is considered to undergo elimination from the eyes, thus resulting in the Equation 6 for drug elimination from the vitreous. Binding may lead to a dose-dependent elimination rate constant because the binding sites may saturate. Thus, at low total drug concentrations (C_t), the drug is mostly bound, and the free concentration (C_f) is much less than the total concentration (C_f ≪ C_t), leading to prolonged vitreal retention. At high drug concentrations, the free concentration approaches total concentration (approximately C_t to C_f), resulting in faster drug elimination based on Equation 6. In the equation, the apparent elimination rate constant of total drug (K_app) approaches the rate constant of the free drug (K_f). The apparent elimination half-life can be calculated as t_1/2,t = Ln2/K_app.  

\begin{eqnarray}{\rm{Ln2/}}{{\rm{t}}_{{\rm{1/2,t}}}} = {{\rm{C}}_{\rm{f}}}{\rm{/}}{{\rm{C}}_{\rm{t}}} \times {\rm{Ln2/}}{{\rm{t}}_{{\rm{1/2,f}}}}\end{eqnarray}

(6)

where t_1/2,t and t_1/2,f are apparent and free drug elimination half-lives, respectively (see also Supplementary Material 1). 

Figure 19.

 

Schematic model of intravitreal pharmacokinetics involving drug binding to the vitreous. The scheme assumes that there is a single binding site.

Dose-dependent vitreal drug elimination is simulated in Figure 20. The elimination half-lives of unbound drug in the human vitreous were 8 days, and the vitreous turnover was zero. Higher binding affinity (K_d = 1 µM) results in more effective binding, smaller fraction of free drug, and prolonged vitreal retention. The simulations can be utilized to assess the potential utility of vitreal binding for prolonged vitreal retention and drug activity. Binding of a drug or a delivery system to the vitreous can be a viable strategy for generating long-acting injections, but the success of this approach depends on the kinetic interplay that is strongly affected by the binding affinity. 

Figure 20.

 

Influence of vitreal binding affinity (K_d 1, 10, 100, and 1000 µM) on drug concentration in the human vitreous (volume 4.5 mL). The assumed drug was a macromolecule (MW = 100 kDa) and elimination half-life was 8 days for free drug. The dose was 20 µmol and the anatomic volume of human vitreous was assumed to be 4.5 mL. The binding capacity of the vitreous (B_max) was 125 µM. (A) The total concentration of the drug in the vitreous. (B) Concentration of free drug in the vitreous. (C) Elimination half-life of total concentration in the vitreous.

In principle, reversible partitioning of drugs to the ocular tissues may induce similar effects as binding (see above). At the level of vitreal PKs, this phenomenon does not seem to be significant. First, the vitreal volume of drug distribution was not increased for lipophilic drugs with high tissue partitioning.⁴ This is probably due to two factors: (1) the tissues have much smaller volumes than the vitreous and (2) intravitreally injected drugs are transferred rather toward blood circulation than back to the vitreous. However, partitioning and binding to the tissues can have significant effects on drug retention and free concentrations in the tissue. Most remarkable effects are due to the melanin binding in the tissues (see the section Melanin Binding). 

Retinal Permeation

Retinal distribution to the extracellular or intracellular space is often the ultimate goal in intravitreal drug administration. In general, retinal permeation of small molecules is easy and also protein drugs are capable of retinal distribution. However, reaching intracellular drug targets of specific retinal cell types is more complicated as reviewed recently.¹⁰⁴ Intra-retinal PK modeling is not yet possible, because there are not enough data and reliable parameter values for modeling. Nevertheless, this may become possible in the future when chemical analytics and imaging at high resolution provide valuable experimental data. 

Other Administration Routes

Intracameral Injections

Intracameral injections are occasionally used to deliver antibiotic solutions and intracameral implant of brimonidine.¹⁰⁵ Intracamerally injected small molecule drugs are eliminated via aqueous humor outflow and blood flow of the iris and ciliary body (see Fig. 1). Top-down curve fitting of intracameral injection data of beta-blocking agents, dexamethasone, and brinzolamide was recently performed using first order one-compartment model.^12,59,106 The clearances from the rabbit aqueous humor ranged from 4.12 to 32.2 µL/min and the distribution volume range was 673 to 1421 µL. Typical half-lives ranged from 0.5 to 3.5 hours. The intracameral compartment has also been used as a component in more complex models for PKs of topical and intravitreal drugs. 

Clearance by aqueous humor outflow is relatively constant and easy to predict, but contributions of corneal distribution and anterior uvea blood flow in drug clearance from the anterior chamber are difficult to predict. This is due to the relatively sparse data that are currently available on intracameral injection PKs. More extensive data would enable more accurate predictions of intracameral clearance and distribution volume early in drug development. 

Subconjunctival Administration

Subconjunctival administration is used to reach higher drug concentrations in the anterior segment as compared to topical eye drop instillation. Our recent studies suggest that subconjunctival injection does not improve the bioavailability of dexamethasone in aqueous humor (0.74%)¹⁰⁷ compared to topical eye drops (0.62–2.15%).¹⁰⁶ However, higher concentrations in the anterior segment may be reached, because larger volume and higher dose can be injected subconjunctivally (even 0.5 mL) as compared to the volume of eye drops (30–40 µL). In order to reach the anterior segment tissues (e.g. the iris, ciliary body, and cornea), a drug should permeate across the relatively leaky sclera, but most of the dose is absorbed to the systemic circulation seriously limiting ocular absorption.^49,107 

Subconjunctival injections have been investigated as a potential way of delivering drugs to the retina. However, several barriers limit retinal bioavailability, including the sclera, conjunctival, and choroidal blood flows, and retinal pigment epithelium. Thus, retinal and vitreal bioavailability after subconjunctival injections is less than 1%, even below 0.1%.^49,107 

A bottom-up model was developed for simulation of subconjunctival drug delivery to the retina and vitreous body.⁴⁹ Due to the sparse experimental data the simulations were based on “typical parameter values” for lipophilic and hydrophilic small molecules and for macromolecules. The simulations indicate a steep concentration gradient from the subconjunctival depot toward the inner eye structures. Based on simulations, subconjunctival injection may be feasible only for the retinal delivery of compounds that are active at low nanomolar concentrations, enabling treatment at 10 mg doses.¹⁰⁸ 

Steady-State Concentration and Daily Dose Estimation

Drug delivery to the eyes is challenging and novel drug delivery systems are needed to treat intraocular tissues. The development of drug delivery systems is time-consuming, and a project's chance of success will be much higher if strong foundations are set. It is essential to verify early on that the dosing requirements are feasible and to understand that they are intimately linked to the site of drug administration and location of the target tissue, to the potency of the selected drug, to its PKs, and intended dosing interval. For this purpose, we have introduced an easy design aid for the calculation of steady-state drug concentrations in the ocular compartments.¹⁰⁹ The required drug dosing of a drug delivery system can be estimated using Equation 7:  

\begin{eqnarray} D = \frac{{C_{\rm{ss,av}} \times CL \times \tau }}{F} \end{eqnarray}

(7)

where D is the drug dose (µg), C_ss,av is the average steady-state concentration at the target site (µg/L), CL is the drug clearance from the target compartment (L/hour), τ is the dosing interval (hour), and F is the bioavailability (a value between 0 and 1). 

To calculate the daily drug dose of a topically administered drug delivery system, typical values of aqueous humor clearance (CL_AH) are 5 to 35 µL/min and bioavailability F_AH approximately 0.005 to 0.05. For example, the timolol maleate ophthalmic gel forming solution (Timoptic XE 0.25%) contains 2.5 mg/mL of timolol (molecular weight 316.421 g/mol) and is administered once daily. Assuming C_ss,av = 0.5 µM, CL_AH =10 µL/min, τ = 24 hours, and F = 0.02, the calculated dose is 0.11 mg, which is a good prediction of the actual dose in a 40 µL eye drop (0.1 mg). The same can be calculated for an intravitreal drug delivery system, where typical values of vitreal clearance (CL_V) in rabbits are 0.05 to 1 mL/hour for small molecules and 0.01 to 0.07 mL/hour for proteins. The drug becomes completely available after intravitreal administration F = 1. Next, we provide two examples of dose calculation: a small molecule drug (fluocinolone acetonide, molecular weight = 452.5 g/mol) and a macromolecule (bevacizumab, molecular weight = 149 kDa). Iluvien is a fluocinolone acetonide intravitreal implant with a total drug dose of 0.19 mg. The implant releases 0.2 µg of fluocinolone acetonide per day and is retained in the vitreous for up to 3 years. The concentration of fluocinolone acetonide in several ocular tissues following intravitreal injection of the Iluvien implant in rabbits has been measured by Kane and Green⁹⁹ and was maximum concentration (C_max) = 1 to 65 ng/g, so we can assume C_ss,av of 0.1 µM. With a vitreal clearance of CL_V = 0.2 mL/hour, τ = 3 years, and F = 1, the calculated total drug dose is 0.24 mg (close to the drug load of 0.19 mg in the implant). In another study,¹¹⁰ the monoclonal antibody bevacizumab was encapsulated in poly(lactic-co-glycolic acid) (PLGA) microspheres and injected intravitreally in rabbit eyes at a total drug dose of 1.25 mg. To reach the target concentration of 0.5 µM and using vitreal clearance CL_V = 0.017 mL/hour, τ = 42 days, and F = 1, the calculated total drug dose is 1.3 mg. Similar calculations can be used during drug development to estimate the required drug loading and release rate to reach target levels in the eye for certain dosing interval. 

In Equation 7, F and CL should be known from prior experimental work or reliably estimated. However, these values are not widely available. It should be noted that saturable transport and metabolism processes may affect ocular absorption (F) and/or elimination (CL) thereby leading to dose-dependent PKs. If dose-dependency is seen at relevant dose levels, misleading estimates for steady-state drug concentrations may be obtained. 

Melanin Binding

Melanin binding of drugs may affect the PKs and drug responses in the pigment containing eye tissues, such as the iris, ciliary body, RPE, and choroid. Melanin is located in the melasomes of pigmented cells.¹¹¹ Therefore, drug distribution into the melanosomes is a prerequisite of melanin binding. Melanin binding can affect the PKs of small molecule drugs that are melanin binders and capable of distributing across plasma and melanosome membranes into the melanosomes.¹¹¹ Melanin binding generates an intra-tissue drug depot thereby leading to prolonged drug retention in the tissues and potentially prolonging drug responses. For example, the retention of levofloxacin,^112,113 beta blockers,^47,114 pilocarpine,^115–117 atropine,¹¹⁸ chloroquine,^119,120 brimonidine and its peptide conjugates,¹²¹ and pazopanib¹²² in pigmented ocular tissues have been demonstrated in animal experiments. Drug retention in the pigmented rat eyes correlated with in vitro melanin binding: the mean residence time in pigmented rat eyes was 0.4 to 24 days after intravenous injections, whereas it was 0.2 to 0.7 days in albino rat eyes.¹¹³ 

From a modeling perspective, the key parameters that are affecting PK impact of melanin binding are as follows. First, parameters affecting binding to melanin, such as affinity, number of binding sites, melanin quantity, and type in the tissue, and equilibrium between free and bound drug in the melanosomes. Second, several parameters may affect the access of drugs to melanosomes, including route of drug administration and access to the target tissue as well as membrane permeability and active transport in the pigmented cells. 

Melanin binding can be estimated in vitro using several methods, such as equilibrium binding assays¹²³ and microscale thermophoresis,¹²⁰ and retention in special chromatographic columns.¹²⁴ Binding studies are done with different sources of melanin, but drug binding to three melanin types (synthetic, Sepia melanin, and isolated porcine melanin) was highly correlated, and dissimilarities were mainly due to differences in particle size and surface area of binding.¹²⁵ Rapid screening methods yield bound percentage values at single drug concentrations; over 3000 compounds were screened for melanin binding in vitro and the obtained results were used to build a machine learning model that predicted high binders (more than 95% bound in vitro) at an accuracy of 90%.⁴⁰ Such a model is useful in early prediction of melanin binding of new structures. However, simple binding percentage is not adequate for in vivo PK model building. Rather, binding affinity (K_d) and binding capacity (B_max) should be obtained using binding algorithms and binding data at several drug concentrations. Melanin binding is characterized by relatively low affinity, high binding capacity, and multiple binding sites: thus, the binding was best described with the Sips algorithm, rather than the Langmuir algorithm. The Sips model takes properly into account multiple sites and binding energies¹²⁶ that are involved in melanin binding of drugs and it can be further used as a component into PBPK models. 

Quantitative binding studies and related modeling have been performed also at the level of RPE cells and re-pigmented ARPE-19 cells.^111,127 Interestingly, drug release from melanin granules is relatively fast, typically less than 1 hour, even for high melanin binders.¹²⁷ However, drug release from melanin containing cells in vivo is much slower (days or weeks). This is explained by the complex equilibrium of melanin bound and free drug in the melanosomes: only the small free drug fraction can cross the melanosomal and plasma membranes that is a prerequisite for drug elimination from the pigmented tissue.^127,127 Unbound drug fraction in pigmented cells was estimated in vitro in pigmented RPE and non-pigmented ARPE-19 cells.¹¹¹ Overall, the unbound drug fraction of 5 test compounds showed a range of 0.00016 to 0.73 (6300-fold) in pigmented cells, whereas the range in non-pigmented cells was only 0.017 to 1.0 (25-fold). The results illustrate the major impact of melanin on binding equilibrium in pigmented cells. The only attempt to determine the free drug fraction in vivo was done in a brimonidine rabbit study.¹²⁹ Shinno et al. did not measure free brimonidine, but modeled the unbound concentration based on in vitro binding studies and melanin content in the retina-choroid. Overall, unbound concentration in the retina-choroid as well as the vitreal concentration were about two orders of magnitude less than the total drug levels in the pigmented retina-choroid. Bottom-up simulations on melanin binding in the RPE were performed by Rimpelä et al.¹²⁸ These simulations showed similar trends as observed in experimental rabbit studies, but true experimental validation was not reported. 

Detailed PK modeling of melanin binding in vivo would benefit from reliable parameter values (e.g. membrane permeability and binding parameters) and high quality in vivo data including analyses of unbound and bound drugh in the pigmented tissues. Currently, the overall retention in pigmented ocular tissues can be roughly predicted with existing information and modeling tools. However, the pharmacological activity depends on unbound drug concentration at the site of action and this cannot be well predicted yet. New analytical methods will be useful in generating such data that may boost in vivo pharmacological modeling of melanin bound drugs in the eyes. 

Interspecies Translation

Quantitative PK translation from the animal to the human eye has been mostly focused on intravitreal drug administration,^8,130 and less comparative data are available for topical and other ocular routes of administration.⁴⁵ Most intravitreal PK data stems from rabbit experiments.^8,50 On the other hand, preclinical pharmacological response studies are usually performed in mice and rats due to availability of various transgenic eye disease models.^131,132 Translation of intravitreal PKs from animal models to humans would foster successful clinical translation. We have previously developed mathematical models and scaling factors for interspecies translation among mouse, rat, rabbit, and human eyes. Table 3 shows intravitreal PKs of fluorescein and FITC-dextran in different species. It is obvious that there are major differences between species. The volume of drug distribution depends on the eye size, and half-life as well as clearance depend on the size of the eye and molecular weight of the injected compound (FITC-dextran ≫ fluorescein). The total drug exposure is characterized with AUC values, but similar AUC leads to different concentration profiles in various species (Fig. 21), that is, steeper profiles are seen in small animal eyes as compared to larger eyes. 

Table 3.

 

The Experimental and Estimated PK Parameters of Intravitreally Injected Fluorescein and FITC-Dextran in Different Species Based on a Previous Publication¹³²

 

Table 4.

 

The Interspecies Comparison of the Dose Required to Have the Same AUC as in Humans

 

Figure 21.

 

Single dose intravitreal kinetics of (A) fluorescein and (B) FITC-dextran (150 kDa) in the eyes of different species. The doses result in similar compound exposure (AUC), but different profiles in various species. The black dotted lines show an assuming effective concentration of 10 µM (A) and 100 nM (B) for small and macromolecules, respectively.

Population Pharmacokinetic/Pharmacodynamic Models

In general, population PK models are recommended by the regulatory agencies (the European Medicines Agency [EMA] and the US Food and Drug Administration [FDA]; Population Pharmacokinetics | FDA¹³³), but they are rarely used in ophthalmology (Table 5). The biggest hurdle is to obtain ocular PK data from patients, because it requires ocular tissue sampling at different time points. Sampling of aqueous humor from patients is possible because the fluid is regenerated naturally, but sampling is ethically justified only from patients in cataract surgery. Vitreous sampling is also invasive and only performed during vitrectomy intervention. Alternatively, plasma data can be collected, but drug levels in plasma are very low, and do not properly report about ocular PK (Table 5).¹⁵ For these reasons, PK studies in translational animal species are needed. 

Table 5.

 

List of the Current Publications of Population Pharmacokinetic and Pharmacokinetic-Pharmacodynamic Models in the Field of Retinal Therapeutics

 

Another clinical modeling approach is to focus on the PD component, and fix PK using typical parameter values for the drug, either from clinical (if available), or from preclinical studies based on rabbit or monkey-to-human scaling factors. The variability in the population model is obtained from noninvasive PD measurements, such as central macular retinal thickness or visual acuity. Audren et al.¹³⁹ investigated the effect of intravitreal triamcinolone acetonide suspension in patients with diabetic macular edema by following the central macular retinal thickness. A similar approach was recently used to model bevacizumab effects in patients with neovascular AMD (nAMD).¹⁸ Even though the nonlinear mixed effect approach can handle sparse data, the time pattern of macular thickness measurements, only done just before the next injection, is problematic in model building. In addition, high response variability among patients is a major challenge for clinical population PK/PD modeling.¹⁸ More complete data would enable building robust and reliable models. Moreover, new PD endpoints for retinal state are investigated. Recently, hyper-reflective foci (HRF), small bright objects in the inner and outer retina, observed in patients with diabetic macular edema and nAMD,¹⁴⁰ have been proposed as imaging biomarkers for the retinal disease state. Such measurements require a special algorithm to obtain accurate quantitative data. 

Future Challenges

Mathematical modeling of ocular PKs and drug delivery has developed remarkably during the last decades, but there are still major areas that need further development. Progress of quantitative and reliable modeling is fueled not only by the progress in mathematical and computational methods, but also by the experimental data that are required in model validation. 

Commercial software for drug development have been launched during the last 2 decades (e.g. GastroPlus and SimCYP Simulator). These software packages utilize both the bottom-up and top-down approaches in parameter estimation for models of per oral, transdermal, and intravenous drug administrations. GastroPlus also includes an ocular PK module. The models are complex and, unfortunately, there are only few published and peer-reviewed reports that assess the real-life validity of those ocular models.^63,65 Among regulatory agencies, the FDA has encouraged and supported PBPK ocular modeling efforts in the context of ophthalmic bioequivalence of complex topical medicinal products. It is important to realize that with large number of parameters, good curve fitting of the data is obtained even if the parameter values are unrealistic. Published unbiased and extensive assessment of commercial software packages would be helpful for the ocular drug delivery community. 

Impact of drug metabolism and active transport are not well understood in ocular drug delivery. Some public data on the active transport in the cornea and blood retina barrier have been published as well as quantitative proteomic assessment of transporters in the RPE cells,¹⁴¹ but modeling of the active transport with experimental data validation is not available. Only some simulation models without experimental verification were published.¹⁴² Based on these simulations active transport may have a substantial effect at cell level, but much less at the ocular tissue level. Although current experimental data are not adequate to prove or disprove this concept, some recent developments in mass spectrometry should enable quantitative analyses of transporter expression and drug levels at cell type level within the ocular tissues (e.g. retina) extending PBPK and finite element modeling to the intra-tissue levels. 

Population PK is an important part of clinical regulatory documentation of new systemic drug products, because it takes into account patient variability and helps to provide dosing recommendations for special patient groups. Unfortunately, this is not the case in ophthalmic drug development, because human PK data are sparse in ophthalmology; an understandable consequence of ethical concerns of ocular sampling. Population PKs cannot be widely used as a clinical tool in ophthalmology unless there will be quantitative analytical methods capable of reporting drug concentrations noninvasively in patients’ eyes. In principle, population PKs can be used in preclinical animal studies, even though it has been only rarely utilized so far.^16,17 

Population PK/PD modeling generates links between PK and drug or biomarker responses. Again, this approach is widely used in systemic pharmacology, but only rarely in human ocular pharmacology (see Table 5) due to the lack of human ocular PK data. As compared to intraocular pressure studies,^148–150 retinal pathologies are more difficult to apply in PK/PD modeling, because reponses to drug treatment are emerging slower and they are not as clearly quantitative. Response data on biomarkers and biological pathways (e.g. transcriptomics, proteomics, and metabolomics) offers a plethora of data that is amenable for modeling. Obviously, this would be of great value in preclinical drug development. Modern analytical methods that will enable the generation of cell type and even single cell data are expected to provide high resolution PK/PD data for further model building. These developments will boost Quantitative Systems Pharmacology (QSP) of the eye. The QSP is a field that aims to link complex biochemical pathways with the PKs and PDs of drugs. Meaningful QSP models are dependent on the availability of quantitative metabolomics and the proteomics data as well as PK/PD results. In the long term, QSP will represent the ultimate model also for ocular pharmacology. 

Artificial intelligence (AI) may play significant role in advancing PKs. Machine learning and AI can be used in the context of PBPK modeling. For example, they help in data collection from publicly available databases, predict PK parameters, and they may be incorporated into PBPK models to predict drug concentration profiles. Machine learning and AI rely on available PK or PD data, whereas the physiology, biology, or pharmacology are not considered in the models. Current recommendation is to use these algorithms as supporting tools for PBPK and PK/PD model development.¹⁵¹ In summary, AI-driven models may enhance our understanding of drug behavior, optimize dosing regimens, and contribute to safer and more effective drug development. 

Overall, modeling is a rapidly developing field that has plenty of useful applications in ocular drug research, discovery, and development. The models help in building scientific understanding, but they are also useful in translational research, benefiting patients who will obtain effective and safer new medications. 

Acknowledgments

The authors thank Veli-Pekka Ranta for insightful comments and suggestions. 

Supported by the Tandem Industry Academia foundation postdoc grant (TIA Postdoc, ASa), the Academy of Finland (grant number 333301, ASu), research support from the strategic funding of the University of Eastern Finland (EMDA). 

Disclosure: A. Sadeghi, None; A. Subrizi, Janssen Pharmaceutica (F); E.M. del Amo, Allen & Overy Shearman Sterling LLP (C), Anidal Pharma (C); A. Urtti, Active Biotech (C), Pharming (C), Allen & Overy Shearman Sterling LLP (C), Ocular Therapeutics (C), UNITHER (C), ReBio Technologies (F), Bayer (F), Roche (F)