A First-Passage Model of Intravitreal Drug Delivery and Residence Time—Influence of Ocular Geometry, Individual Variability, and Injection Location

Abstract

Purpose: Standard of care for various retinal diseases involves recurrent intravitreal injections. This motivates mathematical modeling efforts to identify influential factors for ocular drug residence time, aiming to minimize administration frequency. We sought to describe the vitreal diffusion of therapeutics in nonclinical species frequently used during drug development assessments. In human eyes, we investigated the impact of variability in vitreous cavity size and eccentricity, and in injection location, on drug disposition.

Methods: Using a first-passage time approach, we modeled the transport-controlled distribution of two standard therapeutic protein formats (Fab and IgG) and elimination through anterior and posterior pathways. Anatomical three-dimensional geometries of mouse, rat, rabbit, cynomolgus monkey, and human eyes were constructed using ocular images and biometry datasets. A scaling relationship was derived for comparison with experimental ocular half-lives.

Results: Model simulations revealed a dependence of residence time on ocular size and injection location. Delivery to the posterior vitreous resulted in increased vitreal half-life and retinal permeation. Interindividual variability in human eyes had a significant influence on residence time (half-life range of 5–7 days), showing a strong correlation to axial length and vitreal volume. Anterior exit was the predominant route of drug elimination. Contribution of the posterior pathway displayed a 3% difference between protein formats but varied between species (10%–30%).

Conclusions: The modeling results suggest that experimental variability in ocular half-life is partially attributed to anatomical differences and injection site location. Simulations further suggest a potential role of the posterior pathway permeability in determining species differences in ocular pharmacokinetics.

The eye is a complex organ that varies significantly in size and shape between different species. In the human eye, individual variations of size and shape are common and can cause various vision conditions. With emmetropia describing the absence of refractive error, myopia is generally characterized by an elongated eye,¹ with a larger axial length (AL) compared to an emmetropic eye,² while hypermetropia is often associated with a shorter AL.³ 

Approximately one in three people have some form of disease-induced vision impairment by the age of 65.⁴ A common cause of vision loss among the elderly is age-related macular degeneration (AMD), a progressive disease characterized by damage to the macula.⁴ The wet form of AMD is characterized by upregulation of the vascular endothelial growth factor (VEGF), an angiogenic protein⁵ that induces pathological neovascular growth leading to retinal damage.^6,7 

Among treatment options for wet AMD are intravitreal (IVT) injections of protein therapeutics that bind to VEGF to inhibit its function.⁵ Two standard-of-care therapeutics are ranibizumab, a monoclonal antibody fragment (Fab), and aflibercept, an Fc-fusion protein, with reported hydrodynamic radii (R_h) of 3.0 and 5.2 nm, respectively.^8,9 The latter is comparable to the macromolecular size of bevacizumab (5.0 nm),⁸ a monoclonal full-length IgG1 antibody¹⁰ that is used off-label for the treatment of choroidal neovascularization. These antibodies are administered through IVT injections, and regular administrations were shown to improve visual acuity outcome in the majority of patients.^11–13 However, IVT drugs exhibit suboptimal drug retention, with clinical studies reporting the ocular half-lives of less than 10 days.^14,15 

IVT injections in the human eye are not targeting a specific injection site within the vitreous chamber.¹⁶ Broad guidelines have been specified; for example, the needle tip should be inserted more than 6 mm aiming at the center of the eye¹⁷ with the bevel facing upward,¹⁸ which provides less than a precise target for the injection site. It is also advised to deliver the dosing formulation gently into the vitreous cavity with a slow injection, in order to avoid jet formation and excessive cavitary flow.^17,18 In general, procedures are specified with focus on preventing mechanical damage and infections, with seemingly less attention paid to the potential impact on drug absorption or ocular residence time.^17,19 However, chronic treatment places a burden on patients and health care systems in terms of resources and procedural risks.^20,21 Moreover, it is estimated that the majority of the injected drug is eliminated through the anterior pathway via aqueous humor turnover, with a small proportion permeating the retina (posterior elimination pathway), despite being the target site of action.^22,23 This motivates investigation of drug residence time in the eye due to differences in eye shape and size, drug hydrodynamic radii, and injection locations. 

The translation of results across nonclinical species and patients is crucial for the effective design and characterization of drug candidates. Species commonly used in ocular pharmacokinetic (PK) or pharmacodynamic (PD) studies are the rabbit,^9,24–29 cynomolgus monkey,^30–32 and rat,^33,34 with fewer studies reported in pig^35–37 and mice.^38,39 Previous studies have demonstrated the role of diffusion in determining the vitreal elimination rate of IVT macromolecules, with PK experiments showing a dependence of ocular half-life on both the molecular size and eye size.^8,9,40 In particular, in Caruso et al. (2020),⁸ the ocular half-life was expressed as a linear function of \(R_h \times r_{\rm vit}^2\), with r_vit the vitreal radius (under the assumption of a spherical vitreous chamber), and with a species-specific slope. With the slopes varying between species, it was postulated that other factors than diffusion distance must explain the species-specific half-lives.⁸ For example, the experimental half-life obtained for a given molecule or molecular size is larger in pigs than in humans and in rabbits than in monkeys, although the respective vitreous volumes are smaller. Caruso et al.⁸ and Crowell et al.⁴⁰ postulated that the observed species-specific PK could result from differences in ocular shape and eccentricity or in the contribution of the posterior pathway to drug elimination, among other factors. This motivates modeling efforts in investigating these aspects to support the translational characterization of novel drug candidates. 

Scaling relationships between species have been previously established⁴¹ under the assumption of spherical vitreous chambers, implying simplified ocular anatomies. More anatomically faithful models are also available, such as the computational fluid dynamics works of Missel⁴² and Lamminsalo et al.^43,44 that describe both the posterior and anterior segments in great detail. Besides the complexity and computational cost inherent to such models, the rate of egress of material from the vitreous remains the rate-limiting factor and main determinant of ocular PK, so studying the distribution within the vitreous cavity and at its interfaces in more detail is key. 

As defined in the random walk field, the first passage time is a random variable describing how long it takes for a random walker to reach a given target site.^45,46 Its expected value is called the mean first passage time (MFPT). The MFPT has been described as an effective measure of diffusive transport.^47–49 Recent applications in mathematical biology have been diverse, including applications to animal movement in heterogeneous landscapes,⁵⁰ receptors in the synaptic membrane,⁵¹ and drug molecules crossing the mucus–epithelium interface.⁵² In ocular modeling, an approximation of the MFPT was previously used to define the vitreal diffusion time, the average time for a particle to diffuse from the center of a sphere to its surface, with the vitreous chamber modeled as a sphere in Hutton-Smith et al.⁴¹ However, the first-passage problem was not explicitly solved. In the present MFPT modeling framework, we considered isotropic diffusion and the absence of convection in the distribution of the injected ocular drug, justified by the slow injection of the drug and by previous experiments supporting the absence of flow in the vitreous chamber.^53–57 The MFPT is a measure of the residence time (which does not depend on initial drug concentration), and its framework can be extended to quantify the drug elimination using the distribution of exits, the multidimensional analogue of the splitting probabilities.^47,48,58 This can be used to quantify the amount of drug leaving through each elimination pathway and their relative importance. 

This research aims to investigate the influence of ocular size and shape, interindividual variability, drug molecular features, and injection location on the vitreal kinetics and residence time of IVT macromolecules. We further aim to study whether interspecies differences in vitreous chamber geometry may explain the different pharmacokinetics observed experimentally. To this end, we develop a mathematical model of vitreal drug diffusion based on the first-passage time methodology, deriving equations for the MFPT, the conditional MFPT, and the drug elimination distribution for two standard molecular formats of IVT therapeutics, namely, Fab and IgG. Using literature datasets to construct realistic three-dimensional (3D) geometries, we model human emmetropic, myopic, and hypermetropic eyes for studying the influence of ocular shape on drug retention. We also model the anatomy of the vitreous chamber in the mouse, rat, rabbit, and cynomolgus monkey, aiming to improve the translational understanding of PK in support of drug discovery and development. In order to assess the influence of spatial parameters, we compare the residence time in the different vitreous chamber geometries and derive a scaling relationship between the MFPT, vitreal volume, and AL. We also assess the dependence of retinal absorption on the spatial parameters describing the ocular geometries and on the site of injection within the vitreous cavity, as well as identify the dominant elimination pathway kinetics using the conditional MFPT. 

Methods

Ocular Geometry

The 3D models of the vitreous chamber were built for human and relevant nonclinical species. The posterior cavity was assumed to be an oblate spheroid, obtained by the rotation of an ellipse around its minor axis, which was collinear with the optical axis. The lens protruding into the vitreous humor was similarly defined. The difference between the two determined the vitreous chamber (Fig. 1). Three interfaces were defined, corresponding to the vitreous–lens, vitreous–aqueous humor, and vitreous–retina boundaries. Anatomically, the vitreous–retina interface corresponds to the surface covered by the inner limiting membrane (ILM) and delimited by the ora serrata, and the vitreous-aqueous humor interface corresponds to the zonular fibers and the space of Petit. The parameters used to build the 3D geometries are defined in Figure 1, and the details of the construction of the geometries are summarized in Supplementary S1.2. 

Figure 1.

 

Plane geometry used in the axial rotation to define the 3D ocular model for the human eye, where a and b are the semi-major and semi-minor axes of the vitreous chamber ellipse, l_D and l_T are the lens diameter and thickness, and h_va is the height of the vitreous–aqueous humor interface. The vitreous–lens interface is identified in pink, the vitreous–aqueous humor interface in orange, and the vitreous–retina interface in blue. Parameters not shown: l_p, the proportion of the lens thickness within the vitreous chamber ellipse, and V_vit and A_ret, the volume of the vitreous humor and the area of the retinal surface.

A literature search was conducted to collect the anatomical dimensions of human eyes as well as those of the cynomolgus monkey, rabbit, rat, and mouse. Insufficient anatomical information prevented inclusion of the pig or minipig among modeled species. A summary of the literature data is provided in Table 1. The measurements were performed by various methods, including magnetic resonance imaging (MRI), optical coherence tomography, ultrasound biometry, Scheimpflug photography, and direct measurements of postmortem fixed eyes. The parameter values for the model geometries are also summarized in Table 1 and detailed in Supplementary S1.2. A cross section of the model for each species is displayed in relative scale in Figure 2. We verified the anatomical accuracy of the geometries by comparison with experimental measurements of vitreous volumes and retinal areas (V_vit and A_ret in Table 1) and with in vivo MRI images obtained from the literature, as illustrated in Figure 3. 

Table 1.

 

Literature Values of Ocular Geometry Measures and Ocular Model Dimensions

 

Figure 2.

 

Cross sections of the ocular geometries built using parameters in Table 1 and based on Figure 1, in relative scale. The black bullet represents the injection point P_m, and the gray circle in the rabbit, cynomolgus monkey, and human eye models represents the injection location region considered in the ocular half-life (t_1/2) analysis.

To investigate their influence on ocular PK, different injection site locations within the vitreous chamber were considered. Under the assumption that IVT injections target the central vitreous, equidistantly from the retina and the posterior surface of the lens, a region of interest was defined around the midpoint (P_m) of the vitreous chamber depth. For simplicity, a sphere of diameter corresponding to half of the vitreous chamber depth was defined (Fig. 2). The resulting volume encompasses the vitreous core, arguably representing a conservative estimate for the possible locations of IVT delivery. This injection region was employed to assess the impact on ocular half-life, t_1/2, in human, cynomolgus monkey, and rabbit eyes. In rodents, the lens occupies a significant portion of the posterior cavity (Fig. 3), giving the vitreous chamber a distinctive crescent shape, for which it is more difficult to define the center and reach it with an injection needle. Therefore, in the rat and mouse model, P_m was defined as the midpoint between the retina and the lens, along the vitreous diameter (Fig. 2). 

Additionally, an ensemble of 155 human eye models was built based on AL and vitreous volume measurements obtained from the literature (Fig. 4). The measurements were collected by MRI,^2,59 optical biometry and vitrectomy,⁶⁰ and computed tomography (CT) scan.⁶¹ The AL data collectively cover the range associated with hypermetropic, emmetropic, and myopic eyes,^2,3 and include data for pathological myopia, described as a refractive error of −8 diopters or lower.^59,62 Using the vitreous volume and AL, the eye geometries were constructed assuming a constant lens thickness and anterior chamber depth. The reader is referred to Supplementary S1.2 and Supplementary S2 for further details. 

Figure 3.

 

(A–C, E) Superimposition of the ocular models (blue lines) and in vivo MRI images. The geometries in Figure 2 were scaled to the anatomical features of the MRI images with a constant aspect ratio. (D) No MRI images of the cynomolgus monkey eye could be found in the literature; therefore, a light micrograph of a longitudinal section (grayscale) was used instead. Annotations in (A, C, E) originate from the source images. (C) Reprinted by permission of Taylor & Francis Ltd. from Sawada T, Nakamura J, Nishida Y, Kani K, Morikawa S, Inubushi T. Magnetic resonance imaging studies of the volume of the rabbit eye with intravenous mannitol. Curr Eye Res. 2002;25(3):173–177. (D) Reprinted by permission of Sage Publications from Short BG. Safety evaluation of ocular drug delivery formulations: techniques and practical considerations. Toxicol Pathol. 2008;36(1):49–62.

Figure 4.

 

Literature measurements of vitreous volume and axial length (AL) in 155 human eyes, used to build the ensemble of human eye models. The hypermetropia, emmetropia, and myopia ranges are identified using definitions by Strang et al.³ and Atchison et al.² Black bullet: emmetropic human eye model corresponding to the geometry of Figure 2.

Equations

The equations have been derived by expanding the first-passage time approach^{47–49,97,98} and applying it to vitreal transport. 

Mean First-Passage Time

The MFPT, \(\tau ({\boldsymbol x_0})\), for a particle starting at \({\boldsymbol x_0}\), satisfies the following partial differential equation (PDE) and boundary conditions:  

\begin{eqnarray} \begin{array}{@{}r@{\;}c@{\;}l@{\quad}l@{}} -D \, \nabla ^2 \tau ({\boldsymbol x_0}) &=& 1 &\text{for } {\boldsymbol x_0} \in \Omega ,\\ \nabla \tau ({\boldsymbol x_0}) \cdot {\boldsymbol n} &=& 0 &\text{for } {\boldsymbol x_0} \in \partial \Omega _{vl},\\ -D \, \nabla \tau ({\boldsymbol x_0}) \cdot {\boldsymbol n} &=& \kappa _{va} \, \tau ({\boldsymbol x_0}) & \text{for } {\boldsymbol x_0} \in \partial \Omega _{va},\\ -D \, \nabla \tau ({\boldsymbol x_0}) \cdot {\boldsymbol n} &=& \kappa _{vr} \, \tau ({\boldsymbol x_0}) &\text{for } {\boldsymbol x_0} \in \partial \Omega _{vr},\end{array} \end{eqnarray}

(1)

where \({\boldsymbol n}\) is the outward normal and with parameters defined in Table 2, for the vitreal region Ω and associated lens, anterior and retinal boundaries ∂Ω_vl, ∂Ω_va, and ∂Ω_vr respectively, as illustrated in Figure 1. 

Table 2.

 

Definition of the Drug-Dependent Parameters for the Fab and IgG Molecular Formats

 

The MFPT can be linked to the ocular half-life t_1/2 (i.e., the time required for a quantity that is exponentially decaying to fall to half of its initial value). In this context, it characterizes the rate at which the drug is cleared from the vitreous chamber. Under the assumption that the drug concentration inside the vitreous is decreasing exponentially after an IVT injection (see Supplementary S1.1), the MFPT corresponds to the inverse of the decay rate, and we obtain the relation:  

\begin{eqnarray} t_{1/2}({\boldsymbol x_0}) = \tau ({\boldsymbol x_0}) \ln (2) \end{eqnarray}

(2)

 

for an injection at \({\boldsymbol x_0}\). 

Drug Elimination and Conditional Mean First-Passage Time

Let \(\pi _{va}({\boldsymbol x_0})\) be the proportion of drug leaving through the vitreous–aqueous humor interface (∂Ω_va) and \(\pi _{vr}({\boldsymbol x_0})\) the proportion leaving through the vitreous–retina interface (∂Ω_vr), where \({\boldsymbol x_0}\) is the initial position. Note that these are the only regions in the model where molecules can exit and, therefore, the proportions sum to 1. We define \(T_{va}({\boldsymbol x_0})\) as the MFPT conditional on drug molecules leaving through ∂Ω_va and \(T_{vr}({\boldsymbol x_0})\) the conditional MFPT for molecules leaving through ∂Ω_vr, both functions of the molecules' initial position \({\boldsymbol x_0}\). The proportion of drug exiting through ∂Ω_vr, \(\pi _{vr}({\boldsymbol x_0})\), satisfies the PDE system^48,49,98:  

\begin{eqnarray} \begin{array}{@{}r@{\;}c@{\;}l@{\quad}l@{}} \nabla ^2 \pi _{vr}({\boldsymbol x_0}) &=& 0 &\text{for } {\boldsymbol x_0} \in \Omega ,\\ \nabla \pi _{vr}({\boldsymbol x_0}) \cdot {\boldsymbol n} &=& 0 &\text{for } {\boldsymbol x_0} \in \partial \Omega _{vl},\\ -D \, \nabla \pi _{vr}({\boldsymbol x_0}) \cdot {\boldsymbol n} &=& \kappa _{va} \, \pi _{vr}({\boldsymbol x_0}) &\text{for } {\boldsymbol x_0} \in \partial \Omega _{va}, \\ -D \, \nabla \pi _{vr}({\boldsymbol x_0}) \cdot {\boldsymbol n} &=& - \kappa _{vr} + \kappa _{vr} \, \pi _{vr}({\boldsymbol x_0}) & \text{for } {\boldsymbol x_0} \in \partial \Omega _{vr}, \end{array} \end{eqnarray}

(3)

and \(T_{vr}({\boldsymbol x_0})\) satisfies^48,98:  

\begin{eqnarray} \begin{array}{@{}r@{\;}c@{\;}l@{\quad}l@{}} -D \, \nabla ^2 [\pi _{vr}({\boldsymbol x_0}) \, T_{vr}({\boldsymbol x_0})] &=& \pi _{vr}({\boldsymbol x_0}) & \text{for } {\boldsymbol x_0} \in \Omega ,\\ \nabla [\pi _{vr}({\boldsymbol x_0}) \, T_{vr}({\boldsymbol x_0})] \cdot {\boldsymbol n} &=& 0 &\text{for } {\boldsymbol x_0} \in \partial \Omega _{vl},\\ -D \, \nabla [\pi _{vr}({\boldsymbol x_0}) \, T_{vr}({\boldsymbol x_0})] \cdot {\boldsymbol n} &=& \kappa _{va} \, [\pi _{vr}({\boldsymbol x_0}) \, T_{vr}({\boldsymbol x_0})] &\text{for } {\boldsymbol x_0} \in \partial \Omega _{va}, \\ -D \, \nabla [\pi _{vr}({\boldsymbol x_0}) \, T_{vr}({\boldsymbol x_0})] \cdot {\boldsymbol n} &=& \kappa _{vr} \, [\pi _{vr}({\boldsymbol x_0}) \, T_{vr}({\boldsymbol x_0})] &\text{for } {\boldsymbol x_0} \in \partial \Omega _{vr}.\end{array}\quad \end{eqnarray}

(4)

The conditional MFPT for drug molecules leaving through the vitreous–aqueous humor, \(T_{va}({\boldsymbol x_0})\), was derived following the same method. 

Drug-Dependent Parameters

The drug-dependent parameters present in the PDE systems (Equations (1) to (4)) were set using experimental measures and modeling results found in the literature. The value of the diffusion coefficient D is well defined by experimental studies in the literature and was set to D = 1.07 × 10⁻⁶ cm²/s and D = 0.64 × 10⁻⁶ cm²/s for a Fab and IgG molecular format, respectively.⁸ 

The permeability parameters κ_va and κ_vr, for the vitreous–aqueous humor and vitreous–retina interface, respectively, were more difficult to determine. In addition to varying with the drug molecule size, the permeability parameters are reported to vary between different species.¹⁰⁰ Previous estimates of these parameters for Fab and IgG molecules are summarized in Table 3. To incorporate these estimates into our model, we identified the vitreous–aqueous humor interface as equivalent to the hyaloid membrane described in Hutton-Smith.⁹⁹ For the vitreous–retina interface, we designated its permeability to the lowest permeability between the retinal pigment epithelium (RPE) and the ILM defined in Hutton-Smith et al.^23,99,101 As there was only one estimate for the vitreous–aqueous humor permeability, we set κ_va = 1.91 × 10⁻⁵ cm/s, and we used the estimates from the same reference⁹⁹ to set κ_vr = 1.81 × 10⁻⁷ cm/s, both for a Fab molecule. For the IgG molecule, we followed the same steps and set κ_va = 0.874 × 10⁻⁵ cm/s and κ_vr = 1.19 × 10⁻⁷ cm/s, using the estimations from Hutton-Smith.⁹⁹ We note that the permeability parameters in Table 3 have a high degree of uncertainty, as they cannot be measured directly, and were estimated by fitting mathematical models to rabbit data. 

Table 3.

 

Literature Values of Ocular Permeabilities

 

Numerical Methods

All geometries were built using COMSOL Multiphysics.¹⁰² In constructing each geometry, a mesh for the finite element numerical method was also constructed within COMSOL, with sufficient grid resolution to ensure numerical convergence. This was tested by confirming that further mesh refinement had no impact on example results at the resolution of plotting presented. 

The equations were solved and the figures were generated using COMSOL Multiphysics software,¹⁰² using the implemented stationary solver. Regression lines were obtained with the Scikit-learn library¹⁰³ in Python. The data and the code used to produce the results are available at the GitHub repository: https://github.com/patricia-lamy/MFPT-ocular-drug-delivery. 

Global Sensitivity Analysis

We performed a global sensitivity analysis to identify which geometrical parameters are most important to accurately determine when constructing computational ocular models. We varied the geometrical parameters a, b, l_D, and l_T (see Fig. 1 for their definition) within the range of literature ocular values identified in Table 1 for each species and h_va within ±10% of its base value. We performed this sensitivity analysis for the human, cynomolgus monkey, and rabbit eye models, as the geometries for the rat and mouse models were only well defined for a subset of the parameter combinations. We analyzed the effect of the geometrical parameters on the MFPT for an injection at P_m. This was implemented using the eFAST sensitivity method,^104,105 with the Python SAlib library,^106,107 and with the choice of sampling parameters guided by the methodology proposed in the referenced sources. We set to 4 the number of harmonics to sum in the Fourier series decomposition and to 337 the number of samples to generate. The implementation of the sensitivity analysis sampling was validated by confirming that a dummy variable has sensitivity indices of around zero, demonstrating minimum sampling artifact.¹⁰⁵ See Supplementary S1.3 for more information. 

Results

Mean First-Passage Time

To obtain numerical solutions for the MFPT, we solved Equation (1) with parameter values defined in Tables 1 and 2, using the ocular geometries of each species (Fig. 2). The results for the Fab molecule are illustrated in Figure 5. In each model, the MFPT is maximized for an injection site at the back of the vitreous and decreases for injections closer to the aqueous humor. The same behavior was observed for the MFPT of an IgG molecular format, with overall longer residence times. Figure 6A compares the MFPT in the human model for a Fab and IgG molecule, with the maximum MFPT being 9 days and 14 days, respectively. 

Figure 5.

 

Numerical solution of the mean first-passage time (MFPT) for a Fab molecule in different species, as a function of injection site, with the parameterization of Tables 1 and 2. Contour lines of MFPT are in white, while the colormap indicates the MFPT value at any given point in the vitreous chamber. Black lines are associated with the construction of the model geometry (Fig. 2).

We observed that the MFPT decreases with eye size, with an MFPT of less than 1 day for all injection sites in the mouse and rat models (Fig. 5). The global sensitivity analysis identified that the length of the semi-axis b was the most influential parameter for the MFPT for an injection at P_m (results shown in Supplementary S1.3). The analysis also revealed that the model was sensitive to the permeability parameters when they were varying within their uncertainty range but not when their range varied proportionally with the other parameters (Supplementary S1.3). 

To refine the comparison of the MFPT across species, the MFPT for a Fab and IgG molecular format was solved for an injection at P_m and was plotted against the vitreous chamber depth measure of each species (Fig. 7A). The choice of P_m as a comparison point for the MFPT between species should not significantly influence the results, as the MFPT at a point is a good approximation of the average MFPT of a spherical bolus centered at that point, as long as the radius of the bolus is not too large (see Supplementary S1.4 for more information). The corresponding linear regressions were derived and constrained to go through the origin, as the intercept confidence interval included the origin and as it is the expected theoretical behavior. Following the analysis in Caruso et al. (2020),⁸ where the half-life for each species was plotted as a linear function of \(r_{\rm vit}^2 \times R_h\) with species-specific proportionality constants, the MFPT was also plotted as a function of b² × R_h. In order to represent the higher level of detail in the eye geometries suggested in this work, the semi-axis b was taken to represent the different vitreous sizes (given it was the most sensitive geometrical parameter). The results (shown in Fig. 7A) illustrate proportionality without the need to distinguish between species as all MFPT points are aligned on a linear regression line. A linear relationship for the MFPT across species and molecular formats was obtained, with a regression going through the origin and a slope of 3.81 days/(cm² nm). Equation (2) was used to obtain a slope of 2.64 days/(cm² nm) for t_1/2. 

Figure 6.

 

Numerical solution of the MFPT for Fab and IgG molecules in the human emmetropic eye model (A) and of the proportion of the drug dose exiting through the vitreous–retina interface (B), as a function of injection site, using the parameterization of Tables 1 and 2. The Fab results are also presented with a different color bar scaling in Figures 5 and 8.

Figure 7.

 

Numerical solution (symbols) and linear regressions (lines) of the MFPT for different molecular formats injected at P_m, using the parameterization of Tables 1 and 2. The MFPT is shown for (A) the species-specific geometries of Figure 2 and (B) the ensemble of human eye models of Figure 4.

Considering an injection of a Fab or an IgG at P_m, Figure 7B shows the MFPT as a function of the AL and of vitreous volume for the ensemble of human eye models. We found that the pathological myopia eyes did not significantly affect the trends found in the ensemble of healthy human eyes, with a change of slope of less than 10% for each molecular format when we excluded the eye geometries constructed using the pathological myopia measurements from Zhou et al.⁵⁹ (results presented in Supplementary S1.5). Comparing the two panels of Figure 7B, we see that the MFPT is better predicted by the linear relation with the AL than with the vitreous volume, with the resulting residence times being more sparsely distributed when plotted against the vitreous volume. As mentioned above, these results are not expected to be sensitive to the chosen injection point P_m and should also hold for larger injection regions (see Supplementary S1.4 for more information). 

Using Equation (2), we derived t_1/2 for all species from the MFPT for an injection located at P_m. The results are summarized in Table 4, along with the experimental t_1/2 for each species. For the human eye model, we estimated a range of t_1/2 using the MFPT for hypermetropic to myopic eyes illustrated in Figure 7B, hence excluding pathologically myopic eyes, and found this estimated range to be broadly consistent with the spread of t_1/2 observed in normal eyes. For the human, cynomolgus monkey, and rabbit eye models, a range of t_1/2 was also estimated by solving the MFPT within the injection region (identified in Fig. 2). Agreement between simulation and experiment holds for the human and the rabbit, whereas the model's half-lives are overestimated for the cynomolgus monkey and the rat and underestimated for the mouse. 

Table 4.

 

MFPT and Estimated t_1/2 Modeling Results by Species

 

Drug Elimination

To obtain the numerical solutions for the proportion of drug exiting through the vitreous–retina interface, Equation (3) was solved with the parameters and ocular geometries of Table 2 and Figure 2, respectively. The results for the Fab molecule are illustrated in Figure 8. The maximum contribution of posterior elimination varied across species, with up to 40% of dose permeating the vitreous–retina interface in the human, versus 12% in the mouse, while the posterior elimination from the injection point P_m varied from 10% to 30% across species. Injection sites located at the back of the eye were associated with higher proportions of posterior elimination, which decreased with the distance from the anterior segment. Similar results were obtained for IgG, as visible in Figure 6B for the emmetropic human eye. 

Figure 8.

 

Numerical solution and contour lines of the proportion of drug dose exiting through the vitreous–retina interface (unitless), for a Fab molecule and for each species, as a function of injection site, with the parameterization of Tables 1 and 2, and the geometries in Figure 2.

To study the influence of interindividual anatomical differences, the ensemble of human eye models was solved for the posterior elimination after injection at P_m (Fig. 9). The posterior contribution to ocular elimination showed a strong correlation with the AL and vitreous volume for both Fab and IgG, with poor separation between molecular formats for the latter. 

Figure 9.

 

Numerical solution and regression lines of the proportion of drug dose exiting through the vitreous–retina interface in the ensemble of human eye models (as identified in Fig. 4) for an injection at P_m and with parameters of Table 2, as a function of the AL and vitreous volume.

Conditional MFPT

To obtain numerical solutions for the conditional MFPT, which gives the duration time conditioned on the exit rate, Equations (3) and (4) were solved with parameter values given in Table 2 for a Fab molecule, using the human eye geometry. Figure 10 shows the results for the conditional MFPT. The conditional MFPT for the drug exiting through the vitreous–retina interface has a lower variation range, with exit times varying between 6.5 and 9 days, and has very different contour plots to those of the unconditional MFPT (Figs. 5, 6). In contrast, the conditional MFPT for the drug exiting through the vitreous–aqueous humor interface has a similar range of values and contour plots compared with the unconditional MFPT, indicating that the dynamics in the MFPT solutions could be dominated by the dynamics of the anterior elimination pathway. This behavior was observed in all modeled species (results presented in Supplementary S1.5) and held for the IgG molecular format. 

Figure 10.

 

Numerical solution and contour lines for the MFPT, conditional on exiting through the vitreous–retina (left) and vitreous–aqueous humor (right) interfaces, in the human (emmetropic) eye model for a Fab molecule, as a function of injection site, with the parameterization of Tables 1 and 2.

Discussion

We have given the governing equations for the mean first-passage time, MFPT, the MFPT conditioned on the exit route, and the proportion of drug exiting through subsections of the eye for Fab and IgG molecular formats. We built realistic 3D eye geometries based on ocular measurements and confirmed their anatomical accuracy by comparing them to MRI images, showing that the geometries capture reasonably well the relative size and position of the vitreous chamber and lens (Fig. 3). While the cynomolgus monkey model eccentricity exceeds that of the image, we remark that the image was not obtained in situ and the isolated organ was not subject to the same external forces, that alterations can occur during the tissue processing and fixation, and that all of these causes can affect its appearance. We solved the equations on the ocular geometries, and we analyzed our results to assess the potential influence of spatial parameters on ocular drug residence times. To better study the different geometries, we compared the MFPT at plausible injection points P_m, and we showed that neglecting the bolus aspect of the injection has no significant impact on results (Supplementary S1.4). We linked this analysis to ocular half-life and contrasted it with experimental data from nonclinical species and humans. 

The MFPT approach modeled the distribution and elimination of Fab and IgG molecules after IVT administration as diffusion-driven processes. While this assumption is supported by previous experiments and modeling work,^53–57 the consensus on whether bulk flow plays a significant role for drug distribution in the vitreous is being questioned,¹⁰⁹ with several authors noting its importance in particular cases.^110–113 The approach presented in this work also assumed that no fluid flow was generated during drug injection, a simplification justified by previous modeling work,¹¹⁴ though extensions to consider the prospective impact of fluid flow are an interesting future direction for this study. 

This study's aims included exploring the importance of injection site location and individual variation for IVT administration of protein therapeutics. The model simulations show that the injection location had a significant effect on the MFPT in all species (Fig. 5). This is in line with a previous computational study, where the influence of four different injection locations in a human eye model was investigated,^115,116 concluding that injections at the back of the eye induced higher drug concentrations in the first 24 hours after injection¹¹⁵ and maximal drug absorption at the macula.¹¹⁶ In the rodent models, the MFPT was higher for injection sites centered on the vitreous chamber depth along the optical axis behind the lens in the vitreous, but these may not be accessible due to the curvature of the lens. We postulate that variations in reported ocular half-lives (both in clinical and nonclinical species) could be partially explained by different injection site locations in the vitreous chamber. In the human eye, the results indicate a large posterior region of the vitreous body where IVT injections are expected to yield maximal drug residence time and retinal permeation (Figs. 5, 6, 8). Conversely, injections closer to the anterior segment lead to lower elimination across the retina. It is expected that alternative ocular delivery approaches to IVT injection, such as implants and sustained-release formulations, similarly ought to target the central and posterior vitreous for improved retinal exposure. 

The results of this work agree with previous reports that ocular PK depends on eye and molecular size.^8,40 A linear relation was found between the MFPT and the vitreous chamber depth, for both Fab and IgG molecular formats (Fig. 7A). Also, the t_1/2 was linearly correlated to the product of the eye semi-axis b squared and the protein R_h, with the associated regression line having a slope of 2.64 days/(cm² nm). This is comparable to the range previously reported by Caruso et al.⁸ (1.3–2.4 days/(cm² nm)) for the linear regression of half-life on \(r_{vit}^2 \times R_h\) in different species, where r_vit is the vitreal radius. The experimental estimates of that study suggest the relationship is species-specific, with the minipig exhibiting the steepest slope (2.4 days/(cm² nm)) and the rabbit, human, rat, and monkey displaying shallower lines (2.1, 1.8, 1.6, and 1.3 days/(cm² nm), respectively). Interestingly, this sequence of species is not ordered by vitreous size; for example, the vitreal volume in the rabbit is smaller (and thus, diffusion distance is shorter) than both human and monkey vitreous (Table 1). This suggests that factors other than eye size must play a role in determining the PK differences observed across species. Vitreal chamber shape and eccentricity have been proposed as possible determinants,⁸ a notion that is not supported by the present results as the anatomically realistic description of ocular geometries in this work produced the same linear estimates of residence time across all species (Fig. 7A). 

The model simulations yielded half-life values aligned with interspecies differences in vitreal volume (i.e., longer half-lives in larger eyes) (Table 4). While the human and rabbit t_1/2 estimates are close to the experimental values, a discrepancy is found for the cynomolgus monkey and for the rodents, although the t_1/2 in the latter is not as well established experimentally as in the larger species. We conjecture that the partial mismatch between experimental data and simulation outcomes can be attributed to the assumptions made regarding the permeability parameters. Notably, including species-specific permeability parameters in the present model can significantly influence the relative contribution of the anterior and posterior pathway to drug elimination. The proportion of drug exiting through the posterior pathway is directly linked to the two permeability parameters within the model. The relative contribution in exit pathways has been previously proposed as a potential determinant of t_1/2 in ocular PK.^8,40 Moreover, insights from a prior study on topically applied small molecules indicate variations in ocular tissue permeability across different species.¹⁰⁰ Extending this understanding to IVT macromolecules, species-specific permeability data may be necessary to obtain more accurate modeling results. While a few studies have previously investigated the effect of molecular size on the permeability of ocular tissues,^29,117,118 there is a lack of comparative studies on the permeability of IVT macromolecules across species. 

Our global sensitivity analysis showed that the axial length is the most influential parameter on residence time across species (Supplementary S1.3), confirming the importance of eccentricity in modeling the vitreous chamber. On the other hand, the spherical approximation used in several previous works^41,119–122 implies that the semi-axis b has the same length as semi-axis a. By way of example, if we compare the injection of a Fab molecule at P_m in the human emmetropic eye model (a = 1.1275 cm, b = 0.889 cm) and in a spherical model of the same vitreous volume (a = b = 1.043 cm), we find a meaningful discrepancy in the estimate of the half-life (respectively 5.85 and 7.16 days). Hence, we recommend that future models, aiming to explore PK across various species, do away with the spherical approximation. 

The global sensitivity analysis also identified that, considering the high uncertainty on the permeability parameters, the MFPT results were sensitive to the values chosen, although less sensitive than for the values of the axial length. Otherwise, the sensitivity analysis revealed that the model was not particularly sensitive to the permeability parameters when they were varied proportionally with the other parameters (Supplementary S1.3). While the choice of the permeability parameters will affect the modeled clearance times in absolute terms, relative changes such as the proportion of posterior elimination should be insensitive as long as the ratio between the two permeability parameters remains approximately fixed. 

Previous work has investigated the impact of eye size on PK in diverse animal species. To the best of our knowledge, this is the first report showing a significant impact of eye shape and eccentricity within the same species, namely, humans. Ocular half-life estimates are known to exhibit sizable differences between patients in clinical studies. For example, Avery et al.¹²³ reported a mean t_1/2 of 5.8 days with a standard deviation of 1.8 days for the Fab ranibizumab, and Meyer et al.¹²⁴ reported an average and 95% confidence interval of 11.17 (8.7, 18.2) days following a 3-mg injection of the IgG bevacizumab. On the other hand, a previous study in 41 eyes found no correlation between intraocular drug concentration and AL.¹²⁵ Such contrast of results motivated this research to better understand whether interindividual differences in ocular geometry may impact PK. In the ensemble of human eyes having different volumes and axial lengths, our model showed a large variation in residence time (Fig. 7B), contributing to explain the experimental interindividual variability. The different slopes between the Fab and IgG molecules suggest that the variability in the AL of the eye elongation is more influential on the residence time for molecules with slower diffusion. The linear relation found between MFPT and AL suggests that measuring the AL is sufficient to obtain an estimate of the residence time for any given human eye. This has promising implications for clinical practice, as AL measurements can be obtained in patients with relative ease, more so than the vitreous volume. Optical biometry, both reasonably simple and cost-effective, could serve as a potential stride toward personalized treatment by furnishing insights into individual durability of ocular exposure and pharmacology. 

The model also suggests that the proportion of posterior elimination varies greatly between species (Fig. 8), which appears to be mostly correlated to the distance between the injection site and the vitreous–aqueous humor interface. In the rabbit eye model, the posterior pathway contribution to drug elimination was 19% to 23% (Fig. 8) for an injection within the region identified in Figure 2, which is in line with previous experimental estimates. A prior study in rabbit eyes estimated the posterior clearance as 3% to 20% of the dose administered by IVT injection.⁷⁸ Another computational model calculated the percentage of Fab molecules exiting through the RPE to be 12.7% of the IVT dose in a rabbit experiment.²³ To our knowledge, current estimates are informed by experimental data obtained in rabbits, while Figure 8 highlights clear species differences, making it necessary to further study the elimination pathways in other species. In the ensemble of human eyes, we found that posterior elimination of both Fab and IgG formats is linearly correlated with the axial length and vitreous volume (Fig. 9), strengthening the notion that individual variations in eye shape may influence drug disposition and pharmacology. 

The results of the MFPT conditioned on the exit route have provided further information on the dynamics of the MFPT. In all species, the solutions of the conditional MFPT show that the clearance pathway through the vitreous–aqueous humor interface is dominating the behavior in the MFPT solutions. Furthermore, the model suggests that drug molecules leaving through the retina are spending more time in the vitreous chamber than molecules exiting through the aqueous humor (Fig. 10), and the duration time of drug that exits into the target region is approximately 10% longer than the mean duration time. Therefore, while the half-life underestimates the duration of drug in the vitreous that reaches the target, it is still a good measure of duration. 

In conclusion, the residence time and the posterior elimination were studied across multiple species used in ocular research and drug development, with the aim to strengthen the interspecies translation of pharmacokinetic and pharmacodynamic studies. The anterior pathway was identified as the predominant route of drug elimination, and the contribution of the posterior pathway is predicted to vary significantly across species. The injection location was found to be highly influential in the drug kinetics, and maximum efficacy was obtained for injections in the posterior vitreous. Additionally, we showed that the variability in vitreous chamber size and shape in human eyes can lead to significant differences in drug residence times and proportion of posterior elimination. The methodology developed in this study emerges as a potent framework for characterizing the vitreal transport dynamics of current ocular therapeutics. By combining our methodology with species-specific measurements of posterior permeabilities, it would be possible to investigate the efficacy of emerging ocular therapeutics. 

Acknowledgments

The authors thank Nora Denk for insightful conversations around intravitreal injection procedures. They also thank Norman Mazer for his careful review and input to this manuscript. For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission. 

Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) (grant number 567922), by the Engineering and Physical Sciences Research Council (EPSRC) (grant number EP/S024093/1), and by Roche Pharma Research and Early Development. 

Disclosure: P. Lamirande, Roche (F); E.A. Gaffney, Roche (F), AC Immune (F), GSK (F), AstraZeneca (F); M. Gertz, Roche (E, I); P.K. Maini, Roche (F); J.R. Crawshaw, Roche (F); A. Caruso, Roche (E, I)