Mathematical assessment of drug buildup in the posterior eye following transscleral delivery
 Paola Causin^{1}Email author and
 Francesca Malgaroli^{2}
https://doi.org/10.1186/s1336201600317
© Causin and Malgaroli 2016
Received: 4 February 2016
Accepted: 24 October 2016
Published: 28 October 2016
Abstract
Delivery of drugs to the posterior segment of the eye is a significant challenge in the field of opthalmic pharmaceuticals. Several restrictive barriers hinder drug delivery to this district. Static barriers include tissues and limiting membranes, while dynamic barriers include drug clearance mechanism from blood and lymphatics. Strategies for delivering drugs to the posterior segment most often consist in topical ocular medications or systemic administrations, but dose/response profiles are generally very poor. Intravitreal injections and transscleral delivery are new emerging techniques with promising results. Purpose of this study is to develop a mathematical model to assess drug levels subsequent to a transscleral drug implant. Both computational and analytical techniques are adopted. The model comprises sclera, choroid, retina and vitreous along with the retina pigment epithelium at the choroidretina boundary and the inner blood retinal barrier of the retinal vessels. Darcy equations are used to compute the filtration velocity of the interstitial fluid and a fictitious velocity field is added to model active pumping from the retinal pigmented epithelium. Convectivediffusivereactive equations for drug concentration are then solved. Permeability parameters and partition coefficients simulate the presence of internal membranes and barriers, with possible different values in outward and inward directions. An important result of the model is the evaluation of the roles of the different physical parameters, which offers key points to improve drug delivery techniques. Namely, the sensitivity study suggests that diffusion in tissue, clearance rates, membrane permeabilities and active pumping play important roles in determining drug peak concentration and timetopeak. However, their relative influence can be dramatically different depending on the ratelimiting parameter.
Keywords
1 Introduction
The understanding of drug delivery mechanisms in the posterior segment of the eye (PSE)  including sclera, choroid and retina  is one of the most challenging tasks in the pharmaceutical industry [1]. The efficiency of drug delivery to the PSE is hindered by several barriers. Static barriers consist of physical obstacles to drug diffusion such as the sclera itself, the retinal pigment epithelium (RPE, the socalled outer blood retinal, oBRB) and the retinal vessels (the socalled inner blood retinal barrier, iBRB). Dynamic barriers include drug clearance mechanisms through blood and lymphatic vessels and degradation processes. Drug solubility, lipophilicity, charge, degree of ionization, molecular size and shape affect the penetration rate of the drug across the various barriers [2]. Convection by interstitial fluid filtration can play a certain role especially when considering lowdiffusible molecules [3]. There is also considerable evidence suggesting that active transport (‘pumping’) across the RPE can induce significant effects, sucking out fluid, and thus dissolved drugs, from the retina towards the choroid [4, 5].
The paper is organized as follows. In Section 2, we present the geometrical assumptions and the mathematical model of the PSE. In Section 3, we show the results obtained from numerical solution of the mathematical model and sensitivity analysis. In Section 4, we discuss the significance of the model and the main results relevant for devising new drug formulations and delivery techniques. Eventually, in Section 4, we draw the conclusions of the work. A theoretical analysis is carried out in the Appendix to establish lower and upper bounds for the drug concentration in the retina, an important target for drug delivery.
2 Mathematical model of the posterior segment of the eye
2.1 Geometrical description of the PSE
Compartmental models of the PSE have been presented in [10–12] to describe drug administration via a subconjunctival application or an episcleral hydrogel implant. Consensus has been reached about the necessity of including separate compartments which represent the different anatomical structures in the PSE. In [9], a study has been carried out to evaluate the compartment subdivision which provides the best fitting of experimental data. The most effective identified configuration includes compartments representing site of drug release, periocular tissue, sclera/choroid/RPE, retina and a nonspecified distribution compartment. An analogous identification of relevant independent structures is carried out in [13], where a 1D continuum model describes levels of fluoroscein after periocular administration.

sclera (S), an avascular and largely acellular coat of extracellular matrix relatively permeable to molecules;

choroid (C), a dense network of large and small blood vessels with a relatively sparse population of cells;

retina (R), composition of several layers of densely packed neuronal and glial cells, vascularized by arterioles and venules which run superficially along the retinal inner surface and supply/drain the embedded capillary plexi (see Figure 2, top panel);

vitreous (V), a clear, avascular, gelatinous body which accounts for about 80% of the volume of the eye.
We denote by \(\Omega_{S}=(0,L_{SC})\), \(\Omega_{C}=(L_{SC},L_{CR})\), \(\Omega_{R}=(L_{CR},L_{RV})\) and \(\Omega_{V}=(L_{RV},L_{V})\) the computational domains corresponding to the S, C, R, V layers, respectively. For \(j=S,C,R,V\), we denote by \(n_{j}\) its unit normal vector directed outward. For \(i=S,C,R\) and \(j=C,R,V\), we denote by \(\Gamma_{ij}\) the interface between two adjacent layers i and j.
2.2 Mathematical model of the PSE
Let \(j=S,C,R,V\). For space \(x \in\Omega_{j}\) and time \(t \in(0,T)\), we let \(v_{j}=v_{j}(x)\) (cm/s) be the steady filtration (seepage) velocity in layer j, \((K/\mu)_{j}\) the corresponding hydraulic conductivity (cm^{2}/mmHg/s). We let \(C_{j}=C_{j}(t,x)\) (g/cm^{3}) be the drug concentration in layer j and \(D_{j}\) (cm^{2}/s) and \(k_{j}\) (1/s) the corresponding drug diffusivity and clearance/decay rate, respectively. For \(i=S,C,R\) and \(j=C,R,V\), we let \(\mathcal{R}_{ij}\) (cm/s/mmHg) be the membrane hydraulic permeability at the interface \(\Gamma_{ij}\) and we let \(\mathcal{L}_{ij}\) (cm/s) and \(\mathcal{P}_{ij}\) (⋅) be the drug membrane permeability and the partition coefficient between the two layers.

velocity continuity$$v_{i} \cdot n_{i}= v_{j} \cdot n_{i}; $$

pressure jump condition (reduced KedemKatchalsky conditions for solvent, see e.g., [14])$$v_{i} \cdot n_{i}= \mathcal{R}_{ij}(p_{i}p_{j}). $$
Drug mass balance in the PSE. Drug mass balance is enforced in each layer according to the characteristic features of the layer itself.

drug flux continuity$$D_{i} \frac{\partial C_{i}}{\partial x} \cdot n_{i}= D_{j} \frac {\partial C_{j}}{\partial x} \cdot n_{i}; $$

drug concentration jump condition (reduced KedemKatchalsky conditions for solute, see e.g., [14])where the partition coefficient \(\mathcal{P}_{ij}\) takes into account the possible different hydrophilicity/lipophilicity between layers i and j.$$D_{i} \frac{\partial C_{i}}{\partial x} \cdot n_{i}= \mathcal {L}_{ij}(\mathcal{P}_{ij}C_{i}  C_{j} ), $$
3 Numerical simulations
Value of the model parameters used in the numerical simulations (if not specified otherwise)
Parameter  Value  Unit  Description  Ref. 

\(t_{S}\)  600  μm  Sclera thickness  [13] 
\(t_{C}\)  300  μm  Choroid thickness  [13] 
\(t_{R}\)  246  μm  Retina thickness  [13] 
\(t_{V}\)  15000  μm  Vitreous thickness  [20] 
\((K/\mu)_{S}\)  8.4⋅10^{−7}  cm^{2}/s  Hydraulic conductivity in sclera  [8] 
\((K/\mu)_{C}\)  2.35⋅10^{−11}  cm^{2}/s  Hydraulic conductivity in choroid  [8] 
\((K/\mu)_{R}\)  1.5⋅10^{−11}  cm^{2}/s  Hydraulic conductivity in retina  [8] 
\((K/\mu)_{V}\)  1.5^{−11}  cm^{2}/s  Hydraulic conductivity in vitreous  [8] 
\(\mathcal{R}_{SC}\)  10^{−7}  cm/s/mmHg  Hydraulic permeability at \(\Gamma_{SC}\)  
\(\mathcal{R}_{CR}\)  10^{−7}  cm/s/mmHg  Hydraulic permeability at \(\Gamma_{CR}\)  
\(\mathcal{R}_{RV}\)  10^{−7}  cm/s/mmHg  Hydraulic permeability at \(\Gamma_{RV}\)  [32] 
\(D_{S}\)  4⋅10^{−7}  cm^{2}/s  Drug diffusivity coefficient in sclera  [13] 
\(D_{C}\)  1.6⋅10^{−7}  cm^{2}/s  Drug diffusivity coefficient in choroid  [13] 
\(D_{R}\)  1.17⋅10^{−7}  cm^{2}/s  Drug diffusivity coefficient in retina  [13] 
\(D_{V}\)  6 10^{−6}  cm^{2}/s  Drug diffusivity coefficient in vitreous  [6] 
\(k_{S}\)  3⋅10^{−4}  1/s  Drug clearance/Decay coefficient in sclera  [13] 
\(k_{C}\)  3⋅10^{−4}  1/s  Drug clearance/Decay coefficient in choroid  [13] 
\(k_{Rt}\)  3⋅10^{−4}  1/s  Drug clearance/Decay coefficient in retinal tissue  [13] 
\(k_{Rb}\)  3⋅10^{−4}  1/s  Drug clearance/Decay coefficient in retinal blood  [13] 
\(k_{V}\)  8⋅10^{−5}  1/s  Drug clearance/Decay coefficient in vitreous  [11] 
\(\mathcal{L}_{SC}\)  10^{−4}  cm/s  Permeability coefficient at \(\Gamma_{SC}\)  [21] 
\(\mathcal{L}_{CR}\)  10^{−5}  cm/s  Permeability coefficient at \(\Gamma_{CR}\)  
\(\mathcal{L}_{RV}\)  10^{−5}  cm/s  Permeability coefficient at \(\Gamma_{RV}\)  
\(\mathcal{P}_{CS}\)  1  adim  Partition coefficient at sclera/Choroid interface  [21] 
\(\mathcal{P}_{CR}\)  1/1.33  adim  Partition coefficient at choroid/Retina interface  [13] 
\(\mathcal{P}_{RV}\)  1/10  adim  Partition coefficient at retina/Vitreous interface  [21] 
3.1 Convective field
Solving the Darcy equations yields a constant filtration velocity of about 10^{−7} (cm/s). This value is in accordance with the results of [8, 20]. The corresponding Péclet number based on the layer thickness is definitely less than 1 when considering small weight molecules (diffusivity of the order of 10^{−7} to 10^{−6} (cm^{2}/s)) and of the order 1 when considering large weight molecules (diffusivity of the order of 10^{−8} (cm^{2}/s)). When considering in the retina the active pumping velocity, which is of the order of \(8 \cdot10^{6}\) (cm/s) [8], the filtration velocity turns out to be negligible. The Péclet number of the retina computed with the pumping velocity rises to a value of the order of 20 for small molecules and 200 for large molecules, so that convection is dominating.
3.2 Validation of drug concentration levels
3.3 Sensitivity study
4 Discussion and conclusions
Drug delivery to the PSE is still an open issue in ocular diseases therapy. Many drugs have a narrow concentration window within which they are effective and nontoxic. Currently, about 90% of the treatments of ophthalmic diseases are performed by medications administered topically. However, drugs enter the eye through this pathway at a very limited extent: wash off by various mechanisms (lacrimation, blinking, tear turnover) and low permeability of the corneal epithelial membrane causes less than 5% of the administered drug to effectively reach the posterior targets [22]. Among the other possible drug delivery routes, systemic administration has a poor dose/response profile in the eye. Intravitreous delivery, whilst efficient, can carry significant local complications such as retinal detachment, endophthalmitis, vitreous hemorrhage and cataract formation [23]. Under these premises, sustained drug delivery to the PSE via the alternative transscleral route is gaining increasing importance, due to the easily accessible area, the hypocellularity and permeability of the sclera to relatively large molecules, and, importantly, to the degree of acceptance of patients [24, 25]. Pharmacokinetics of drugs in the PSE following transscleral delivery is an emerging issue [4]. The reported data are very sparse and, for the most part, refer only to the vitreous, which is easily accessible in experiments, to draw comparisons.
Transscleral drug delivery has been analyzed in this paper using a 1D continuum model including diffusive, convective and reactive mechanisms and comprehending the sclera, choroid, retina and vitreous domains. The presence of internal membranes has been kept into account by appropriate interface boundary conditions and/or a fictitious advective field for RPE active pumping. Drug concentration levels in the retina have been modeled distinguishing the tissue phase from the blood phase.
In [13], the partial failure in reproducing with the mathematical model the experimental results is ascribed to the discrepancy of the values of the parameters estimated from basic physical considerations vs values of the parameters obtained by fitting the model to experimental data. This fact suggested to explore ranges of parameter values to identify characteristic model sensitivities. The authors found a relatively low sensitivity of peak concentration and timetopeak in the retina to the diffusion and clearance coefficients and a higher sensitivity to drug resistance (inverse of the drug permeability) of the episcleral layers with respect to peak concentration and to RPE resistance. However, while it is clearly recognized that a major fraction of the drug is lost from episcleral lymphatics and blood vessels [11], one can think to eliminate the exceedingly large influence of such a mechanism and just focus on the area from sclera to vitreous. Varying the parameters in the same range as in [13] yields in this work sensitivities which reflect comparable, if not more important, roles of diffusion and clearance rates.
As for the convective field, it is apparent that the filtration velocity is too low to induce significant changes (see Figure 8). Different is the case of the active pumping velocity. Accordingly to what found in [8] and [21] (notice that in these references, the authors represent this same mechanism in a different way), active pumping plays a very relevant role, if one looks specifically at the retina and possibly at the vitreous. It is interesting to observe the shape of the curves with respect to drug permeability. Both for peak concentration and timetopeak, the curves results relatively flat even varying the parameter of 1 or 2 decades in log scale. This is probably due to permeability not being a limiting phenomenon in these regions. The iBRB drug permeability does not seem to have a major role except for a region of diameter spanning about 1 decade at the left and at the right of the baseline value, where the slope of the curve is comparable to the other ones. The mild influence of iBRB might due to having neglected the complexity of active/carriedmediated transport mechanisms across these interfaces and having considered a single partition coefficient across the vessel walls in both (inward and outward) directions. Representing active pumping as a fictitious velocity and not as a membrane effect, hinders the role of RPE (see [21] for this choice).
An important result of the present analysis is the way in which a 30% increase of the drug peak concentration in choroid and retina can be obtained from the baseline values, for example to meet a therapeutic threshold. Assuming continuous dependence of the solution on data  a property which can be inferred via a mathematical analysis not much different to the one performed in the Appendix  it is found that such an increase can be obtained acting on (i) scleral biophysical parameters and then on choroidal and retinal ones; (ii) active pumping. As for (i), in the neighborhood of baseline conditions, diffusion and clearance appear to play a comparable role, because comparable variations in the scleral diffusion (increase) or scleral clearance (decrease) are required to increase drug peak concentration (notice the slope of the curves in Figure 6). The sclera is permeable to hydrophilic compounds, even macromolecules, but the permeability in the RPE/choroid/Bruch’s membrane is 12 orders of magnitude lower than in the sclera [16]. Moreover, the trend line of decreasing choroidBruch’s membrane permeability with increasing solute lipophilicity and/or molecular radius appears to be steeper than the sclera. However, the simulations suggest that the slope of the sensitivity curves is always higher for the scleral parameters that for the choroidal ones. Moreover, sensitivity in a certain layer is higher to properties of layers located at its left side than at its right side. This is a natural consequence of the positioning of the source. In particular, referring to the theoretical analysis carried out for the retinal domain, this implies that the drug concentration in the choroid is always responsible for the upper bound. As for (ii), prodrugs have been envisaged as carriers able to favorably enhance drug delivery. This a very advanced issue in pharmacokinetics, we refer to [26] for a general review.
The present model does not allow to provide data regarding the 3D spatial distribution of the drug on the eye globe. While this aspect is very important and has been recently considered in a few mathematical models [8, 17, 21], it remains very difficult to compare results from such models with experiments. Unknown boundary conditions in the 3D models, on the one hand, and tissue homogenization after explant with loss of spatial dependence, on the other, are just examples of such problems.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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