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Effect of choroidal blood flow on transscleral retinal drug delivery using a porous medium model
International Journal of Heat and Mass Transfer 55 (2012) 5665–5672
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of choroidal blood flow on transscleral retinal drug delivery using a porous medium model Arunn Narasimhan ⇑, Ramanathan Vishnampet Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Received 23 March 2011 Received in revised form 16 April 2012 Accepted 19 April 2012 Available online 16 June 2012 Keywords: Drug delivery Biofluids Human eye Porous medium Numerical Choroidal blood flow Transscleral Anecortave acetate
a b s t r a c t Transscleral drug delivery is one of the methods of depositing drug in the posterior segment (comprising retina, choroid, sclera and macula) of the human eye, to treat diseases such as age related macular degeneration (AMD). In this study, the effect of choroidal blood flow on transscleral drug delivery to the retina is investigated using a porous medium model of the sclera and the choroid. A two-dimensional geometrical model of the human eye is constructed from available measurements and determination of physicochemical properties of the sclera and the choroid, such as their effective diffusivity D and porous medium permeability K. Position and time dependent concentrations of the drug in the sclera and the choroid are predicted and the relative magnitudes of the periocular, vitreous and circulation losses are compared for various blood flow velocities U b . The simulations also predict the transient mean plasma concentration C of the drug anecortave desacetate in the choroid and the effect of choroidal blood flow on the peak mean plasma concentration C max . Comparison of predicted C with available experimental results is good. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In the human eye, effective treatment of posterior segment (comprising retina, choroid, sclera and macula) diseases such as age related macular degeneration (AMD) requires adequate levels of drugs to reach those tissues. Methods of drug delivery include topical administration of eye drops [1,2], systemic administration, and intravitreal implants and they attempt to deposit therapeutic concentration levels of drugs at the choroid and the retina. Intravitreal implants have been successfully used for antibody based treatment of AMD. However, it is an invasive, potentially hazardous and undesirable method of treatment. Due to the large and accessible surface of the sclera, and its high and age-independent permeability coefficient P s to macromolecules and water-soluble drugs, transscleral drug delivery is a favourable alternative [3,4]. In transscleral drug delivery, the drug is placed at a periocular site in the posterior segment from which it diffuses through the sclera to the target tissues. Transscleral drug delivery can be classified into three types depending on the route taken by the drug to reach the target tissue: (1) anterior chamber route (2) systemic circulation route and (3) direct penetration pathway. In the preferred direct penetration pathway type of delivery, the target tissue is
⇑ Corresponding author. E-mail address: [email protected] (A. Narasimhan). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.060
located in the posterior segment of the eye and the drug diffuses through the sclera, choroid and the retina into the vitreous humor. Several recent studies of transscleral drug delivery have attempted to derive pharmacokinetic models to explain the transfer rates of the drug through the posterior eye tissues [5–7]. The permeation of drugs through the sclera, choroid and the retinal tissues are affected by the barrier to diffusion imposed by these tissues and losses through periocular spaces, systemic circulation, and vitreous absorption. The chief parameter that governs the barrier to drug diffusion is the scleral permeability coefficient Ps . An experimental study of human scleral permeability determined that the in vitro permeability coefficient of the sclera P s is inversely related to the molecular weight of the drug compound MWt. [8]. In another recent study, the permeability coefficient of the choroid Pc has been found to be in the same range as that of the sclera [9]. The kinetics of the various losses has shown that the transfer rate of the direct permeation is a small fraction ð0:05 0:2%Þ of the rate constant of periocular losses [10]. This fraction has been labeled the bioavailability of the drug in the vitreous BAv . Pharmacokinetic models succeed in predicting overall properties such as the permeability coefficient Ps [11] of a drug or the average concentrations of the drug in various tissues [12]. Simulation models using established numerical methods to solve blood flow and drug species equations and based on the geometry and physicochemical characteristics of the eye are more comprehensive. They also predict time and position dependent concentrations
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Nomenclature C C D Ps K t U
v x; y
mean plasma concentration, kg m-3 Concentration, kg m-3 Effective diffusion coefficient, m2/s permeability coefficient of the sclera, Pa Permeability, m2 time, s Flow velocity, m/s Darcy velocity, m s-1 Cartesian co-ordinates, m
q l /
density, kg m-3 viscosity, N s m-2 surface porosity
Subscripts b blood c choroid max maximum value s sclera
Greek symbols / porosity
of the drug across the eye. Such models have been attempted for drug delivery through systemic administration [13] and intravitreal implants [14,15]. A drawback of these models is their inability to account for the effect of choroidal blood flow on systemic loss of the drug. In a recent study [4], a pharmacokinetic simulation model based on the scleral permeability coefficient Ps was developed, which accounts for the circulation loss and predicts the overall permeation flux through the sclera. However, this simulation was based on steady-state calculations and the concentration on the surface of the sclera was assumed to be a constant. A 3D porous medium approach using finite element method for studying transscleral drug delivery was proposed in another recent study [16], which accounted for the diffusion and convection losses. However, the choroidal blood flow effect on the drug delivery is lumped as a sink term and assumed linear. There have been instances reported in medical literature [17,18] in the pre-drug delivery stages, usage of scleral buckle, a surgical procedure used to treat retinal detachment, alters the choroidal blood flow rate. In this study, the effect of choroidal blood flow on transscleral drug delivery to the retina is investigated using a porous medium model of the sclera and the choroid. The permeation of the drug through the direct penetration pathway is modeled as a diffusion process and studied using Fick’s second law of diffusion in conjunction with an effective diffusivity for the porous media. Using the developed model, the transient mean plasma concentration C of the drug anecortave desacetate in the choroid is predicted. The effect of choroidal blood flow on the transient peak mean plasma concentration C max is also studied and compared with available experiments [19].
curvature of 7.79 mm [22]. The thickness of the sclera varies from 588 lm at the limbus down to 491 lm at the equator and reaches a maximum of 996 lm at the posterior pole of the normal human eye [23]. The average thickness of the sclera is taken as 670 lm. The thickness of the choroid is taken to be 287 lm [24]. The optic nerve is assumed to be a cylindrical segment with an axis inclined at 30 to the pupillary axis. The optic nerve sheath diameter (ONSD) is about 4.8 mm [25] and the optic disc diameter is about 1.83 mm [26]. The diameter of the optic nerve where it meets the posterior eye is assumed to be the average of the ONSD and the optic disc diameter, or about 3.32 mm.
2. Drug diffusion model
2.3. Properties of the sclera and the choroid
Fig. 1 shows a schematic of the vertical cross-section of the human eye indicating the posterior juxtascleral region (PJD). Anecortave acetate, one of the standard drug base used, is administered on the sclera at a PJD and takes a direct penetration route to the vitreous through the sclera, choroid, and the retina. The diffusive transport of anecortave acetate through the ocular tissues is impeded by blood flowing through choroidal vessels. Choroidal blood flow adversely convects the drug and lowers the drug concentration that can reach the target tissue in the retina.
The surface porosity of the sclera /s is determined from its transmission electron microscopy (TEM) images. Digitized TEM images of the sclera are obtained from various sources [27–30].
2.2. Conservation equations In the computational domain, the sclera and the choroid are treated as homogeneous isotropic porous media. The conservation equations for porous media are solved to obtain the concentration distribution within the domain. The volume averaged mass conservation, and momentum and species transport equations for porous media in the sclera and the choroid are written as,
r v b ¼ 0;
@v /l qb b þ ðvb $Þvb ¼ $P þ lb r2 vb c b vb ; @t Kc @C 2 ¼ Ds r C; /s @t @C /c þ ðv b $ÞC ¼ Dc r2 C: @t
ð1Þ ð2Þ ð3Þ ð4Þ
2.1. Geometry of the eye Fig. 2 (a) shows the geometrical model of the eye used in the present simulations. The diameter of the eye along the pupillary axis is about 23.58 mm [20]. The posterior surface of the eye is almost spherical. The anterior surface of the cornea is a segment of a sphere which is about 6 mm [21] in diameter and has a radius of
Fig. 1. Schematic of the vertical cross-section of the human eye indicating the posterior juxtascleral region.
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A
Fig. 2. (a) Computational domain and grid (mesh element size 80 lm) used in the present simulations indicating the boundary conditions and depot location, and (b) region A shown at high resolution (mesh element size 10 lm).
The images are in the form of two-dimensional arrays of pixels with gray level 0 6 g 6 255. Light regions of the image correspond to pores and dark regions correspond to the solid matrix. A threshold gray level of 128 is used to distinguish pores from the solid. Annotations are removed from the image prior to the analysis. An iterative process was used in which the size of the window is increased till the porosity no longer changes. The microscopic details are averaged over an REV just as it is done in any porous continuum model. The visual resolution level of all TEM images were roughly equal and of lm order. The surface porosity from each image is computed as the ratio of the area of light pixels ðg > 128Þ to the area of dark pixels ðg 6 128Þ. Using this method on several sclera images, the mean surface porosity of the sclera was obtained as /s ¼ 0:39 0:09 ðn ¼ 9Þ. The porosity of the adjacent choroid /c is taken to be equal to the porosity of the sclera /s . The in vitro permeability coefficient of the sclera to various hydrophilic compounds has been experimentally studied [8]. It has been found that the mean scleral permeability coefficient Ps is inversely related to the molecular weight MWt. of the drug compound. Using the correlation log Ps ¼ 3:642 0:3753 log MWt:, 1 where Ps is in cm s1 and MWt. is in g mol , that predicts the experimental values in [8] with R2 ¼ 0:9719, for anecortave acetate ðMWt: ¼ 386:48ÞPs is found to be 2:4429 105 cm s1 . The effective diffusivity of anecortave acetate through the sclera is given by Ds ¼ Ps ts where ts is the thickness of sclera. Using t s ¼ 670 lm gives Ds ¼ 1:64 1010 m2 =s, which is more accurate than the value used in [8]. It is assumed that the effective diffusivity of the drug through the sclera Ds is equal to the effective diffusivity through the choroid Dc . The present model do not lump the sclera and the choroid into a single region as done in [8]. This allows the hydraulic conductivity of both tissues to be assumed same permitting blood flow rate in the choroid to be substantially different from that in the sclera. In transscleral drug delivery, the drug penetration is through the sclera and the choroid before it reaches the retina. The mass transfer resistances of the sclera and the choroid are up to an order higher than those of the retina and the vitreous. Hence modeling the drug transport beyond the retinal layer and inside the vitreous with a continuum model-like resolution is not relevant in the current context. The permeability of the choroid K c is assumed to be equal to the hydrodynamic permeability of the sclera. The scleral permeability K s is obtained from experimentally measured values of the hydraulic conductivity of the sclera [31]. At normal hydration of the sclera, it is found that K s ¼ 1:3 nm2 at 37 C. The density and viscosity of blood are assumed to be constant, qb ¼ 1060 kg=m3 and lb ¼ 2:7 103 N s=m2 .
3. Numerical procedure and solution methodology The conservation equations are numerically solved using the commercial software FluentÒ 6.3.26 which employs the Finite Volume Method (FVM). A semi-implicit time marching scheme with first-order discretization is used for the transient terms. Fig. 2 (a) shows the computational domain and grid used in the present simulations. The domain is created using GambitÒ 2.3.16. Quadrilateral and triangular finite volume (surface) elements of uniform size are used for meshing the eye domain. Fig. 2 (b) shows region A in Fig. 2 (a) at a higher resolution. The size of the mesh elements ð 10 lmÞ is larger than the representative elemental volume (REV) of the porous media ð 1 lmÞ. 3.1. Boundary conditions At the interface between the sclera and the choroid, a no-slip boundary condition is used for the blood velocity. The concentration and its gradient normal to the surface are assumed to be continuous. An impermeable boundary condition, ð$CÞ n ¼ 0 is used at the interfaces between the sclera and the cornea, and the sclera and the optic nerve respectively. Blood is assumed to enter the choroid through the interface between the choroid and the optic nerve with a physical or fluid inlet velocity of U b and exit through the interface between the choroid and the cornea. In reality, choroidal vessels anastomose [32] and blood flows in through the arteries and out through the veins both of which leave the eye through the optic nerve sheath. However, it is assumed that the convective effect of blood flow on the diffusion process is captured sufficiently accurately in this alternate simpler flow configuration. It is also assumed that the flow exit is sufficiently far away from the region of interest that its effect is negligible. Further, the blood flow through the choroidal vessels is assumed to be uniform. In reality, blood flow pulsates between a peak systolic velocity (PSV) of 10:15 cm s1 and an end diastolic velocity (EDV) of 2:85 cm s1 . The effect of a pulsating blood flow is neglected in the present simulations. At the interface between the choroid and the optic nerve, the concentration of the drug is assumed to be zero. A transient concentration profile is assumed to exist on the outer surface of the sclera, written as
" Cðx; y; tÞ ¼
C 00 ekt
exp
ðx xD Þ2 þ ðy yD Þ2
r2
# ;
ð5Þ
where C 00 is the initial depot concentration of the drug and ðxD ; yD Þ are the Cartesian coordinates of the point of injection. Due to loss
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of drug from periocular spaces, C 00 < C 0 , the dose concentration. The relation 4r ¼ d is used to obtain r, where d is the diameter of the needle used for the injection. The relation k ¼ tln1=22 is used to obtain k, where t 1=2 is the half-life of the depot concentration which is determined experimentally. The loss of drug from the periocular spaces is assumed to be equal to the change in the average concentration gradient at the outer surface of the sclera due to the use of C 00 as the initial depot concentration in place of C 0 . The initial depot concentration C 00 is obtained from the relation,
C 00 ¼ b10 C 0 ;
ð6Þ
where b10 is the dimensionless periocular loss coefficient. b10 is related to the bioavailability of the drug in the vitreous BAv . For a given b10 , the bioavailability BAv of the drug is the ratio of the average concentration gradients normal to the surface of the sclera obtained when the simulation is performed with depot concentrations C 00 and C 0 respectively. BAv has been typically found to be very low ð 0:2%Þ for subconjuctival injection of prednisolone in rabbits [10]. However, it has been suggested that this is an overestimate since circulation losses are ignored [4]. At the inner surface of the choroid, the concentration flux of the drug diffusing into the vitreous is assumed to be proportional to the difference between the concentration at the surface and the concentration in the vitreous. The vitreous is treated as a sink with infinitely large capacity. Hence, the concentration of the drug in the vitreous is assumed to be negligible. The boundary condition on this surface is written as,
sffiffiffiffiffiffi k ð$CÞ n ¼ b20 Cðx; y; tÞ; Dc
ð7Þ
where b20 is the dimensionless vitreous coefficient. 3.2. Anecortave acetate case study The present simulations have been performed for a posterior juxtascleral depot (PJD) administration of anecortave acetate ðMWt: ¼ 386:48 g=molÞ using a 19-gauge needle ðd ¼ 0:686 mmÞ. To study the nature of pharmacokinetics, three different doses of concentration 3, 15 and 30 mg anecortave acetate in 0.5 ml water are used. The half-life of the PJD concentration of anecortave acetate for 15 and 30 mg doses have been previously measured as 3.3 and 4.5 days respectively [19]. For the 3 mg dose, an observable concentration of anecortave desacetate has been measured up to 2 weeks after the dose. Assuming that an observable concentration corresponds to 1% of the initial dose, it is estimated that t1=2 for the 3 mg dose is about 2.1 days. On the other hand, fitting a linear model for the half-life versus dose concentration gives t1=2 ¼ 2:3 days. The mean of the two estimates, t 1=2 ¼ 2:2 days is used. Anecortave acetate is normally administered using a 56 cannula inserted through an incision made at 8 mm from the limbus measured along the sclera in the posterior direction [33]. The present simulations employ a Cartesian coordinate system where the origin is located at the center of the orbit and the posterior direction corresponds to the positive x-axis. Measured from the positive x-axis, the polar angle hD at which the drug is injected is approximately 51 . The coordinates of the point of injection are obtained as xD ¼ R cos hD and yD ¼ R sin hD and where R is approximately 11.25 mm. All simulations correspond to a single dose of anecortave acetate. The periocular loss coefficient b10 for anecortave acetate must be determined from a knowledge of its bioavailability BAv . Since BAv of anecortave acetate has not been previously measured, it is assumed to be about 1:3%. This value has been chosen so that the results for C max predicted by the present model for a normal choroidal blood flow velocity and a 15 mg dose matches
the results measured experimentally. The vitreous loss coefficient b20 is assumed to be 1. Simulations are performed for a range of values of b10 and b20 in order to check the sensitivity of the results to these coefficients. These values are chosen because these parameters appear in the boundary conditions. The thermophysical properties of the sclera and the choroid, for instance, have been directly measured in experiments under in vitro conditions. The in vitro properties can be used, without further sensitivity checks, in conjunction with a porous continuum approach and is the standard approach taken in earlier models of drug delivery. In the present simulations, the mean plasma concentration of the drug C is reported as the average concentration over the choroid. Immediately following PJD administration C increases from 0 to a maximum value C max . The peak mean plasma concentration C max is an important indicator of the therapeutic effectiveness of a drug. For a given dose concentration, it indicates whether the drug is bioavailable at the retina. Due to the loss of drug through periocular, vitreous and circulation routes, the mean plasma concentration C eventually decreases and ultimately becomes 0 again. The duration of time for which a traceable concentration of anecortave desacetate exists in the choroid varies from 2 to 6 weeks. 3.3. Grid and time dependence studies Table 1 shows the peak mean plasma concentration C max following a 15 mg PJD administration of anecortave acetate for various blood flow velocities U b ¼ 2:85, 6.5 and 10:15 cm s1 . Due to the inherently unsteady nature of the problem, transient simulations are performed to study grid and time dependence. The results for each mesh element size are shown for various values of the time-step used. The results for C max using a time-step of 10 s differ from the results using a time-step of 1 s by < 1% for all mesh element sizes. Similarly, the results using a mesh element size of 20 lm differ from the results using a mesh element size of 10 lm by < 2% for time-steps 100, 10 and 1 s. All further simulations use a mesh element size of 20 lm. A variable time-stepping scheme is used, with the time-step being 10 seconds for the first 5 h (of flow time), 100 s for the next 50 h and 1000 for the last 500 h. Overall, drug transport across the sclera and choroid for a period of 23 days is simulated. 4. Results and discussion At t ¼ 0, a steep concentration gradient exists just near the depot which causes the drug to diffuse passively through the sclera. As the drug reaches the choroid, the transport is through
Table 1 Grid and time dependence study of C max for U b ¼ 2:85, 6.5 and 10:15 cm s1 following a 15 mg PJD administration. Mesh element size (lm)
Time step (s) 1000
100
10
1
2.85
80 40 20 10
4.67 4.87 4.97 4.98
4.73 4.94 4.99 4.92
4.74 4.96 5.02 5.00
4.74 4.97 5.04 5.05
6.5
80 40 20 10
2.05 2.11 2.08 1.86
2.07 2.17 2.17 2.13
2.08 2.18 2.21 2.21
2.08 2.18 2.22 2.23
10.15
80 40 20 10
1.31 1.35 1.33 1.19
1.33 1.39 1.39 1.37
1.33 1.40 1.42 1.42
1.34 1.40 1.42 1.43
U b ðcm s1 Þ
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Table 2 Effect of U b on C max and AUCinf following a 3, 15 and 30 mg PJD administration. U b ðcm s1 Þ
2.85 4.07 5.28 6.5 7.72 8.93 10.15
Fig. 3. Transient mean plasma concentration C of anecortave acetate following a 15 mg PJD administration for various blood flow velocities U b .
both convection and diffusion. The mass diffusion Peclet number Pe given by,
Pe ¼
/s U b t s ; Ds
ð8Þ
indicates the relative magnitude of convection over diffusion. Assuming that the blood flow velocity U b is uniform and equal to the average of the PSV and the EDV, the Peclet number is determined as Pe ¼ 1:48 105 , which indicates that convection is the dominant mode of transport. As the blood flow velocity U b increases, Pe
C max ðng ml
1
AUCinf ðng day ml
Þ
1
Þ
3 mg
15 mg
30 mg
3 mg
15 mg
30 mg
1.00 0.70 0.54 0.44 0.37 0.32 0.28
5.02 3.52 2.72 2.21 1.86 1.61 1.42
10.06 7.06 5.45 4.43 3.73 3.23 2.84
3.85 2.72 2.08 1.69 1.43 1.23 1.09
25.90 18.09 14.38 11.92 10.15 8.70 7.73
66.42 47.31 36.69 29.68 25.52 22.32 19.58
increases and the desired diffusion of drug towards the posterior segment is further suppressed by the undesired convective loss which takes the drug away from the target tissue. Fig. 3 shows the transient mean plasma concentration C of anecortave acetate following a 15 mg PJD administration for various blood flow velocities U b . The present numerical study predicts the observed trend of the transient variation of C. As U b increases, C max decreases as expected. An observable concentration of anecortave acetate persists in the choroid for up to 3 weeks. The peak mean plasma concentration C max occurs on the first day of the dose. The mean plasma concentra1 tion C is in the range 2 5 ng ml for the normal range of blood flow velocities U b ð2:85 cm s1 6 U b 6 10:15 cm s1 Þ. Table 2 shows the peak mean plasma concentration C max and the total area under the mean plasma concentration curve between t ¼ 0 and t ¼ 23 days 1 AUCinf . The AUCinf of 11:92 ng ml day predicted by the present simulations for a 15 mg dose and U b ¼ 6:5 cm s1 is nearly twice
Fig. 4. Contours of the concentration of anecortave acetate near the depot at t ¼ tmax for (a) U b ¼ 0:01 cm=s, (b) U b ¼ 0:1 cm=s, (c) U b ¼ 1 cm=s, and (d) U b ¼ 10 cm=s, following a 15 mg PJD administration.
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Table 3 tmax and Dtres for U b ¼ 6:5 cm s1 following a 3, 15 and 30 mg PJD administration. Dose
t max (h)
Dt res (h)
3 mg 15 mg 30 mg
1.03 0.83 0.83
5.19 8.78 11.17
Fig. 6. Effect of blood flow velocity U b on peak overall plasma concentration C max of anecortave acetate following a 3, 15 and 30 mg PJD administration.
Fig. 5. Comparison of peak mean plasma concentrations C max predicted by the present model with previous experimental results for U b ¼ 2:85, 6.5 and 10:15 cm s1 following a 3, 15 and 30 mg PJD administration.
1
the result of 5:28 2:40 ng ml day obtained from a previous experimental study. This could be the result of two possible reasons. Firstly, the experimental result is based on measurements of C at just 9 time instances. Secondly, the numerical result is based on an experimentally determined value of t 1=2 ¼ 3:3 days for which a standard deviation of 0:6 days is reported. Fig. 4 shows the contours of the concentration of anecortave acetate for blood flow velocities U b ¼ 0:01, 0.1, 1 and 10 cm s1 respectively. The levels used to plot these contours are exponentially distributed in order to represent the sharp decrease in the local concentration of the drug in the direction of diffusion. For U b ¼ 0:01 cm s1 , a thick streak of closely spaced lines is observed adjacent to the interface between the sclera and the choroid across which there is a steep concentration gradient. As U b increases, the streak becomes thinner and moves away from the choroid-vitreous boundary. The presence of the streak is due to the choroidal blood flow which carries the drug away from the posterior segment as a result of which the lines of higher concentration are shifted away from the posterior direction. The contours have been shown at the time instant t ¼ tmax at which the mean plasma concentration reaches its maximum value. tmax is difficult to measure experimentally and is expected to result in high measurement errors. This is due to the fact that the mean plasma concentration remains within 5% (say) of the peak mean plasma concentration C max for a duration of up to 8 10 times t max . In the present study, the time interval beyond tmax up to which the value of C remains within 5% of C max is defined as the residence time Dt res . An experimental setup which gives up to 5% error in the measurement of C could predict the time at which C reaches the value C max as any time instant between ðtmax þ 0:5Dtres Þ 0:5Dt res . Table 3 shows the present results for tmax and Dtres for a 3, 15 and 30 mg dose. Comparing the result for tmax þ 0:5Dtres predicted by the present simulations with the experimental results for t max we find that this is indeed likely. For a 15 mg and 30 mg dose, ðt max þ 0:5Dtres Þ predicted by the present simulations for U b ¼ 6:5 cm s1 are 0.22 and 0.27 days respectively compared to experimental values of 0:27 0:16 and 0:36 0:19 days respectively.
Fig. 7. Average plasma concentration along the radial coordinate C 31 6 h 6 71 and various U b .
for
Table 4 Sensitivity of C max to b10 for 2:85 6 U b 6 10:15 cm s1 . U b ðcm s1 Þ
b10 ð%Þ 0.1
0.2
0.5
1
2
2.85 4.07 5.28 6.5 7.72 8.93 10.15
0.39 0.27 0.21 0.17 0.14 0.12 0.11
0.77 0.54 0.42 0.34 0.29 0.25 0.22
1.93 1.36 1.05 0.85 0.72 0.62 0.54
3.86 2.71 2.09 1.70 1.43 1.24 1.09
7.73 5.42 4.18 3.40 2.87 2.48 2.18
Fig. 5 shows the results for C max predicted by the present model and the experimentally measured peak mean plasma concentrations for a 3, 15 and 30 mg dose. The present result for C max has been matched with the experimental result for a 15 mg dose. Fig. 5 also shows the maximum and minimum values of C max as U b varies between the PSV and the EDV. The comparison may seem redundant but is done to highlight the proposed model is tenable. In reality, the choroidal blood flow rate is not fixed and can vary with time, even after the drug is applied. The experimental results subsume such flow rate variations as they are obtained at the vitreous over a time period. The comparison in Fig. 5 shows that the modelling is sensitive to flow rate variations. It can be used for predictions at even shorter time scales where the choroidal blood flow rate could be significantly different. Experimental results of drug concentrations are not available at such time periods.
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U b ðcm s1 Þ 2.85
0.01 0.1 1 10 100
6.5
C max
AUCinf
C max
5.02 5.02 5.02 5.02 5.02
25.54 25.42 25.90 25.87 25.89
2.21 2.21 2.21 2.21 2.21
10.15 AUCinf 12.07 12.00 11.92 11.95 12.03
C max
AUCinf
1.42 1.42 1.42 1.42 1.42
7.73 7.73 7.73 7.73 7.73
port was studied for U b between 0:01 cm s1 and 10 cm s1 in this paper. From Fig. 8, it is clear that reducing U b to below 0:4 cm s1 is not very useful because vitreous losses begin to dominate for U b < 0:4 cm s1 .
5. Conclusions
Fig. 8. Effect of U b on vitreous clearance ratio for 0:1 6 U b 6 1 cm s1 .
v2 and circulation clearance ratio v3
Fig. 6 shows the effect of U b on C max for a 3, 15 and 30 mg dose. For the 3 mg dose, the relative change in C max with U b is lesser compared to the change for the 30 mg dose. This is because the convective term in Eq. (2) scales as the local concentration C which is higher for the 30 mg dose. Fig. 7 shows the average concentration of anecortave acetate C avg along a radial line at polar angle h for U b ¼ 0:01, 0.1, 1 and 10 cm s1 . C avg is maximum at the depot angle hD as expected. It decreases sharply in the posterior direction indicating the convective effect of the choroidal blood flow. The periocular and vitreous losses have been characterized using a periocular coefficient b10 and a vitreous coefficient b20 respectively. Table 4 shows the results for C max obtained from the present simulations for various values of b10 . The results show that C max is sensitive to b10 and proportionately changes as b10 is changed. Table 5 shows the results for C max and AUCinf obtained from the present simulations for 5 orders of magnitude of b20 . The results show that both C max and AUCinf are not sensitive to b20 . Hence, the assumption of using b20 ¼ 1 appears to be justified. Periocular losses constitute 98:7% of the total loss of anecortave acetate in transscleral retinal delivery ðb10 ¼ 1:3%Þ. The remaining routes of loss of anecortave acetate are through vitreous and systemic circulation losses. Fig. 8 shows the relative magnitudes of the vitreous and circulation losses. The vitreous loss is determined as the average diffusive flux at the interface between the choroid and the vitreous. The circulation loss is determined as the average convective flux at a cross section of the choroid. The ratio of the circulation loss to the sum of the circulation and vitreous losses is denoted by v3 and the corresponding ratio for the vitreous loss is denoted by v2 ðv2 þ v3 ¼ 1Þ. As U b increases, the circulation loss increases until it becomes equal to the vitreous loss for U b 0:4 cm s1 . Typically U b is between the PSV ð2:85 cm s1 Þ and EDV ð10:15 cm s1 Þ. In this range, the circulation loss is the dominant route of intraocular loss of anecortave acetate. However, it is possible to surgically control and reduce the choroidal blood flow in order to reduce the circulation loss. It is for this reason that the drug trans-
A porous medium model has been proposed for simulating transscleral retinal drug delivery. The effective diffusivity in the porous medium model are determined by correlating previous experimental results for the scleral permeability coefficient to various hydrophilic compounds with their molecular weights. The porosity of the sclera and the choroid are obtained by image processing of their available TEM images. Using the numerical model along with a transient boundary condition for the concentration profile of the administered drug anecortave acetate on the surface of the sclera, periocular and viterous drug losses have been predicted. The results for the peak mean plasma concentration C max , the time tmax at which the mean plasma concentration C attains its peak value and the area under the mean plasma concentration curve AUCinf obtained from the present simulations have been compared with a previous experimental study. The present simulations have been found to predict C max accurately within the range of experimentally observed values for 3, 15 and 30 mg doses. The porous medium model used, predicts the overall trend of the transient variation of the mean plasma concentration and the time up to which an observable concentration of anecortave acetate persists in the choroid. The advantage of using the present model over available compartmentalized model with averaged macroscopic quantities is that, the present simulations solve discretized conservation equations within finite mesh elements. As a result, spatial and temporal variation of the drug concentration are recovered. The peak mean plasma concentration C max , which can be measured by regular plasma sampling, is a useful indication of the reliability of the present model since it is sensitive to the choroidal blood flow velocity U b . The effect of choroidal blood flow velocity U b (a patient-specific quantity) on the C max has also been studied. The peak mean plasma concentration C max is found to decrease by about 70% as U b increases from EDV to PSV. Decreasing U b would lead to higher concentration levels of anecortave acetate in the choroid and may achieve bioavailable delivery of the drug at the retina. Finally, the relative magnitudes of the vitreous and circulation losses have been compared for various blood flow velocities U b . For the normal range of U b between the PSV and the EDV, the circulation loss is found to be the dominant route of intraocular drug loss.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of choroidal blood flow on transscleral retinal drug delivery using a porous medium model Arunn Narasimhan ⇑, Ramanathan Vishnampet Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai 600 036, India
a r t i c l e
i n f o
Article history: Received 23 March 2011 Received in revised form 16 April 2012 Accepted 19 April 2012 Available online 16 June 2012 Keywords: Drug delivery Biofluids Human eye Porous medium Numerical Choroidal blood flow Transscleral Anecortave acetate
a b s t r a c t Transscleral drug delivery is one of the methods of depositing drug in the posterior segment (comprising retina, choroid, sclera and macula) of the human eye, to treat diseases such as age related macular degeneration (AMD). In this study, the effect of choroidal blood flow on transscleral drug delivery to the retina is investigated using a porous medium model of the sclera and the choroid. A two-dimensional geometrical model of the human eye is constructed from available measurements and determination of physicochemical properties of the sclera and the choroid, such as their effective diffusivity D and porous medium permeability K. Position and time dependent concentrations of the drug in the sclera and the choroid are predicted and the relative magnitudes of the periocular, vitreous and circulation losses are compared for various blood flow velocities U b . The simulations also predict the transient mean plasma concentration C of the drug anecortave desacetate in the choroid and the effect of choroidal blood flow on the peak mean plasma concentration C max . Comparison of predicted C with available experimental results is good. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction In the human eye, effective treatment of posterior segment (comprising retina, choroid, sclera and macula) diseases such as age related macular degeneration (AMD) requires adequate levels of drugs to reach those tissues. Methods of drug delivery include topical administration of eye drops [1,2], systemic administration, and intravitreal implants and they attempt to deposit therapeutic concentration levels of drugs at the choroid and the retina. Intravitreal implants have been successfully used for antibody based treatment of AMD. However, it is an invasive, potentially hazardous and undesirable method of treatment. Due to the large and accessible surface of the sclera, and its high and age-independent permeability coefficient P s to macromolecules and water-soluble drugs, transscleral drug delivery is a favourable alternative [3,4]. In transscleral drug delivery, the drug is placed at a periocular site in the posterior segment from which it diffuses through the sclera to the target tissues. Transscleral drug delivery can be classified into three types depending on the route taken by the drug to reach the target tissue: (1) anterior chamber route (2) systemic circulation route and (3) direct penetration pathway. In the preferred direct penetration pathway type of delivery, the target tissue is
⇑ Corresponding author. E-mail address: [email protected] (A. Narasimhan). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.05.060
located in the posterior segment of the eye and the drug diffuses through the sclera, choroid and the retina into the vitreous humor. Several recent studies of transscleral drug delivery have attempted to derive pharmacokinetic models to explain the transfer rates of the drug through the posterior eye tissues [5–7]. The permeation of drugs through the sclera, choroid and the retinal tissues are affected by the barrier to diffusion imposed by these tissues and losses through periocular spaces, systemic circulation, and vitreous absorption. The chief parameter that governs the barrier to drug diffusion is the scleral permeability coefficient Ps . An experimental study of human scleral permeability determined that the in vitro permeability coefficient of the sclera P s is inversely related to the molecular weight of the drug compound MWt. [8]. In another recent study, the permeability coefficient of the choroid Pc has been found to be in the same range as that of the sclera [9]. The kinetics of the various losses has shown that the transfer rate of the direct permeation is a small fraction ð0:05 0:2%Þ of the rate constant of periocular losses [10]. This fraction has been labeled the bioavailability of the drug in the vitreous BAv . Pharmacokinetic models succeed in predicting overall properties such as the permeability coefficient Ps [11] of a drug or the average concentrations of the drug in various tissues [12]. Simulation models using established numerical methods to solve blood flow and drug species equations and based on the geometry and physicochemical characteristics of the eye are more comprehensive. They also predict time and position dependent concentrations
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Nomenclature C C D Ps K t U
v x; y
mean plasma concentration, kg m-3 Concentration, kg m-3 Effective diffusion coefficient, m2/s permeability coefficient of the sclera, Pa Permeability, m2 time, s Flow velocity, m/s Darcy velocity, m s-1 Cartesian co-ordinates, m
q l /
density, kg m-3 viscosity, N s m-2 surface porosity
Subscripts b blood c choroid max maximum value s sclera
Greek symbols / porosity
of the drug across the eye. Such models have been attempted for drug delivery through systemic administration [13] and intravitreal implants [14,15]. A drawback of these models is their inability to account for the effect of choroidal blood flow on systemic loss of the drug. In a recent study [4], a pharmacokinetic simulation model based on the scleral permeability coefficient Ps was developed, which accounts for the circulation loss and predicts the overall permeation flux through the sclera. However, this simulation was based on steady-state calculations and the concentration on the surface of the sclera was assumed to be a constant. A 3D porous medium approach using finite element method for studying transscleral drug delivery was proposed in another recent study [16], which accounted for the diffusion and convection losses. However, the choroidal blood flow effect on the drug delivery is lumped as a sink term and assumed linear. There have been instances reported in medical literature [17,18] in the pre-drug delivery stages, usage of scleral buckle, a surgical procedure used to treat retinal detachment, alters the choroidal blood flow rate. In this study, the effect of choroidal blood flow on transscleral drug delivery to the retina is investigated using a porous medium model of the sclera and the choroid. The permeation of the drug through the direct penetration pathway is modeled as a diffusion process and studied using Fick’s second law of diffusion in conjunction with an effective diffusivity for the porous media. Using the developed model, the transient mean plasma concentration C of the drug anecortave desacetate in the choroid is predicted. The effect of choroidal blood flow on the transient peak mean plasma concentration C max is also studied and compared with available experiments [19].
curvature of 7.79 mm [22]. The thickness of the sclera varies from 588 lm at the limbus down to 491 lm at the equator and reaches a maximum of 996 lm at the posterior pole of the normal human eye [23]. The average thickness of the sclera is taken as 670 lm. The thickness of the choroid is taken to be 287 lm [24]. The optic nerve is assumed to be a cylindrical segment with an axis inclined at 30 to the pupillary axis. The optic nerve sheath diameter (ONSD) is about 4.8 mm [25] and the optic disc diameter is about 1.83 mm [26]. The diameter of the optic nerve where it meets the posterior eye is assumed to be the average of the ONSD and the optic disc diameter, or about 3.32 mm.
2. Drug diffusion model
2.3. Properties of the sclera and the choroid
Fig. 1 shows a schematic of the vertical cross-section of the human eye indicating the posterior juxtascleral region (PJD). Anecortave acetate, one of the standard drug base used, is administered on the sclera at a PJD and takes a direct penetration route to the vitreous through the sclera, choroid, and the retina. The diffusive transport of anecortave acetate through the ocular tissues is impeded by blood flowing through choroidal vessels. Choroidal blood flow adversely convects the drug and lowers the drug concentration that can reach the target tissue in the retina.
The surface porosity of the sclera /s is determined from its transmission electron microscopy (TEM) images. Digitized TEM images of the sclera are obtained from various sources [27–30].
2.2. Conservation equations In the computational domain, the sclera and the choroid are treated as homogeneous isotropic porous media. The conservation equations for porous media are solved to obtain the concentration distribution within the domain. The volume averaged mass conservation, and momentum and species transport equations for porous media in the sclera and the choroid are written as,
r v b ¼ 0;
@v /l qb b þ ðvb $Þvb ¼ $P þ lb r2 vb c b vb ; @t Kc @C 2 ¼ Ds r C; /s @t @C /c þ ðv b $ÞC ¼ Dc r2 C: @t
ð1Þ ð2Þ ð3Þ ð4Þ
2.1. Geometry of the eye Fig. 2 (a) shows the geometrical model of the eye used in the present simulations. The diameter of the eye along the pupillary axis is about 23.58 mm [20]. The posterior surface of the eye is almost spherical. The anterior surface of the cornea is a segment of a sphere which is about 6 mm [21] in diameter and has a radius of
Fig. 1. Schematic of the vertical cross-section of the human eye indicating the posterior juxtascleral region.
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A
Fig. 2. (a) Computational domain and grid (mesh element size 80 lm) used in the present simulations indicating the boundary conditions and depot location, and (b) region A shown at high resolution (mesh element size 10 lm).
The images are in the form of two-dimensional arrays of pixels with gray level 0 6 g 6 255. Light regions of the image correspond to pores and dark regions correspond to the solid matrix. A threshold gray level of 128 is used to distinguish pores from the solid. Annotations are removed from the image prior to the analysis. An iterative process was used in which the size of the window is increased till the porosity no longer changes. The microscopic details are averaged over an REV just as it is done in any porous continuum model. The visual resolution level of all TEM images were roughly equal and of lm order. The surface porosity from each image is computed as the ratio of the area of light pixels ðg > 128Þ to the area of dark pixels ðg 6 128Þ. Using this method on several sclera images, the mean surface porosity of the sclera was obtained as /s ¼ 0:39 0:09 ðn ¼ 9Þ. The porosity of the adjacent choroid /c is taken to be equal to the porosity of the sclera /s . The in vitro permeability coefficient of the sclera to various hydrophilic compounds has been experimentally studied [8]. It has been found that the mean scleral permeability coefficient Ps is inversely related to the molecular weight MWt. of the drug compound. Using the correlation log Ps ¼ 3:642 0:3753 log MWt:, 1 where Ps is in cm s1 and MWt. is in g mol , that predicts the experimental values in [8] with R2 ¼ 0:9719, for anecortave acetate ðMWt: ¼ 386:48ÞPs is found to be 2:4429 105 cm s1 . The effective diffusivity of anecortave acetate through the sclera is given by Ds ¼ Ps ts where ts is the thickness of sclera. Using t s ¼ 670 lm gives Ds ¼ 1:64 1010 m2 =s, which is more accurate than the value used in [8]. It is assumed that the effective diffusivity of the drug through the sclera Ds is equal to the effective diffusivity through the choroid Dc . The present model do not lump the sclera and the choroid into a single region as done in [8]. This allows the hydraulic conductivity of both tissues to be assumed same permitting blood flow rate in the choroid to be substantially different from that in the sclera. In transscleral drug delivery, the drug penetration is through the sclera and the choroid before it reaches the retina. The mass transfer resistances of the sclera and the choroid are up to an order higher than those of the retina and the vitreous. Hence modeling the drug transport beyond the retinal layer and inside the vitreous with a continuum model-like resolution is not relevant in the current context. The permeability of the choroid K c is assumed to be equal to the hydrodynamic permeability of the sclera. The scleral permeability K s is obtained from experimentally measured values of the hydraulic conductivity of the sclera [31]. At normal hydration of the sclera, it is found that K s ¼ 1:3 nm2 at 37 C. The density and viscosity of blood are assumed to be constant, qb ¼ 1060 kg=m3 and lb ¼ 2:7 103 N s=m2 .
3. Numerical procedure and solution methodology The conservation equations are numerically solved using the commercial software FluentÒ 6.3.26 which employs the Finite Volume Method (FVM). A semi-implicit time marching scheme with first-order discretization is used for the transient terms. Fig. 2 (a) shows the computational domain and grid used in the present simulations. The domain is created using GambitÒ 2.3.16. Quadrilateral and triangular finite volume (surface) elements of uniform size are used for meshing the eye domain. Fig. 2 (b) shows region A in Fig. 2 (a) at a higher resolution. The size of the mesh elements ð 10 lmÞ is larger than the representative elemental volume (REV) of the porous media ð 1 lmÞ. 3.1. Boundary conditions At the interface between the sclera and the choroid, a no-slip boundary condition is used for the blood velocity. The concentration and its gradient normal to the surface are assumed to be continuous. An impermeable boundary condition, ð$CÞ n ¼ 0 is used at the interfaces between the sclera and the cornea, and the sclera and the optic nerve respectively. Blood is assumed to enter the choroid through the interface between the choroid and the optic nerve with a physical or fluid inlet velocity of U b and exit through the interface between the choroid and the cornea. In reality, choroidal vessels anastomose [32] and blood flows in through the arteries and out through the veins both of which leave the eye through the optic nerve sheath. However, it is assumed that the convective effect of blood flow on the diffusion process is captured sufficiently accurately in this alternate simpler flow configuration. It is also assumed that the flow exit is sufficiently far away from the region of interest that its effect is negligible. Further, the blood flow through the choroidal vessels is assumed to be uniform. In reality, blood flow pulsates between a peak systolic velocity (PSV) of 10:15 cm s1 and an end diastolic velocity (EDV) of 2:85 cm s1 . The effect of a pulsating blood flow is neglected in the present simulations. At the interface between the choroid and the optic nerve, the concentration of the drug is assumed to be zero. A transient concentration profile is assumed to exist on the outer surface of the sclera, written as
" Cðx; y; tÞ ¼
C 00 ekt
exp
ðx xD Þ2 þ ðy yD Þ2
r2
# ;
ð5Þ
where C 00 is the initial depot concentration of the drug and ðxD ; yD Þ are the Cartesian coordinates of the point of injection. Due to loss
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of drug from periocular spaces, C 00 < C 0 , the dose concentration. The relation 4r ¼ d is used to obtain r, where d is the diameter of the needle used for the injection. The relation k ¼ tln1=22 is used to obtain k, where t 1=2 is the half-life of the depot concentration which is determined experimentally. The loss of drug from the periocular spaces is assumed to be equal to the change in the average concentration gradient at the outer surface of the sclera due to the use of C 00 as the initial depot concentration in place of C 0 . The initial depot concentration C 00 is obtained from the relation,
C 00 ¼ b10 C 0 ;
ð6Þ
where b10 is the dimensionless periocular loss coefficient. b10 is related to the bioavailability of the drug in the vitreous BAv . For a given b10 , the bioavailability BAv of the drug is the ratio of the average concentration gradients normal to the surface of the sclera obtained when the simulation is performed with depot concentrations C 00 and C 0 respectively. BAv has been typically found to be very low ð 0:2%Þ for subconjuctival injection of prednisolone in rabbits [10]. However, it has been suggested that this is an overestimate since circulation losses are ignored [4]. At the inner surface of the choroid, the concentration flux of the drug diffusing into the vitreous is assumed to be proportional to the difference between the concentration at the surface and the concentration in the vitreous. The vitreous is treated as a sink with infinitely large capacity. Hence, the concentration of the drug in the vitreous is assumed to be negligible. The boundary condition on this surface is written as,
sffiffiffiffiffiffi k ð$CÞ n ¼ b20 Cðx; y; tÞ; Dc
ð7Þ
where b20 is the dimensionless vitreous coefficient. 3.2. Anecortave acetate case study The present simulations have been performed for a posterior juxtascleral depot (PJD) administration of anecortave acetate ðMWt: ¼ 386:48 g=molÞ using a 19-gauge needle ðd ¼ 0:686 mmÞ. To study the nature of pharmacokinetics, three different doses of concentration 3, 15 and 30 mg anecortave acetate in 0.5 ml water are used. The half-life of the PJD concentration of anecortave acetate for 15 and 30 mg doses have been previously measured as 3.3 and 4.5 days respectively [19]. For the 3 mg dose, an observable concentration of anecortave desacetate has been measured up to 2 weeks after the dose. Assuming that an observable concentration corresponds to 1% of the initial dose, it is estimated that t1=2 for the 3 mg dose is about 2.1 days. On the other hand, fitting a linear model for the half-life versus dose concentration gives t1=2 ¼ 2:3 days. The mean of the two estimates, t 1=2 ¼ 2:2 days is used. Anecortave acetate is normally administered using a 56 cannula inserted through an incision made at 8 mm from the limbus measured along the sclera in the posterior direction [33]. The present simulations employ a Cartesian coordinate system where the origin is located at the center of the orbit and the posterior direction corresponds to the positive x-axis. Measured from the positive x-axis, the polar angle hD at which the drug is injected is approximately 51 . The coordinates of the point of injection are obtained as xD ¼ R cos hD and yD ¼ R sin hD and where R is approximately 11.25 mm. All simulations correspond to a single dose of anecortave acetate. The periocular loss coefficient b10 for anecortave acetate must be determined from a knowledge of its bioavailability BAv . Since BAv of anecortave acetate has not been previously measured, it is assumed to be about 1:3%. This value has been chosen so that the results for C max predicted by the present model for a normal choroidal blood flow velocity and a 15 mg dose matches
the results measured experimentally. The vitreous loss coefficient b20 is assumed to be 1. Simulations are performed for a range of values of b10 and b20 in order to check the sensitivity of the results to these coefficients. These values are chosen because these parameters appear in the boundary conditions. The thermophysical properties of the sclera and the choroid, for instance, have been directly measured in experiments under in vitro conditions. The in vitro properties can be used, without further sensitivity checks, in conjunction with a porous continuum approach and is the standard approach taken in earlier models of drug delivery. In the present simulations, the mean plasma concentration of the drug C is reported as the average concentration over the choroid. Immediately following PJD administration C increases from 0 to a maximum value C max . The peak mean plasma concentration C max is an important indicator of the therapeutic effectiveness of a drug. For a given dose concentration, it indicates whether the drug is bioavailable at the retina. Due to the loss of drug through periocular, vitreous and circulation routes, the mean plasma concentration C eventually decreases and ultimately becomes 0 again. The duration of time for which a traceable concentration of anecortave desacetate exists in the choroid varies from 2 to 6 weeks. 3.3. Grid and time dependence studies Table 1 shows the peak mean plasma concentration C max following a 15 mg PJD administration of anecortave acetate for various blood flow velocities U b ¼ 2:85, 6.5 and 10:15 cm s1 . Due to the inherently unsteady nature of the problem, transient simulations are performed to study grid and time dependence. The results for each mesh element size are shown for various values of the time-step used. The results for C max using a time-step of 10 s differ from the results using a time-step of 1 s by < 1% for all mesh element sizes. Similarly, the results using a mesh element size of 20 lm differ from the results using a mesh element size of 10 lm by < 2% for time-steps 100, 10 and 1 s. All further simulations use a mesh element size of 20 lm. A variable time-stepping scheme is used, with the time-step being 10 seconds for the first 5 h (of flow time), 100 s for the next 50 h and 1000 for the last 500 h. Overall, drug transport across the sclera and choroid for a period of 23 days is simulated. 4. Results and discussion At t ¼ 0, a steep concentration gradient exists just near the depot which causes the drug to diffuse passively through the sclera. As the drug reaches the choroid, the transport is through
Table 1 Grid and time dependence study of C max for U b ¼ 2:85, 6.5 and 10:15 cm s1 following a 15 mg PJD administration. Mesh element size (lm)
Time step (s) 1000
100
10
1
2.85
80 40 20 10
4.67 4.87 4.97 4.98
4.73 4.94 4.99 4.92
4.74 4.96 5.02 5.00
4.74 4.97 5.04 5.05
6.5
80 40 20 10
2.05 2.11 2.08 1.86
2.07 2.17 2.17 2.13
2.08 2.18 2.21 2.21
2.08 2.18 2.22 2.23
10.15
80 40 20 10
1.31 1.35 1.33 1.19
1.33 1.39 1.39 1.37
1.33 1.40 1.42 1.42
1.34 1.40 1.42 1.43
U b ðcm s1 Þ
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Table 2 Effect of U b on C max and AUCinf following a 3, 15 and 30 mg PJD administration. U b ðcm s1 Þ
2.85 4.07 5.28 6.5 7.72 8.93 10.15
Fig. 3. Transient mean plasma concentration C of anecortave acetate following a 15 mg PJD administration for various blood flow velocities U b .
both convection and diffusion. The mass diffusion Peclet number Pe given by,
Pe ¼
/s U b t s ; Ds
ð8Þ
indicates the relative magnitude of convection over diffusion. Assuming that the blood flow velocity U b is uniform and equal to the average of the PSV and the EDV, the Peclet number is determined as Pe ¼ 1:48 105 , which indicates that convection is the dominant mode of transport. As the blood flow velocity U b increases, Pe
C max ðng ml
1
AUCinf ðng day ml
Þ
1
Þ
3 mg
15 mg
30 mg
3 mg
15 mg
30 mg
1.00 0.70 0.54 0.44 0.37 0.32 0.28
5.02 3.52 2.72 2.21 1.86 1.61 1.42
10.06 7.06 5.45 4.43 3.73 3.23 2.84
3.85 2.72 2.08 1.69 1.43 1.23 1.09
25.90 18.09 14.38 11.92 10.15 8.70 7.73
66.42 47.31 36.69 29.68 25.52 22.32 19.58
increases and the desired diffusion of drug towards the posterior segment is further suppressed by the undesired convective loss which takes the drug away from the target tissue. Fig. 3 shows the transient mean plasma concentration C of anecortave acetate following a 15 mg PJD administration for various blood flow velocities U b . The present numerical study predicts the observed trend of the transient variation of C. As U b increases, C max decreases as expected. An observable concentration of anecortave acetate persists in the choroid for up to 3 weeks. The peak mean plasma concentration C max occurs on the first day of the dose. The mean plasma concentra1 tion C is in the range 2 5 ng ml for the normal range of blood flow velocities U b ð2:85 cm s1 6 U b 6 10:15 cm s1 Þ. Table 2 shows the peak mean plasma concentration C max and the total area under the mean plasma concentration curve between t ¼ 0 and t ¼ 23 days 1 AUCinf . The AUCinf of 11:92 ng ml day predicted by the present simulations for a 15 mg dose and U b ¼ 6:5 cm s1 is nearly twice
Fig. 4. Contours of the concentration of anecortave acetate near the depot at t ¼ tmax for (a) U b ¼ 0:01 cm=s, (b) U b ¼ 0:1 cm=s, (c) U b ¼ 1 cm=s, and (d) U b ¼ 10 cm=s, following a 15 mg PJD administration.
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Table 3 tmax and Dtres for U b ¼ 6:5 cm s1 following a 3, 15 and 30 mg PJD administration. Dose
t max (h)
Dt res (h)
3 mg 15 mg 30 mg
1.03 0.83 0.83
5.19 8.78 11.17
Fig. 6. Effect of blood flow velocity U b on peak overall plasma concentration C max of anecortave acetate following a 3, 15 and 30 mg PJD administration.
Fig. 5. Comparison of peak mean plasma concentrations C max predicted by the present model with previous experimental results for U b ¼ 2:85, 6.5 and 10:15 cm s1 following a 3, 15 and 30 mg PJD administration.
1
the result of 5:28 2:40 ng ml day obtained from a previous experimental study. This could be the result of two possible reasons. Firstly, the experimental result is based on measurements of C at just 9 time instances. Secondly, the numerical result is based on an experimentally determined value of t 1=2 ¼ 3:3 days for which a standard deviation of 0:6 days is reported. Fig. 4 shows the contours of the concentration of anecortave acetate for blood flow velocities U b ¼ 0:01, 0.1, 1 and 10 cm s1 respectively. The levels used to plot these contours are exponentially distributed in order to represent the sharp decrease in the local concentration of the drug in the direction of diffusion. For U b ¼ 0:01 cm s1 , a thick streak of closely spaced lines is observed adjacent to the interface between the sclera and the choroid across which there is a steep concentration gradient. As U b increases, the streak becomes thinner and moves away from the choroid-vitreous boundary. The presence of the streak is due to the choroidal blood flow which carries the drug away from the posterior segment as a result of which the lines of higher concentration are shifted away from the posterior direction. The contours have been shown at the time instant t ¼ tmax at which the mean plasma concentration reaches its maximum value. tmax is difficult to measure experimentally and is expected to result in high measurement errors. This is due to the fact that the mean plasma concentration remains within 5% (say) of the peak mean plasma concentration C max for a duration of up to 8 10 times t max . In the present study, the time interval beyond tmax up to which the value of C remains within 5% of C max is defined as the residence time Dt res . An experimental setup which gives up to 5% error in the measurement of C could predict the time at which C reaches the value C max as any time instant between ðtmax þ 0:5Dtres Þ 0:5Dt res . Table 3 shows the present results for tmax and Dtres for a 3, 15 and 30 mg dose. Comparing the result for tmax þ 0:5Dtres predicted by the present simulations with the experimental results for t max we find that this is indeed likely. For a 15 mg and 30 mg dose, ðt max þ 0:5Dtres Þ predicted by the present simulations for U b ¼ 6:5 cm s1 are 0.22 and 0.27 days respectively compared to experimental values of 0:27 0:16 and 0:36 0:19 days respectively.
Fig. 7. Average plasma concentration along the radial coordinate C 31 6 h 6 71 and various U b .
for
Table 4 Sensitivity of C max to b10 for 2:85 6 U b 6 10:15 cm s1 . U b ðcm s1 Þ
b10 ð%Þ 0.1
0.2
0.5
1
2
2.85 4.07 5.28 6.5 7.72 8.93 10.15
0.39 0.27 0.21 0.17 0.14 0.12 0.11
0.77 0.54 0.42 0.34 0.29 0.25 0.22
1.93 1.36 1.05 0.85 0.72 0.62 0.54
3.86 2.71 2.09 1.70 1.43 1.24 1.09
7.73 5.42 4.18 3.40 2.87 2.48 2.18
Fig. 5 shows the results for C max predicted by the present model and the experimentally measured peak mean plasma concentrations for a 3, 15 and 30 mg dose. The present result for C max has been matched with the experimental result for a 15 mg dose. Fig. 5 also shows the maximum and minimum values of C max as U b varies between the PSV and the EDV. The comparison may seem redundant but is done to highlight the proposed model is tenable. In reality, the choroidal blood flow rate is not fixed and can vary with time, even after the drug is applied. The experimental results subsume such flow rate variations as they are obtained at the vitreous over a time period. The comparison in Fig. 5 shows that the modelling is sensitive to flow rate variations. It can be used for predictions at even shorter time scales where the choroidal blood flow rate could be significantly different. Experimental results of drug concentrations are not available at such time periods.
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U b ðcm s1 Þ 2.85
0.01 0.1 1 10 100
6.5
C max
AUCinf
C max
5.02 5.02 5.02 5.02 5.02
25.54 25.42 25.90 25.87 25.89
2.21 2.21 2.21 2.21 2.21
10.15 AUCinf 12.07 12.00 11.92 11.95 12.03
C max
AUCinf
1.42 1.42 1.42 1.42 1.42
7.73 7.73 7.73 7.73 7.73
port was studied for U b between 0:01 cm s1 and 10 cm s1 in this paper. From Fig. 8, it is clear that reducing U b to below 0:4 cm s1 is not very useful because vitreous losses begin to dominate for U b < 0:4 cm s1 .
5. Conclusions
Fig. 8. Effect of U b on vitreous clearance ratio for 0:1 6 U b 6 1 cm s1 .
v2 and circulation clearance ratio v3
Fig. 6 shows the effect of U b on C max for a 3, 15 and 30 mg dose. For the 3 mg dose, the relative change in C max with U b is lesser compared to the change for the 30 mg dose. This is because the convective term in Eq. (2) scales as the local concentration C which is higher for the 30 mg dose. Fig. 7 shows the average concentration of anecortave acetate C avg along a radial line at polar angle h for U b ¼ 0:01, 0.1, 1 and 10 cm s1 . C avg is maximum at the depot angle hD as expected. It decreases sharply in the posterior direction indicating the convective effect of the choroidal blood flow. The periocular and vitreous losses have been characterized using a periocular coefficient b10 and a vitreous coefficient b20 respectively. Table 4 shows the results for C max obtained from the present simulations for various values of b10 . The results show that C max is sensitive to b10 and proportionately changes as b10 is changed. Table 5 shows the results for C max and AUCinf obtained from the present simulations for 5 orders of magnitude of b20 . The results show that both C max and AUCinf are not sensitive to b20 . Hence, the assumption of using b20 ¼ 1 appears to be justified. Periocular losses constitute 98:7% of the total loss of anecortave acetate in transscleral retinal delivery ðb10 ¼ 1:3%Þ. The remaining routes of loss of anecortave acetate are through vitreous and systemic circulation losses. Fig. 8 shows the relative magnitudes of the vitreous and circulation losses. The vitreous loss is determined as the average diffusive flux at the interface between the choroid and the vitreous. The circulation loss is determined as the average convective flux at a cross section of the choroid. The ratio of the circulation loss to the sum of the circulation and vitreous losses is denoted by v3 and the corresponding ratio for the vitreous loss is denoted by v2 ðv2 þ v3 ¼ 1Þ. As U b increases, the circulation loss increases until it becomes equal to the vitreous loss for U b 0:4 cm s1 . Typically U b is between the PSV ð2:85 cm s1 Þ and EDV ð10:15 cm s1 Þ. In this range, the circulation loss is the dominant route of intraocular loss of anecortave acetate. However, it is possible to surgically control and reduce the choroidal blood flow in order to reduce the circulation loss. It is for this reason that the drug trans-
A porous medium model has been proposed for simulating transscleral retinal drug delivery. The effective diffusivity in the porous medium model are determined by correlating previous experimental results for the scleral permeability coefficient to various hydrophilic compounds with their molecular weights. The porosity of the sclera and the choroid are obtained by image processing of their available TEM images. Using the numerical model along with a transient boundary condition for the concentration profile of the administered drug anecortave acetate on the surface of the sclera, periocular and viterous drug losses have been predicted. The results for the peak mean plasma concentration C max , the time tmax at which the mean plasma concentration C attains its peak value and the area under the mean plasma concentration curve AUCinf obtained from the present simulations have been compared with a previous experimental study. The present simulations have been found to predict C max accurately within the range of experimentally observed values for 3, 15 and 30 mg doses. The porous medium model used, predicts the overall trend of the transient variation of the mean plasma concentration and the time up to which an observable concentration of anecortave acetate persists in the choroid. The advantage of using the present model over available compartmentalized model with averaged macroscopic quantities is that, the present simulations solve discretized conservation equations within finite mesh elements. As a result, spatial and temporal variation of the drug concentration are recovered. The peak mean plasma concentration C max , which can be measured by regular plasma sampling, is a useful indication of the reliability of the present model since it is sensitive to the choroidal blood flow velocity U b . The effect of choroidal blood flow velocity U b (a patient-specific quantity) on the C max has also been studied. The peak mean plasma concentration C max is found to decrease by about 70% as U b increases from EDV to PSV. Decreasing U b would lead to higher concentration levels of anecortave acetate in the choroid and may achieve bioavailable delivery of the drug at the retina. Finally, the relative magnitudes of the vitreous and circulation losses have been compared for various blood flow velocities U b . For the normal range of U b between the PSV and the EDV, the circulation loss is found to be the dominant route of intraocular drug loss.
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