Contribution of Saccadic Motion to Intravitreal Drug Transport: Theoretical Analysis

ABSTRACT

Purpose

The vitreous humor liquefies with age and readily sloshes during eye motion. The objective was to develop a computational model to determine the effect of sloshing on intravitreal drug transport for transscleral and intra-vitreal drug sources at various locations

Methods

A finite element model based on a telescopic implicit envelope tracking scheme was developed to model drug dispersion. Flow velocities due to saccadic oscillations were solved for and were used to simulate drug dispersion.

Results

Saccades induced a three-dimensional flow field that indicates intense drug dispersion in the vitreous. Model results showed that the time scale for transport decreased for the sloshing vitreous when compared to static vitreous. Macular concentrations for the sloshing vitreous were found be much higher than that for the static vitreous. For low viscosities the position of the intravitreal source did not have a big impact on drug distribution.

Conclusion

Model results show that care should be taken when extrapolating animal data, which are mostly done on intact vitreous, to old patients whose vitreous might be a liquid. The decrease in drug transport time scales and changes in localized concentrations should be considered when deciding on treatment modalities and dosing strategies.

APPENDIX

A description and the value of the parameters used in the equations in this section are provided in Table III. For more details on the values and description of the parameters, see (31). c₂ and c₃ in the following equations represent the concentrations in the retina and the choroid-sclera, respectively.

Table III Parameters Used for Evaluating Flux at the Retinal Surface

A. Flux of Drug from the Vitreous Out of the Eye

The species balance equations in the posterior tissues are given by Eqs. A1 and A2, and the corresponding solution to the equations are given by Eqs. A3 and A4. The flux in the tissues is given by Eqs. A5 and A6 (Fig. 3).

$$ \begin{array}{*{20}{c}} {{D_r}\frac{{{d^2}{c_2}}}{{d{x^2}}} = {K_a}\frac{{d{c_2}}}{{dx}}} \hfill & {0 < x < {L_1}} \hfill \\\end{array} $$

(A1)

$$ \begin{array}{*{20}{c}} {{D_{{cs}}}\frac{{{d^2}{c_3}}}{{d{x^2}}} = \gamma {c_3}} \hfill & {{L_1} < x < {L_2}} \hfill \\\end{array} $$

(A2)

$$ {c_2} = {A_1} + {A_2}{e^{{ax}}} $$

(A3)

$$ {c_3} = {A_3}{e^{{\beta x}}} + {A_4}{e^{{ - \beta x}}} $$

(A4)

$$ flu{x_2} = - {D_r}\frac{{d{c_2}}}{{dx}} + {K_a}{c_2} = {K_a}{A_1} + {A_2}\left( {{K_a}{e^{{ax}}} - {D_r}\alpha {e^{{ax}}}} \right) $$

(A5)

$$ flu{x_3} = - {D_{{cs}}}\frac{{d{c_3}}}{{dx}} = - {D_{{cs}}}\alpha \left( {{A_3}{e^{{\beta x}}} - {A_4}{e^{{ - \beta x}}}} \right) $$

(A6)

A₁, A₂, A₃, and A₄ are constants, and alpha and beta in the above equations are defined as

$$ \alpha \equiv \frac{{{K_a}}}{{{D_r}}}{\hbox{and}}\,\beta \equiv \sqrt {{\frac{\gamma }{{{D_{{cs}}}}}}} $$

(A7)

The boundary conditions that were solved simultaneously for A₁, A₂, A₃, and A₄, are defined as follows:

$$ \begin{array}{*{20}{c}} {{\hbox{value}},\,{\hbox{x}} = 0,\,{{\left. {{c_2}} \right|}_{{x = 0}}} = {c_v}} \hfill \\{{\hbox{value,}}\,{\hbox{x = }}{{\hbox{L}}_1},\,{{\left. {{c_2}} \right|}_{{x = {L_1}}}} = {{\left. {{c_3}} \right|}_{{x = {L_1}}}}} \hfill \\{{\hbox{flux}},\,{\hbox{x}} = {{\hbox{L}}_1},\,{{\left. {flu{x_2}} \right|}_{{x = {L_1}}}} = {{\left. {flu{x_3}} \right|}_{{x = {L_1}}}}} \hfill \\{{\hbox{flux}},\,{\hbox{x}} = {{\hbox{L}}_2},\,{{\left. {flu{x_2}} \right|}_{{x = {L_2}}}} = {{\left. {{k_{{sc}}}{c_3}} \right|}_{{x = {L_2}}}}} \hfill \\\end{array} $$

The flux of drug at x = 0 which was used in the model (Eq. 8) was evaluated by substituting the constants into either Eqs. A5 or A6.

B. Flux of Drug from a Constant Concentration Transscleral Source into the Vitreous

The choroid and retina were assumed to be ablated under the transscleral drug source. Hence, the species balance equations for transport in the posterior tissues listed above were modified to accommodate the changes as follows:

$$ \begin{array}{*{20}{c}} {{D_r}\frac{{{d^2}{c_2}}}{{d{x^2}}} = 0} \hfill & {{L_1} < x < {L_2}} \hfill \\\end{array} $$

(A8)

$$ \begin{array}{*{20}{c}} {{D_{{cs}}}\frac{{{d^2}{c_3}}}{{d{x^2}}} = 0} \hfill & {0 < x < {L_1}} \hfill \\\end{array} $$

(A9)

$$ {c_2} = {B_1}x + {B_2} $$

(A10)

$$ {c_3} = {B_3}x + {B_4} $$

(A11)

$$ flu{x_2} = - {D_r}\frac{{d{c_2}}}{{dx}} = - {D_r}{B_1} $$

(A12)

$$ flu{x_3} = - {D_{{cs}}}\frac{{d{c_3}}}{{dx}} = - {D_{{cs}}}{B_3} $$

(A13)

As in the previous case, the boundary conditions were solved simultaneously for the constants B₁, B₂, B₃, and B₄. The boundary conditions are defined as below:

$$ \begin{array}{*{20}{c}} {{\hbox{value}},\,{\hbox{x}} = 0,\,{{\left. {{c_3}} \right|}_{{x = 0}}} = {c_0}} \hfill \\{{\hbox{value}},\,{\hbox{x}} = {{\hbox{L}}_1},\,{{\left. {{c_3}} \right|}_{{x = {L_1}}}} = {{\left. {{c_2}} \right|}_{{x = {L_1}}}}} \hfill \\{{\hbox{flux}},\,{\hbox{x}} = {{\hbox{L}}_1}\,{{\left. {flu{x_3}} \right|}_{{x = {L_1}}}} = {{\left. {flu{x_2}} \right|}_{{x = {L_1}}}}} \hfill \\{{\hbox{value}},\,{\hbox{x}} = {{\hbox{L}}_2}\,{{\left. {{c_2}} \right|}_{{x = {L_2}}}} = {c_v}} \hfill \\\end{array} $$

The flux of drug at x = L₂ was evaluated by substituting the coefficients into either Eq. 19 or 20.