**Purpose**:
Intravitreal injection of drugs is commonly used for treatment of chorioretinal ocular pathologies, such as age-related macular degeneration. Injection causes a transient increase in the intraocular volume and, consequently, of the intraocular pressure (IOP). The aim of this work is to investigate how intravitreal flow patterns generated during the post-injection eye deflation influence the transport and distribution of the injected drug.

**Methods**:
We present mathematical and computational models of fluid motion and mass transport in the vitreous chamber during the transient phase after injection, including the previously unexplored effects of globe deflation as ocular volume decreases.

**Results**:
During eye globe deflation, significant fluid velocities are generated within the vitreous chamber, which can possibly contribute to drug transport. Pressure variations within the eye globe are small compared to IOP.

**Conclusions**:
Even if significant fluid velocities are generated in the vitreous chamber after drug injection, these are found to have negligible overall effect on drug distribution.

^{1}Such injections are now very widely used; for example, one review, drawing from a single US database, identified more than 800,000 intravitreal injections being administered over an 18 month period to treat neovascular age-related macular degeneration.

^{2}

^{3}

^{–}

^{7}in vitro experiments,

^{8}

^{,}

^{9}and in vivo measurements.

^{10}Various authors

^{3}

^{–}

^{5}have used numerical simulations to show that, even if the velocities associated with this bulk percolation flow are very small (\(\sim{10^{ - 8}}\) m/s),

^{11}advection is an important transport mechanism, especially in the case of large molecules with small diffusivities. Further, various authors have shown that vitreous motion induced by eye rotations can also play a key role in transport processes in the vitreous chamber.

^{9}

^{,}

^{12}

^{–}

^{14}

^{15}showed with a mathematical model that fluid flow can also be generated by another mechanism. The tissue is locally pressurized at the injection site, which drives fluid away. This fluid motion can be explained making use of Biot’s consolidation model, which is a classic theory in soil mechanics.

^{16}Basser’s work for transport in the brain is based on the assumption that the brain tissue is a porous material of infinite extent. Chan et al.

^{17}have speculated that a similar mechanism can take place in the eye after a vitreoretinal injection, even if Basser’s theory cannot be directly applied in this case as the eye has finite size and an intraocular injection will lead to scleral expansion and a pressure elevation.

^{18}

^{–}

^{20}which is related to the compliance of the ocular globe.

^{21}

^{,}

^{22}This IOP increase resolves as fluid drains from the eye at a greater than normal rate. Although several studies have considered intraocular flow patterns associated with an intravitreal injection,

^{23}

^{–}

^{25}they have not included the effects of transient globe expansion and deflation. In this work, we present a mathematical model to study fluid motion and mass transport within the vitreous chamber after an intravitreal drug injection. Our aim is to assess whether and to what extent this flow may contribute to transporting the injected drug within the vitreous chamber. The problem is of clinical relevance since understanding drug transport processes within the vitreous body is key to predicting the fate of the drug in the eye, particularly the regions of the retina that will preferentially receive the treatment and the associated time scales.

*V*= 50 µL and an injection flow rate of

_{inj}*Q*= 1 mL/min,

_{inj}^{26}we can estimate the injection time as

*t*=

_{inj}*V*/

_{inj}*Q*≈ 3 s. This is very rapid compared to the subsequent dynamics (verified a posteriori); hence, we here simply model eye globe deflation, assuming that the injection is effectively instantaneous. We therefore take time zero to be immediately after the injection, when an increase in ocular volume (equal to the injection volume) has produced a pressure increase, which, in turn, drives flow out of the eye during deflation.

_{inj}*t*time,

*V*=

_{o}*V*(

_{o}*t*) the total volume of the eye,

*Q*the rate of aqueous humor production by the ciliary processes minus rate of pressure-independent outflow, and

_{in}*Q*the rate of pressure-dependent outflow. We assume that

_{out}*Q*is constant and equal to 2.2 µL/min.

_{in}^{27}

*P*) is related to globe volume

*V*through ocular compliance, and

_{o}*C*=

*dV*/

_{o}*dP*, which is assumed to be constant and equal to 1 µL

*/*mm Hg.

^{21}This quantifies the capacity of the eye to change its volume in response to an IOP change. Moreover, we describe pressure-dependent outflow by

*P*is the episcleral venous pressure, and \(\mathcal{R}\) is the hydrodynamic resistance to fluid drainage, assumed to be constant. Episcleral venous pressure depends on posture and other factors; we here assume

_{epv}*P*= 8 mm Hg.

_{epv}^{28}

^{–}

^{30}

*P*(0), which in turn depends on the volume of the injected fluid

*V*through the expression \(P( 0 ) = {V_{inj}}/C + \hat P\). The resulting analytical solution is

_{inj}*V*) represents the eye globe, while the inner sphere (with volume

_{o}*V*) represents the bolus of injected fluid (see Fig. 1). The space between the two spheres is occupied by the vitreous humor, whereas the inner sphere contains only the injected fluid (including the drug). We assume that the injected fluid displaces the vitreous body, creating an inner fluid bolus, without changing vitreous porosity. Thus, at the initial time, the volume of the inner sphere coincides with the injected volume, \({V_i}( 0 ) = {V_{inj}} = {V_o}( 0 ) - \hat V\).

_{i}^{31}

^{,}

^{32}We use this fact, as well as the formula for the expansion of a thick-walled internally pressurized elastic incompressible sphere, to compute the relative expansion expected from 1) the vitreous alone containing an internally pressured bolus of fluid and 2) the scleral shell alone, which is acted upon internally by the IOP. Specifically, if the sphere has external and internal undeformed radii of

*a*and

*b*, respectively, then \({\rm{\Delta }}a = \frac{{3\hat Pa}}{{4E}}( {\frac{1}{{{{( {a/b} )}^3} - 1}}} )\), where \(\hat P\) is the internal pressure (IOP in physiologic conditions),

*E*is the Young's modulus of the sphere wall, and Δ

*a*is the expansion of the outer surface of the sphere.

^{33}Using superscripts

*S*and

*V*to denote sclera alone and vitreous, the internal radius for the vitreous is that of the injected fluid bolus (

*b*≈ 2.3 mm) and the external radius is approximately the radius of the eye (

^{V}*a*≈ 1.1 cm for a human eye). In the case of the sclera, the internal radius is approximately

^{V}*b*=

^{S}*a*, while the external radius

^{V}*a*=

^{S}*b*+

^{S}*h*, where

*h*is the scleral thickness (≈ 670 µm). Noting that (

*a*/

^{V}*b*)

^{V}^{3}≫ 1 and

*h*/

*b*≪ 1, the ratio of the expansion of the sclera alone to the vitreous alone can be expressed as

^{S}*a*) is the physiologic radius of the eye. This ratio is approximately ( ≈ 10

^{V}^{−5})( ≈ 5)( ≈ 120) ≈ 0.006 ≪ 1. This shows that the contribution of the vitreous in withstanding the internal pressure is negligible.

^{31}) and thus much shorter than the deflation time of the eye of interest in our model, which is of tens of minutes.

*dV*/

_{o}*dt*=

*dV*/

_{i}*dt*. The radii of the two spheres are computed as

*V*(

_{o}*t*) is given by Equation (4). We note that, since we assume that both the injected fluid and the vitreous body are incompressible materials, the increase in pressure within the domain during injection is uniform and instantaneous.

*r*denoting the radial coordinate,

*u*denoting the radial component of vitreous displacement, and over dots indicating differentiation with respect to time.

_{r}*is fluid velocity in the fixed frame,*

**v***is tissue displacement,*

**u***k*is vitreous permeability to water, μ is the dynamic viscosity of the percolating fluid, and

*p*(

*,*

**x***t*) is the pressure, which is now a function of both time and position. Note that the above equation accounts for the fact that also the solid matrix of the tissue can possibly move. To distinguish the pressure used in Darcy's law from the IOP in the zero-dimensional model, we here denote pressure with a small

*p*, as opposed to the capital

*P*used in the previous section. Taking the divergence of the above expression, assuming that both fluid and solid matrix are incompressible and that

*k*does not vary with space, we find that the pressure satisfies Laplace's equation, which is a statement of mass conservation, and reads

*c*is the concentration of injected drug and

*D*is the diffusion coefficient (assumed constant).

*P*(0).

- •
*Case A.*Fluid outflow is assumed to be spherically symmetric and to thus involve the whole corneoscleral surface, as shown in Figure 1, top row, and Figure 2A. - •
*Case B*. Fluid outflow occurs entirely through the trabecular meshwork and is thus localized to a specific region of the corneoscleral surface, as depicted in Figure 1, bottom row, and Figure 2B.

*Case A:*At the scleral surface (\({\mathcal{B}_s}\)), we impose a constant water flux

*is the outward unit normal vector to the scleral surface, which is purely radial. The value of the fluid flux imposed (right hand side of Equation (11)) is derived from the zero-dimensional solution of the Zero-Dimensional Model of Eye Globe Deflation section.*

**n***v*is proportional to

_{r}*r*

^{−2}, similar to the dependence of vitreous displacement on

*r*. Equation (7) and condition (11) immediately show that

*r*and

*t*, and this equation reads

*= −*

**q**_{s}*D*∇

*c*+

*c*, and in this case,

**v***=*

**v****0**. Thus, the above equation reads \( - D\frac{{dc}}{{dr}} = 0\) on \(\;{\mathcal{B}_s}.\) Condition (14) could be replaced by an adsorption condition at the retina. However, over the time scale of our simulations, the concentration of the drug at the boundary is negligible, which means that condition (14) is effectively correct.

*P*(

*t*) is determined from Equation (4), under the assumption that the pressure drop across the vitreous is small compared to the IOP (justified a posteriori). Moreover,

*c*

_{0}is the drug concentration in the injected fluid.

*Case B:*Here the scleral surface is subdivided into two regions \({\mathcal{B}_s} = {\mathcal{B}_{s1}} \cup {\mathcal{B}_{s2}}\), where \({\mathcal{B}_{s1}}\) is the surface from which fluid exits the domain (the trabecular meshwork region) and \({\mathcal{B}_{s2}}\) is the rest of the eye globe surface now considered impermeable, as shown in Figure 2 (case B). Using spherical coordinates, \({\mathcal{B}_{s1}}\) is the region defined by θ

_{1}≤ θ ≤ θ

_{2}and 0 ≤ ϕ < 2π, with θ and ϕ being the polar and azimuthal angular coordinates, respectively. We impose

*A*

_{s1}is the area of the outlet region \({\mathcal{B}_{s1}}\), which is equal to \(2\pi R_o^2( {\cos {\theta _2} - \cos {\theta _1}} )\). For modeling solute transport, we impose zero flux across the sclera on \({\mathcal{B}_{s2}}\), as for case A, and zero diffusive flux on \({\mathcal{B}_{s1}}\).

^{7}elements/m

^{2}. We conducted mesh invariance tests using eight meshes, increasing the number of elements by ∼10% in each mesh and sampling the computed pressure at five fixed locations. The mesh used for production runs showed a variation of pressure of less than 1% with respect to the most refined one.

_{2}→ θ

_{1}, the condition (16) reads

_{1}) denotes the Dirac delta function, centered at θ

_{1}, and \(Q( t ) = - {\dot V_o}( t )\) is the fluid outflow rate from the globe. The above condition satisfies the requirement that the total outflow rate is

*Q*(

*t*). At the inner spherical surface,

*r*=

*R*, we still use the Dirichlet condition on the pressure (15).

_{i}^{34}in the form

*P*is the Legendre polynomial of degree

_{l}*l*and we have not included the modified Legendre polynomials, since they are singular at the poles, which is not physical in our case. The coefficients α

_{l}and β

_{l}can be determined by imposing the boundary conditions at

*R*and

_{i}*R*and taking advantage of the orthogonality properties of the Legendre polynomials to obtain

_{o}_{1}shown in Figure 2 has been estimated using the maximum depth

*H*and maximum diameter

*L*of the anterior chamber and the radius of the eye \({\hat R_o}\) (Fig. 2), according to the following relationship:

*P*

_{epv}= 8 mm Hg.

^{28}

^{–}

^{30}The aqueous humor flow rate minus the pressure insensitive outflow rate is taken equal to 2.2 µL/min.

^{38}

^{,}

^{39}Based on these values, we obtain \(\mathcal{R} = 3.18\;\)mm Hg min µL

^{−1}. In Goel's work,

^{40}an estimated value for the resistance of the conventional aqueous drainage tissues ranges between 3 and 4 mm Hg min µL

^{−1}, which is consistent with our choice.

*k*is Boltzmann's constant (1.38 × 10

_{B}^{−23}J/K),

*T*is the absolute temperature (310 K), and

*a*denotes the hydrodynamic radius of the diffusing species. For bevacizumab, we have

*a*= 4.58 × 10

^{−9}m,

^{41}which yields

*D*= 7 × 10

^{−11}m

^{2}s

^{−1}, in line with the value derived by Hutton-Smith et al.

^{42}The question then arises as to the appropriate value of

*D*in the vitreous, where some degree of diffusive hindering may occur. The major components of the vitreous humor are collagen and hyaluronan, and we here estimate the pore sizes in the vitreous from knowledge of their concentrations. We first consider collagen, whose concentration in the bovine vitreous is 1.1 × 10

^{−2}weight %,

^{43}which corresponds to a collagen solid fraction of ϕ = 8 × 10

^{−5}, considering a density of collagen fiber of ρ

_{c}= 1420 kg/m

^{3}.

^{44}Idealizing collagen fibrils in the vitreous as uniformly distributed cylinders of radius

*a*= 12.5 nm,

_{c}^{45}the solid fraction can be expressed as \(\phi = {\rm{\pi }}{( {\frac{{{a_c}}}{b}} )^2}\), where

*b*is the side of a “squared unit cell” around each collagen fiber,

^{46}which is equal to half the characteristic interfiber spacing. Based on this approach, we compute a characteristic interfiber spacing of 2.5 × 10

^{−6}m, which is far larger than the hydrodynamic radius of bevacizumab, from which we conclude that collagen in the vitreous does not appreciably hinder the diffusion of bevacizumab. A similar calculation can be repeated for hyaluronan, using a concentration of 2 × 10

^{−2}weight % (ϕ = 6.1 × 10

^{−5}, with hyaluronan density ρ

_{h}≈ 1800 kg/m

^{3}) and fiber radius of 0.5 nm, yielding a characteristic interfiber spacing of 1.1 × 10

^{−7}m, which again is much larger than the hydrodynamic radius of bevacizumab. We therefore expect that, to a first approximation, diffusive hindrance due to the vitreous is modest, and we therefore use the free solution value for the diffusion coefficient of bevacizumab in our calculations, namely,

*D*= 7 × 10

^{−11}m

^{2}s

^{−1}. All the parameters used in this work are summarized in Table.

*| of the fluid velocity relative to the vitreous. Both plots are at the initial time. As expected, the pressure decreases from the inner bolus (where we impose the pressure*

**q***P*computed from the lumped parameter model) to the region where the fluid exits the domain via the trabecular meshwork, \({\mathcal{B}_{s1}}\). We note that pressure variations within the vitreous are small compared to

*P*, which justifies our choice of imposing

*P*at the inner boundary. The isobars, reported in the plot, are orthogonal to the boundary in the region \({\mathcal{B}_{s2}}\), which is impermeable to fluid. Relatively strong, localized pressure gradients exist close to the outer boundary, at the region \({\mathcal{B}_{s1}}\).

*t*> 0), the pressure imposed at the margin of the fluid bolus,

*P*(

*t*), asymptotically decreases to the physiologic pressure of 15 mm Hg. At all times, the spatial distribution of the pressure remains qualitatively very similar to that shown in the figure, and the spatial pressure drop across the vitreous remains small compared to IOP.

*is depicted by color contours in Figure 4c, overlain by the streamlines of*

**v***, again at the initial time. It is interesting to notice that fluid moves from the posterior to the anterior region of the vitreous chamber and, by doing so, crosses the fluid bolus. In other words, fluid enters the inner sphere from the posterior region and exits it from its anterior side. Fluid velocity generated in the vitreous chamber at the initial time reaches a maximum value of approximately 6 µm/s at the outlet and 0.8 µm/s in the region of the injection. This flow can potentially affect the transport and distribution of the injected drug since the velocity*

**v***appears in the advection-diffusion Equation (10).*

**v***are not orthogonal any more to the boundary of the domain since there is fluid leakage out of it. The main conclusions drawn commenting Figure 4 relative to case B also hold in this case.*

**q**^{38}although this fraction is less in humans. Unfortunately, in humans, only indirect estimates of unconventional outflow are available, which provide quite sparse data (see Table 4 of Johnson et al.

^{47}).

^{3}

^{–}

^{5}

^{,}

^{11}and this motivates our interest in such a flow and its potential role on drug transport. Indeed, the Pèclet number associated with this flow,

*Pe*=

*UL*/

*D*, is approximately 100 at the beginning of the ocular deflation phase (here we use a characteristic velocity

*U*taken close to the fluid bolus and the length scale

*L*equal to the radius of the eye). This confirms the hypothesis that advection dominates drug transport at the initial times after injection. However, our results show that, overall, this flow has very little effect on drug delivery to the retina, as is clearly demonstrated by Figure 4d, which shows that advection is not capable of transporting the drug far away from the injection site. This is because the deflation phase has a duration that is too short for advection to effectively contribute to drug transport.

^{38}Thus, the pressure decay computed in the Zero-Dimensional Model of Eye Globe Deflation section seems to be in contradiction with our assumptions in case A, where outflow largely occurs by uptake into the choroid and sclera (i.e., by pressure-insensitive unconventional outflow). However, Johnson et al.

^{47}highlight the fact that the unconventional drainage might to some extent depend on IOP. This partially justifies our choice of considering case A. In reality, the drug transport in the eye likely lies between the predictions of cases A and B, and this situation has also been considered (see Fig. 5). The limiting cases A and B are, however, of particular interest as they cover the most extreme conditions, and in both cases, the conclusion is that fluid motion due to globe deflation does not appreciably affect drug transport.

^{48}in which the authors injected tritiated water into the vitreous of living rabbits and sampled vortex veins and aqueous humor in the anterior chamber to assess whether outflow occurred through the choroid or the conventional pathway. Interestingly, they found that the vast majority of injectate was recovered in the vortex veins, implying that outflow through the choroid was dominant. Of course, we note that the transport of water considered by Moseley et al.

^{48}differs from the transport of a large molecule like bevacizumab, since the Pèclet number relevant to water transport is much smaller (Pe ≈ 2) than for bevacizumab transport (Pe ≈ 98), that is, diffusive transport of water is much more significant than diffusive transport of drug.

*k*in Equation (8) would be replaced by a permeability tensor. This, however, would require significant speculation about the values of this tensor for which experimental data are sparse or entirely lacking. Moreover, our simulations show the advective drug transport, which is related to water flow, is largely irrelevant, and this conclusion would not be modified by inclusion of the anisotropy of vitreous permeability.

^{49}These authors experimentally measured the diffusion of bevacizumab in the rabbit vitreous, obtaining a fitted diffusivity of

*D*= 12 ± 6 × 10

^{−11}m

^{2}s

^{−1}, slightly higher than our value of

*D*= 7 × 10

^{−11}m

^{2}s

^{−1}. Zhang et al.

^{49}did not account for convection in their data-fitting process, and thus their diffusivity may be a slight overestimate of

*D*. In any case, it appears that the value of

*D*that we utilized is reasonable.

^{9}

^{,}

^{12}

^{–}

^{14}

**A. Ruffini**, None;

**A. Casalucci**, None;

**C. Cara**, None;

**C.R. Ethier**, None;

**R. Repetto**, None

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