In Vivo Experimental and Analytical Studies for Bevacizumab Diffusion Coefficient Measurement in the Rabbit Vitreous Humor

Abstract

In order to measure the effective diffusion coefficient $D$ of Bevacizumab (Avastin, Genentech) in the vitreous humor, a new technique is developed based on the “contour method” and in vivo optical coherence tomography measurements. After injection of Bevacizumab-fluorescein conjugated compound solution into the rabbit eye, the contours of drug concentration distribution at the subsurface of injection were tracked over time. The 2D contours were extrapolated to 3D contours using reasonable assumptions and a numerically integrated analytical model was developed for the theoretical contours for the irregularly shaped drug distribution in the experimental result. By floating the diffusion coefficient, different theoretical contours were constructed and the least-squares best fit to the experimental contours was performed at each time point to get the best fit solution. The approach generated consistent diffusion coefficient values based on the experiments on four rabbit eyes over a period of 3 h each, which gave $D=1.2\pm 0.6\times {10}^{-6}\hspace{0.17em}{\mathrm{cm}}^2/\mathrm{s}$, and the corresponding theoretical contours matched well with the experimental contours. The quantitative measurement of concentration using optical coherence tomography and fluorescein labeling gives a new approach for the “noncontact” in vivo drug distribution measurement within vitreous.

1 Introduction

Bevacizumab (Avastin, Genentech) is a recombinant monoclonal antibody that inhibits human vascular endothelial growth factor and it has been administered intravitreally in vascular endothelial growth factor-mediated diseases such as wet age-related macular degeneration and proliferative diabetic retinopathy, which are among the leading causes of visual impairment in older people in developed countries [1–3]. In clinical practice, this drug is injected intravitreally which encounters more concerns, and thus puts the limitation on the dose and frequency of usage. The transscleral drug delivery, although less invasive compared to intravitreal injection method, is not efficient enough for Bevacizumab considering the low partition coefficient, low permeability coefficient and long transport lag time because of the large molecular size and molecular weight of 149 kDa [4]. Another approach is the use of implants to setup sustained-release drug delivery system [5,6]. With intravitreal injection and implant-release being applied and a narrow concentration range in which it is effective, it is critical to know the concentration distribution after intravitreal injection. It is imperative therefore that diffusion coefficient determination is important parameter for modeling and analysis of drug delivery process.

For in vivo system, both diffusion and convection processes are present. Even though they are driven by separate mechanisms [7–11], it is difficult to isolate diffusion from convection. Therefore, the focus of this paper is on the effective diffusion coefficient, based on both diffusion and convection of Bevacizumab in the vitreous humor. Accurate measurement of the diffusion coefficient is one of the necessary parameters needed for predictive pharmacokinetic mathematical model that can be applied to various ocular drug candidates.

Among the diffusion coefficient measurement techniques developed over the last 30-40 years (see [12] for a recent review), imaging methods have the advantage of providing a snapshot of concentration distribution profile without extracting the vitreous. The method proposed in this work focuses on the concentration distribution in the whole volume as opposed to measurements with a cut slice of the vitreous (for example, Ohtori and Tojo [13] and Gisladottir et al. [14] with the disk method using the diffusion chamber). In this regard, magnetic resonance imaging (MRI) and optical coherence tomography (OCT) are two commonly used techniques [12] with MRI being particularly useful for providing concentration over numerous slices of the subject eye while with fluorescence imaging, because of the limitation of our equipment, only produces a two-dimensional cross-sectional image [15]. Previous work on diffusion coefficient measurements have been carried out for gadolinium-based contrast agents, such as Gd-DTPA, Prohance, Gd-Alb and IgG [16–18]. This method is based on solving the inverse problem. Experimentally, this process starts with intravitreal injection of a gadolinium-based surrogate, allowing it to spread by diffusion and tracking the concentrations contours by MRI for different time points. In parallel, an analytical model was developed for theoretical construct of gadolinium surrogate diffusion. By comparing experimental and analytical results, both consisting of spheroidal contours, the value of $D$ was obtained through minimizing the deviation between theory and experiment for different time points while floating the unknown diffusion coefficient $D$.

Since gadolinium conjugation of Bevacizumab has been challenging to accomplish, our approach for in vivo measurements of the diffusion coefficient is limited to OCT using fluorescein-conjugated Bevacizumab. Based on low-coherence interferometry, optical coherence tomography is capable of scanning a two-dimensional section of the eye structure given the signals outside this plane are canceled. Once fluorescein-labeled Bevacizumab is injected, the emitted fluorescent light from the molecules at each pixel point is measured with a light sensitive detector. Since the fluorescent light intensity in our range of concentration is linear with the concentration, the distribution of labeled Bevacizumab in the particular focal plane can be measured and tracked with time. Based on the same idea, two sets of results, consisting of concentration contours, while planar, can be compared and the diffusion coefficient $D$ is obtained by minimizing the deviation between theory and experiment.

2 Methods and Approach

2.1 Experiment.

Experiments were carried out on six in vivo rabbit eyes. The rabbits were anesthetized with a mixture of 35-50 mg/kg ketamine and 5–10 mg/kg xylazine, Intramuscular injections, with a frequency as needed to maintain surgical anesthesia, using a 30–gauge needle, in accordance with animal welfare rules and regulations. By means of syringe pump, total volume of $20\hspace{0.17em}\mathrm{\mu }\mathrm{L}$, $6.86\times {10}^{-6}\mathrm{M}$ Bevacizumab was administered for a period of 10 min. The ocular characteristics of the rabbit eye, namely, transparency of the cornea, lens, and vitreous humor and physical capabilities of SPECTRALIS HRA + OCT (Heidelberg Engineering) such as low-coherence light, allowed capturing optical coherence tomography (OCT) images of the subsurface of injection at 15 min intervals for a total period of 3 h. The experiment setup are shown in Fig. 2. The illumination of fluorescence could light up the area with Bevacizumab and a linear relationship is expected between the signal intensity and concentration of Bevacizumab. The concentration distribution profile of Bevacizumab of the particular subsurface was then tracked throughout the whole experimental process.

Fig. 2.

Experiment setup

Fig. 1.

The absorbance of Bevacizumab-fluorescein (blue) illustrates the absorption contribution from Bevacizumab at ${\lambda }_{\max }=280\hspace{0.17em}\mathrm{nm}$ and fluorescein dye at ${\lambda }_{\max }=494\hspace{0.17em}\mathrm{nm}$. The absorbance of fluorescein (black) illustrates the contribution only from the dye.

Bevacizumab Conjugation Protocol.

For the protein labeling, Fluorescein-EX Protein Labeling Kit (Invitrogen, Cat #F10240), was used, and a brief description of the process is provided here. The final concentration of Bevacizumab in labeling reaction was 2 mg/mL. About 50 μL of 1 M bicarbonate was added to 0.5 mL of 2 mg/mL protein solution. Reaction vial containing dye was warmed to the room temperature before the addition of the protein. Then, reaction vial was incubated at room temperature for one hour with constant agitation. Purification column was packed with reagent C. Reaction solution was then added to the column and labeled protein was eluted with 1× phosphate buffered saline buffer. Pure labeled protein was collected into a clean vial and degree of the labeling was determined by using fluorimeter. With 1:10 dilution of the stock solution and the moles of protein per dye ratio, we determined one Bevacizumab molecules to be bound to five fluorescein molecules.

Absorbance Measurements.

The Bevacizumab-fluorescein conjugate compound was synthesized using the Fluorescein-EX Protein Labeling Kit (F10240) from Molecular Probes. The stock solution was then diluted 1:10 using phosphate buffered saline buffer as a solvent. The UV-visible spectra were recorded on a Hewlett-Packard 4853 diode array spectrometer.

The absorbance of the Bevacizumab-fluorescein conjugate was measured to obtain two parameters: the protein concentration $\left(M\right)$ and the fluorophore to protein ratio. The equations used to determine these two parameters are standard relationships for determining the protein concentration and degree of labeling (moles of dye per moles of protein), and are shown below for completeness:

$$\mathrm{Protein}\hspace{0.17em}\hspace{0.17em}\mathrm{concentration}\hspace{0.17em}\left(\mathrm{M}\right)=\frac{[A_{280}-(A_{494}\hspace{0.17em}\times \hspace{0.17em}0.20\left)\right]\hspace{0.17em}}{\mathrm{molecular}\hspace{0.17em}\hspace{0.17em}\mathrm{extinction}\hspace{0.17em}\hspace{0.17em}\mathrm{of}\hspace{0.17em}\hspace{0.17em}\mathrm{protein}}$$ (1)

$$\text{Moles}\hspace{0.17em}\hspace{0.17em}\text{of}\hspace{0.17em}\hspace{0.17em}\text{dye}\hspace{0.17em}\hspace{0.17em}\text{per}\hspace{0.17em}\hspace{0.17em}\text{mole}\hspace{0.17em}\hspace{0.17em}\text{of}\hspace{0.17em}\hspace{0.17em}\text{protein}=\frac{A_{494}\times \mathrm{dilution}\hspace{0.17em}\hspace{0.17em}\mathrm{factor}}{68,000\hspace{0.17em}\times \hspace{0.17em}\mathrm{protein}\hspace{0.17em}\hspace{0.17em}\mathrm{concentration}\hspace{0.17em}\left(\mathrm{M}\right)}$$ (2)

where A₂₈₀ and A₄₉₄ are the absorbance value at 280 nm and 494 nm, respectively, 0.20 is the correction factor for dye absorption at 280 nm and molar extinction coefficient for the conjugate is $68,000\mathrm{c}{\mathrm{m}}^{-1}{\mathrm{M}}^{-1}$. Using a molar extinction of $203,000\mathrm{c}{\mathrm{m}}^{-1}{\mathrm{M}}^{-1}$, the protein concentration was calculated to be $6.86\times {10}^{-6}\mathrm{M}$. The moles of dye per mole of protein value was calculated to be five moles of dye per mole of protein, where an optimal labeling can be achieved with 4-8 moles of fluorescein per mole of protein.

To obtain near-spherically symmetric distribution of Bevacizumab, the injection was carried out at a very low rate. Nevertheless, in in vivo rabbit experiments, the distribution of the injected bolus still came out somewhat distorted compared to intravitreally injected surrogates in ex vivo bovine eyes [16,17]. The injected bolus is small enough and thus vitreous boundaries (retina, lens, and other boundary tissues of eye) can be neglected and only the vitreous humor, treated as porous medium [19], is considered.

2.2 Modeling and Data Processing.

The model starts with the exact solution for an infinite medium of distribution of point source which is spherically symmetric.

$$c_{\mathrm{s}}\left(r,t\right)=\frac{M}{8{\left(\pi Dt\right)}^{\frac{3}{2}}}{e}^{-\frac{r^2}{4Dt}}$$ (3)

where $M$ is total amount of the drug injected, $r$ is the radial distance from the point of injection, $t$ is time, $D$ is diffusion coefficient, and $c_s\left(r,\hspace{0.17em}t\right)$ is the concentration. For a unit-mass point source at an arbitrary location ${\mathit{r}}^{\prime }=\left(x^{\prime },y^{\prime },z^{\prime }\right)$, and time $t^{\prime }$, the concentration distribution is effectively the Green's function

$$G\left(\mathit{r},{\mathit{r}}^{\prime },t,t^{\prime }\right)=\frac{e^{-\frac{{\left|\mathit{r}-{\mathit{r}}^{\prime }\right|}^2}{4D\left(t-t^{\prime }\right)}}}{{8\left[\pi D\left(t-t^{\prime }\right)\right]}^{\frac{3}{2}}}$$ (4)

It should be noted that the concentration is taken in units of fluorescence intensity since the relationship between concentration and intensity is taken to be linear. Furthermore, the diffusion process is also linear and therefore the concentration units can be scaled out. In other words, it is not necessary to convert intensity data to actual concentration values to obtain the diffusion coefficient. By considering the bolus as a continuous set of infinitesimal point sources, the point-source model works as the fundamental solution which is integrated numerically in order to obtain the distribution for irregular shaped initial distribution. In fact, the evolution of the distribution after every measured time point can be considered as a new starting point of known distribution. The concentration distribution can thus be expressed as

$$c\left(x,y,z,t\right)=\int \iint \frac{c_0\left(x^{\prime },y^{\prime },z^{\prime },t^{\prime }\right)}{{8\left[\pi D\left(t-t^{\prime }\right)\right]}^{\frac{3}{2}}}{e}^{-\frac{{\left(x-x^{\prime }\right)}^2+{\left(y-y^{\prime }\right)}^2+{\left(z-z^{\prime }\right)}^2}{4D\left(t-t^{\prime }\right)}}d{x}^{\prime }d{y}^{\prime }d{z}^{\prime },$$ (5)

where $\left(x\text{'},\hspace{0.17em}y\text{'},\hspace{0.17em}z\text{'}\right)$ is the position of the point source, $\left(x,\hspace{0.17em}y,\hspace{0.17em}z\right)$ is the position of the point studied,$\hspace{0.17em}{c}_0\left(x^{\prime },y^{\prime },z^{\prime },t^{\prime }\right)$ is the concentration at the new starting point $t^{\prime }$. Treating the initial injection as a distributed nonzero-radius bolus instead of a point source significantly increases the accuracy of the result as discussed in a previous paper [16].

2.2.1 Axial Symmetry.

In each experiment, as mentioned before, the drug distribution at every time point is treated as a new starting point for the subsequent time points. Thus, the distribution at a time point $t^{\prime }$ is fed as a continuous set of infinitesimal point sources that are numerically integrated as per Eq. (4) to predict the distribution at the time points that follow. In parallel, these distributions are also measured experimentally, for the sake of providing comparative data, to determine the unknown $D$. One of the obstacles with the current experiment is that the fluorescence imaging data are two-dimensional in the focal plane of the camera. Since only 2-D contours are obtained, while the actual dispersion is 3-D, a reasonable assumption of axial symmetry needs to be made to study the concentration distribution in the 3-D volume (see more discussion in Sec. 2.4.4 on “Data Fitting”). As a result, Eq. (5) becomes:

$$c\left(\rho ,y,t\right)=\frac{2\pi c_0\left({\rho }^{\prime },y^{\prime },t\right)}{8{\left(\pi D\left(t-t^{\prime }\right)\right)}^{\frac{3}{2}}}\int e^{-\frac{{\left(y-y^{\prime }\right)}^2+{\rho }^2}{4D\left(t-t^{\prime }\right)}}d{y}^{\prime }\int e^{-\frac{{{\rho }^{\prime }}^2}{4Dt}}{I}_0\left(\frac{\rho {\rho }^{\prime }}{2D\left(t-t^{\prime }\right)}\right){\rho }^{\prime }d{\rho }^{\prime }$$ (6)

where ${\rho }^{\prime }$ is the radius of the ring pattern source, $y^{\prime }$ is the distance of the source from the origin along the axis, $\rho$ is the radial distance of the inspection point from the axis, and $y$ is the distance of the inspection point from the origin along the axis. The $\left(\rho ,y\right)$ coordinate system in relation to the eye is shown in Fig. 3.

Fig. 3.

Coordinate system orientation

2.2.2 Nonzero-Radius Bolus, Infinite Vitreous Size (Without Wall Effect).

A major concern is that as the diffusion progresses with time, the walls of the eyeball, including vitreous and lens, could perhaps significantly influence the concentration distribution compared with an infinite medium model and numerical integration of the off-centered point source might make this influence even more prominent. In order to analyze the wall effect, a spherically-symmetric model with the injection of a spherical bolus at the center of a spherical shell is made and studied for different boundary conditions and approaches.

To begin with, the diffusion equation of the spherically symmetric model was used,

$$\frac{1}{D}\frac{\partial c}{\partial t}=\frac{{\partial }^{2}c}{{\partial r}^2}+\frac{2}{r}\frac{\partial c}{\partial r}$$ (7)

assuming nonpenetration boundary condition of the outer wall

$${\frac{\partial c}{\partial t}|}_{r=R}=0.$$ (8)

Initially injecting bolus of radius $r_0$ and concentration $c_0$ determined the initial condition

$$c\left(r,0\right)=\left\{\begin{array}{c}{c}_0,\hspace{0.17em}0\le r\le r_0,\\ 0,\hspace{0.17em}{r}_0<r\le R\end{array}\right.$$ (9)

The solution to Eq. (7) with given boundary/initial conditions (8) and ($9$) is:

$$c\left(r,t\right)=c_0\left[\frac{{r_0}^3}{R^3}+2\sum _{n=1}^{\mathrm{\infty }}\frac{\text{sin}\left({\lambda }_n{r}_0\right)-{\lambda }_n{r}_0\text{cos}\left({\lambda }_n{r}_0\right)}{{\lambda }_{n}R{\text{sin}}^2\left({\lambda }_{n}R\right)}{e}^{-D{{\lambda }_n}^{2}t}\frac{\text{sin}\left({\lambda }_{n}r\right)}{{\lambda }_{n}r}\right]$$ (10)

where ${\lambda }_n$ satisfies the transcendental equation

$${\lambda }_{n}R=\text{tan}\left({\lambda }_{n}R\right)$$ (11)

With $20\hspace{0.17em}\mathrm{\mu }\mathrm{m}$ volume of injection, $r_0$ is calculated to be $0.168\hspace{0.17em}\mathrm{cm}$. Considering the size of the rabbit eye, the spherical equivalent radius of the vitreous region assumed to be $R=7.5\hspace{0.17em}\text{mm}$.

Another case considered is the nonzero-radius bolus in an infinite medium. Two different approaches were studied for comparison. One is to solve for a similar model with a sink boundary condition ${\left.c\right|}_{r=R}=0$ instead of using Eq. (8) . By giving a value big enough to $R$, in this case 10 cm, the model is a good approximation of infinite medium. The concentration distribution is then given by:

$$\left(r,t\right)={2c}_0\sum _{n=1}^{\mathrm{\infty }}\left[\text{sin}\left(\frac{n\pi r_0}{R}\right)-\frac{n\pi r_0}{R}\text{cos}\left(\frac{n\pi r_0}{R}\right)\right]e^{-D{\left(\frac{n\pi }{R}\right)}^{2}t}\frac{\text{sin}\left(\frac{n\pi r}{R}\right)}{{\left(n\pi \right)}^2\left(\frac{r}{R}\right)}.$$ (12)

Alternately, one can solve a similar model with the same initial condition but a boundary condition at infinity, ${\left.c\right|}_{r\to \infty }=0$. The solution is [12,20]

$$c\left(r,t\right)=c_0\left\{\frac{1}{2}\left[\mathrm{erf}\left(\frac{r+a}{\sqrt{4Dt}}\right)-\mathrm{erf}\left(\frac{r-a}{\sqrt{4Dt}}\right)\right]-\sqrt{\frac{Dt}{\pi r^2}}\left[e^{-\frac{{\left(r-a\right)}^2}{4Dt}}-e^{-\frac{{\left(r+a\right)}^2}{4Dt}}\right]\right\}$$ (13)

Equivalently, in the limit of $R\to \infty$ in Eq. ($12$) leads to

$$c\left(r,t\right)=\frac{{2c}_0}{\pi }{\int }_{\lambda =0}^{\infty }\left[\frac{\text{sin}\left(\lambda r_0\right)}{{\lambda }^2}-\frac{r_0}{\lambda }\text{cos}\left(\lambda r_0\right)\right]e^{-D{\lambda }^{2}t}\frac{\text{sin}\left(\lambda r\right)}{r}$$ (14)

The other approach is integrating the point-source model in order to get the concentration distribution with nonzero-radius-bolus source without wall effect. This will undoubtedly lead to the above results in Eqs. (12), (13). To be more representative, the analysis starts with the 2D contour of size 50 by 50 pixels with $\Delta x=1.818\times {10}^{-2}\mathrm{c}\mathrm{m}$. The concentration can be calculated by solving Eq. (6) with $c_0\left({\rho }^{\prime },y^{\prime },t\right)$ satisfying Eq. (8). All three approaches use the diffusion coefficient $D=6\times {10}^{-7}{\mathrm{cm}}^2/\mathrm{s}$ as an estimate based on an initial analysis of the data using the model with nonzero-radius bolus in an infinite medium as described by Eq. (13). This analysis has illustrated that for the timescales involved in the present set of experiments, the wall effect can be neglected and integrated point-source approach which applies to infinite media is accurate enough for the current purpose.

The results given in Fig. 4 show a good match of the concentration distribution in the radial direction considering the three different approaches for the first 5 h after injection. By comparing the solution of Eqs. (10) and (13), it is shown that the boundary (wall effect) does not significantly influence the concentration distribution at early time ($\le 5\hspace{0.17em}\mathrm{h}$). By taking the solution given by Eq. (6) and numerically integrating the point-source model at a low resolution (50 by 50), it is found to be reliable for obtaining the concentration distribution of an axially symmetric model, and thus adequate for the analysis.

Fig. 4.

Concentration distribution of theoretically-based axisymmetric model. The plot corresponds to a bolus source of $20\hspace{0.17em}\mu \mathrm{m}$ volume at t = 0.5 h, 1 h and 5 h with $D=6\times {10}^{-7}{\mathrm{cm}}^2/\mathrm{s}$. : Eq. ($10$); : Eq. (13); : Eq. (6).

2.2.3 Processing of Experimental Data

2.2.3.1 2-D Gaussian Smoothing.

High-frequency noise is expected in the images obtained from OCT, which might change the value of deviation from least square fitting of 2D contours and thus introduce error to the diffusion coefficient obtained. Gaussian smoothing is a commonly used low pass filter in image processing for dealing with such noise. To test the effectiveness of applying such filter, a theoretical model with a point source in an infinite medium is created using Eq. (3). After applying Gaussian noise to the theoretical model, it becomes very close to the images captured by OCT as shown in Figs. 5(a) and 6(a).

Fig. 5.

Intensity image

Fig. 6.

Normalized contour

In order to examine the influence brought by the Gaussian filter, the concentration distribution based on Eq. (3) at time 2 h is used as the absolute solution. By adding Gaussian noise to the theoretically-based solution, the sum of square error (SSE) based on Eq. (15) compared to the absolute solution can be obtained.

$$\mathit{SSE}=\hspace{0.17em}\sum _{i=1}^N{\left(c-c_{\mathrm{ref}}\right)}^2$$ (15)

where $c$ is the concentration studied and $c_{\mathrm{ref}}$ is the corresponding reference result, where these are solutions with Gaussian noise and absolute solution, respectively. Since a concentration value is obtained for each discrete data point (or pixel) at a particular time point, a summation of square errors for each pixel is calculated to measure the difference between contours.

Then a Gaussian filter of different standard deviation $\sigma$ is applied to the solution with noise to smooth out the result. The sum of square error between Gaussian-smoothed solution and absolute solution is also calculated. A decrease of more than 70% of SSE is noticed for $\sigma$ within the range of 4 to 10 as shown in Fig. 7.

Fig. 7.

Decrease of sum of square error caused by Gaussian filter

Besides, considering that Gaussian smoothing is taking the weighted average for selected region, the same Gaussian filter is applied to the analytical result as well before comparing with experimental result. The effect of Gaussian smoothing on the calculation of diffusion coefficient is minimized by applying Gaussian smoothing to both experimental and analytical results.

2.2.3.2 Decreasing resolution.

In order to decrease the computation time while not introducing significant error, the resolution of the contour is decreased by extracting a value at every 5 points both horizontally and vertically as shown in Figs. 5 and 6(b)–6(c). This decreases the computational time to 1/625 of its original time. A test has been carried out based on the theoretical model from Eq. (3), at time t = 1 h. A 769 by 769 sized 2D contour of the subsurface of point source with $\Delta x=6.7\times {10}^{-3}\mathrm{c}\mathrm{m}$ is obtained and given a decrease of resolution by a factor of 5. The new 2D contour is then used to get the concentration distribution 15 min later based on Eq. (6). By comparing with the analytical solution at 1 h 15 min, it is found that only an average squared error of 0.3 is introduced considering the maximum value of the contour to be about 140. Decreasing the resolution is also helpful for lowering the influence of high frequency noise introduced by the OCT imaging.

2.2.3.3 Needle distortion.

A dark tunnel at the position of the needle is noticed in the intensity image obtained from OCT as shown in Fig. 5. The contour line of intensity, which is linearly related with concentration, is expected to be perpendicular to the boundary of the needle given the fact that it worked as a nonpenetrating wall, yet it is not quite like it from the contour shown in Fig. 6. A possible reason is light scattering. Besides, since only a 2-D distribution is known and the axis of symmetry is not always collinear with the needle, the existence of such blank tunnel will introduce more error to the analysis compared with simply neglecting it. Therefore, the approach of filling up the black tunnel is taken in order to obtain the best utility of the data. The result of filling is shown in Fig. 8. It should also be mentioned that in the experiments, the needle was left in place after injection so as to minimize the disturbance to the eye.

Fig. 8.

Comparison between raw data with needle-distorted data

2.2.4 Data Fitting.

For the experimental part, the concentration distribution of the subsurface of injection is obtained by focusing on the needle tip when adjusting the focus of OCT. A slight variation of the position and orientation is observed when dealing with the images from OCT. In order to get rid of the errors introduced, the weighted center based on concentration and the major axis of the contour is also calculated and aligned.

For the analytical part, since only a 2-D image is obtained for each time point, an assumption of axial symmetry is needed in order to acquire the 3-D distribution. In this case, because of the irregular shape of the 2-D contour, a varying-center approach is applied. The weighted center based on the concentration is obtained for each row of pixels as shown in Fig. 9. Based on the axis of symmetry for each row, the concentration distribution with a thin-plate ring pattern source can be applied by using Eq. (6) as shown in Fig. 10 thus the 3-D concentration distribution based on the whole 2-D image can be calculated by numerical integration. Undoubtedly, this introduces errors since we are making an assumption about an unknown 3D distribution and forcing an axisymmetric distribution about the varying “centerline.” However, this appears to be the best possible approach to extract the data given that the equipment is only capable of acquiring planar data within the focal plane. The ensuing approximation resulting from this assumption is duly noted.

Fig. 9.

Concentration contour with weighted center for each row of pixels. : after applying axial symmetry; : raw data; : weighted center

Fig. 10.

Rotation with respect to the weighted center to get the thin plate for each row

The golden section search has been used to determine coefficient $D$ through the minimization of the sum of square error (SSE) based on Eq. (15) within the region of $2\times {10}^{-7}\sim 3\times {10}^{-6}{\mathrm{cm}}^2/\mathrm{s}$ with the tolerance of ${10}^{-8}{\mathrm{cm}}^2/\mathrm{s}$, i.e.,

3 Results

The normalized and modified planar concentration distribution of Bevacizumab was obtained after reconstructing the OCT signal intensity data using the procedure described in the previous section under “Processing of experimental data.” The application of the algorithm resulted in close resemblance of the experimental data with the theoretical contours, and representative results are plotted in Fig. 11 where the concentration contours are given for one of the rabbit eyes for the time points at $t$ = 1 and 2 h. The irregular shape is well matched because of the usage of the distribution with irregular shape at a corresponding to the experimental data of the previous time point.

Fig. 11.

Concentration contour from experiment and analysis at t = 1 h and 2 h. : experiment; : analysis

The diffusion coefficient obtained by applying the least sum of squares error described in under “Data Fitting” section, for each bovine eye are listed in Table 1. Based on these results, the obtained diffusion coefficient $D$ is $1.2\pm 0.6\times {10}^{-6}{\mathrm{cm}}^2/\mathrm{s}$ where the error is estimated based on the standard deviation over the 4 samples at all time-points. In this process, since not all minimum SSE can be found within the region of $2\times {10}^{-7}\sim 3\times {10}^{-6}{\mathrm{cm}}^2/\mathrm{s}$, the $D$ returned outside the tolerance difference with respect to the boundary are removed, which takes out 10% of the whole solution sets.

Table 1.

Diffusion coefficient with standard deviation for different eye samples

Rabbit eye sample	Diffusion coefficient $D$ (${10}^{-7}{\mathrm{cm}}^2/\mathrm{s}$)	Standard deviation for $D$ (${{10}^{-7}\mathrm{c}\mathrm{m}}^2/\mathrm{s}$)
1	12.3	4.6
2	9.5	4.0
3	9.2	4.9
4	16.4	7.3
Average	12.0	6.2

The histogram of diffusion coefficient $D$ from experiment is shown in Fig. 12 and compared with the normal distribution based on the calculated mean and standard deviation of $D$ and the total number of $D$ values.

Fig. 12.

Diffusion coefficient histogram compared with normal distribution

4 Discussion

The relationship between diffusion coefficient and molecular weight for different drugs has been examined by Penkova et al. [12] to develop a comprehensive technique for$\hspace{0.17em}D\hspace{0.17em}$measurements. Earlier studies of different drug diffusion in the vitreous body of different species have been explored by other authors [13,14,20–22]. A power-law best fit gives the relationship as follows [12]:

$$D=3.8{\left(\mathrm{MW}\right)}^{-0.544}$$ (16)

where $D$ in in units of ${10}^{-6}{\mathrm{cm}}^2/\mathrm{s}$ while MW is in kDa. The diffusion coefficient values of some of the molecules of comparable size are given in Table 2 below

Table 2.

Diffusion coefficient of compounds with different molecular weight

Drug/Surrogate	Mol. Wt. (kDa)	Species	$D$(${10}^{-6}{\mathrm{cm}}^2/\mathrm{s}$)
GalbuminTM [19]	74	Rabbit (in vivo)	0.8 ± 0.3
GalbuminTM [17]	74	Bovine	0.227 ± 0.02
Gadolinium-immunoglobulin [20]	160	Bovine	0.213 ± 0.05

Based on Eq. (16), with a molecular weight of around 149 kDa, the diffusion coefficient of Bevacizumab is close to $2.5\times {10}^{-7}{\mathrm{cm}}^2/\mathrm{s}$. One notable point is that the relationship is obtained given a number of available data points within the molecular weight range 0.1-1.0 kDa while no available result in the 1-74 kDa region. For large molecular weight, only two more data points were given for ex vivo rabbit and bovine, respectively, at a 74 kDa molecular weight and one more data point a 160 kDa for ex vivo bovine eye as shown in Table 2. By looking at the two diffusion coefficient values for Galbumin, whose molecular weight is 74 kDa, Molokhia et al. [19] reported a substantially higher value based on experiment with in vivo rabbit eye (at 37 °C) compared with the result from Rattanakijsuntorn et al. [17] which are acquired with experiments with ex vivo bovine eye at 20 °C. According to the work done by Penkova et al. [20], the diffusion coefficient of Gadolinium-immunoglobulin, whose molecular weight is 160 kDa, is very close to the result for Galbumin in ex vivo bovine eyes [17]. Thus, given an approximate molecular weight of 149 kDa for Bevacizumab, its diffusion coefficient can be expected to be close the result obtained by Molokhia et al. [19], which is $8\pm 3\times {10}^{-7}{\mathrm{cm}}^2/\mathrm{s}$. Considering this, the result from the present work can be given considerable credence.

Fluorescence imaging has been widely used to obtain cross-sectional images for structure evaluation and also for detecting the existence of a certain fluorescein-conjugated drugs. This mode of imaging has expanded the variety of drugs for obtaining the concentration distribution, given that gadolinium-based labeling is limited to MRI, and not easily achievable for Bevacizumab. However, disadvantages with the current approach still exist in the sense that the developed 3D distribution is an extrapolated from the plane distribution and the middle cross section focal plane is hard to find without a distinct reference (needle in this paper). Undoubtedly, this represents a considerable approximation in the measurement and opens up room for improvement. Further possible errors can be introduced during data process described in Sec. 2.2.3 including Gaussian smoothing, decreasing resolution, and needle-distortion data adjustment but insignificant compared with the disadvantages mentioned above. Nevertheless, the current work represents the first diffusion coefficient measurement for Bevacizumab in the vitreous humor and the results will be highly useful in developing the comprehensive model for ocular fluid dynamics and transport in relation to retinal drug delivery.

Nomenclature

$A_{280},\hspace{0.17em}{A}_{494}$ =

absorbance value (a.u.)

$c$ =

concentration

$c_0$ =

initial concentration

$D$ =

diffusion coefficient (cm²/s)

$c$ =

concentration developed from Green's function

$G(\mathit{r},{\mathit{r}}^{\prime },t,t^{\prime })$ =

Green's function (Eq. (4))

$M$ =

total amount of the drug injected

$\hspace{0.17em}\mathit{r}=\left(x,y,z\right)$ =

location (cm)

$r,\hspace{0.17em}r\prime$ =

radial distance/ radial coordinate (cm)

$r_0$ =

radius of injected bolus (cm)

$R$ =

radius of the boundary (cm)

$t,t\prime$ =

time (s)

$x,x\prime$ =

horizontal coordinate (cm)

$y,y\prime$ =

vertical coordinate (cm)

$z,z\prime$ =

coordinate vertical to 2-D image (cm)

$\mathrm{MW}$ =

molecular weight (kDa)

$\mathrm{SSE}$ =

sum of squares error

$\mathrm{SD}$ =

standard deviation

${\lambda }_n,\hspace{0.17em}\lambda$ =

Fourier expansion parameters (cm^–1)

$\rho$ =

radial coordinate (cm)