Team:NCKU Tainan/Model
Overview
Experiment alone is not enough to prove the effectiveness of Eye kNOw. To build a bridge between experimental data from wet team and dry team results, we built three models that fully describe the whole working process of Eye kNOw, including the Contact Lens Deformation Model, NO Ocular Diffusion Model, and effectiveness of Eye kNOw.
By establishing these mathematical models with the combination of scientific theories and our team’s experimental data, we’re able to understand three critical part of Eye kNOw:
Quantitative result of contact lens deformation caused by IOP elevation.
Obtain the initial parameters needed for Eye kNOw design by calculation, and describe NO ocular diffusion with quantitative simulation.
Combining Model 1, Model 2, and results from wet team and dry team experiments, to precisely calculate the time needed for Eye kNOw to lower IOP.
Background
This year, We aim to provide a real-time treatment that can release ocular hypotensive agents (nitric oxide) according to patients’ IOP in a contact lens system—Eye kNOw. Eye kNOw contains a designed chamber filled with L-arginine, IPTG, and our engineered bacteria. Here are the three main principles we used to design Eye kNOw:
bNOS produced by our engineered bacteria turns L-arginine into NO.
Different concentrations of IPTG induces different production rates of bNOS.
IOP elevation causes corneal radius of curvature to change, leading to deformation of Eye kNOw.
Below are the workflows of Eye kNOw:
IOP elevates, causing the cornea's radius of curvature to change.
Chamber’s volume decreased due to the deformation of the cornea.
The semipermeable chamber only allows water and gas to pass. Therefore, the concentration of IPTG inside the chamber elevates.
Elevated IPTG leads to higher production rate of NOS by engineered bacteria.
The more NOS produced, the more L-arginine turned into NO.
More NO is released into the eyes, relaxing trabecular meshwork.
Finally, IOP will be lowered to normal level.
We establish three models to simulate the whole process above: contact lens deformation model and NO ocular diffusion model. Model 3 combines model 1, model 2 and experimental data, which precisely calculate the effectiveness of Eye kNOw.
Our goals
To build a model that describes the deformation of contact lens quantitatively.
To establish a model that simulates nitric oxide ocular diffusion system.
To precisely calculate the effectiveness of Eye kNOw by taking both experimental data and model theories into consideration.
Model 1: Deformation of Contact Lens
This model is used to describe the deformation of the contact lens under varying intraocular pressure (IOP) with quantitative estimation. Research suggested that the radius of cornea curvature increases linearly with the increment of IOP[1]. However, the pressure exerted on the contact lens while being worn is not fully studied, not to mention the volume difference of contact lens with respect to different IOP. Therefore, to build this model from scratch, we first made the following assumptions:
The contact lens fits on the cornea perfectly no matter it deforms or not.
The contact lens remains in the shape of a sphere during the whole process of deformation.
Deformation of the contact lens is not changing with time, which is a static model.
Table 1A shows the parameters of cornea that will be used in our model.
| Description | Value | |
|---|---|---|
| ECornea | Elastic modulus of the cornea | 15.3 mPa |
| tCornea | Thickness of the cornea | 0.666 mm |
| RCornea_0 | Corneal radius of curvature | 8.45 mm |
Considering the force equilibrium equation between the pressure of fluid and the membrane (cornea) in the section, as you can see in the figure below, we would be able to obtain equation (1)
According to Hooke’s Law, we have (2)
Under proportional limit, the radius difference - ΔR, is given by (3)
From (1), (2), (3), we obtain (4)
For a thick membrane, which means the thickness of the membrane is too large to be neglected in comparison to the radius. Thus, we have a constant term that is related to the poisson ratio, ν, in the equation (5). [1] [2]
E: Elastic modulus of the membrane (cornea of porcine in this case).
t: Thickness of the membrane (cornea of porcine ).
Notice that the p in (4) and (5) represents the pressure difference with respect to the initial pressure (IOP difference) instead of the pressure itself. We adopted the initial IOP to be 10 mmHg.
By equation (5), we predict the relation between IOP and the radius difference, or the strain, is linear.(See Proof of concept: step 1)
The volume of the ring-like compartment in Eye kNOw is calculated by the same method of the volume of revolution in calculus. Volume of compartment equals 4.4 cubic millimeters.
The volume difference is given by the elongation of the part above the compartment in meridional direction, which is given by the formula list below:
while
and
By doing so, the ratio of volume difference is approximated to be 1% when the IOP has an increment of 1 mmHg.
Model 2: Nitric Oxide Diffusion Model
Overview
Eye is a complicated organ with various tissues. In other ocular delivery models, they consider the whole eye as one compartment since their drugs are stable and large enough that the difference of tissues can be ignored. However, we use NO as our drug, which is small and unstable. Thus, we need to set up a new model to simulate NO ocular delivery system.
Background
Eye structure can be divided into three main compartments: cornea, anterior chamber, and posterior chamber. Target site of NO is trabecular meshwork, which is located at the posterior angle of anterior chamber. We simplify the eye structures into one dimension as below:
Since the tissue characteristics of the three compartments varies a lot, we need to calculate them separately by different partial differentiation equations (PDEs).
Model
First, we calculated the initial state of NO distribution in the eye before the production of NO by Eye kNOw. After ensuring the initial state, we can calculate the NO distribution after the production of NO by Eye kNOw, which is a formula of position and time. Before all these things, we should set up the PDEs that can describe the distribution of NO. Refer table below for abbreviations and variables:
1. Setting up PDEs:
PDEs are used to describe the factors that will affect the concentration of NO. Factors are listed below:
(1) Degradation
The two main factors that will cause degradation of nitric oxide are Hemoglobin and oxygen. Hemoglobin doesn’t exist in the eye system, thus we only need to consider oxygen to calculate the amount of nitric oxide degradation.
According to papers, the formula can be written as below:
(2) Diffusion
By Fick's second law, the diffusion formula can be written as below:
(3) Endogenous production
Since aqueous humor has an initial concentration of NO around 53.4uM, there should exist endogenous NO production. There are three different types of NOS in human eyes: e-NOS, n-NOS, and i-NOS.
e-NOS and n-NOS work constantly, which are expressed on iris, cornea, vascular endothelium, etc. i-NOS only works when inflammation happens, which is not a common symptom for glaucoma, thus we can ignore it. We assume that these NOS produce NO in a constant rate, so the formula can be written as below:
(4) Convection
There’s a flow of aqueous humor in the anterior chamber, which is 2.4 uL/min. Convection will affect the distribution of nitric oxide, which can be written as below:
Combining the factors listed above, the PDE for describing the distribution of NO can be written as below:
To make the calculation easier, we need to simplify Equation 1 into homogeneous form.
First, since the concentration of oxygen can be maintained by oxygen in the air( oxygen in the air can dissolve and penetrate into our eyes), we can set k[Oxygen] as a constant.
Second, we want to simplify C^2 into C. In order to prevent over-estimation of nitric oxide concentration, we change C^2 into C(max)*C, where C(max) is the max concentration of nitric oxide in the compartment. Since C(max)*C is always larger than (or same as) C^2, we can make sure that we won’t over-estimate the concentration of nitric oxide. Since C(max) is a constant, we can set constant B = k[Oxygen]C(max).
So, the modified form of PDE will be:
2. Parameters:
Parameters used are listed below. We collected them from different papers.
We're still lacking some of the parameters. We can estimate them by solving the steady state of NO distribution.
3. Calculating steady state:
We start from the contact lens compartment. When in steady state, concentration of NO won't change by time. Thus, Equation 2.6 can be written as below:
Where subscript 1 stands for the parameters and variables of the contact lens compartment. By solving this equation, we can get the solution of C1 in steady state:
We use the approximation:
Thus turning equation 2.8 into linear form:
Since C1(0)=0 (one of the boundary conditions), the final form of C1 in steady state can be written as below:
By the same method, we can simplify the concentration of NO in cornea and aqueous humor compartments as a linear formula of position. To simplify the formulas, we set the outer surface of every compartment as the starting position of that compartment. For example, for cornea compartment, the interface of cornea and contact lens is considered the starting position, namely x=0.
As for the boundary conditions, the interfaces should have only one concentration. For example, the formula of C1 at x=0.02 should have the same value as C2 at x=0. Combining these conditions, we can get the function of C2 and C3:
Now we need to solve the slope of NO concentration, namely a1, a2, a3. Here, we use Fick’s first law as boundary conditions, that is at the interfaces, input of NO from one compartment should be the same as output of another compartment. For example, at the interface of cornea and contact lens, input of NO to contact lens should be the same as output of NO from cornea.
The input of NO to contact lens can be written as below, according to Fick’s first law:
The output of NO from cornea can be written as below:
Applying the solution we’d found before, we can get the relationship of slopes of each compartment:
Therefore, we can draw the graph of NO concentration in steady state:
Red line represents the NO concentration in contact lens, blue one represents cornea compartment, and green one represents aqueous humor compartment.
Now, there are just B and Pro not determined yet. Recall that B = k[Oxygen]C(max), we can calculator them. As for Pro, we can calculate each compartment’s total input and output, they should be the same since we’re dealing with steady state. After some calculations, the table can be filled completely:
4. Calculating NO concentration after production of NO by Eye kNOw (non-steady state):
After confirming all the parameters, we went on to calculate the NO concentration of each compartment when Eye kNOw senses the change of IOP. We assume that the production rate of Eye kNOw raised j% compared with steady state.
Start from the contact lens compartment. Below are the PDE, initial conditions, and boundary conditions:
After taking Laplace transform and inverse Laplace transform, we obtained the solutions for each compartment:
Notice that there is an undefined parameter r in our solutions. Parameter r is correlated with the interfaces’ properties in our eyes, namely the interface of cornea and contact lens, and the interface of cornea and aqueous humor. Since we cannot do experiments on living animals, we cannot obtain the value of r. In fact, the value of r varies individually, so we just pick some of the r to demonstrate the result.
We use MATLAB to plot the final graph of NO distribution in our eyes according to time:
Model 3: Effectiveness of Eye kNOw
This model is a combination of model 1 and model 2, aiming to calculate the time needed from IOP elevation to NO reaching minimum effective concentration
We assume that contact lens deformation happens immediately after IOP elevation, and [IPTG] elevation happens immediately after contact lens deformation, too. Therefore, the only two steps we should take in concern are how fast will NOS be produced after [IPTG] elevation, and how fast will [NO] at trabecular meshwork be raised to minimum effective concentration (MEC). We’ll discuss them one by one.
First, we calculate the speed of NOS production after [IPTG] elevation. By wet team’s results (NO kinetics experiment and IPTG induction experiment), we can calculate the NO production rate according to [IPTG] and time:
Notice that the unit of equation 3.1 is nmol/hr, and the reaction volume is 60uL. Since the volume of contact lens’ chamber is 4.4uL, the production rate in Eye kNOw with certain concentration of IPTG can be written as below:
Notice that the production rate depends on not only the efficiency of NOS, but also the concentration of NOS in the solution. We set IOP = 15mmHg, [IPTG] = 0.1mM as initial conditions. By model 2, the initial production rate should be 7umol/s to maintain steady state. Therefore, by elevating bacteria concentration in Eye kNOw, we can rewrite equation as below:
Since in initial state, NOS has already been made for a long time, so we can neglect the time-related term in equation 3.2. Assume that patient’s IOP raised i mmHg, we can calculate the production rate of NO by Eye kNOw after IPTG induction as below:
Second, we calculate the time needed to reach MEC of NO at trabecular meshwork. We use MATLAB to plot out the concentration change of NO at trabecular meshwork according to time after induction. Below are the results:
We can obtain that time needed for Eye kNOw to raise trabecular meshwork’s NO concentration up to MEC is very short, which theoretically proves that Eye kNOw can treat glaucoma with high efficiency and accuracy.
References
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