Team:NCKU Tainan/Model
Overview
There are three critical parts in our project that crucially affect our final design of Eye kNOw: IOP-elevation induced contact lens deformation, NO diffusion efficiency in ocular system, and seeking for higher effectiveness. Experiment alone is not enough to provide the answers. To build a bridge between experimental data and scientific theories, we built three models that fully describe the whole working process of Eye kNOw:
Contact Lens Deformation Model: Provide quantitative result of contact lens deformation caused by IOP elevation.
NO Ocular Diffusion Model: Obtain the initial parameters needed for Eye kNOw design, and describe NO ocular diffusion with quantitative simulation.
NO Ocular Diffusion Model: Utilize model 1, model 2 and experimental data to precisely calculate the effectiveness of Eye kNOw.
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:
NOS produced by our engineered bacteria turns L-arginine into NO.
Different concentrations of IPTG induces different production rates of NOS.
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 |
First, we need to obtain the relationship between IOP and radius Eye kNOw. Considering the force equilibrium equation between the pressure of fluid and the membrane (cornea) in the section, we obtain Equation 1.1:
According to Hooke’s Law, we have Equation 1.2:
Under proportional limit, the radius difference (ΔR), is given by Equation 1.3:
From the equations listed above, we obtain Equation 1.4:
Notice that the p in Equation 1.4 and Equation 1.5 represents the pressure difference with respect to the initial pressure instead of the pressure itself. We assumed the initial IOP to be 10 mmHg. By Equation 1.5, we predict the relation between IOP and the radius difference, or the strain, is linear, which fits the results of digital image correlation experiment. (See Proof of concept: step 1 )
Second, we calculate the relationship between IOP and volume difference of the chamber in Eye kNOw. Table 1B lists all the parameters of Eye kNOw used in the following calculation.
| Description | Value | |
|---|---|---|
| REye_kNOw_0 | Inner radius of curvature | 8.2 mm |
| EEye_kNOw | Elastic modulus of Eye kNOw | 3．106 dyne/mm2 |
| d | Diameter of Eye kNOw | 14 mm |
| d1 | Inner diameters of the chamber | 10 mm |
| d2 | Outer diameters of the chamber | 12 mm |
| tEye_kNOw | Thickness of Eye kNOw | 2．10-4 mm |
The volume of the chamber in Eye kNOw , which is a ring-like compartment, is calculated by the same method of the volume of revolution in calculus. After calculation, the volume of the chamber equals 4.4 cubic millimeters initially.
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
The ratio of volume difference is given by Equation 1.9:
After calculation with the parameters listed above, the result is approximately 1% when the IOP has an increment of 1 mmHg.
Model 2: Nitric Oxide Diffusion Model
Overview
Eye is a complicated organ consisting of various tissues with different physical properties. In most of the ocular delivery models, they consider the whole eye as one compartment since their drugs are stable and large enough that the difference of tissue characteristics 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.
This model divides the eye system into three different compartment according to the tissues differences, and provide quantitative NO concentration profiles dynamically. This model can be applied on ocular drug delivery system of other small and unstable drugs.
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 simulate the behavior of NO in them separately, by different partial differential equations (PDEs).
Model
First, we establish the initial state of NO distribution in the eye before Eye kNOw being triggered. Second, we build the dynamic NO concentration profile according to time and distance after Eye kNOw activation. We use several partial differential equations(PDEs) to describe the behavior of NO. Table 2A lists all the abbreviations used in PDEs.
| Full Name | Unit | |
|---|---|---|
| C | Concentration of NO | M |
| x | distance (from outside of the eye to inside) | cm |
| t | time (t=0 resembles the time when Eye kNOw start to produce NO) | s |
| D | Diffusion coefficient | cm2/s |
| R | Flow rate of liquid | L/s |
| k | Degradation coefficient of NO | M-2/s |
| Pro | Production rate of NO in the compartment | mol/s |
| V | Volume of the compartment | L |
1. Setting up PDEs:
PDEs take the factors that may affect the concentration of NO into consideration to describe the behavior of NO. Factors are listed below:
(1) Degradation
The two main factors that will cause degradation of nitric oxide in the human body are hemoglobin and oxygen. Hemoglobin doesn’t exist in the eye system, thus we only need to consider oxygen as the factor causing NO degradation.
According to current studies, the formula can be written as below:
(2) Diffusion
By Fick's second law, the diffusion of chemicals can be described as below:
(3) Endogenous production
Glaucoma patients’ aqueous humor has an initial average concentration of NO around 53.4uM[7], which is maintained by endogenous NO produced by NO synthase (NOS). There are three different types of NOS in human eyes: e-NOS, n-NOS, and i-NOS[7].
e-NOS and n-NOS work constantly, which are expressed on iris, cornea, vascular endothelium, etc. i-NOS only works when ocular 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[8]. 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:
Equation 2.5 is a heterogeneous PDE, which is very hard to solve. We need to simplify Equation 2.5 into homogeneous form for further calculation.
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 the C2 term into C. In order to prevent over-estimation of nitric oxide concentration, we change C2 into Cmax．C, where Cmax is the max concentration of nitric oxide in the compartment. Since Cmax．C is always larger than (or same as) C2, we can make sure that we won’t over-estimate the concentration of nitric oxide. Since Cmax is a constant, we can consider constant B = k[Oxygen]Cmax.
Therefore, the modified form of PDE will be:
2. Obtaining parameters from current studies:
Parameters used are listed in table 2B. We collected them from current studies.
| Contact Lens | Cornea | Aqueous Humor | Unit | |
|---|---|---|---|---|
| D | 1.8．10-6[9] | 2.81．10-5[10] | 3．10-5[11] | cm2/s |
| B | ? | ? | ? | 1/s |
| R | 0 | 0 | 4．10-8[8] | L/s |
| V | 2.77．10-5 | 1.6．10-5[12] | 2.5．10-4[13] | L |
| Pro | ? | ? | ? | mol/s |
| Initial Average [NO] | ? | ? | 5.34．10-5[6] | M |
| Maximum Distance | 0.02 | 0.055[14] | 1.122[15][16] | cm |
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, NO concentration 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 Equation 2.7, 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 x. 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 obtained, we can get the relationship of slopes of each compartment:
Therefore, we can draw the steady-state-graph of NO distribution in the eye:
Now, there are just B and Pro not determined yet. Recall that B = k[Oxygen]Cmax, we can calculate 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 calculations, Table 2B can be filled completely as Table 2C.
| Contact Lens | Cornea | Aqueous Humor | Unit | |
|---|---|---|---|---|
| D | 1.8．10-6 | 2.81．10-5 | 3．10-5 | cm2/s |
| B | 2.84．10-2 | 1.03．10-2 | 1.566．10-2 | 1/s |
| R | 0 | 0 | 4．10-8 | L/s |
| V | 2.77．10-5 | 1.6．10-5 | 2.5．10-4 | L |
| Pro | 7．10-6 | 1.52．10-5 | 1.239．10-3 | mol/s |
| Initial Average [NO] | C1 = 933.7x | C2 = 59.818x + 18.674 | C3 = 56.063x + 21.964 | uM |
| Maximum Distance | 0.02 | 0.055 | 1.122 | cm |
4. Calculating NO concentration after production of NO by Eye kNOw (non-steady state):
After confirming all the parameters, we continue to calculate the NO concentration of each compartment when Eye kNOw is activated by elevation of IPTG concentration. We assume that the production rate of Eye kNOw raised j% after activation compared with steady state. Table 2D. Shows the PDEs, initial conditions, and boundary conditions:
| PDE | Initial Condition | Boundary Condition | |
|---|---|---|---|
| Contact Lens |
|
|
|
| Cornea |
|
|
|
| Aqueous Humor |
|
|
|
However, we found the PDEs hard to solve by current methods, including Fourier transform and separation of variables. After lots of effort, we finally solved the PDEs by means of double Laplace transform and inverse double Laplace transform.
We developed two MATLAB programs, one of them functioning as PDE solver utilizes double Laplace transform, and another one working as Laplace transform estimator for functions which cannot be transformed by current method. These programs are available on the internet with clear documentation, see modeling software for more information.
The dynamic NO concentration profiles can be described by three formulas as follow:
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 dynamic NO concentration profile of each compartment according to time:
Model 3: Effectiveness of Eye kNOw
As a new developing treatment of Glaucoma, it's important to show that the effectiveness of Eye kNOw is better than current treatment. Effectiveness reflects how a drug actually works in the patient's body, which can be evaluated by time and drug amount needed to cause therapeutic effect. Particularly, the minimum concentration of a drug that can cause therapeutic effect is called minimum effective concentration (MEC).
To prove that Eye kNOw can help treat glaucoma more effectively, this model combines model 1, model 2, and experimental data, aiming to precisely quantify Eye kNOw's MEC and the time needed to reach it. Noted that there's no MEC data of NO since current treatment utilizes NO donor instead of NO directly, but estimation can be made by adopting data from current NO prodrug. Here, we adopt data of Latanoprostene Bunod to estimate the MEC of NO, approximately 1.182．10-8M.[17][18][19]
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 concern are how fast will NOS be produced after [IPTG] elevates, and how fast will [NO] at trabecular meshwork reach 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 of the experiments is 60uL. Since the volume of Eye kNOw’s chamber is 4.4uL, the production rate in Eye kNOw 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 7．10-6 mol/s to maintain steady state. Therefore, by elevating bacteria concentration in Eye kNOw, the initial production rate of NOS in Eye kNOw can be written as Equation 3.3:
Since in initial state, NOS has already been made, so we can neglect the time-related term in Equation 3.2. Assume that patient’s IOP raises i mmHg, the production rate of NOS can be written as below:
Second, we calculate the time needed for Eye kNOw to raise the [NO] at the trabecular meshwork to MEC once the IOP is elevated. 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.
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