Team:TAS Taipei/Model
Model
Cataract prevention occurs over 20 – 50 years, so we cannot perform experiments on the long-term impact of adding GSR or CH25H. However, computational biology allows us to predict cataract development in the long-term. These models allow our team to understand the impact of adding GSR-loaded nanoparticles into the lens over a 50 year period, and to design a full treatment plan on how to prevent and treat cataracts with our project. Therefore, the results of our model are essential in developing a functional prototype.
For sake of clarity, we will discuss each model in detail with respect to prevention (using GSR) only. At the end, we explain these results to treatment. In addition, we include collapsibles for interested readers and judges, in order to fully document our modeling work (eg. assumptions, mathematics) while keeping the main page clear with basic points only.
Introduction
- How much GSR to maintain in the lens?
- How to maintain that amount of GSR using nanoparticles and eyedrops?
Focus of Models
Since our construct is not directly placed into the eyes, how our synthesized protein impacts the eye after it is separately transported into the lens is of greater importance. As a result, we create models with the intent on understanding how GSR and CH25H impacts the eye.
Relation to Experiments
We use measurements from the cataract lens experiment to create the first model, and extend the results with the second model to find GSR concentrations, which our experiments could not find.
We modelled nanoparticle degradation before performing actual experiments, to estimate the optimal amount of protein to load into the nanoparticles. After performing the experiments, we can directly compare them to see the efficiency of our drug-loading. Although we could not experimentally test an eyedrop prototype, model 4 extends the results of the nanoparticle degradation to better understand how to use eyedrops to deliver nanoparticles.
Relation to Overall Project
Although experiments could validate a prototype, it is through modeling that we find the optimal designs for these prototypes. By building a full calculator, we can envision how the results of this project can be applied clinically. A patient finds his or her LOCS score from the physician, and then enters it into the calculator, which returns a full treatment plan, using standardized cataract prevention and treatment eyedrops. Our nanoparticles are fully customizable, to allow physicians to alter the design to best fit a patient’s specific cataract conditions.
Model 1: Crystallin Damage
The amount of damage to crystallin by H2O2 determines the severity of a cataract. We relate the amount of crystallin damage to the corresponding rating on the LOCS scale, used by physicians to rate cataract severity. Our goal is to lower LOCS to below 2.5, the threshold for surgery. Through literature research as well as our own experimental data, we find the maximum allowable crystallin damage to prevent a LOCS 2.5 cataract from developing.
Measurement of Cataract Severity
There are four ways of measuring cataract severity, each used for a different purpose.
- Lens Optical Cataract Scale (LOCS): Physicians use this scale, from 0 – 6, to grade the severity of cataracts.
- Absorbance at 397.5 nm: This is the experimental method, used by our team in the lab (c.d.)
- Crystallin Damage: This is a chemical definition. We quantify cataract severity as a function of how much oxidizing agents there are, as well as how long crystalline is exposed to oxidizing agents.
Measurement of Cataract Severity
Numerous studies show how absorbance measurements can be converted to the LOC scale that physicians use. With the results of ________ and ________, we construct the first two columns in Table 2.
Absorbance Equivalence to Crystallin Damage: Experimental Data
We use experimental data from our team’s Cataract Lens Model (link). They induced an amount of crystallin damage, and measured the resulting absorbance. With this relation in Figure 2, we calculate the equivalent crystallin damage of each LOCS rating and absorbance.
Conclusion
To guarantee that surgery is not needed for 50 years, we need to limit crystallin damage to 0.9981 units. If crystallin damage goes above this threshold, then surgery is needed. This is the crystallin damage threshold for a LOCS 2.5 cataract.
Model 2: GSR Pathway
Now that we know how much GSR we need to limit crystallin damage to LOCS 2.5, we model the naturally occurring GSR Pathway in the lens of a human eye. For prevention (2A), we calculate the necessary GSR concentration to be maintained over 50 years so that the resulting cataract is below LOCS 2.5.
Chemical Kinetics Model: Differential Equations
By the Law of Mass Action, Michaelis-Menten Enzyme kinetics, Ping-pong mechanism, and the Law of Passive Diffusion, we build a system of 10 differential equations based on 6 chemical reactions. All parameters, constants, and initial conditions are based off literature data. Estimates made are also shown with assumptions and reasoning. The details are shown in the collapsible for interested readers.
Blackbox Approach: Testing GSR Impact
We vary the input, Initial GSR concentration, holding all other variables constant, and numerically solve for the amount of hydrogen peroxide over time. With this graph, we can find the amount of crystallin damage accumulated over 20 to 50 years if different levels of GSR is maintained.
From this graph, we can find the GSR concentration needed for the LOCS 2.5 threshold and the LOCS 1.0 threshold.
Crystallin Damage vs. GSR Level
According to literature data and our model, the naturally occurring GSR concentration is 10 uM. All curves show crystallin damage decreasing as GSR levels are increased, which supports both research and experimental data, and suggests that this prototype is effective in preventing crystallin damage. However, GSR levels need to be raised significantly, up to 40+ uM from the natural 10 uM of GSR in order to show long-term protection.
Table 3 shows the amount of GSR we need to maintain for 50 years in order to prevent a LOCS cataract of a certain severity. The row of interest is LOCS 2.5, the threshold for surgery. Notice that we say “maintain” the level of GSR. This level needs to be constant at all times for 50 years for full prevention. The delivery of GSR to maintain this level is discussed in Model 3.
Conclusion
We need to maintain (NOT add) 43.5 uM of GSR in the lens so that the crystallin damage recorded over 50 years is below the LOCS 2.5 threshold.
Model 3: Nanoparticle Protein Delivery
From Models 1-2, we have found the necessary concentration of GSR that needs to be maintained in the lens. Now we design nanoparticles that will maintain those amounts. To maximize delivery efficiency to the lens, we encapsulate GSR in chitosan nanoparticles. We create a model of how GSR concentration changes in the lens over time after GSR-loaded are delivered into the lens. From this model, we can find how nanoparticles release GSR at appropriate rates to control the amount of GSR in the lens, and find the best engineered design for nanoparticles.
Nanoparticle Diffusion
Nanoparticles are diffused to the lens through the cornea. Nanoparticle Diffusion can be modelled with the Noyes-Whitney equation, which takes into account the volume, surface area, and radius of nanoparticles, the temperature and viscosity of the medium, and the thickness of the diffusion layer (which is related to the type of nanoparticle). Most parameters are found through literature, while some are estimated (see collapsible).
Using these parameters, we build a differential equation model for the impact of a single dose of nanoparticles over time. We get two curves, concentration of GSR in the nanoparticles, and GSR release from nanoparticles, over time.
In addition, all protein degrade over time, so we apply an degradation factor to the amount of GSR diffused out of the nanoparticle into the aqueous humor before it enters the lens. using knowledge of turnover rates in the aqueous humor. After applying the degradation factor, the results of the single dose model are shown in Figure 1.
To generalize the model, instead of using absolute concentrations, we use relative concentration, with respect to the natural amount, or initial amount of GSR in the lens.
Single Dose: Change in GSR Concentration
We alter the concentration of GSR inside the nanoparticle, to understand the resulting change in GSR concentration in the aqueous humor over time. As shown in Figure 2, for the first few days, GSR diffusion is fast due to the high concentration gradient between the inside and outside of the nanoparticle. Then, as diffusion slows, degradation causes the concentration to be restored back to the initial. The concentration inside the nanoparticle only impacts the maximum increase in the aqueous humor, but does not alter the rate equilibrium is restored.
Multiple Dose: Change in GSR Concentration
In Figure ____, all curves approach equilibrium, after which the concentration oscillates about equilibrium. We have three goals, in order of importance for best nanoparticle design:
- GSR equilibrium concentration equal to amount we desire (i.e. 43.5 uM from Model 2)
- Stability of concentration at equilibrium (Model 4 goes into deeper depth regarding sensitivity)
- Time to reach equilibrium (time for full prevention to come into effect)
To do so, we can alter different variables: GSR concentration in nanoparticles, nanoparticle radius, and dose frequency. For a full analysis of how each variable impacts the concentration function, see the collapsible. Below is a summary of the results:
| Independent Variable | Time to Reach Equilibrium | Equilibrium Concentration | Stability |
|---|---|---|---|
| Concentration | No impact | Proportional Increase | Slight Increase |
| Radius | No impact | Decrease | No impact |
| Frequency | No impact | Decrease | Increase |
We find the optimal combination of parameters is: _____________.
A Two Stage Eyedrop Approach
As shown in Table ____, we cannot alter the time to reach equilibrium, or reach full prevention. As supported by ______, the time to reach equilibrium is a property of the lens that we cannot change. However, we propose a two-step eyedrop approach, of two differing concentrations, to decrease the time needed for full prevention. A full explanation is found in the collapsible.
Generalized Nanoparticles: Customizer
Although the model we created was made for drug delivery into the eye, by changing relevant parameters, the model can be generalized for any type of drug delivery. With the analysis performed in the single and multiple nanoparticle dose models, we build the following calculators:
Conclusion
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Model 4: Eyedrop Prototype
We have found a nanoparticle design to deliver GSR. We also need to model the function of eyedrops, to determine the concentration of GSR-loaded nanoparticles to put in eyedrops, and analyze how sensitive the resulting system is.
Bioavailability of GSR Delivery
The eye is well protected from foreign material attempting to enter the eye. The corneal epithelium is the most essential barrier against topical drugs in eyedrops, and as a result, much of drugs in eyedrops are lost in tear drainage.
Bioavailability describes the proportion of the drug that reaches the site of action, regardless of the route of administration. For example, it is estimated that only 1-5% of an active drug with small solutes in an eyedrop penetrates the cornea (Schoenwald 1997). In the case of nanoparticles, which are much larger than chemical molecules, more is lost. (Clinical Ocular Toxicology)
The results show that the bioavailability of nanoparticles is about 8.35 x 10-5%, which means that for every gram of GSR (or any drug) we place into nanoparticles, approximately 0.835 ug of the drug reach the aqueous humor. The variance is 0.125 ug/g.
Necessary Adjustments in Eyedrops
To ensure that sufficient concentrations of GSR are delivered, we must place an excess of GSR. To determine how much, we simply divide the concentration of GSR in nanoparticles we found in Model 3 by the fraction of GSR that reaches the aqueous humor.
[Calculations]
We conclude that we need ________ mM of GSR in nanoparticles to maintain 43.5 uM GSR and thus 2.5 LOCS.
Sensitivity Analysis: Revisiting Nanoparticles Model
The mechanism for eyedrop delivery is complex, and there are variances in the bioavailability depending on the conditions of the eye. The thickness of the cornea, lens, other eye diseases, age, and even time of day may impact the bioavailability of the drug. We use a stochastic model to simulate Model 3 again, but this time, add a degree of variance.
The variance is impacted by the frequency of eyedrops. By giving eyedrops more frequently with less amounts given each time, the variance is decreased.
Ideally, we wish to deliver 100% of the GSR concentration of the amount found in Model 2 (43.5 uM). Because of variance, the actual amount maintained in the lens is different, shown in Figure 5. The full details and mathematics of the stochastic model can be found in the collapsible.
Conclusion
We need to maintain (NOT add) 43.5 uM of GSR in the lens so that the crystallin damage recorded over 50 years is below the LOCS 2.5 threshold.
Citations
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