For every iGEM team, modeling is a crucial part to link wet team experimental results and dry team theories together. In most of the cases, we need to simulate the behavior of chemicals or bacteria, such as distribution or concentration of them. Scientists often utilize partial differential equations (PDEs) , namely differential equations containing two or more variables, to describe the target substances’ behavior. However, PDEs usually don’t have solutions due to the lack of strategies to deal with them. Here, we introduce a method that might help future iGEM teams to deal with PDEs—double Laplace transform.

The definition of the single-variable Laplace transform, which is what we normally adopted, is defined as below:

The definition of the double-variable Laplace transform (or, double Laplace transform) has a similar form as single-variable Laplace transform:

Notice that for single variable Laplace transform, the x domain is defined: { x | x > 0 }, so analogously, the domain of double Laplace transform, (x, y), is defined to be in the first quadrant, namely { (x, y) | x, y > 0 }.

The general properties of double Laplace transform are listed below ^[1]:

We performed a solution to one example of a partial differential equation using double Laplace transform.

Find the general solution of the equation below:

With boundary and initial conditions:

By adopting double Laplace transform method, we can transform the original equation:

Solving F:

By operating inverse double Laplace transform on F, we can get the solution of C:

We have demonstrated that double Laplace transform is a useful tool to deal with PDEs. However, double Laplace transform is not easy to calculate, since the process involves integrations that might not end up with a certain answer. Therefore, development of the program that can conduct double Laplace transform will be very helpful in solving PDEs.

MATLAB has a function called “laplace(F,x,s)” that can conduct a normal Laplace transform. However, if the function is not transformable, then the function cannot get the result. During the development of our models, we have built two MATLAB programs that has the function below:

1. Conducting double Laplace transform of PDEs.

2. Making Laplace transform estimation of those who cannot be directly transformed by MATLAB’s function.

What’s more, we make these two programs available on the internet with well documentation. Below is the link of the two programs:

On Github: Double_Laplace_Transform_and_Laplace_Estimation

Code

Result

Figures below show the running result of Numerical Laplace Transform Estimator. We plot out the approximation of target function's Laplace transformation utilize our program, and the real solution of target function's Laplace transformation. Result shows that our approximation is very close to the real situation.

Code

Result

Figure below shows the running result of Double Laplace Transform PDE Solver. By applying initial conditions, boundary conditions and target PDE, we sucessfully plot out the double Laplace transform of target function. Result shows that the program can act as PDE solver.

In order to create a personalized IOP tracking system, we designed an app - Eye Cloud that works with Eye Screen. Eye Cloud not only displays the IOP value on the phone through the Bluetooth connection but also upload each measurement value to ThingSpeak’s personal account that is convenient for long-term tracking and observation. In addition, it can also be used as a tool for large-scale data collection to assist the development of Eye kNOw or other research related to intraocular pressure.

ThingSpeak is an IoT analytics platform service that allows us to aggregate, visualize, and analyze live data streams in the cloud. We can easily send data to ThingSpeak from Eye Screen, allowing users to upload and record IOP value through a URL (your personal API Keys)

1. Sign up for ThingSpeak.

2. Click Channels > My Channels. Create a new channel.

3. Check the boxes next to Fields 1–3. Enter these channel setting values:

Name: Eye Screen

Field 1: IOP (mV)

4. Click Save Channel at the bottom of the settings.

5. Click API Keys tab and copy the write URL.

1. This is the homepage of Eye Cloud, with our lovely logo. Click the “START” button to start.

2. Then the next screen will ask if you have applied for a ThingSpeak account.

Click “Yes! Ready to start” button to next page and wait for your IOP receiving.

Click “No, click to apply” button to link to ThingSpeak homepage.

Click “No,thanks” button to start directly

3. Click “Bluetooth ” button to connect with Eye Screen when reaching this page.

4. Bluetooth connection screen.

After the IOP value shows on the screen, click the “Record” button to upload data to your ThingSpeak’s personal account, monitoring daily or monthly measurement records.