Pharmacokinetic Data Modeling and Visualization Tool

Pharmacokinetics is the study of drug absorption, distribution, metabolism, and excretion over time. We often want to apply pharmacokinetic principles to the safe and effective therapeutic management of drugs in patients, but it may be difficult to measure drug concentrations in specific tissues. While we can easily measure drug concentrations in places like blood, urine, saliva, it can be much harder or impossible to measure drug concentrations in other tissues like in the subcutaneous space. However, easily obtained measurements can be leveraged to develop pharmacokinetic models to predict drug concentrations in other tissues where we cannot feasibly collect data from. Compartment models can be generated by dividing the body into compartments that describe the absorption, distribution and elimination. Pharmacokinetic models can be extremely valuable to drug research and development, shedding light to how a drug candidate interacts with the target organism.

Introduction

This application is designed to aid researchers who are currently developing and researching drugs for deployment in animal models or other subjects. This tool consists of two parts that operate entirely independently, although both can be used at the same time. Enjoy and best of luck with your research!

Note: you will notice some stutters as packages load in as soon as you press the "Load-in Packages" button in part 2. This is completely normal. Additionally, the Calculate k-values button in part 2 is grayed out until said packages load - this can take up to 30 seconds. The Rate Constant Calculator runs ONLY in Chrome and Firefox.

• Part 1 is the Pharmacokinetic Data Modeling Calculator and Visualizer
  • The user inputs rate constants (k-values) and initial values for each compartment and the application solves the system of differential equations and displays mass vs. time or concentration vs. time curves for each compartment.
• Part 2 is the Rate Constant Calculator
  • The user uploads mass vs. time or concentration vs. time data in a csv file and the application solves for best fit rate constants (k-values) for the system of differential equations for the data.

Python Files and Jupyter Notebook

It should also be noted that supporting files for this application including a commented version of the Python code in this web application and a sample Jupyter Notebook for download that mimics the functionality of this application (allowing for greater end-user customizability) is available here.

Table of Contents

• Installation
• Compartment Model Example
• Part 1
• Part 2
• Methodologies
• Troubleshooting

Installation

This application can be run in multiple ways.

Github Pages

• Navigate over to the Github Page for this repository where the latest version is deployed.

Supramolecular Biomaterials

• This calculator was developed under the mentorship and guidance of members in the Stanford Appel Lab. There is a prettier version of what I have on my Github Page on my lab's website here.

Downloading the repository

• Clone the project:

git clone https://github.com/peytoncchen/PK-Visualization

• Or: download the source code into a zip file from the green button labelled 'Code'
• Click on index.html and the application will run in your default browser. You may need to switch to Chrome or Firefox, especially if you would like to run part 2 (Rate Constant Calculator).

Compartment Model and Differential Equations Example

Compartment Model

This example system illustrates the model set-up for both parts of the application. In both parts, the user will be able to define the number of compartments in their model system. Each compartment starts with an initial mass and the application uses first-order kinetics to model flow in and out of each compartment. We have chosen to use a 3 compartment model as an example for the inputs and corresponding system of differential equations. The flow is dictated by the rate constants labeled as k₁, k₂, and k₃ in our example picture. Initial mass in each compartment is given by X_1,0, X_2,0, X_3,0. X₁, X₂, and X₃ correspond to the mass in compartments 1, 2, and 3 at a given time t.

Differential equations

The differential equations used in the calculator, resemble the following, where each compartment's derivative is modified by the incoming mass and outgoing mass, where X represents the mass of a compartment.

Part 1

The Pharmacokinetic Data Modeling Calculator and Visualizer allows users to set up a system of ordinary differential equations (ODEs) describing a simple compartment model with first-order kinetics and then solve and plot the drug profiles in each compartment over time.

Inputs:

• Number of Compartments:
  • The number of compartments you have in your model. The compartment model example above has 3 (three) compartments.
• Compartment info:
  • Name (optional): for use in downloadable csv file of generated data
  • Initial values: Mass at t = 0 in each compartment
  • k-value: rate constant for each compartment. The k-value input field in row 1 represents the k-value from Compartment 1 to 2 (k₁ in the compartment model example), k-value in row 2 represents the k-value from Compartment 2 to 3 (k₂ in the compartment model example) and so on. k-values should have consistent units of time with the time range specified below. k-values should have units of 1/t.
• Time range and units:
  • Specify the time range (0 to t) for the generated data and the associated units. Default is 250 minutes.
• Number of steps:
  • Specify the number of steps that you would like the integrator to take. Default is 100,000. Warning: too many steps will result in a lot of RAM usage and may take a long time.
• Mass or concentration:
  • Specify whether you want the y-values generated to be in mass or concentration. The default output is mass (of the same units as initial values). If you would like to have outputs given as concentrations, another row will appear called the "Animal Model Constant" prompting you to fill out a constant and select which compartments you want it to affect.

Animal Model Constant: While modeling mass transfer between systems is typically easiest, often to compare the amount of drug in the body to the efficacious therapeutic dose (or cytotoxic dose) a concentration is required. The Animal Model Constant should be a conversion factor that allows you to convert mass to concentration (typically (blood volume)^-1 or (distribution volume)^-1. At present this application only supports one conversion factor and so for systems where one would need to use a different conversion factor for each compartment we recommend only converting to concentration for the compartment of interest or keeping all compartments in terms of mass. You are allowed to put in mathematical expressions or numbers here.

After inputting the parameters for your system, you can press the Calculate & Graph button which will solve the system of ODEs using a 4th order Runge-Kutta ODE solver and generate mass vs. time or concentration vs. time curves for each compartment. The results will be graphed and can also be downloaded as a csv to be plotted by the user. See methodologies and troubleshooting for more information.

The download field will also enable at this point. You can input a filename (default is export) and select which compartment(s) you want to export to a CSV file. This file will download at your default download location. An example of what will download can be seen here.

Part 2

The Rate Constant Calculator allows users to set up a system of ordinary differential equations (ODEs) describing a simple compartment model and input a CSV containing experimental data, then have an optimizing algorithm run to fit the data to a set of k-values.

Inputs:

• Press the "Load-in packages" button! (to the right of the Rate Constant Calculator title)
• Choose CSV File to upload:
  • Clicking this field will bring up an input dialog for you to upload a CSV. Please follow the format of the CSV here. Essentially, you will want the first row to be headers, your first column to be time, and your second column to be the dependent variable. Any additional columns are not supported currently and will produce errors. Please ensure that you delete any data that you do not want to consider any entry (time, value) that is missing either value will just result in the entire row being ignored. Please note that you must utilize normalized data.
• Number of Compartments, Mass or Concentration, Animal Model Constant
  • Same as part 1
• CSV data represents
  • Please input what compartment your CSV data represents. In the example case, our data represented Compartment 3.
• Compartment Info
  • Please indicate the bounds for k-values that you are unsure of and the initial value that the compartment starts off at. If you do know the value, please tick the "Constrain" box and fill in what the k-value is for that compartment.

Finally, you can press the Calculate k-values which will parse your inputs and return final k-value results for each compartment. For specifics as to how this is done check-out the Python files which the code from this app is based on and the methodologies section below. There is also an example Jupyter Notebook that can be customized to your specific situation. The link to that repository is here.

Methodologies

Below is a quick summary of the math that runs behind the scenes.

Ordinary Differential Equation (ODE) solver

We have adapted a module from Ricky Reusser (2015) for use in this web application. The code integrates a system of ODEs using the Fourth Order Runge-Kutta method. More information as well as formulas that the code is based on can be viewed here.

Optimization Algorithm and set-up

For commented Python code (and a customizable Jupyter Notebook!), visit here. Apart from retrieving and parsing the inputs, the core of the algorithm is handed over to pyodide and Python code to deliver the final result. First, the CSV data array is converted into a dictionary that we can manipulate with time as the keys. That dictionary is then matched to a fit dictionary that is normalized. Then, the sum of squared errors (SSE) between the two dictionaries are minimized through scipy's optimize.brute function, which returns an array one for each compartment in your model.

Troubleshooting

If you get an error message:

• Check your inputs were of the correct type and that they were not impossible.
• Check your k-values - extremely large k-values are known to cause errors as we are using a 4th order Runge-Kutta ODE solver. See methodologies for more information.

If you get outputs that don’t seem to make sense:

• Check your units for rate constants, time range, and initial mass values to make sure units are consistent.
• Check if you have correctly selected mass or concentration as an output and inputted a relevant Animal Model Constant
• Check you have correctly selected mass or concentration as an output and inputted a relevant Animal Model Constant If you’re using the Rate Constant Calculator, check if your csv file follows the example format.

Built With

• HTML/CSS/JS
• pyodide (which allowed us to run Python in the browser) and the Python packages scipy and numpy (go WebAssembly!)

Feedback/Development

The code for this application is open source and can be downloaded and modified as you wish.

Licensing

This application is licensed under the MIT License. See LICENSE.txt for more details.

Developers

• Peyton Chen

Acknowledgments

• Caitlin Maikawa, Stanford Bioengineering Ph.D.
  • For all the math help and documentation help!
• Joseph Mann, Stanford Materials Science Ph.D.
  • The original idea and help with optimization algorithms
• Eric Appel, Assistant Professor of Materials Science and Engineering at Stanford University