insulin.ipynb

Kernel: Python 3 (ipykernel)

The Insulin Minimal Model

Modeling and Simulation in Python

Copyright 2021 Allen Downey

License: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International

# install Pint if necessary

try:
    import pint
except ImportError:
    !pip install pint

# download modsim.py if necessary

from os.path import basename, exists

def download(url):
    filename = basename(url)
    if not exists(filename):
        from urllib.request import urlretrieve
        local, _ = urlretrieve(url, filename)
        print('Downloaded ' + local)
    
download('https://github.com/AllenDowney/ModSimPy/raw/master/modsim.py')

# import functions from modsim

from modsim import *

The following cells download and read the data.

download('https://raw.githubusercontent.com/AllenDowney/' +
         'ModSim/main/data/glucose_insulin.csv')

data = pd.read_csv('glucose_insulin.csv', index_col='time');

In Chapter 17 I present the glucose minimal model; in Chapter 18 we implemented it using run_simulation and run_solve_ivp. In this case study, we'll implement the other half of the Minimal Model, which describes the concentration of insulin.

In the insulin minimal model, the concentration of insulin, $I$, is governed by this differential equation:

$\frac{dI}{dt} = -k I(t) + \gamma (G(t) - G_T) t$

where $G(t)$ is the concentration of glucose at time $t$, and $k$, $\gamma$, and $G_T$ are positive-valued parameters:

• $k$ controls the rate of insulin disappearance.

• $G_T$ determines the glucose threshold: when $G(t)$ exceeds this threshold, it causes insulin to appear; when $G(t)$ is below this threshold, it causes insulin to disappear.

• $\gamma$ controls how quickly insulin appears or disappears when the concentration of glucose is elevated or depressed.

Notice that this equation depends on time, $t$, since the initial injection. It has the effect of increasing glucose sensitivity over time. If you are familiar with control systems, the effect of this term is similar to the integral term in a PID controller.

In addition to the three parameters in the equation, we will also consider the initial concentration of insulin, $I_0$, to be a free parameter; that is, we will choose the value of $I_0$, and the other parameters, that best fit the data.

Exercise: Write a version of make_system that takes the parameters of the model (I0, k, gamma, and G_T) as parameters, along with a DataFrame containing the measurements, and returns a System object suitable for use with run_solve_ivp.

Use it to make a System object with the following parameters:

I0 = 360   
k = 0.25
gamma = 0.004
G_T = 80

params = I0, k, gamma, G_T

# Solution goes here

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Exercise: Write a slope function that takes a time stamp, a State object, and a System object, and returns the derivative of I with respect to time. Test your function with the initial conditions from system.

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Exercise: Run run_solve_ivp with your System object and slope function, and plot the results, along with the measured insulin levels. Use the keyword argument t_eval=data.index so the results are evaluated as the same time stamps as the data.

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Exercise: Write an error function that takes a sequence of parameters as an argument, along with the DataFrame containing the measurements. It should make a System object with the given parameters, call run_solve_ivp, and compute the difference between the results of the simulation and the measured values. Test your error function by calling it with the parameters from the previous exercise.

Hint: As we did with the glucose model, you might want to drop the first 2-3 elements from the sequence of errors.

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Exercise: Use leastsq to find the parameters that best fit the data. Make a System object with those parameters, run it, and plot the results along with the measurements.

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Exercise: Using the best parameters, estimate the sensitivity to glucose of the first and second phase pancreatic responsivity:

$\phi_1 = \frac{I_{max} - I_b}{k (G_0 - G_b)}$

$\phi_2 = \gamma \times 10^4$

For $G_0$, use the best estimate from the glucose model, 272 mg/dL. For $G_b$ and $I_b$, use the initial measurements from the data. For $I_{max}$ is the maximum measurement of insulin concentration.

# Solution goes here

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According to Pacini and Bergman, here are the normal ranges for these quantities.

phi_1_interval = 2, 4
phi_1_interval

phi_2_interval = 20, 35
phi_2_interval

Do your estimates fall in these ranges?