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# Configure Jupyter so figures appear in the notebook
%matplotlib inline
# Configure Jupyter to display the assigned value after an assignment
%config InteractiveShell.ast_node_interactivity='last_expr_or_assign'
# import functions from the modsim.py module
from modsim import *
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data = pd.read_csv('data/glucose_insulin.csv', index_col='time');
Exercise: Write a version of make_system that takes the parameters of this 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_simulation or run_odeint.
Use it to make a System object with the following parameters:
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params = Params(I0 = 360,
k = 0.25,
gamma = 0.004,
G_T = 80)
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# Solution
def make_system(params, data):
# params might be a Params object or an array,
# so we have to unpack it like this
I0, k, gamma, G_T = params
init = State(I=I0)
t_0 = get_first_label(data)
t_end = get_last_label(data)
G=interpolate(data.glucose)
system = System(I0=I0, k=k, gamma=gamma, G_T=G_T, G=G,
init=init, t_0=t_0, t_end=t_end, dt=1)
return system
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# Solution
system = make_system(params, data)
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Exercise: Write a slope function that takes state, t, system as parameters and returns the derivative of I with respect to time. Test your function with the initial condition $I(0)=360$.
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# Solution
def slope_func(state, t, system):
[I] = state
k, gamma = system.k, system.gamma
G, G_T = system.G, system.G_T
dIdt = -k * I + gamma * (G(t) - G_T) * t
return [dIdt]
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# Solution
slope_func(system.init, system.t_0, system)
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Exercise: Run run_ode_solver with your System object and slope function, and plot the results, along with the measured insulin levels.
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# Solution
results, details = run_ode_solver(system, slope_func)
details
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# Solution
results.tail()
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# Solution
plot(results.I, 'g-', label='simulation')
plot(data.insulin, 'go', label='insulin data')
decorate(xlabel='Time (min)',
ylabel='Concentration ($\mu$U/mL)')
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, run it, 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 in a previous exercise, you might want to drop the errors for times prior to t=8.
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# Solution
def error_func(params, data):
"""Computes an array of errors to be minimized.
params: sequence of parameters
actual: array of values to be matched
returns: array of errors
"""
print(params)
# make a System with the given parameters
system = make_system(params, data)
# solve the ODE
results, details = run_ode_solver(system, slope_func)
# compute the difference between the model
# results and actual data
errors = (results.I - data.insulin).dropna()
return TimeSeries(errors.loc[8:])
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# Solution
error_func(params, data)
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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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# Solution
best_params, details = leastsq(error_func, params, data)
print(details.mesg)
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# Solution
system = make_system(best_params, data)
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# Solution
results, details = run_ode_solver(system, slope_func, t_eval=data.index)
details
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# Solution
plot(results.I, 'g-', label='simulation')
plot(data.insulin, 'go', label='insulin data')
decorate(xlabel='Time (min)',
ylabel='Concentration ($\mu$U/mL)')
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, 290. For $G_b$ and $I_b$, use the inital measurements from the data.
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# Solution
I0, k, gamma, G_T = best_params
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# Solution
I_max = data.insulin.max()
Ib = data.insulin[0]
I_max, Ib
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# Solution
# The value of G0 is the best estimate from the glucose model
G0 = 289
Gb = data.glucose[0]
G0, Gb
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# Solution
phi_1 = (I_max - Ib) / k / (G0 - Gb)
phi_1
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# Solution
phi_2 = gamma * 1e4
phi_2
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