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that outputs graphs of concentration of tracer over time." />
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<header>
<h1 class="title">Module <code>pk_two_comp</code></h1>
</header>
<section id="section-intro">
<p>The pk2Comp object is a two compartment PK model
that outputs graphs of concentration of tracer over time.</p>
<details class="source">
<summary>
<span>Expand source code</span>
</summary>
<pre><code class="python">#!/usr/bin/env python
# coding: utf-8

# In[1]:


&#34;&#34;&#34;The pk2Comp object is a two compartment PK model
    that outputs graphs of concentration of tracer over time.
&#34;&#34;&#34;
#!/usr/bin/env python
# coding: utf-8

# In[1]:
import pathlib
import os
import csv
import re
import math
import numpy as np
import matplotlib.pyplot as plt

from scipy.integrate import solve_ivp
from scipy.optimize import fmin

class pk_two_comp:
    &#34;&#34;&#34;The pk2Comp object is a two compartment PK model
    that outputs graphs of concentration of tracer over time.
    &#34;&#34;&#34;

    def __init__(self, wd=pathlib.Path(&#39;Data&#39;).absolute(), filename=&#39;CTPERF005_stress.csv&#39;):
        &#34;&#34;&#34;Initializes the model with default parameter values for flow, Vp, Visf, and PS.
        Parameters
        ----------
        wd : path
            Absolute path name to current directory. Defaults to ./Data
        filename : String
            Name of the data file you&#39;d like to access

        Attributes
        -----------
        time : double[]
            list of all timepoints
        aorta : double[]
            concentration of tracer in aorta (input function)
        myo : double[]
            concentration of tracer in myocardial tissue (conc_isf)
        Flow : double
            Flow is the flow of plasma through the blood vessel in mL/(mL*min). Defaults to 1/60.
        Vp : double
            Vp is the volume of plasma in mL. Defaults to 0.05.
        Visf : double
            Visf is the volume of interstitial fluid in mL. Defaults to 0.15.
        PS : double
            PS is the permeability-surface area constant in mL/(g*min). Defaults to 1/60.
        ymax : int
            Magnitude of Gamma-var peak.
        tmax : double
            Time of which highest peak in Gamma-var appears
        alpha : double
            Scale factor
        delay : double
            Delay to start Gamma-var curve.

        &#34;&#34;&#34;
        # Subject Data
        if os.path.basename(os.path.normpath(pathlib.Path().absolute())) != &#39;Data&#39;:
            self.wd = pathlib.Path(&#39;Data&#39;).absolute()
        else:
            self.wd = wd

        if not isinstance(filename, str):
            raise ValueError(&#39;Filename must be a string&#39;)

        self.filename = filename
        self.time = []
        self.aorta = []
        self.myo = []

        # Declare Variables for initial conditions
        # Compartment variables to be fitted
        self.flow = 1/60
        self.visf = 0.15
        self.baseline = 60

        # Other Compartmental Modelvariables
        self.perm_surf = 0.35
        self.vol_plasma = 0.10

        # Solved ode
        self.sol = []

        # Gamma variables
        self.ymax = 250
        self.tmax = 6.5
        self.alpha = 2.5
        self.delay = 0

        self.deriv_sol = np.array([])
        self.fit_myo = np.array([])

    def get_data(self, filename):
        &#34;&#34;&#34;Imports data from all .csv files in directory.
        Parameters
        ----------
        filename : string
            Name of the data file you&#39;d like to access
        wd : str
            wd is the working directory path

        Attributes
        ----------
        time : double[]
            list of all timepoints
        aorta : double[]
            concentration of tracer in aorta (input function)
        myo : double[]
            concentration of tracer in myocardial tissue (conc_isf)

        Returns
        -------
        time : double[]
            list of all timepoints
        aorta : double[]
            concentration of tracer in aorta (input function)
        myo : double[]
            concentration of tracer in myocardial tissue (conc_isf)
        &#34;&#34;&#34;
        self.time = []
        self.aorta = []
        self.myo = []

        os.chdir(self.wd)
        # File not found error
        if not os.path.isfile(filename):
            raise ValueError(
                &#34;Input file does not exist: {0}. I&#39;ll quit now.&#34;.format(filename))

        data = list(csv.reader(open(filename), delimiter=&#39;\t&#39;))

        for i in range(12):
            self.time.append(
                float(re.compile(r&#39;\d+[.]+\d+|\d+&#39;).findall(data[i+1][0])[0]))
            self.aorta.append(
                float(re.compile(r&#39;\d+[.]+\d+|\d+&#39;).findall(data[i+1][1])[0]))
            self.myo.append(
                float(re.compile(r&#39;\d+[.]+\d+|\d+&#39;).findall(data[i+1][2])[0]))

        return self.time, self.aorta, self.myo

    # gamma_var distribution curve
    def gamma_var(self, time=np.arange(0, 25), ymax=10, tmax=10, alpha=2, delay=0):
        &#34;&#34;&#34;Creates a gamma variate probability density function with given alpha,
        location, and scale values.
        Parameters
        ----------
        t : double[]
            array of timepoints
        ymax : double
            peak y value of gamma distribution
        tmax : double
            location of 50th percentile of function
        alpha : double
            scale parameter
        delay : double
            time delay of which to start gamma distribution

        Returns
        -------
        y : double[]
            probability density function of your gamma variate.
        &#34;&#34;&#34;
        # Following Madsen 1992 simplified parameterization for gamma variate
        t = time
        self.ymax = ymax
        self.tmax = tmax
        self.alpha = alpha
        self.delay = delay

        y = np.zeros(np.size(t))  # preallocate output

        # For odeint, checks if t input is array or float
        if isinstance(t, (list, np.ndarray)):
            for i in range(np.size(y)):
                if t[i] &lt; delay:
                    y[i] = 0
                else:
                    y[i] = round((ymax*tmax**(-alpha)*math.exp(alpha))*(t[i]-delay)
                                 ** alpha*math.exp(-alpha*(t[i]-delay)/tmax), 3)
        else:
            y = round((ymax*tmax**(-alpha)*math.exp(alpha))*(t-delay)
                      ** alpha*math.exp(-alpha*(t-delay)/tmax), 3)
        return y

    # gamma_var_error
    def input_mse(self, guess=[10, 10, 2, 5]):
        &#34;&#34;&#34;Calculates Mean squared error (MSE) between data and
        gamma variate with given parameters values.
        Parameters
        ----------
        param : ndarray[]
            time : double[]
                array of timepoints
            ymax : double
                peak y value of gamma distribution
            tmax : double
                location of 50th percentile of function
            alpha : double
                scale parameter
            delay : double
                time delay of which to start gamma distribution

        Returns
        -------
        MSE : double
            Mean squared error
        &#34;&#34;&#34;
        if len(guess) &lt; 1:
            self.ymax = 10
            self.tmax = 10
            self.alpha = 2
            self.delay = 5
        elif len(guess) &lt; 2:
            self.ymax = guess[0]
            self.tmax = 10
            self.alpha = 2
            self.delay = 5
        elif len(guess) &lt; 3:
            self.ymax = guess[0]
            self.tmax = guess[1]
            self.alpha = 2
            self.delay = 5
        elif len(guess) &lt; 4:
            self.ymax = guess[0]
            self.tmax = guess[1]
            self.alpha = guess[2]
            self.delay = 5
        else:
            # Mean squared error (MSE) between data and gamma variate with given parameters
            self.ymax = guess[0]
            self.tmax = guess[1]
            self.alpha = guess[2]
            self.delay = guess[3]

        mse = 0

        if self.tmax &lt;= 0 or self.ymax &lt;= 10 or self.delay &lt; 0 or self.alpha &lt; 0 \
            or self.alpha &gt; 1000 or self.tmax &gt; 1000:
            mse = 1000000  # just return a big number

        else:
            model_vals = self.gamma_var(
                self.time, self.ymax, self.tmax, self.alpha, self.delay)

            for i in range(len(self.aorta)):
                mse = (self.aorta[i] - model_vals[i])**2 + mse
            mse = mse / len(self.aorta)
        return round(mse, 3)

    def input_func_fit(self, initGuesses):
        &#34;&#34;&#34;Uses fmin algorithm (Nelder-Mead simplex algorithm) to
        minimize loss function (MSE) of input function.
        Parameters
        ----------
        initGuesses : ndarray[]
            Array of initial guesses containing:
                time : double[]
                    array of timepoints
                ymax : double
                    peak y value of gamma distribution
                tmax : double
                    location of 50th percentile of function
                alpha : double
                    scale parameter
                delay : double
                    time delay of which to start gamma distribution
        Returns
        -------
        opt : double[]
            optimized parameters
        &#34;&#34;&#34;
        # Mean squared error (MSE) between data and gamma variate with given parameters
        opt = fmin(self.input_mse, initGuesses, maxiter=1000)

        self.ymax = opt[0]
        self.tmax = opt[1]
        self.alpha = opt[2]
        self.delay = opt[3]

        return opt.round(2)

    # Derivative function
    def derivs(self, time, curr_vals):
        &#34;&#34;&#34;Finds derivatives of ODEs.

        Parameters
        ----------
        curr_vals : double[]
            curr_vals it he current values of the variables we wish to
            &#34;update&#34; from the curr_vals list.

        time : double[]
            time is our time array from 0 to tmax with timestep dt.

        Returns
        -------
        dconc_plasma_dt : double[]
            contains the derivative of concentration in plasma with respect to time.
        dconc_isf_dt : double[]
            contains the derivative of concentration in interstitial fluid with respect to time.
        &#34;&#34;&#34;
        # Unpack the current values of the variables we wish to &#34;update&#34; from the curr_vals list
        conc_plasma, conc_isf = curr_vals

        # Define value of input function conc_in
        conc_in = self.gamma_var(time, self.ymax, self.tmax,
                                 self.alpha, self.delay)

        # Right-hand side of odes, which are used to computer the derivative
        dconc_plasma_dt = (self.flow/self.vol_plasma)*(conc_in - conc_plasma) + \
            (self.perm_surf/self.vol_plasma)*(conc_isf - conc_plasma)
        dconc_isf_dt = (self.perm_surf/self.visf)*(conc_plasma - conc_isf)
        return dconc_plasma_dt, dconc_isf_dt

    def output_mse(self, guess):
        &#34;&#34;&#34;Calculates Mean squared error (MSE) between data and
        output derivs with given parameters values.
        Parameters
        ----------
        guess : ndarray[]

            Flow : double
                Flow is the flow of plasma through the blood vessel in mL/(mL*min).
                Defaults to 1/60.

            Vp : double
                Vp is the volume of plasma in mL. Defaults to 0.05.

            Visf : double
                Visf is the volume of interstitial fluid in mL. Defaults to 0.15.

            PS : double
                PS is the permeability-surface area constant in mL/(g*min). Defaults to 1/60.
        Returns
        -------
        MSE : double
            Mean squared error
        &#34;&#34;&#34;
        self.flow = guess[0]
        self.visf = guess[1]
        self.baseline = guess[2]

        mse = 0

        if self.flow &lt;= 0 or self.flow &gt;= 25 or self.visf &gt; 100 \
            or self.visf &lt; 0 or self.baseline &gt; 150 or self.baseline &lt; 0:
            mse = 100000  # just return a big number

        else:
            sol = solve_ivp(self.derivs, [0, 30], [0, 0], t_eval=self.time)
            MBF = sol.y[0] + sol.y[1]

            temp = np.asarray(self.myo) - self.baseline

            for i in range(len(self.myo)):
                mse = (temp[i] - MBF[i])**2 + mse

            mse = mse / len(self.myo)
        return mse

    def output_func_fit(self, initGuesses):
        &#34;&#34;&#34;Uses fmin algorithm (Nelder-Mead simplex algorithm) to minimize
        loss function (MSE) of output function.
        Parameters
        ----------
        initGuesses : ndarray[]
            Array of initial guesses containing:
                flow : double
                    Flow is the flow of plasma through the blood vessel in mL/(mL*min).
                    Defaults to 1/60.

            Vp : double
                Vp is the volume of plasma in mL. Defaults to 0.05.

            Visf : double
                Visf is the volume of interstitial fluid in mL. Defaults to 0.15.

            PS : double
                PS is the permeability-surface area constant in mL/(g*min). Defaults to 1/60.
        Returns
        -------
        opt : double[]
            optimized parameters
        &#34;&#34;&#34;
        # Mean squared error (MSE) between data and gamma variate with given parameters
        opt1 = fmin(self.output_mse, initGuesses, maxiter=10000)

        self.flow = opt1[0].round(4)
        self.visf = opt1[1].round(4)
        self.baseline = opt1[2].round(4)

        return opt1.round(4)

    def main(self):

        # Gets data from file
        self.get_data(self.filename)

        # Plots original data
        plt.plot(self.time, self.aorta, &#39;bo&#39;)
        plt.plot(self.time, self.myo, &#39;ro&#39;)

        # Fit gamma_var input function and plots it
        opt = self.input_func_fit([250, 7, 4, 0])
        plt.plot(np.arange(0, 25, 0.01), self.gamma_var(np.arange(0, 25, 0.01),
                                                        opt[0], opt[1], opt[2], opt[3]), &#39;k-&#39;)

        # Fit uptake function and plot it
        opt2 = self.output_func_fit([.011418, .62, self.myo[0]])
        self.deriv_sol = solve_ivp(self.derivs, [0, 30],
                                   [0, 0], t_eval=self.time)
        self.fit_myo = self.deriv_sol.y[0] + self.deriv_sol.y[1]
        plt.plot(self.time, self.fit_myo + self.baseline, &#39;m-&#39;)
        </code></pre>
</details>
</section>
<section>
</section>
<section>
</section>
<section>
</section>
<section>
<h2 class="section-title" id="header-classes">Classes</h2>
<dl>
<dt id="pk_two_comp.pk_two_comp"><code class="flex name class">
<span>class <span class="ident">pk_two_comp</span></span>
<span>(</span><span>wd=WindowsPath('C:/Users/Ethan/OneDrive - Michigan State University/MSU/Classwork/Computational Modeling/Models/Data'), filename='CTPERF005_stress.csv')</span>
</code></dt>
<dd>
<section class="desc"><p>The pk2Comp object is a two compartment PK model
that outputs graphs of concentration of tracer over time.</p>
<p>Initializes the model with default parameter values for flow, Vp, Visf, and PS.
Parameters</p>
<hr>
<dl>
<dt><strong><code>wd</code></strong> :&ensp;<code>path</code></dt>
<dd>Absolute path name to current directory. Defaults to ./Data</dd>
<dt><strong><code>filename</code></strong> :&ensp;<code>String</code></dt>
<dd>Name of the data file you'd like to access</dd>
</dl>
<h2 id="attributes">Attributes</h2>
<dl>
<dt><strong><code>time</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>list of all timepoints</dd>
<dt><strong><code>aorta</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>concentration of tracer in aorta (input function)</dd>
<dt><strong><code>myo</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>concentration of tracer in myocardial tissue (conc_isf)</dd>
<dt><strong><code>Flow</code></strong> :&ensp;<code>double</code></dt>
<dd>Flow is the flow of plasma through the blood vessel in mL/(mL*min). Defaults to 1/60.</dd>
<dt><strong><code>Vp</code></strong> :&ensp;<code>double</code></dt>
<dd>Vp is the volume of plasma in mL. Defaults to 0.05.</dd>
<dt><strong><code>Visf</code></strong> :&ensp;<code>double</code></dt>
<dd>Visf is the volume of interstitial fluid in mL. Defaults to 0.15.</dd>
<dt><strong><code>PS</code></strong> :&ensp;<code>double</code></dt>
<dd>PS is the permeability-surface area constant in mL/(g*min). Defaults to 1/60.</dd>
<dt><strong><code>ymax</code></strong> :&ensp;<code>int</code></dt>
<dd>Magnitude of Gamma-var peak.</dd>
<dt><strong><code>tmax</code></strong> :&ensp;<code>double</code></dt>
<dd>Time of which highest peak in Gamma-var appears</dd>
<dt><strong><code>alpha</code></strong> :&ensp;<code>double</code></dt>
<dd>Scale factor</dd>
<dt><strong><code>delay</code></strong> :&ensp;<code>double</code></dt>
<dd>Delay to start Gamma-var curve.</dd>
</dl></section>
<details class="source">
<summary>
<span>Expand source code</span>
</summary>
<pre><code class="python">class pk_two_comp:
    &#34;&#34;&#34;The pk2Comp object is a two compartment PK model
    that outputs graphs of concentration of tracer over time.
    &#34;&#34;&#34;

    def __init__(self, wd=pathlib.Path(&#39;Data&#39;).absolute(), filename=&#39;CTPERF005_stress.csv&#39;):
        &#34;&#34;&#34;Initializes the model with default parameter values for flow, Vp, Visf, and PS.
        Parameters
        ----------
        wd : path
            Absolute path name to current directory. Defaults to ./Data
        filename : String
            Name of the data file you&#39;d like to access

        Attributes
        -----------
        time : double[]
            list of all timepoints
        aorta : double[]
            concentration of tracer in aorta (input function)
        myo : double[]
            concentration of tracer in myocardial tissue (conc_isf)
        Flow : double
            Flow is the flow of plasma through the blood vessel in mL/(mL*min). Defaults to 1/60.
        Vp : double
            Vp is the volume of plasma in mL. Defaults to 0.05.
        Visf : double
            Visf is the volume of interstitial fluid in mL. Defaults to 0.15.
        PS : double
            PS is the permeability-surface area constant in mL/(g*min). Defaults to 1/60.
        ymax : int
            Magnitude of Gamma-var peak.
        tmax : double
            Time of which highest peak in Gamma-var appears
        alpha : double
            Scale factor
        delay : double
            Delay to start Gamma-var curve.

        &#34;&#34;&#34;
        # Subject Data
        if os.path.basename(os.path.normpath(pathlib.Path().absolute())) != &#39;Data&#39;:
            self.wd = pathlib.Path(&#39;Data&#39;).absolute()
        else:
            self.wd = wd

        if not isinstance(filename, str):
            raise ValueError(&#39;Filename must be a string&#39;)

        self.filename = filename
        self.time = []
        self.aorta = []
        self.myo = []

        # Declare Variables for initial conditions
        # Compartment variables to be fitted
        self.flow = 1/60
        self.visf = 0.15
        self.baseline = 60

        # Other Compartmental Modelvariables
        self.perm_surf = 0.35
        self.vol_plasma = 0.10

        # Solved ode
        self.sol = []

        # Gamma variables
        self.ymax = 250
        self.tmax = 6.5
        self.alpha = 2.5
        self.delay = 0

        self.deriv_sol = np.array([])
        self.fit_myo = np.array([])

    def get_data(self, filename):
        &#34;&#34;&#34;Imports data from all .csv files in directory.
        Parameters
        ----------
        filename : string
            Name of the data file you&#39;d like to access
        wd : str
            wd is the working directory path

        Attributes
        ----------
        time : double[]
            list of all timepoints
        aorta : double[]
            concentration of tracer in aorta (input function)
        myo : double[]
            concentration of tracer in myocardial tissue (conc_isf)

        Returns
        -------
        time : double[]
            list of all timepoints
        aorta : double[]
            concentration of tracer in aorta (input function)
        myo : double[]
            concentration of tracer in myocardial tissue (conc_isf)
        &#34;&#34;&#34;
        self.time = []
        self.aorta = []
        self.myo = []

        os.chdir(self.wd)
        # File not found error
        if not os.path.isfile(filename):
            raise ValueError(
                &#34;Input file does not exist: {0}. I&#39;ll quit now.&#34;.format(filename))

        data = list(csv.reader(open(filename), delimiter=&#39;\t&#39;))

        for i in range(12):
            self.time.append(
                float(re.compile(r&#39;\d+[.]+\d+|\d+&#39;).findall(data[i+1][0])[0]))
            self.aorta.append(
                float(re.compile(r&#39;\d+[.]+\d+|\d+&#39;).findall(data[i+1][1])[0]))
            self.myo.append(
                float(re.compile(r&#39;\d+[.]+\d+|\d+&#39;).findall(data[i+1][2])[0]))

        return self.time, self.aorta, self.myo

    # gamma_var distribution curve
    def gamma_var(self, time=np.arange(0, 25), ymax=10, tmax=10, alpha=2, delay=0):
        &#34;&#34;&#34;Creates a gamma variate probability density function with given alpha,
        location, and scale values.
        Parameters
        ----------
        t : double[]
            array of timepoints
        ymax : double
            peak y value of gamma distribution
        tmax : double
            location of 50th percentile of function
        alpha : double
            scale parameter
        delay : double
            time delay of which to start gamma distribution

        Returns
        -------
        y : double[]
            probability density function of your gamma variate.
        &#34;&#34;&#34;
        # Following Madsen 1992 simplified parameterization for gamma variate
        t = time
        self.ymax = ymax
        self.tmax = tmax
        self.alpha = alpha
        self.delay = delay

        y = np.zeros(np.size(t))  # preallocate output

        # For odeint, checks if t input is array or float
        if isinstance(t, (list, np.ndarray)):
            for i in range(np.size(y)):
                if t[i] &lt; delay:
                    y[i] = 0
                else:
                    y[i] = round((ymax*tmax**(-alpha)*math.exp(alpha))*(t[i]-delay)
                                 ** alpha*math.exp(-alpha*(t[i]-delay)/tmax), 3)
        else:
            y = round((ymax*tmax**(-alpha)*math.exp(alpha))*(t-delay)
                      ** alpha*math.exp(-alpha*(t-delay)/tmax), 3)
        return y

    # gamma_var_error
    def input_mse(self, guess=[10, 10, 2, 5]):
        &#34;&#34;&#34;Calculates Mean squared error (MSE) between data and
        gamma variate with given parameters values.
        Parameters
        ----------
        param : ndarray[]
            time : double[]
                array of timepoints
            ymax : double
                peak y value of gamma distribution
            tmax : double
                location of 50th percentile of function
            alpha : double
                scale parameter
            delay : double
                time delay of which to start gamma distribution

        Returns
        -------
        MSE : double
            Mean squared error
        &#34;&#34;&#34;
        if len(guess) &lt; 1:
            self.ymax = 10
            self.tmax = 10
            self.alpha = 2
            self.delay = 5
        elif len(guess) &lt; 2:
            self.ymax = guess[0]
            self.tmax = 10
            self.alpha = 2
            self.delay = 5
        elif len(guess) &lt; 3:
            self.ymax = guess[0]
            self.tmax = guess[1]
            self.alpha = 2
            self.delay = 5
        elif len(guess) &lt; 4:
            self.ymax = guess[0]
            self.tmax = guess[1]
            self.alpha = guess[2]
            self.delay = 5
        else:
            # Mean squared error (MSE) between data and gamma variate with given parameters
            self.ymax = guess[0]
            self.tmax = guess[1]
            self.alpha = guess[2]
            self.delay = guess[3]

        mse = 0

        if self.tmax &lt;= 0 or self.ymax &lt;= 10 or self.delay &lt; 0 or self.alpha &lt; 0 \
            or self.alpha &gt; 1000 or self.tmax &gt; 1000:
            mse = 1000000  # just return a big number

        else:
            model_vals = self.gamma_var(
                self.time, self.ymax, self.tmax, self.alpha, self.delay)

            for i in range(len(self.aorta)):
                mse = (self.aorta[i] - model_vals[i])**2 + mse
            mse = mse / len(self.aorta)
        return round(mse, 3)

    def input_func_fit(self, initGuesses):
        &#34;&#34;&#34;Uses fmin algorithm (Nelder-Mead simplex algorithm) to
        minimize loss function (MSE) of input function.
        Parameters
        ----------
        initGuesses : ndarray[]
            Array of initial guesses containing:
                time : double[]
                    array of timepoints
                ymax : double
                    peak y value of gamma distribution
                tmax : double
                    location of 50th percentile of function
                alpha : double
                    scale parameter
                delay : double
                    time delay of which to start gamma distribution
        Returns
        -------
        opt : double[]
            optimized parameters
        &#34;&#34;&#34;
        # Mean squared error (MSE) between data and gamma variate with given parameters
        opt = fmin(self.input_mse, initGuesses, maxiter=1000)

        self.ymax = opt[0]
        self.tmax = opt[1]
        self.alpha = opt[2]
        self.delay = opt[3]

        return opt.round(2)

    # Derivative function
    def derivs(self, time, curr_vals):
        &#34;&#34;&#34;Finds derivatives of ODEs.

        Parameters
        ----------
        curr_vals : double[]
            curr_vals it he current values of the variables we wish to
            &#34;update&#34; from the curr_vals list.

        time : double[]
            time is our time array from 0 to tmax with timestep dt.

        Returns
        -------
        dconc_plasma_dt : double[]
            contains the derivative of concentration in plasma with respect to time.
        dconc_isf_dt : double[]
            contains the derivative of concentration in interstitial fluid with respect to time.
        &#34;&#34;&#34;
        # Unpack the current values of the variables we wish to &#34;update&#34; from the curr_vals list
        conc_plasma, conc_isf = curr_vals

        # Define value of input function conc_in
        conc_in = self.gamma_var(time, self.ymax, self.tmax,
                                 self.alpha, self.delay)

        # Right-hand side of odes, which are used to computer the derivative
        dconc_plasma_dt = (self.flow/self.vol_plasma)*(conc_in - conc_plasma) + \
            (self.perm_surf/self.vol_plasma)*(conc_isf - conc_plasma)
        dconc_isf_dt = (self.perm_surf/self.visf)*(conc_plasma - conc_isf)
        return dconc_plasma_dt, dconc_isf_dt

    def output_mse(self, guess):
        &#34;&#34;&#34;Calculates Mean squared error (MSE) between data and
        output derivs with given parameters values.
        Parameters
        ----------
        guess : ndarray[]

            Flow : double
                Flow is the flow of plasma through the blood vessel in mL/(mL*min).
                Defaults to 1/60.

            Vp : double
                Vp is the volume of plasma in mL. Defaults to 0.05.

            Visf : double
                Visf is the volume of interstitial fluid in mL. Defaults to 0.15.

            PS : double
                PS is the permeability-surface area constant in mL/(g*min). Defaults to 1/60.
        Returns
        -------
        MSE : double
            Mean squared error
        &#34;&#34;&#34;
        self.flow = guess[0]
        self.visf = guess[1]
        self.baseline = guess[2]

        mse = 0

        if self.flow &lt;= 0 or self.flow &gt;= 25 or self.visf &gt; 100 \
            or self.visf &lt; 0 or self.baseline &gt; 150 or self.baseline &lt; 0:
            mse = 100000  # just return a big number

        else:
            sol = solve_ivp(self.derivs, [0, 30], [0, 0], t_eval=self.time)
            MBF = sol.y[0] + sol.y[1]

            temp = np.asarray(self.myo) - self.baseline

            for i in range(len(self.myo)):
                mse = (temp[i] - MBF[i])**2 + mse

            mse = mse / len(self.myo)
        return mse

    def output_func_fit(self, initGuesses):
        &#34;&#34;&#34;Uses fmin algorithm (Nelder-Mead simplex algorithm) to minimize
        loss function (MSE) of output function.
        Parameters
        ----------
        initGuesses : ndarray[]
            Array of initial guesses containing:
                flow : double
                    Flow is the flow of plasma through the blood vessel in mL/(mL*min).
                    Defaults to 1/60.

            Vp : double
                Vp is the volume of plasma in mL. Defaults to 0.05.

            Visf : double
                Visf is the volume of interstitial fluid in mL. Defaults to 0.15.

            PS : double
                PS is the permeability-surface area constant in mL/(g*min). Defaults to 1/60.
        Returns
        -------
        opt : double[]
            optimized parameters
        &#34;&#34;&#34;
        # Mean squared error (MSE) between data and gamma variate with given parameters
        opt1 = fmin(self.output_mse, initGuesses, maxiter=10000)

        self.flow = opt1[0].round(4)
        self.visf = opt1[1].round(4)
        self.baseline = opt1[2].round(4)

        return opt1.round(4)

    def main(self):

        # Gets data from file
        self.get_data(self.filename)

        # Plots original data
        plt.plot(self.time, self.aorta, &#39;bo&#39;)
        plt.plot(self.time, self.myo, &#39;ro&#39;)

        # Fit gamma_var input function and plots it
        opt = self.input_func_fit([250, 7, 4, 0])
        plt.plot(np.arange(0, 25, 0.01), self.gamma_var(np.arange(0, 25, 0.01),
                                                        opt[0], opt[1], opt[2], opt[3]), &#39;k-&#39;)

        # Fit uptake function and plot it
        opt2 = self.output_func_fit([.011418, .62, self.myo[0]])
        self.deriv_sol = solve_ivp(self.derivs, [0, 30],
                                   [0, 0], t_eval=self.time)
        self.fit_myo = self.deriv_sol.y[0] + self.deriv_sol.y[1]
        plt.plot(self.time, self.fit_myo + self.baseline, &#39;m-&#39;)</code></pre>
</details>
<h3>Methods</h3>
<dl>
<dt id="pk_two_comp.pk_two_comp.derivs"><code class="name flex">
<span>def <span class="ident">derivs</span></span>(<span>self, time, curr_vals)</span>
</code></dt>
<dd>
<section class="desc"><p>Finds derivatives of ODEs.</p>
<h2 id="parameters">Parameters</h2>
<dl>
<dt><strong><code>curr_vals</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>curr_vals it he current values of the variables we wish to
"update" from the curr_vals list.</dd>
<dt><strong><code>time</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>time is our time array from 0 to tmax with timestep dt.</dd>
</dl>
<h2 id="returns">Returns</h2>
<dl>
<dt><strong><code>dconc_plasma_dt</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>contains the derivative of concentration in plasma with respect to time.</dd>
<dt><strong><code>dconc_isf_dt</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>contains the derivative of concentration in interstitial fluid with respect to time.</dd>
</dl></section>
<details class="source">
<summary>
<span>Expand source code</span>
</summary>
<pre><code class="python">def derivs(self, time, curr_vals):
    &#34;&#34;&#34;Finds derivatives of ODEs.

    Parameters
    ----------
    curr_vals : double[]
        curr_vals it he current values of the variables we wish to
        &#34;update&#34; from the curr_vals list.

    time : double[]
        time is our time array from 0 to tmax with timestep dt.

    Returns
    -------
    dconc_plasma_dt : double[]
        contains the derivative of concentration in plasma with respect to time.
    dconc_isf_dt : double[]
        contains the derivative of concentration in interstitial fluid with respect to time.
    &#34;&#34;&#34;
    # Unpack the current values of the variables we wish to &#34;update&#34; from the curr_vals list
    conc_plasma, conc_isf = curr_vals

    # Define value of input function conc_in
    conc_in = self.gamma_var(time, self.ymax, self.tmax,
                             self.alpha, self.delay)

    # Right-hand side of odes, which are used to computer the derivative
    dconc_plasma_dt = (self.flow/self.vol_plasma)*(conc_in - conc_plasma) + \
        (self.perm_surf/self.vol_plasma)*(conc_isf - conc_plasma)
    dconc_isf_dt = (self.perm_surf/self.visf)*(conc_plasma - conc_isf)
    return dconc_plasma_dt, dconc_isf_dt</code></pre>
</details>
</dd>
<dt id="pk_two_comp.pk_two_comp.gamma_var"><code class="name flex">
<span>def <span class="ident">gamma_var</span></span>(<span>self, time=array([ 0,
1,
2,
3,
4,
5,
6,
7,
8,
9, 10, 11, 12, 13, 14, 15, 16,
17, 18, 19, 20, 21, 22, 23, 24]), ymax=10, tmax=10, alpha=2, delay=0)</span>
</code></dt>
<dd>
<section class="desc"><p>Creates a gamma variate probability density function with given alpha,
location, and scale values.
Parameters</p>
<hr>
<dl>
<dt><strong><code>t</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>array of timepoints</dd>
<dt><strong><code>ymax</code></strong> :&ensp;<code>double</code></dt>
<dd>peak y value of gamma distribution</dd>
<dt><strong><code>tmax</code></strong> :&ensp;<code>double</code></dt>
<dd>location of 50th percentile of function</dd>
<dt><strong><code>alpha</code></strong> :&ensp;<code>double</code></dt>
<dd>scale parameter</dd>
<dt><strong><code>delay</code></strong> :&ensp;<code>double</code></dt>
<dd>time delay of which to start gamma distribution</dd>
</dl>
<h2 id="returns">Returns</h2>
<dl>
<dt><strong><code>y</code></strong> :&ensp;<code>double</code>[]</dt>
<dd>probability density function of your gamma variate.</dd>
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