{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "*This notebook contains course material from [CBE30338](https://jckantor.github.io/CBE30338)\n", "by Jeffrey Kantor (jeff at nd.edu); the content is available [on Github](https://github.com/jckantor/CBE30338.git).\n", "The text is released under the [CC-BY-NC-ND-4.0 license](https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode),\n", "and code is released under the [MIT license](https://opensource.org/licenses/MIT).*" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "< [Fitting First Order plus Time Delay to Step Response](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/03.04-Fitting-First-Order-plus-Time-Delay-to-Step-Response.ipynb) | [Contents](toc.ipynb) | [Second Order Models](http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/03.06-Second-Order-Models.ipynb) >
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "# One Compartment Pharmacokinetics" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "## Summary\n", "\n", "Pharmacokinetics is a branch of pharmacology that studies the fate of chemical species in living organisms. The diverse range of applications includes the administration of drugs and anesthesia in humans. This notebook introduces a one compartment model for pharmacokinetics, and shows how it can be used to determine strategies for the intravenous administration of an antibiotic.\n", "\n", "The notebook demonstrates the simulation and analysis of systems modeled by a single first-order linear differential equation." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "## Antibiotics\n", "\n", "Let's consider the administration of an antibiotic to a patient. Concentration $C$ refers to the concentration of the antibiotic in blood plasma with units [mg/liter]. \n", "\n", "**Minimum Inhibitory Concentration (MIC)** The minimum concentration of the antibiotic that prevents growth of a particular bacterium.\n", "\n", "**Minimum Bactricidal Concentration (MBC)** The lowest concentration of the antibiotic that kills a particular bacterium.\n", "\n", "Extended exposure to an antibiotic at levels below MBC leads to [antibiotic resistance](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4378521/)." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "## Model Description\n", "\n", "A simple pharmacokinetic model has the same form as a model for the dilution of a chemical species in a constant volume stirred-tank mixer. For a stirred-tank reactor with constant volume $V$, volumetric outlet flowrate $Q$, and inlet mass flow $u(t)$,\n", "\n", "$$V \\frac{dC}{dt} = u(t) - Q C(t)$$\n", "\n", "where $C$ is concentration in units of mass per unit volume. In this pharacokinetics application, $V$ refers to blood plasma volume, and $Q$ to the clearance rate." ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "## Problem Statement 1\n", "\n", "The minimum inhibitory concentration (MIC) of a particular organism to a particular antibiotic is 5 mg/liter, the minimum bactricidal concentration (MBC) is 8 mg/liter. Assume the plasma volume $V$ is 4 liters with a clearance rate $Q$ of 0.5 liters/hour. \n", "\n", "An initial intravenous antibiotic dose of 64 mg results in an initial plasma concentration $C_{initial}$ of 64mg/4 liters = 16 mg/liter. How long will the concentration stay above MBC? Above MIC?" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "## Solution Strategy 1: Simulation from a Known Initial Condition\n", "\n", "For this first simulation we compute the response of the one compartment model due starting with an initial condition $C_{initial}$, and assuming input $u(t) = 0$. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### Step 1. Initialization\n", "\n", "Generally the first steps in any Jupyter notebook are to \n", "\n", "1. Initialize the plotting system.\n", "2. Import the `numpy` library for basic mathematical functions.\n", "3. Import the `matplotlib.pyplot` library for plotting.\n", "\n", "In addition, for this application we also import `odeint` function for solving differential equations from the `scipy.integrate` library." ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "slideshow": { "slide_type": "-" } }, "outputs": [], "source": [ "%matplotlib inline\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from scipy.integrate import odeint" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### Step 2. Enter Parameter Values" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [], "source": [ "V = 4 # liters\n", "Q = 0.5 # liters/hour\n", "MIC = 5 # mg/liter\n", "MBC = 8 # mg/liter\n", "\n", "Cinitial = 16 # mg/liter" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### Step 3. A Function the RHS of the Differential equation\n", "\n", "$$\\frac{dC}{dt} = \\frac{1}{V}u(t) - \\frac{Q}{V}C$$\n", "\n", "where $u(t) = 0$." ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [], "source": [ "def u(t):\n", " return 0\n", "\n", "def deriv(C,t):\n", " return u(t)/V - (Q/V)*C" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "-" } }, "source": [ "### Step 4. Solution and Visualization" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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