{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"# Uncertainty quantification and sensitivity analysis for arterial wall models\n",
"\n",
" \n",
"**Vinzenz Gregor Eck**, Expert Analytics, Oslo \n",
"\n",
" **Jacob Sturdy**, Department of Structural Engineering, NTNU\n",
"\n",
"Date: **Jul 13, 2018**"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# ipython magic\n",
"%matplotlib notebook\n",
"%load_ext autoreload\n",
"%autoreload 2"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# plot configuration\n",
"import matplotlib\n",
"import matplotlib.pyplot as plt\n",
"plt.style.use(\"ggplot\")\n",
"# import seaborn as sns # sets another style\n",
"matplotlib.rcParams['lines.linewidth'] = 3\n",
"fig_width, fig_height = (7.0,5.0)\n",
"matplotlib.rcParams['figure.figsize'] = (fig_width, fig_height)\n",
"\n",
"# font = {'family' : 'sans-serif',\n",
"# 'weight' : 'normal',\n",
"# 'size' : 18.0}\n",
"# matplotlib.rc('font', **font) # pass in the font dict as kwar"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"import chaospy as cp\n",
"import numpy as np\n",
"from monte_carlo import generate_sample_matrices_mc\n",
"from monte_carlo import calculate_sensitivity_indices_mc\n",
"unit_mmhg_pa = 133.3\n",
"unit_pa_mmhg = 1./unit_mmhg_pa\n",
"unit_cm2_m2 = 1. / 100. / 100.\n",
"unit_m2_cm2 = 1. / unit_cm2_m2"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Introduction\n",
"\n",
"The arterial wall models we are investigating in this part, are used to describe the (visco-)elastic behaviour of arteries in one-dimensional\n",
"simulations of the cardiovascular system.\n",
"\n",
"# Arterial Wall Models\n",
"\n",
"\n",
"The elastic wall models are simplified algebraic functions $A(P)$ ([[eck2015stochastic;@boileau_benchmark_2015]](#eck2015stochastic;@boileau_benchmark_2015)),\n",
"which state the arterial lumen area $A$ as function of transmural pressure $P$.\n",
"\n",
"For the calibration of the applied wall models, the wave speed in an arterial segment is required.\n",
"The wave speed is given from fluid dynamics equations for one-dimensional arteries ([[eck2015stochastic]](#eck2015stochastic)):"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"\\label{eq:defwaveSpeed} \\tag{1}\n",
"c(P) = \\sqrt{\\frac{A(P)}{\\rho\\ C(P)}},\n",
"\\end{equation}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"with blood density $\\rho= 1050\\ [kg\\ m^{-3}]$ and compliance $C(P) = dA / dP$.\n",
"\n",
"## *Quadratic* model\n",
"The *Quadratic* area-pressure relationship ([[sherwin2003]](#sherwin2003)) is defined as:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"A(P) = \\left((P-P_s) \\frac{A_s}{\\lambda} + \\sqrt{A_s} \\right)^2,\n",
"\\label{eq:lapArea} \\tag{2}\n",
"\\end{equation}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"where $\\lambda$ is referred to as the stiffness coefficient and $A_s$\n",
"is the area at the reference pressure $P_s$.\n",
"\n",
"The stiffness coefficient $\\lambda$ is defined as:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"\\lambda = \\frac{2 \\rho c_s^2 A_s}{\\sqrt{A_s}}\n",
"\\label{_auto1} \\tag{3}\n",
"\\end{equation}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The model is implemented in the following function:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# begin quadratic area model\n",
"def quadratic_area_model(pressure_range, samples):\n",
" \"\"\"\n",
" calculate the arterial vessel wall for a set pressure range\n",
" from 75 to 140 mmHg for a given set of reference wave speeds\n",
" area and pressure.\n",
"\n",
" :param\n",
" pressure_range: np.array\n",
" pressure range for which to calculate the arterial area\n",
"\n",
" samples: np.array with shape (4:n_samples)\n",
" sample matrix where\n",
" first indices correspond to\n",
" a_s: reference area\n",
" c_s: reference waves seed\n",
" p_s: reference pressure\n",
" rho: blood density\n",
" and n_samples to the number of samples\n",
"\n",
" :return:\n",
" arterial pressure : np.array\n",
" of size n_samples\n",
" \"\"\"\n",
" pressure_range = pressure_range.reshape(pressure_range.shape[0], 1)\n",
" a_s, c_s, p_s, rho = samples\n",
" beta = 2*rho*c_s**2/np.sqrt(a_s)*a_s\n",
" #C_Laplace = (2. * ((P - Ps) * As / betaLaplace + np.sqrt(As))) * As / betaLaplace\n",
" return ((pressure_range - p_s)*a_s/beta + np.sqrt(a_s))**2."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## *Logarithmic* model\n",
"The *Logarithmic* area-pressure relationship ([[Hayashi1980]](#Hayashi1980)) is defined as:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"A(P) = A_s \\left( 1 + \\frac{1}{\\beta} ln \\left(\\frac{P}{P_s}\\right) \\right)^2,\n",
"\\label{eq:expArea} \\tag{4}\n",
"\\end{equation}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"where $\\beta$ is called the stiffness coefficient and $A_s$ is the\n",
"area at the reference pressure $P_s$.\n",
"\n",
"The stiffness coefficient $\\beta$ is defined as:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
"\n",
"\n",
"$$\n",
"\\begin{equation}\n",
"\\beta = \\frac{2\\ \\rho c_s^2}{P_s}\n",
"\\label{_auto2} \\tag{5}\n",
"\\end{equation}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The model is implemented in the following function:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# begin exponential area model\n",
"def logarithmic_area_model(pressure_range, samples):\n",
" \"\"\"\n",
" calculate the arterial vessel wall for a set pressure range\n",
" from 75 to 140 mmHg for a given set of reference wave speeds\n",
" area and pressure.\n",
"\n",
" :param\n",
" pressure_range: np.array\n",
" pressure range for which to calculate the arterial area\n",
"\n",
" samples: np.array with shape (4:n_samples)\n",
" sample matrix where\n",
" first indices correspond to\n",
" a_s: reference area\n",
" c_s: reference waves seed\n",
" p_s: reference pressure\n",
" rho: blood density\n",
" and n_samples to the number of samples\n",
"\n",
" :return:\n",
" arterial pressure : np.array\n",
" of size n_samples\n",
" \"\"\"\n",
" pressure_range = pressure_range.reshape(pressure_range.shape[0], 1)\n",
" a_s, c_s, p_s, rho = samples\n",
" beta = 2.0*rho*c_s**2./p_s\n",
" #C_hayashi = 2.0 * As / betaHayashi * (1.0 + np.log(P / Ps) / betaHayashi) / P\n",
" return a_s*(1.0 + np.log(pressure_range / p_s)/beta) ** 2.0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Comparison\n",
"\n",
"For a comparison of the wall models, we set the reference values to:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"A_s &= 5.12 \\;\\mathrm{cm^2} \\\\\n",
"c_s &= 6.256 \\;\\mathrm{m/s} \\\\\n",
"P_s &= 100 \\;\\mathrm{mmHg} \\\\\n",
"\\rho &= 1045 \\; \\mathrm{kg/m^3}\n",
"\\end{align*}\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The two wall models give almost the same result around the reference area and pressure, for which the model was calibrated."
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# start deterministic comparison\n",
"pressure_range = np.linspace(45, 180, 100) * unit_mmhg_pa\n",
"a_s = 5.12 * unit_cm2_m2\n",
"c_s = 6.25609258389\n",
"p_s = 100 * unit_mmhg_pa\n",
"rho = 1045.\n",
"\n",
"plt.figure()\n",
"for model, name in [(quadratic_area_model, 'Quadratic model'), (logarithmic_area_model, 'Logarithmic model')]:\n",
" y_area = model(pressure_range, (a_s, c_s, p_s, rho))\n",
" plt.plot(pressure_range * unit_pa_mmhg, y_area * unit_m2_cm2, label=name)\n",
" plt.xlabel('Pressure [mmHg]')\n",
" plt.ylabel('Area [cm2]')\n",
" plt.legend()\n",
"plt.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Uncertainty Quantification and Sensitivity Analysis\n",
"\n",
"\n",
"## Definition of uncertain input parameters\n",
"For each model input, we define a uniform distributed random variable which varies with a deviation of $5\\%$ around the deterministic value from above:"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"$$\n",
"\\begin{align*}\n",
"A_s &= 5.12 \\; \\mathrm{cm^2} \\quad Z_0 = \\mbox{U}(4.864, 5.376) \\\\\n",
"c_s &= 6.256 \\; \\mathrm{m/s} \\quad Z_1 = \\mbox{U}(5.943, 6.569) \\\\\n",
"P_s &= 100 \\; \\mathrm{mmHg} \\quad Z_2 = \\mbox{U}(95.0, 105.0) \\\\\n",
"\\rho &= 1045 \\; \\mathrm{kg/m^3} \\quad Z_3 = \\mbox{U}(992.75, 1097.25) \n",
"\\end{align*}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# Create marginal and joint distributions\n",
"dev = 0.05\n",
"a_s = 5.12 * unit_cm2_m2\n",
"A_s = cp.Uniform(a_s * (1. - dev), a_s*(1. + dev))\n",
"\n",
"c_s = 6.25609258389\n",
"C_s = cp.Uniform(c_s * (1. - dev), c_s*(1. + dev))\n",
"\n",
"p_s = 100 * unit_mmhg_pa\n",
"P_s = cp.Uniform(p_s * (1. - dev), p_s*(1. + dev))\n",
"\n",
"rho = 1045.\n",
"Rho = cp.Uniform(rho * (1. - dev), rho*(1. + dev))\n",
"\n",
"jpdf = cp.J(A_s, C_s, P_s, Rho)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Scatter plots and Samples"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# scatter plots\n",
"pressure_range = np.linspace(45, 180, 100) * unit_mmhg_pa\n",
"Ns = 200\n",
"sample_scheme = 'R'\n",
"samples = jpdf.sample(Ns, sample_scheme)\n",
"\n",
"for model, name, color in [(quadratic_area_model, 'Quadratic model', '#dd3c30'), (logarithmic_area_model, 'Logarithmic model', '#2775b5')]:\n",
" # evaluate the model for all samples\n",
" Y_area = model(pressure_range, samples)\n",
"\n",
" plt.figure()\n",
" plt.title('{}: evaluations Y_area'.format(name))\n",
" plt.plot(pressure_range * unit_pa_mmhg, Y_area * unit_m2_cm2)\n",
" plt.xlabel('Pressure [mmHg]')\n",
" plt.ylabel('Area [cm2]')\n",
" plt.ylim([3.5,7.])\n",
"\n",
" plt.figure()\n",
" plt.title('{}: scatter plots'.format(name))\n",
" for k in range(len(jpdf)):\n",
" plt.subplot(len(jpdf)/2, len(jpdf)/2, k+1)\n",
" plt.plot(samples[k], Y_area[0]*unit_m2_cm2, '.', color=color)\n",
" plt.ylabel('Area [cm2]')\n",
" plt.ylim([3.45, 4.58])\n",
" xlbl = 'Z' + str(k)\n",
" plt.xlabel(xlbl)\n",
" plt.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Monte Carlo\n",
"\n",
"We apply the Monte Carlo method for both models as discussed before:"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# start Monte Carlo\n",
"pressure_range = np.linspace(45, 180, 100) * unit_mmhg_pa\n",
"Ns = 50000\n",
"sample_method = 'R'\n",
"number_of_parameters = len(jpdf)\n",
"\n",
"# 1. Generate sample matrices\n",
"A, B, C = generate_sample_matrices_mc(Ns, number_of_parameters, jpdf, sample_method)\n",
"\n",
"fig_mean = plt.figure('mean')\n",
"ax_mean = fig_mean.add_subplot(1,2,1)\n",
"ax_dict = {}\n",
"figs = {}\n",
"for name in ['Quadratic model', 'Logarithmic model'] :\n",
" figs[name] = plt.figure(name)\n",
" ax = ax_dict[name]= figs[name].add_subplot(1,2,1)\n",
" \n",
"print(\"\\n Uncertainty measures (averaged)\\n\")\n",
"print('\\n E(Y) | Std(Y) \\n')\n",
"for model, name, color in [(quadratic_area_model, 'Quadratic model', '#dd3c30'),\n",
" (logarithmic_area_model, 'Logarithmic model', '#2775b5')]:\n",
" # evaluate the model for all samples\n",
" Y_A = model(pressure_range, A.T)\n",
" Y_B = model(pressure_range, B.T)\n",
"\n",
" Y_C = np.empty((len(pressure_range), Ns, number_of_parameters))\n",
" for i in range(number_of_parameters):\n",
" Y_C[:, :, i] = model(pressure_range, C[i, :, :].T)\n",
"\n",
" # calculate statistics\n",
" plotMeanConfidenceAlpha = 5.\n",
" Y = np.hstack([Y_A, Y_B])\n",
" expected_value = np.mean(Y, axis=1)\n",
" variance = np.var(Y, axis=1)\n",
" std = np.sqrt(np.var(Y, axis=1))\n",
" prediction_interval = np.percentile(Y, [plotMeanConfidenceAlpha / 2., 100. - plotMeanConfidenceAlpha / 2.], axis=1)\n",
"\n",
" print('{:2.5f} | {:2.5f} : {}'.format(np.mean(expected_value)* unit_m2_cm2, np.mean(std)* unit_m2_cm2, name))\n",
"\n",
" # 3. Approximate the sensitivity indices\n",
" # Better to centralize data before calculating sensitivities\n",
" expected_value_col = expected_value.copy()\n",
" expected_value_col.shape = (expected_value.shape[0],1)\n",
" Y_Ac = Y_A - expected_value_col\n",
" Y_Bc = Y_B - expected_value_col\n",
" Y_Cc = Y_C.transpose() - expected_value\n",
" Y_Cc = Y_Cc.transpose()\n",
" S, ST = calculate_sensitivity_indices_mc(Y_A, Y_B, Y_C)\n",
" Sc, STc = calculate_sensitivity_indices_mc(Y_Ac, Y_Bc, Y_Cc)\n",
"\n",
" ## Plots\n",
" fig = fig_mean\n",
" ax_mean = ax_mean\n",
" ax_mean.plot(pressure_range * unit_pa_mmhg, expected_value * unit_m2_cm2, label=name, color=color)\n",
" ax_mean.fill_between(pressure_range * unit_pa_mmhg, prediction_interval[0] * unit_m2_cm2,\n",
" prediction_interval[1] * unit_m2_cm2, alpha=0.3, color=color)\n",
" ax_mean.set_xlabel('Pressure [mmHg]')\n",
" ax_mean.set_ylabel('Area [cm2]')\n",
" ax_mean.legend()\n",
" fig.tight_layout()\n",
"\n",
" fig = figs[name]\n",
" ax = ax_dict[name]\n",
" ax.set_title('{}: sensitvity indices'.format(name))\n",
" colorsPie = ['yellowgreen', 'gold', 'lightskyblue', 'lightcoral']\n",
" labels = ['Area', 'wave speed', 'pressure', 'density']\n",
" N = 4\n",
" ind = np.arange(N) # the x locations for the groups\n",
" width = 0.35 # the width of the bars\n",
"\n",
" #firstOrderIndicesTimeAverage = S[:, 50]\n",
" #totalIndicesTimeAverage = ST[:, 50]\n",
" #rects1 = plt.bar(ind, firstOrderIndicesTimeAverage, width, color=colorsPie, alpha=0.5)\n",
" #rects2 = plt.bar(ind + width, totalIndicesTimeAverage, width, color=colorsPie, hatch='.')\n",
"\n",
" firstOrderIndicesTimeAverage = np.mean(S*variance,axis=1)/np.mean(variance)\n",
" totalIndicesTimeAverage = np.mean(ST*variance,axis=1)/np.mean(variance)\n",
" rects1 = ax.bar(ind, firstOrderIndicesTimeAverage, width, color=colorsPie, alpha = 0.5)\n",
" rects2 = ax.bar(ind + width, totalIndicesTimeAverage, width, color=colorsPie, hatch = '.')\n",
"\n",
" # add some text for labels, title and axes ticks\n",
" ax.set_ylabel('Si')\n",
" ax.set_xticks(ind + width)\n",
" ax.set_xticklabels(labels)\n",
" ax.set_xlim([-width, N + width])\n",
" ax.set_ylim([0, 1])\n",
" ax.spines['top'].set_visible(False)\n",
" ax.tick_params(axis='x', top='off')\n",
" ax.spines['right'].set_visible(False)\n",
" ax.tick_params(axis='y', right='off')\n",
"\n",
" fig.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Polynomial chaos\n",
"\n",
"We apply the Polynomial Chaos method for both models as discussed before:"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {
"collapsed": false
},
"outputs": [],
"source": [
"# start polynomial chaos\n",
"# orthogonal C_polynomial from marginals\n",
"polynomial_order = 3\n",
"Ns = 2*cp.bertran.terms(polynomial_order, len(jpdf))\n",
"\n",
"print(\"\\n Uncertainty measures (averaged)\\n\")\n",
"print('\\n E(Y) | Std(Y) \\n')\n",
"\n",
"\n",
"\n",
"ax_mean = fig_mean.add_subplot(1,2,2)\n",
"ax_dict = {}\n",
"for name in ['Quadratic model', 'Logarithmic model'] :\n",
" figs[name] = plt.figure(name)\n",
" ax = ax_dict[name]= figs[name].add_subplot(1,2,2)\n",
" \n",
"for model, name, color in [(quadratic_area_model, 'Quadratic model', '#dd3c30'),\n",
" (logarithmic_area_model, 'Logarithmic model', '#2775b5')]:\n",
" sample_scheme = 'R'\n",
" # create samples\n",
" samples = jpdf.sample(Ns, sample_scheme)\n",
" # create orthogonal polynomials\n",
" orthogonal_polynomials = cp.orth_ttr(polynomial_order, jpdf)\n",
" # evaluate the model for all samples\n",
" Y_area = model(pressure_range, samples)\n",
" # polynomial chaos expansion\n",
" polynomial_expansion = cp.fit_regression(orthogonal_polynomials, samples, Y_area.T)\n",
"\n",
" # calculate statistics\n",
" plotMeanConfidenceAlpha = 5\n",
" expected_value = cp.E(polynomial_expansion, jpdf)\n",
" variance = cp.Var(polynomial_expansion, jpdf)\n",
" standard_deviation = cp.Std(polynomial_expansion, jpdf)\n",
" prediction_interval = cp.Perc(polynomial_expansion,\n",
" [plotMeanConfidenceAlpha / 2., 100 - plotMeanConfidenceAlpha / 2.],\n",
" jpdf)\n",
" print('{:2.5f} | {:2.5f} : {}'.format(np.mean(expected_value) * unit_m2_cm2, np.mean(std) * unit_m2_cm2, name))\n",
"\n",
" # compute sensitivity indices\n",
" S = cp.Sens_m(polynomial_expansion, jpdf)\n",
" ST = cp.Sens_t(polynomial_expansion, jpdf)\n",
"\n",
" fig = fig_mean\n",
" ax_mean = ax_mean\n",
" ax_mean.plot(pressure_range * unit_pa_mmhg, expected_value * unit_m2_cm2, label=name, color=color)\n",
" ax_mean.fill_between(pressure_range * unit_pa_mmhg, prediction_interval[0] * unit_m2_cm2,\n",
" prediction_interval[1] * unit_m2_cm2, alpha=0.3, color=color)\n",
" ax_mean.set_xlabel('Pressure [mmHg]')\n",
" ax_mean.set_ylabel('Area [cm2]')\n",
" ax_mean.legend()\n",
" fig.tight_layout()\n",
"\n",
" fig = figs[name]\n",
" ax = ax_dict[name]\n",
" ax.set_title('{}: sensitvity indices'.format(name))\n",
" colorsPie = ['yellowgreen', 'gold', 'lightskyblue', 'lightcoral']\n",
" labels = ['Area', 'wave speed', 'pressure', 'density']\n",
" N = 4\n",
" ind = np.arange(N) # the x locations for the groups\n",
" width = 0.35 # the width of the bars\n",
"\n",
" #firstOrderIndicesTimeAverage = S[:, 50]\n",
" #totalIndicesTimeAverage = ST[:, 50]\n",
" #rects1 = plt.bar(ind, firstOrderIndicesTimeAverage, width, color=colorsPie, alpha=0.5)\n",
" #rects2 = plt.bar(ind + width, totalIndicesTimeAverage, width, color=colorsPie, hatch='.')\n",
"\n",
" firstOrderIndicesTimeAverage = np.mean(S*variance,axis=1)/np.mean(variance)\n",
" totalIndicesTimeAverage = np.mean(ST*variance,axis=1)/np.mean(variance)\n",
" rects1 = ax.bar(ind, firstOrderIndicesTimeAverage, width, color=colorsPie, alpha = 0.5)\n",
" rects2 = ax.bar(ind + width, totalIndicesTimeAverage, width, color=colorsPie, hatch = '.')\n",
"\n",
" # add some text for labels, title and axes ticks\n",
" ax.set_ylabel('Si')\n",
" ax.set_xticks(ind + width)\n",
" ax.set_xticklabels(labels)\n",
" ax.set_xlim([-width, N + width])\n",
" ax.set_ylim([0, 1])\n",
" ax.spines['top'].set_visible(False)\n",
" ax.tick_params(axis='x', top='off')\n",
" ax.spines['right'].set_visible(False)\n",
" ax.tick_params(axis='y', right='off')\n",
"\n",
" fig.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# References\n",
"\n",
" 1. **V. G. Eck, J. Feinberg, H. P. Langtangen and L. R. Hellevik**. \n",
" Stochastic Sensitivity Analysis for Timing and Amplitude of Pressure waves in the Arterial System,\n",
" *International Journal for Numerical Methods in Biomedical Engineering*,\n",
" 31(4),\n",
" 2015.\n",
"\n",
" 2. **E. Boileau, P. Nithiarasu, P. J. Blanco, L. O. Mueller, F. E. Fossan, L. R. Hellevik, W. P. Donders, W. Huberts, M. Willemet and J. Alastruey**. \n",
" A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling,\n",
" *International Journal for Numerical Methods in Biomedical Engineering*,\n",
" 31(10),\n",
" pp. n/a-n/a,\n",
" 2015.\n",
"\n",
" 3. **S. Sherwin, V. Franke, J. Peir\\'o and K. Parker**. \n",
" One-Dimensional Modelling of Vascular Network in Space-Time Variables,\n",
" *Journal of Engineering Mathematics*,\n",
" 47(3-4),\n",
" pp. 217-233,\n",
" 2003.\n",
"\n",
" 4. **K. Hayashi, H. Handa, S. Nagasawa, A. Okumura and K. Moritake**. \n",
" Stiffness and Elastic Behavior of Human Intracranial and Extracranial arteries,\n",
" *Journal of Biomechanics*,\n",
" 13(2),\n",
" pp. 175-184,\n",
" 1980."
]
}
],
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"nbformat_minor": 2
}