# Size of variable arrays: sizeAlgebraic = 21 sizeStates = 6 sizeConstants = 40 from math import * from numpy import * def createLegends(): legend_states = [""] * sizeStates legend_rates = [""] * sizeStates legend_algebraic = [""] * sizeAlgebraic legend_voi = "" legend_constants = [""] * sizeConstants legend_voi = "time in component environment (second)" legend_constants[0] = "R_mt in component heart_parameters (kPa_second_per_liter)" legend_constants[1] = "R_av in component heart_parameters (kPa_second_per_liter)" legend_constants[2] = "R_tc in component heart_parameters (kPa_second_per_liter)" legend_constants[3] = "R_pv in component heart_parameters (kPa_second_per_liter)" legend_constants[4] = "R_pul in component heart_parameters (kPa_second_per_liter)" legend_constants[5] = "R_sys in component heart_parameters (kPa_second_per_liter)" legend_constants[6] = "HR in component heart_parameters (dimensionless)" legend_constants[7] = "V_tot in component heart_parameters (liter)" legend_constants[8] = "P_pl in component heart_parameters (kPa)" legend_algebraic[1] = "e_t in component driver_function (dimensionless)" legend_constants[9] = "A in component driver_function (dimensionless)" legend_constants[10] = "B in component driver_function (dimensionless)" legend_constants[11] = "C in component driver_function (dimensionless)" legend_algebraic[0] = "tau in component driver_function (second)" legend_constants[12] = "period in component driver_function (dimensionless)" legend_algebraic[2] = "V_pcd in component pericardium (liter)" legend_algebraic[3] = "P_pcd in component pericardium (kPa)" legend_algebraic[4] = "P_peri in component pericardium (kPa)" legend_states[0] = "V_lv in component left_ventricle (liter)" legend_states[1] = "V_rv in component right_ventricle (liter)" legend_constants[13] = "P_0_pcd in component pericardium (kPa)" legend_constants[14] = "V_0_pcd in component pericardium (liter)" legend_constants[15] = "lambda_pcd in component pericardium (per_liter)" legend_algebraic[9] = "V_lvf in component left_ventricle (liter)" legend_algebraic[10] = "P_lvf in component left_ventricle (kPa)" legend_algebraic[18] = "P_lv in component left_ventricle (kPa)" legend_algebraic[11] = "V_spt in component septum (liter)" legend_algebraic[12] = "P_es_lvf in component lvf_calculator (kPa)" legend_algebraic[13] = "P_ed_lvf in component lvf_calculator (kPa)" legend_algebraic[6] = "P_pu in component pulmonary_vein (kPa)" legend_algebraic[7] = "P_ao in component aorta (kPa)" legend_constants[16] = "E_es_lvf in component lvf_calculator (kPa_per_liter)" legend_constants[17] = "V_d_lvf in component lvf_calculator (liter)" legend_constants[18] = "P_0_lvf in component lvf_calculator (kPa)" legend_constants[19] = "lambda_lvf in component lvf_calculator (per_liter)" legend_constants[20] = "V_0_lvf in component lvf_calculator (liter)" legend_algebraic[14] = "V_rvf in component right_ventricle (liter)" legend_algebraic[15] = "P_rvf in component right_ventricle (kPa)" legend_algebraic[19] = "P_rv in component right_ventricle (kPa)" legend_algebraic[16] = "P_es_rvf in component rvf_calculator (kPa)" legend_algebraic[17] = "P_ed_rvf in component rvf_calculator (kPa)" legend_algebraic[5] = "P_pa in component pulmonary_artery (kPa)" legend_algebraic[8] = "P_vc in component vena_cava (kPa)" legend_constants[21] = "E_es_rvf in component rvf_calculator (kPa_per_liter)" legend_constants[22] = "V_d_rvf in component rvf_calculator (liter)" legend_constants[23] = "P_0_rvf in component rvf_calculator (kPa)" legend_constants[24] = "lambda_rvf in component rvf_calculator (per_liter)" legend_constants[25] = "V_0_rvf in component rvf_calculator (liter)" legend_algebraic[20] = "P_sept in component septum (kPa)" legend_constants[26] = "E_es_spt in component septum (kPa_per_liter)" legend_constants[27] = "V_d_spt in component septum (liter)" legend_constants[28] = "P_0_spt in component septum (kPa)" legend_constants[29] = "lambda_spt in component septum (per_liter)" legend_constants[30] = "V_0_spt in component septum (liter)" legend_constants[31] = "one in component septum (dimensionless)" legend_constants[32] = "E_es_pa in component pulmonary_artery (kPa_per_liter)" legend_states[2] = "V_pa in component pulmonary_artery (liter)" legend_constants[33] = "V_d_pa in component pulmonary_artery (liter)" legend_constants[34] = "E_es_pu in component pulmonary_vein (kPa_per_liter)" legend_states[3] = "V_pu in component pulmonary_vein (liter)" legend_constants[35] = "V_d_pu in component pulmonary_vein (liter)" legend_constants[36] = "E_es_ao in component aorta (kPa_per_liter)" legend_states[4] = "V_ao in component aorta (liter)" legend_constants[37] = "V_d_ao in component aorta (liter)" legend_constants[38] = "E_es_vc in component vena_cava (kPa_per_liter)" legend_states[5] = "V_vc in component vena_cava (liter)" legend_constants[39] = "V_d_vc in component vena_cava (liter)" legend_rates[0] = "d/dt V_lv in component left_ventricle (liter)" legend_rates[1] = "d/dt V_rv in component right_ventricle (liter)" legend_rates[2] = "d/dt V_pa in component pulmonary_artery (liter)" legend_rates[3] = "d/dt V_pu in component pulmonary_vein (liter)" legend_rates[4] = "d/dt V_ao in component aorta (liter)" legend_rates[5] = "d/dt V_vc in component vena_cava (liter)" return (legend_states, legend_algebraic, legend_voi, legend_constants) def initConsts(): constants = [0.0] * sizeConstants; states = [0.0] * sizeStates; constants[0] = 0.06 constants[1] = 1.4 constants[2] = 0.18 constants[3] = 0.48 constants[4] = 19 constants[5] = 140 constants[6] = 80 constants[7] = 5.5 constants[8] = -0.533289474 constants[9] = 1 constants[10] = 80 constants[11] = 0.27 constants[12] = 0.405 states[0] = 0.005 states[1] = 0.005 constants[13] = 0.067 constants[14] = 0.2 constants[15] = 30 constants[16] = 454 constants[17] = 0.005 constants[18] = 0.17 constants[19] = 15 constants[20] = 0.005 constants[21] = 87 constants[22] = 0.005 constants[23] = 0.16 constants[24] = 15 constants[25] = 0.005 constants[26] = 6500 constants[27] = 0.002 constants[28] = 0.148 constants[29] = 435 constants[30] = 0.002 constants[31] = 1 constants[32] = 45 states[2] = 0.16 constants[33] = 0.16 constants[34] = 0.8 states[3] = 0.2 constants[35] = 0.2 constants[36] = 94 states[4] = 0.8 constants[37] = 0.8 constants[38] = 1.5 states[5] = 2.83 constants[39] = 2.83 return (states, constants) def computeRates(voi, states, constants): rates = [0.0] * sizeStates; algebraic = [0.0] * sizeAlgebraic algebraic[2] = states[0]+states[1] algebraic[3] = constants[13]*(exp(constants[15]*(algebraic[2]-constants[14]))-1.00000) algebraic[4] = algebraic[3]+constants[8] algebraic[0] = voi % constants[12] algebraic[1] = constants[9]*exp(-constants[10]*(power((algebraic[0]*constants[6])/60.0000-constants[11], 2.00000))) rootfind_0(voi, constants, rates, states, algebraic) algebraic[18] = algebraic[10]+algebraic[4] algebraic[6] = constants[34]*(states[3]-constants[35]) algebraic[7] = constants[36]*(states[4]-constants[37]) rates[0] = custom_piecewise([less(algebraic[6]-algebraic[18] , 0.00000) & less(algebraic[18]-algebraic[7] , 0.00000), 0.00000 , less(algebraic[6]-algebraic[18] , 0.00000), -(algebraic[18]-algebraic[7])/constants[1] , less(algebraic[18]-algebraic[7] , 0.00000), (algebraic[6]-algebraic[18])/constants[0] , True, (algebraic[6]-algebraic[18])/constants[0]-(algebraic[18]-algebraic[7])/constants[1]]) algebraic[5] = constants[32]*(states[2]-constants[33]) rates[3] = custom_piecewise([less(algebraic[5]-algebraic[6] , 0.00000) & less(algebraic[6]-algebraic[18] , 0.00000), 0.00000 , less(algebraic[5]-algebraic[6] , 0.00000), -(algebraic[6]-algebraic[18])/constants[0] , less(algebraic[6]-algebraic[18] , 0.00000), (algebraic[5]-algebraic[6])/constants[4] , True, (algebraic[5]-algebraic[6])/constants[4]-(algebraic[6]-algebraic[18])/constants[0]]) algebraic[8] = constants[38]*(states[5]-constants[39]) rates[4] = custom_piecewise([less(algebraic[18]-algebraic[7] , 0.00000) & less(algebraic[7]-algebraic[8] , 0.00000), 0.00000 , less(algebraic[18]-algebraic[7] , 0.00000), -(algebraic[7]-algebraic[8])/constants[5] , less(algebraic[7]-algebraic[8] , 0.00000), (algebraic[18]-algebraic[7])/constants[1] , True, (algebraic[18]-algebraic[7])/constants[1]-(algebraic[7]-algebraic[8])/constants[5]]) algebraic[19] = algebraic[15]+algebraic[4] rates[1] = custom_piecewise([less(algebraic[8]-algebraic[19] , 0.00000) & less(algebraic[19]-algebraic[5] , 0.00000), 0.00000 , less(algebraic[8]-algebraic[19] , 0.00000), -(algebraic[19]-algebraic[5])/constants[3] , less(algebraic[19]-algebraic[5] , 0.00000), (algebraic[8]-algebraic[19])/constants[2] , True, (algebraic[8]-algebraic[19])/constants[2]-(algebraic[19]-algebraic[5])/constants[3]]) rates[2] = custom_piecewise([less(algebraic[19]-algebraic[5] , 0.00000) & less(algebraic[5]-algebraic[6] , 0.00000), 0.00000 , less(algebraic[19]-algebraic[5] , 0.00000), -(algebraic[5]-algebraic[6])/constants[4] , less(algebraic[5]-algebraic[6] , 0.00000), (algebraic[19]-algebraic[5])/constants[3] , True, (algebraic[19]-algebraic[5])/constants[3]-(algebraic[5]-algebraic[6])/constants[4]]) rates[5] = custom_piecewise([less(algebraic[7]-algebraic[8] , 0.00000) & less(algebraic[8]-algebraic[19] , 0.00000), 0.00000 , less(algebraic[7]-algebraic[8] , 0.00000), -(algebraic[8]-algebraic[19])/constants[2] , less(algebraic[8]-algebraic[19] , 0.00000), (algebraic[7]-algebraic[8])/constants[5] , True, (algebraic[7]-algebraic[8])/constants[5]-(algebraic[8]-algebraic[19])/constants[2]]) return(rates) def computeAlgebraic(constants, states, voi): algebraic = array([[0.0] * len(voi)] * sizeAlgebraic) states = array(states) voi = array(voi) algebraic[2] = states[0]+states[1] algebraic[3] = constants[13]*(exp(constants[15]*(algebraic[2]-constants[14]))-1.00000) algebraic[4] = algebraic[3]+constants[8] algebraic[0] = voi % constants[12] algebraic[1] = constants[9]*exp(-constants[10]*(power((algebraic[0]*constants[6])/60.0000-constants[11], 2.00000))) algebraic[18] = algebraic[10]+algebraic[4] algebraic[6] = constants[34]*(states[3]-constants[35]) algebraic[7] = constants[36]*(states[4]-constants[37]) algebraic[5] = constants[32]*(states[2]-constants[33]) algebraic[8] = constants[38]*(states[5]-constants[39]) algebraic[19] = algebraic[15]+algebraic[4] algebraic[20] = algebraic[18]-algebraic[19] return algebraic initialGuess0 = None def rootfind_0(voi, constants, rates, states, algebraic): """Calculate values of algebraic variables for DAE""" from scipy.optimize import fsolve global initialGuess0 if initialGuess0 is None: initialGuess0 = ones(9)*0.1 if not iterable(voi): soln = fsolve(residualSN_0, initialGuess0, args=(algebraic, voi, constants, rates, states), xtol=1E-6) initialGuess0 = soln algebraic[9] = soln[0] algebraic[10] = soln[1] algebraic[11] = soln[2] algebraic[12] = soln[3] algebraic[13] = soln[4] algebraic[14] = soln[5] algebraic[15] = soln[6] algebraic[16] = soln[7] algebraic[17] = soln[8] else: for (i,t) in enumerate(voi): soln = fsolve(residualSN_0, initialGuess0, args=(algebraic[:,i], voi[i], constants, rates[:i], states[:,i]), xtol=1E-6) initialGuess0 = soln algebraic[9][i] = soln[0] algebraic[10][i] = soln[1] algebraic[11][i] = soln[2] algebraic[12][i] = soln[3] algebraic[13][i] = soln[4] algebraic[14][i] = soln[5] algebraic[15][i] = soln[6] algebraic[16][i] = soln[7] algebraic[17][i] = soln[8] def residualSN_0(algebraicCandidate, algebraic, voi, constants, rates, states): resid = array([0.0] * 9) algebraic[9] = algebraicCandidate[0] algebraic[10] = algebraicCandidate[1] algebraic[11] = algebraicCandidate[2] algebraic[12] = algebraicCandidate[3] algebraic[13] = algebraicCandidate[4] algebraic[14] = algebraicCandidate[5] algebraic[15] = algebraicCandidate[6] algebraic[16] = algebraicCandidate[7] algebraic[17] = algebraicCandidate[8] resid[0] = (algebraic[9]-(states[0]-algebraic[11])) resid[1] = (algebraic[10]-(algebraic[1]*algebraic[12]+(1.00000-algebraic[1])*algebraic[13])) resid[2] = (algebraic[12]-constants[16]*(algebraic[9]-constants[17])) resid[3] = (algebraic[13]-constants[18]*(exp(constants[19]*(algebraic[9]-constants[20]))-1.00000)) resid[4] = (algebraic[14]-(states[1]+algebraic[11])) resid[5] = (algebraic[15]-(algebraic[1]*algebraic[16]+(1.00000-algebraic[1])*algebraic[17])) resid[6] = (algebraic[16]-constants[21]*(algebraic[14]-constants[22])) resid[7] = (algebraic[17]-constants[23]*(exp(constants[24]*(algebraic[14]-constants[25]))-1.00000)) resid[8] = (algebraic[10]-((algebraic[1]*constants[26]*(algebraic[11]-constants[27])+(constants[31]-algebraic[1])*constants[28]*(exp(constants[29]*(algebraic[11]-constants[30]))-constants[31]))-algebraic[15])) return resid def custom_piecewise(cases): """Compute result of a piecewise function""" return select(cases[0::2],cases[1::2]) def solve_model(): """Solve model with ODE solver""" from scipy.integrate import ode # Initialise constants and state variables (init_states, constants) = initConsts() # Set timespan to solve over voi = linspace(0, 10, 500) # Construct ODE object to solve r = ode(computeRates) r.set_integrator('vode', method='bdf', atol=1e-06, rtol=1e-06, max_step=1) r.set_initial_value(init_states, voi[0]) r.set_f_params(constants) # Solve model states = array([[0.0] * len(voi)] * sizeStates) states[:,0] = init_states for (i,t) in enumerate(voi[1:]): if r.successful(): r.integrate(t) states[:,i+1] = r.y else: break # Compute algebraic variables algebraic = computeAlgebraic(constants, states, voi) return (voi, states, algebraic) def plot_model(voi, states, algebraic): """Plot variables against variable of integration""" import pylab (legend_states, legend_algebraic, legend_voi, legend_constants) = createLegends() pylab.figure(1) pylab.plot(voi,vstack((states,algebraic)).T) pylab.xlabel(legend_voi) pylab.legend(legend_states + legend_algebraic, loc='best') pylab.show() if __name__ == "__main__": (voi, states, algebraic) = solve_model() plot_model(voi, states, algebraic)