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Mathematical Modeling

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14-Nov-2014
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Universit degli Studi di Catania Facolt di Ingegneria Corso di Laurea Specialistica in Ingegneria dellAutomazione e del Controllo dei sistemi complessiCorso di Fondamenti di Bioingegneria Elettronica MATHEMATICAL MODELING OF BIOLOGICAL SYSTEMSLEPISCOPO GAETANO MAZZARA BOLOGNA GIUSEPPE ANNO ACCADEMICO 2006-2007 ELETTRICAL MODELS FOR BIOLOGICAL SYSTEMS Biological System Physical Laws Electrical Parameters State-Space Equations Transfer Function RESISTANCEResistance Resistive and dissipative properties of systemV I R Ohms Law V=RI V I Potential Current Ohms Law can be generalized, applying it to other systems Generalized Ohms Law y=Rz y z Generalized effort Generalized flow RESISTANCE EXAMPLESApplications of generalized Ohms law Mechanics damping law F=Rmv Fluidic Poiseuilles law P=RtQ1 Q P Fouriers Thermal transfer law =RtQ 2 Q 1 2 Q Chemical Ficks law of Diffusion =RcQ CAPACITANCECapacitance Capacitance Law Storage properties of systemV 1 V = idt C VI C Potential Current I Also Capacitance Law can be generalized, applying it to other systems Generalized Capacitance Law 1 y = zdt C y z Generalized effort Generalized flow CAPACITANCE EXAMPLESApplications of generalized Capacitance law Hookes Mechanics Compliance law 1 F= vdt CM Fluidic Compliance law V=CfPx F V P Thermal Heat storage law Q=Ct 1 2 =1-2 INERTANCEInertance Inductance Law Inertial properties of systemV L dI V =L dt V I Potential Current I Also inertance Law can be generalized, applying it to other systems Generalized Inductance Law dz y=L dt y z Generalized effort Generalized flow INERTANCE EXAMPLESApplications of generalized inertance law Newtons second lawma m F dv F =m dt Fluidic inertance law dQ P = LF dt Q P There is no element that represents inertance in thermal and chemical systems Exercise 1: 5-element Windkessel Model of aortic and arterial hemodynamics Rao Lao Cao Rp//Cp Viscous resistance of aortic wall Inertance to flow through aorta Compliance of aortic wall Modeling of the rest of arterial vasculature Exercise 1: 5-element Windkessel Model of aortic and arterial hemodynamicsState space equations:dQ R V P = ao Q C + ao dt Lao Lao LaodVC Q Pao = dt Cao + C P Rao ( Cao + C P )Pao Q, VC input Q output VC State space variables Transfer function:Q S ( RP Cao + RPC P ) + 1 = 2 Pao S ( RPCao Lao + RPC P Lao ) + S ( Lao + Rao R p Cao + Rao RPC P ) + Rao + R P Exercise 2: Equivalent electrical circuit of Hodgkin-Huxley model of neuronal electrical activity C Rk,Na,C1 Ek,Na,C1 Membrane capacitance Resistance of membrane to K,Na,C1 Nernst Potential of membrane for K,Na,C1 Exercise 2: Equivalent electrical circuit of Hodgkin-Huxley model of neuronal electrical activityEquations are:dV 1 1 1 E E E V + K Na + Cl I =C + + + dt RK RNa RCl RK RNa RCl IK = V + EK RK I Na V E Na = RNa V + ECl RCl I Cl = Exercise 3: Analysis of the respiratory mechanics model with effect of inertance to gas flow in central airwaysRP RC, LC, CS Cw CL Rc Lc Cs Resistance of central airways Compliance of central airways Rp Cw CL Resistance of peripheral airways Compliance of chest-wall Compliance of lung Inertance through central airways Exercise 3: Analysis of respiratory mechanics model with effect of inertance to gas flow in central airways Equations are:Pao = RC Q + LC dQ 1 + dt C S ( Q Q ) dtA 1 1 1 QA dt = RP QA + + C CS L CW ( Q Q ) dtA Reducing two equations to one, we obtain:d 2 Pao 1 dPao d 3Q LC d 2Q 1 R dQ 1 1 1 2 + + C + + = LC 3 + RC + 2 dt C R C dt R C C + C Q dt RPCT dt dt RP CT P T P S W L SWhere: 1 1 1 CT = + + C CW CS L 1 Exercise 3: Analysis of respiratory mechanics model with effect of inertance to gas flow in central airwaysUsed Simulink model is: Input Pao=sin(2b*60-1*t) cm H2O where b = breaths/min Output Q and Volume Fixed values for system parameters are: RC= 1 cm H2O L-1 RP= 0.5 cm H2O L-1 CL= 0.2 L cm H2O CW= 0.2 L cm H2O CS= 0.005 L cm H2O Exercise 3: Analysis of respiratory mechanics model with effect of inertance to gas flow in central airwaysSimulation for LC=0 cm H2O s2 L-1 Neglecting inertance Peak to peak amplitudes at 15 breaths/min: Q=0.127 L/s Volume=0.502 L Peak to peak amplitudes at 60 breaths/min: Q=0.504 L/s Volume=0.496 L Exercise 3: Analysis of respiratory mechanics model with effect of inertance to gas flow in central airwaysSimulation for LC=0.01 cm H2O s2 L-1 Taking inertance into account Peak to peak amplitudes at 15 breaths/min: Q=0.129 L/s Volume=0.515 L Peak to peak amplitudes at 60 breaths/min: Q=0.512 L/s Volume=0.509 L Exercise 3: Analysis of respiratory mechanics model with effect of inertance to gas flow in central airwaysSimulation for LC=0.01 cm H2O s2 L-1, CL=0.4 L cm H2O-1, RP=7.5 cm H2O s L-1 Subject with emphysema (higher lung compliance and higher peripheral airway resistance) Peak to peak amplitudes at 15 breaths/min: Q=0.166 L/s Volume=0.661 L Peak to peak amplitudes at 60 breaths/min: Q=0.457 L/s Volume=0.496 L Exercise 3: Analysis of respiratory mechanics model with effect of inertance to gas flow in central airwaysSummary of all simulations Conclusions: Both inertance-complete model and neglecting-inertance one have quite similar trends at all input frequencies. Emphysema model, instead, has a very different trend, particularly at high frequencies; infact its peak-to-peak amplitude both for air flow and for air volume is smaller as emphysema features outline.

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