Mathematical Modeling

MATHEMATICAL MODELINGMATHEMATICAL MODELINGOF BIOLOGICAL SYSTEMSOF BIOLOGICAL SYSTEMS

L’EPISCOPO GAETANOL’EPISCOPO GAETANO

MAZZARA BOLOGNA GIUSEPPEMAZZARA BOLOGNA GIUSEPPE

ANNO ACCADEMICO 2006-2007

Università degli Studi di CataniaUniversità degli Studi di CataniaFacoltà di IngegneriaFacoltà di Ingegneria

Corso di Laurea Specialistica in Ingegneria Corso di Laurea Specialistica in Ingegneria dell’Automazione e del Controllo dei sistemi dell’Automazione e del Controllo dei sistemi

complessicomplessi

Corso di Fondamenti di Bioingegneria ElettronicaCorso di Fondamenti di Bioingegneria Elettronica

ELETTRICAL MODELS FOR BIOLOGICAL SYSTEMSELETTRICAL MODELS FOR BIOLOGICAL SYSTEMS

BiologicalBiological SystemSystem

Physical LawsPhysical Laws

Electrical ParametersElectrical Parameters

State-SpaceState-SpaceEquationsEquations

TransferTransferFunctionFunction

RESISTANCERESISTANCE

Ohm’s LawV=RI

ResistanceResistance Resistive and dissipative properties of system

V

I

Potential

Current

Generalized Ohm’s Lawy=Rz

y

z

“Generalized effort”

“Generalized flow”

Ohm’s Law can be generalized, applying it to other systems

I

V

R

RESISTANCE EXAMPLESRESISTANCE EXAMPLES

Mechanics damping law Mechanics damping law F=Rmv

Fluidic Poiseuille’s law Fluidic Poiseuille’s law ΔP=RtQ

Fourier’s Thermal transfer law Fourier’s Thermal transfer law Δθ=RtQ

Q ΔP

Q

θ1 θ2

Applications of generalized Ohm’s lawApplications of generalized Ohm’s law

Q

φ1 φ2

Chemical Fick’s law of Diffusion Chemical Fick’s law of Diffusion ΔΦ=RcQ

CAPACITANCECAPACITANCE

Capacitance LawCapacitance Law

Generalized Capacitance LawGeneralized Capacitance Law

CapacitanceCapacitance Storage properties of system

idtC

V1

zdtC

y1

V

I

Potential

Current

y

z



I

V

C

Also Capacitance Law can be generalized, applying it to other systems

CAPACITANCE EXAMPLESCAPACITANCE EXAMPLES

Hooke’s Mechanics Compliance lawHooke’s Mechanics Compliance law

Fluidic Compliance law Fluidic Compliance law ΔV=CfΔP

Thermal Heat storage law Thermal Heat storage law Q=Ct Δθ

vdtC

FM

1 F

x

ΔV

ΔP

Applications of generalized Capacitance lawApplications of generalized Capacitance law

θ1 θ2

Δ θ=θ1-θ2

INERTANCEINERTANCE

Inductance LawInductance Law

Generalized Inductance LawGeneralized Inductance Law

InertanceInertance Inertial properties of system

V

I

Potential

Current

y

z



dt

dILV

dt

dzLy

Also inertance Law can be generalized, applying it to other systems

I

V

L

INERTANCE EXAMPLESINERTANCE EXAMPLES

Newton’s second lawNewton’s second law

Fluidic inertance lawFluidic inertance law

There is no element that represents inertance in thermal and chemical systems There is no element that represents inertance in thermal and chemical systems

Applications of generalized inertance lawApplications of generalized inertance law

dt

dvmF

Fm

ma

dt

dQLP F

Q ΔP

Exercise 1: 5-element Windkessel Exercise 1: 5-element Windkessel Model of aortic and arterial Model of aortic and arterial

hemodynamicshemodynamics

RaoViscous resistance of aortic wall

Inertance to flow through aorta

Rp//Cp

Compliance of aortic wall

Modeling of the rest of arterial vasculature

Lao

Cao

Exercise 1: 5-element Windkessel Exercise 1: 5-element Windkessel Model of aortic and arterial Model of aortic and arterial

hemodynamicshemodynamics

State space equations:State space equations:

ao

ao

ao

C

ao

ao

L

P

L

VQ

L

R

dt

dQ

Paoao

ao

Pao

C

CCR

P

CC

Q

dt

dV

PaoPPaoaopaoaoaoPPaoaoP

PPaoP

ao RRCRRCRRLSLCRLCRS

CRCRS

P

Q

2

1

VC

Transfer function:Transfer function:

QPao input output

Q, VC State space variables

Exercise 2: Equivalent electrical circuit of Exercise 2: Equivalent electrical circuit of Hodgkin-Huxley model of neuronal electrical Hodgkin-Huxley model of neuronal electrical

activityactivity

CRk,Na,C1

Membrane capacitance

Resistance of membrane to K,Na,C1

Nernst Potential of membrane for K,Na,C1 Ek,Na,C1

Exercise 2: Equivalent electrical circuit of Exercise 2: Equivalent electrical circuit of Hodgkin-Huxley model of neuronal electrical Hodgkin-Huxley model of neuronal electrical

activityactivity

Equations are:Equations are:

K

KK R

EVI

Na

NaNa R

EVI

Cl

ClCl R

EVI

Cl

Cl

Na

Na

K

K

ClNaK R

E

R

E

R

EV

RRRdt

dVCI

111

Exercise 3: Analysis of the respiratory mechanics Exercise 3: Analysis of the respiratory mechanics model with effect of inertance to gas flow in central model with effect of inertance to gas flow in central

airwaysairways

Rc

Lc

Resistance of central airways

Inertance through central airways

CL

Rp

Compliance of chest-wall

Resistance of peripheral airways

Cs

Cw

Compliance of lungCompliance of central airways

RC, LC, CS

Rp

RP

Cw

CL

Exercise 3: Analysis of respiratory mechanics Exercise 3: Analysis of respiratory mechanics model with effect of inertance to gas flow in central model with effect of inertance to gas flow in central

airwaysairways

Equations are:Equations are:

QCCCRdt

dQ

CR

R

Cdt

Qd

CR

LR

dt

QdL

dt

dP

CRdt

Pd

LWSPTP

C

STP

CCC

ao

TP

ao

111112

2

3

3

2

2

1111

SWLT CCCCWhere:Where:

dtQQCdt

dQLQRP A

SCCao

1

dtQQ

CdtQ

CCQR A

SA

WLAP

111

Reducing two equations to one, we obtain:Reducing two equations to one, we obtain:


airwaysairways

Used Simulink model is:Used Simulink model is:

InputInput Pao=sin(2πb*60-1*t) cm H2O

OutputOutput Q and Volume

Fixed values for system Fixed values for system parameters are:parameters are:

RC= 1 cm H2O L-1

RP= 0.5 cm H2O L-1

CL= 0.2 L cm H2OCW= 0.2 L cm H2OCS= 0.005 L cm H2O

where b = breaths/min


airwaysairways

Simulation for LC=0 cm H2O s2 L-1 Neglecting inertance

Peak to peak amplitudes Peak to peak amplitudes at 15 breaths/min:at 15 breaths/min:Q=0.127 L/sVolume=0.502 L



airwaysairways

Simulation for LC=0.01 cm H2O s2 L-1 Taking inertance into account


Peak to peak amplitudes Peak to peak amplitudes at 60 breaths/min: at 60 breaths/min: Q=0.512 L/sVolume=0.509 L


airwaysairwaysSimulation 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 Peak to peak amplitudes at 60 breaths/min: at 60 breaths/min: Q=0.457 L/sVolume=0.496 L


airwaysairways

Summary of all simulationsSummary of all simulations

Conclusions: 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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Mathematical Modeling