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
“Generalized effort”
“Generalized flow”
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
“Generalized effort”
“Generalized flow”
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:
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
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
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
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
Peak to peak amplitudes Peak to peak amplitudes at 60 breaths/min:at 60 breaths/min:Q=0.504 L/sVolume=0.496 L
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
Simulation for LC=0.01 cm H2O s2 L-1 Taking inertance into account
Peak to peak amplitudes Peak to peak amplitudes at 15 breaths/min:at 15 breaths/min:Q=0.129 L/sVolume=0.515 L
Peak to peak amplitudes Peak to peak amplitudes at 60 breaths/min: at 60 breaths/min: Q=0.512 L/sVolume=0.509 L
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
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 15 breaths/min:at 15 breaths/min:Q=0.166 L/sVolume=0.661 L
Peak to peak amplitudes Peak to peak amplitudes at 60 breaths/min: at 60 breaths/min: Q=0.457 L/sVolume=0.496 L
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
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.