Modelling of Mammalian Lungs with Hysteresis operators Denis Flynn - - PowerPoint PPT Presentation

Modelling of Mammalian Lungs with Hysteresis

• perators

Denis Flynn

MURPHYS-HSFS 2016 WORKSHOP

Overview

— Mammalian lungs — Hysteresis in the Pressure-Volume Rela9onship — Mathema9cal Modelling — Results — Further work

Physiology of the lungs

Source: Wikicommons: http://upload.wikimedia.org/wikipedia/commons/d/db/Illu_bronchi_lungs.jpg

Hysteresis:

Volume-Pressure curves of the respiratory system

Adapted from: Harris R.S. Pressure-Volume curves of the respiratory system, Respiratory care 50 (2005), 78-79

Motivations for studying the PV relationship

• Help with Diagnosis
• Acute Lung Injury
• Acute Respiratory Distress Syndrome
• Set parameters for ventilators
• Aid understanding physiology of the lungs

Possible mechanisms giving rise to Hysteresis in Lungs

• Opening and closing of alveoli and/or

airways

• Non elastic behavior of the lung tissue
• Surfactant lining the alveoli
• Disease states:
• Hysteresis is more pronounced in certain

diseases.

Surfactant lining the alveoli

Without Surfactant With Surfactant

b

R 1 R 2

a

1 2 1 2 1 2

0.1mm, 0.05mm T 0.72N/m, 0.72N/m 1440N/m, 2880N/m R R T P P = = = = = =

1 2 1 2 1 2

0.1mm, 0.05mm T 0.02N/m, 0.01N/m 400N/m, 400N/m R R T P P = = = = = =

2 :

recoil

Young Laplace T P r − =

Reference: http://www.medicine.mcgill.ca/physio/resp-web/TEXT3.htm

Evidence that surfactant plays the major role in hysteresis

Source: Harris R.S. Pressure-Volume curves of the respiratory system, Respiratory care 50 (2005), 78-79

Mathema9cal Modelling

• Venegas’ four parameter model:

Parameters: a,b,c and d. V is volume and p is pressure.

V (p) = a + b (1 + e(p−c)/d)

Preisach Model Density

• Can easily recover f(α):
• Provided

max

'( ) ( ) f α ρ α β α = −

max

( ) ( )

x

f x d d

β α

ρ α β α = ∫ ∫

(0) f =

Preisach Model

• Exhalation curve:

Vmax = f (αmax) = ρ(α)

α βmax

∫

αmax

∫

dβ dα

β α β=α (α , β )

max max

Preisach Model

• Following the exhalation curve f(x)

max

( ) ( )

x

f x d d

β α

ρ α β α = ∫ ∫

β α x(t) β=α (α , β )

max max

Preisach Model

• On switching direction, we can calculate

inhalation curve:

1

inhalation 1

( ) ( ) ( )

x x

f x f d d

α α

α ρ α β α = + ∫∫

max max

x(t) β α

1

(α ,β )

1

β=α (α , β )

Restricted Preisach Model

ρ(α) = f '(α) βmax − w(α)

f (x) = ρ(α)

w(α) βmax

∫

x

∫

dβ dα

w(α)

Restricted Preisach Model

ρ(α) = f '(α) βmax − w(α)

f (x) = ρ(α)

w(α) βmax

∫

x

∫

dβ dα

• Chose simple w(α):

w(α) =α +e

Model 1

Exhaling: Preisach density: Where:

f (x) = a + b 1+e

c−x d

ρ(x) = f '(x) (βmax − x)

ρ(x) = bSech2(c − x 2d ) 4d(βmax − x)

Model 2 (restricted Preisach)

Exhaling: Preisach density: Where:

f (x) = a + b 1+e

c−x d

ρ(x) = f '(x) (βmax −(x +e))

ρ(x) = bSech2 c−x

2d

• 4d(x + e − βmax)

Model 1 Model 2

Results

Cannine Lungs Murnine Lungs Model 1: 0.47 0.58 Model 2 0.32 0.45

Error = SSE VMax

Further work

• Modelling rate dependent hysteresis
• Development of alterna9ve operators

Rate Dependence: Time series

Source: Harris R.S. Pressure-Volume curves of the respiratory system, Respiratory care 50 (2005), 78-79

Rate dependence: PV Loops

Source: Harris R.S. Pressure-Volume curves of the respiratory system, Respiratory care 50 (2005), 78-79

Rate Dependence

• Speed of input changes PV loops
• Previous work by Bates & Irvin [2]
• Approach using Dynamic Preisach Model[1]

[1] Mayergoyz, Isaak D. Mathematical models of hysteresis and their applications. Academic Press, 2003. [2] Bates, Jason HT, and Charles G. Irvin. "Time dependence of recruitment and derecruitment in the lung: a theoretical model." Journal of Applied Physiology 93.2 (2002): 705-713.

Dynamic Preisach Model

• Density:
• Series expansion:
• ρ(α, β, dV

dt )

ρ(α, β, p(t), ˙ V (t)) = ρ0(α, β, p(t)) + ˙ V (t)ρ1(α, β, p(t)) + · · ·

V (t) = P0[p(t)] + ˙ V P1[p(t)]

Other Hysteresis Operators

Piecewise defini9on of operator

V (x) = { A +

B e−Kp+1,

η = 1 V,

• therwise

Discrete Model: Rate Independent

50 hysterons

• Forcing func9on
• Forcing func9on

20(0.9+ Sin(3t - π /2)) 20(0.9+ Sin(0.2 t - π /2))

Thank you!