le transport dans les voies aériennes respiratoires,

ou la géométrie fractale au service des fonctions physiologiques

Marcel Filoche

Physique de la Matière Condensée

Ecole Polytechnique, CNRS, Palaiseau

Séminaire Cristolien d’Analyse Multifractale, 21 février 2013

The pulmonary structure

Rat

Human

(Weibel)

At the end of each of the 30.000 bronchioles, one acinus

2D Cut

(Weibel)

300 million pulmonary alveoli

Red cell

1/4 mm

oxygen

(Weibel)

Our needs in oxygen

PD

D = diffusion constant of O2 in water,

= membrane thickness,

= Henry’s constant,

P = O2 partial pressure difference

12812

4

65

113125

s.mol.cm10.4s.mol.cm1010

10.3,110.3

atm760

80150.atmmol.l10.3,1

μm1

s.cm10.3

11

max, Kg.minml.402

OV !!m6010.4,2210.460

8040

10.4,22

60

80

2

383

max,2

OV

In humans, the alveolar surface is

about 100-150 m2 !

Our needs in oxygen

Gas transfer: diffusion across a 1 micron barrier (water)

Courtesy of E.R. Weibel)

A few simple calculations

100 m2 stored in about 10 L?

μm100

1

μm1010

μm10100

L10

m100

335

2262

2222m24.0m2.06cm206 S

Entrance velocity?

3

10

01.03

100m100tocm3from

6

2

22

11-

6

1- s.μm3m.s10

3tom.s1from

Limited time ??m.s100 -1Diffusion

Cross section:

Air velocity:

1 acinus ~ 8 generations 10 000 alveoli Gas exchange

O2

O2 O2

Geometrical complexity in the lung airway system

What are the transport properties of this system?

Why these specific shape and sizes?

We see the answer,

but what was the question?

A large number of small diffusion units

accessed by convection

30,000 acini

A conductive tree to access these acini

The tracheobronchial tree (Weibel)

Trees as optimal distribution systems

11

The scale invariance of the human airway system

The tracheobronchial tree

- purely conducting pipes, convective transport - dead space volume (DSV)

D(g) = D0 h0g

h0= 2-1/3≈ 0.79

(Weibel, 1963)

Simplest model: from Weibel’s “A” model - Cylindrical pipes - Self-similarity: uniform scaling ratio - Dichotomous and symmetric branching

Horse

Front

Large Dog Front

Cat Front

Cat Back

Pig Front

Pig Back

Pig Zoom

Camel Zoom

(Courtesy of E.R.Weibel)

The transport in the lung airway system

pulsatile,

scale invariant,

asymmetric

3 keywords:

Minimize energy in a vessel for a given flow rate

W.R.Hess, 1913 ; C.D. Murray, 1927; T.F. Sherman, 1981

The Murray-Hess law

79.02 31

0

1

D

Dhc

2

4

22 D

LLDRV

2

5420

D

LLD !!3 D

3

2

3

1

3

0210 DDD

In bifurcations:

The volume of the conductive tree

N

n

nN

n

nnn hVL

DV

0

30

0

2

24

2

ch1 0.5

h V

The airway resistance

ch1 0.5

Jean-Louis-Marie Poiseuille

(1797-1869)

N

nn

n

nN

nn

h

RD

LR

03

040 2

1

2

1128

R

4

128

D

LP

The transit time across the conductive tree

ch1 0.5

N

n

nnN

n nn

nnN

n n

n hDU

DLR

U

L

0

30

02

2

0

0

2

s06.0m.s1

cm61-0

s106.016 chh

The protection against external environment

The alveolar surface: About 100 m2 topologically outside!

Can one say anything simple on the particle capture?

Zhang et al., J. Aerosol Science (2001) Farkas et al., J. Aerosol Science (2007)

Comer et al., J. Fluid Mech. (2001) Ashgarian et al., J. Areosol Sci. (2006)

Understand and predict particle deposition?

The capture in a 4 generations tree

The multiplicative process

= 3x ?

4 generations 7 generations !

Re = 50

The mathematical model

Steady state Navier-Stokes equation

0)( udiv

Uncompressible flow

Puuu

Boundary conditions

- Non slip condition on the walls

- Uniform velocity profile at the entrance

- Free outlets with identical pressure

-3Kg.m18.1

-1-15 sKg.m1079.1

Particle transport : Stokes drag

ppdrag uudF

3

The dimensionless parameters

DuRe

ReD

d

D

udSt

pppp

22

1818

cm2

m.s1

Kg.m10.2

sKg.m1079.1

Kg.m18.1

1-

3-3

1-1-5

-3

D

u

p

Reynolds number:

Stokes number:

μmin ,103 24pp ddSt

Finite element simulations

A few 100,000 tetrahedra

Velocity and pressure map in the bifurcation

Velocity Pressure

Fluent™ software

Universal collapse

Tests on model geometries

Weibel’s “A” model of the tracheobronchial tree

4 (7) generation trees → 3 bifurcations

(including inertial effects and secondary flows)

Tuning parameters of the geometry:

- Scaling ratio, (default h = 0.79)

- Azimuthal angle, (default = 90°)

- Entrance velocity, (default Re=50)

The influence of the scaling ratio h

The influence of the branching angle

Independence between successive bifurcations

de Vasconcelos et al., J Appl Physiol, 2011

The structure as seen by the capture process

Inhomogeneous tree

60

90

790

3

2

1

.h

.h

.h

Azimuthal angle in inhomogeneous tree

Asymmetrical tree

“Analytical toolbox” of the particle capture

At each bifurcation: probability E(St) to cross the

bifurcation without being captured

1

1

1

N

i

iicapture StEP132

1 i

i

i Sth

St

Constant scaling ratio h :

Critical scaling ratio:

1

113

0

2

1

N

iicapture

h

StEP

101

Ncapture StEP

i

ipp

iD

udSt

18

2

31

2

ch

The protection against external environment

D

UdSt

pp

18

2

The alveolar surface: About 100 m2 topologically outside!

nnn

nn

n

nn

hD

DU

D

USt

33

2

2

1

Filtering inertial particles by impaction: chh

Multi constrained system

ch1 0.5

Volume too large

Resistance too large Time too long

Poor filtering

ch1 0.5

An optimal bronchial tree may be dangerous

Mauroy et al., Nature (2004)

Physicist’s “asthma”

If one adds a small layer of constant thickness to all bronchi

nnn

n

n

n

n

n

Dhh

DD

hD

D

hD

D

Dh

2

12

12

2

21

nD

hh

2

1

If h=0.85, then Δh=-0.05 is achieved for %166

1

15.0

05.0

2

1

nD

Variability of Raw

biological variability of the scaling factor h

Randomization

at each bifurcation

ch1 0.5

0.85

Too many constraints?

ch1 0.5 ch

1 0.5

50

Branching asymmetry in the tracheobronchial tree

hmin

hmax

[Majumdar et al., PRL 2005]

What are the ventilation properties of an asymmetric tree ?

hmin ≈ 0.67

hmax ≈ 0.87

Air flow ≠ Oxygen flow

Airway size distribution

Symmetric Weibel model:

Generation 7

0( ) g

tracheaD g h D

Identical diameters for all airways

in the same generation.

Identical generation number for

all airways of a given diameter.

Spreading the distribution of airway sizes in a given generation g :

Branching asymmetry

Stochastic fluctuations of the scaling ratios.

D = 2 mm

Generation D (mm)

tracheamgm DhhgD

minmax)(

A more precise description of the lung airway system

Proximal airways

Terminal bronchioles

Morphometry:

distribution of terminal generations

mm5.0D

Average diameter of terminal bronchioles

431514 GG

A realistic model of the airway sizes

Specific geometry of the proximal airways Small DSV, admissible transit time to acinar regions.

Self-similar & asymmetric intermediate TB tree Small hydrodynamic resistance, airway size distribution. Terminal airway: diameter of the terminal bronchioles D = 0.5 mm

Systematic branching asymmetry Florens et al., J Appl Physiol, 2011

Agreement with morphometric data

Teminal airways: 9 ≤ g ≤ 23 & <g> ≈ 15-16 ± 2 (15 ± 2-4)

Acini: ≈ 23000 (30000: 15000 – 61000) DSV: 153 mL (150 – 170 mL)

Weibel (1963) Horsfield,

J Appl Physiol (1971)

References

Flow

Time

Ventilation model

1- The air flow entering each acinus is assumed uniform and constant during inspiration (hydrodynamic pump).

2- Transit time from the entrance of the mouth to the entrance of the acinus.

3- Volume of fresh air delivered to the acinus.

acini

tracheaacinus

N

transit

term

t

g

g

gextraninspiratiooxygen tttt

0

g

g

i

itrachea

g

hV

t

31

0

otherwise0

0if oxygenoxygenacinusfresh

ttV

Symmetry versus asymmetry

Symmetric tree

Asymmetric tree

Similar delivery time Similar oxygenation of the acini

tox = 1 ± 0.07 s

tox = 1 s

100% of the acini are ventilated with fresh air during inspiration. Total ventilation in fresh air (250 mL) is close to the average physiological data (240 mL).

Asymmetry level: parameter α

Same tree volume for all α (DSV):

Tuning the asymmetry

Specific geometry of the proximal airways (L/D)

3 3 3

max min 0h h h

Terminal airways: diameter of the terminal bronchioles D = 0.5 mm (Different pathways may have different number of generations)

α = 36 % 87.01

67.0131

0min

310min

hh

hh

“Lung efficiency” vs asymmetry level

Asymmetry level in the human lung = Maximal asymmetry level that allows to feed all acini.

Asymmetry level: parameter α

31

0min

310max

1

1

hh

hh

Florens et al., PRL, 2011

Patchiness of the flux distribution

Patchiness of the flux distribution

79.0 chh 85.0h

10 generations tree, 5% fluctuation of the parameter h

The distribution of fluxes is inherently patchy

Ventilation heterogeneity

Ventilation heterogeneity is intrinsic of the lung structure.

Multifractal distribution of the ventilation at infinite generation.

Distribution of fresh air volumes delivered to the acini

Spatial distribution of fresh volumes delivered at generation 10

First level of 3D representation

Asymmetric branching

Branching angle: 180°

Angle of rotation of the branching planes: 90°

3D representation: volume of fresh air delivered by each terminal airway (mm3)

Multiplicative distribution in the TB Tree

Volume of polarized gas (mm3)

Distribution of polarized gas

Comparison: Model & Real Lung Images

From LKB and U2R2M (1999)

Human: Volume of fresh air (mm3)

S. Bayat, EJR (2008)

Rabbit: Ventilation

The heterogeneity of the gas distribution results from the lung structure, namely from the uneven splitting at each generation.

Geometrical complexity in the lung airway system

The system is highly constrained

All deterministic constraints come to the same (!)

critical value, which corresponds to Murray’s law.

The fitness landscape is much narrower than one could

think → very high sensitivity in the system.

When variability is added, no simple scale invariant

tree is able to fulfill all constraints → such a system

would require regulation.

For pulsatile distribution trees of uniform depth,

branching asymmetry reduces the average delivery

time, and smoothens the oxygen delivery.

Asymmetry increases the robustness vs structure

variability.

The asymmetry level seems to be set at the maximum

value that allows to feed all acini.

Intrinsic patchiness due to the tree structure and

heavily depends on the working conditions.

B. Sapoval, A. Foucquier, A. Soualah (Ecole Polytechnique)

E.R. Weibel (Bern Universität, Switzerland)

B. Mauroy, M. Florens, L. Desvillettes, A. Moussa (CMLA, ENS Cachan)

T. Similowski, C. Straus, A. Pradel (ER10, Université Pierre et Marie Curie)

J.S. Andrade Jr., T. Felipe de Vasconcelos (Universidade Federal do Ceara, Brazil)

J.B. Grotberg (Univ. of Michigan), T. Hossein (Univ. of Akron)

Daniel Isabey, B. Louis (Université Paris-Est, INSERM)

Joint work with

This work is supported by the ANR program SAMOVAR ANR-2010-BLAN-1119.