Diffusion, reaction, and spinecho signal attenuation in branched structures

1
Diffusion, reaction, and spin-echo signal
attenuation in branched structures

• Denis S. Grebenkov
• Laboratoire de Physique de la Matière Condensée
• CNRS Ecole Polytechnique, Palaiseau, France

Workshop IV Optimal Transport in the Human Body
Lungs and Blood , 22 May 2008, Los Angeles, USA
2
Outline of the talk

• Branched structure of the lung acinus
• Oxygen diffusion and lung efficiency
• Toward lung imaging and understanding

E. Weibel
H. Kitaoka and co-workers
B. Sapoval and M. Filoche
M. Felici (PhD thesis)
G. Guillot
3
Pulmonary system
4
Pulmonary system

• Dichotomic branching
• Densely filling the volume
• With a large surface area for oxygen transfer to
  blood

5
Geometrical model of the acinus

• Dichotomic branching
• Filling of a given volume
• Controlled surface area and other physiological
  scales
• Random realizations
• Simplicity for numerical use

6
Kitaoka model
Idea to fill densely a given volume with a
branching structure
7
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
the previously current box takes the largest
index 1
the new box takes the largest index 2
8
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
9
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
10
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
11
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
12
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
the previously current box takes the largest
index 1
the new box takes the largest index 2
13
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
2
the previously current box takes the largest
index 1
the new box takes the largest index 2
14
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
2
the previously current box takes the largest
index 1
the new box takes the largest index 2
15
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
2?1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
16
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
17
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
3
2
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
18
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
2
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
19
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
2
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
20
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
2
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
21
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
2
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
22
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
2?1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
23
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
24
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
2
the previously current box takes the largest
index 1
the new box takes the largest index 2
25
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
3
2
the previously current box takes the largest
index 1
the new box takes the largest index 2
26
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
3
2
the previously current box takes the largest
index 1
the new box takes the largest index 2
27
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
1
3
2
the previously current box takes the largest
index 1
the new box takes the largest index 2
28
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
3
when chosen, suppress 1, shift other indexes by
-1
2
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
29
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
3
when chosen, suppress 1, shift other indexes by
-1
2
1
the previously current box takes the largest
index 1
the new box takes the largest index 2
30
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
2
when chosen, suppress 1, shift other indexes by
-1
1
3
the previously current box takes the largest
index 1
the new box takes the largest index 2
31
Kitaoka model
Idea to fill densely a given volume with a
branching structure
the current box is always 1
when chosen, suppress 1, shift other indexes by
-1
the previously current box takes the largest
index 1
the new box takes the largest index 2
32
Kitaoka model
2D labyrinth
its skeleton
Felici et al., PRL 92, 068101 (2004) Grebenkov
et al., PRL 94, 050602 (2005)
33
Kitaoka model
2D labyrinth
its skeleton
Felici et al., PRL 92, 068101 (2004) Grebenkov
et al., PRL 94, 050602 (2005)
34
Kitaoka model
35
Outline of the talk

• Branched structure of the lung acinus
• Oxygen diffusion and lung efficiency
• Toward lung imaging and understanding

36
Finite element resolution
in the bulk
?c 0
D ?nc Wc
on the boundary
at the entrance
c c0
solving the discretized equations
Felici et al. J. Appl. Physiol. 94, 2010 (2003)
Felici et al. Phys. Rev. Lett. 92, 068101 (2004)
Felici et al. Resp. Physiol. Neurob. 145, 279
(2005)
37
Diffusion on a skeleton tree
2D labyrinth
its skeleton
Felici et al., Phys. Rev. Lett. 92, 068101 (2004)
38
A step further analytical theory
Diffusion-reaction on a tree can be solved
analytically using a branch-by-branch
computation
Grebenkov et al., PRL 94, 050602 (2005).
39
One branch analysis
Continuous problem
Discrete problem
D?nc Wc
½(ck-1ck1)-ck ?ck
?c 0
c0
c0
a
?ext Wcl1
Grebenkov et al., PRL 94, 050602 (2005)
40
One branch analysis
Continuous problem
Discrete problem
D?nc Wc
½(ck-1ck1)-ck ?ck
?c 0
c0
c0
a
?ext Wcl1
Grebenkov et al., PRL 94, 050602 (2005)
41
Branch-by-branch computation

0
1
2
l1-1
l1
l11

l-1
l
l1

1
2
l2-1
l2
l21
0
Grebenkov et al., PRL 94, 050602 (2005)
42
Branch-by-branch computation

1
2
l1-1
l1
l11
0

l-1
l
l1

1
2
l2-1
l2
l21
At branching point
Grebenkov et al., PRL 94, 050602 (2005)
43
Branch-by-branch computation
?ext D cl1/?

l-1
l
l1
At branching point
Grebenkov et al., PRL 94, 050602 (2005)
44
Symmetric trees
m2
?1 f?l (?)
?
?
?
?
?
?
?
?
Grebenkov et al., PRL 94, 050602 (2005)
45
Symmetric trees
m2
?1 f?l (?)
?2 f?l (?1)
?1
?1
?1
?1
Grebenkov et al., PRL 94, 050602 (2005)
46
Symmetric trees
m2
?1 f?l (?)
?2 f?l (?1)
?2
?2

?n f?l (f?l (f?l (f?l (?))))
Grebenkov et al., PRL 94, 050602 (2005)
47
Application to human acini
Haefeli-Bleuer and Weibel, Anat. Rec. 220, 401
(1988)
48
Human acinus
Approximation by a symmetric tree

• of the same total area
• of the same average length of branches
• of the same branching order (m2)

Grebenkov et al., PRL 94, 050602 (2005)
49
Human acinus
Grebenkov et al., PRL 94, 050602 (2005)
50
Outline of the talk

• Branched structure of the lung acinus
• Oxygen diffusion and lung efficiency
• Toward lung imaging and understanding

51
Schematic principle of NMR
Static magnetic field B0
90 rf pulse
Grebenkov, Rev. Mod. Phys. 79, 1077 (2007)
52
Schematic principle of NMR
Static magnetic field B0
Grebenkov, Rev. Mod. Phys. 79, 1077 (2007)
53
Schematic principle of NMR
Static magnetic field B0
Grebenkov, Rev. Mod. Phys. 79, 1077 (2007)
54
Monte Carlo simulations

• Spin trajectory Xt is modeled as a sequence of
  normally distributed random jumps, with
  reflections on the boundary of the acinus

Grebenkov et al., JMR 184, 143 (2007)
55
Healthy acinus
Fixed gradient direction
Grebenkov et al., JMR 184, 143 (2007)
56
Healthy acinus
Fixed gradient direction
Grebenkov et al., JMR 184, 143 (2007)
57
Healthy acinus
Averaged gradient direction
Grebenkov et al., JMR 184, 143 (2007)
58
Emphysematous acini
How can one model emphysematous acini?
59
Emphysematous acini
Emphysema may lead to partial destruction of the
internal alveolar tissue
60
Emphysematous acini
? 0.6
? 0.5
? 0.4
? 0.3
? 0
61
Conclusions
?Branching structures present peculiar properties
which have to be taken into account to understand
the lungs
?The Kitaoka model of the acinus is geometrically
realistic and particularly suitable for numerical
simulations
62
Conclusions
?Oxygen diffusion can be studied on the skeleton
tree of the model or realistic human acinus a
crucial simplification!
?Diffusion-reaction on a tree can be solved using
a branch-by-branch trick which allows for very
fast computation and derivation of analytical
results
63
Conclusions
?Tree structures are robust against the change of
the permeability (mild edema)
?Partial destruction of branched structure by
emphysema can potentially be detected in
diffusion-weighted magnetic resonance imaging
experiments with HP helium-3
64
Perspectives
Further theoretical, numerical and experimental
study of restricted diffusion in branched or
porous structures are important
65
Thank you for your attention!!!
denis.grebenkov_at_polytechnique.edu
http//pmc.polytechnique.fr/pagesperso/dg
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Lung imaging with helium-3
Normal volunteer
Healthy smoker
Patient with severe emphysema
van Beek et al. JMRI 20, 540 (2004)
69
Human acinus
Haefeli-Bleuer and Weibel, Anat. Rec. 220, 401
(1988)