Denis S. Grebenkov Laboratoire de Physique de la Matière Condensée

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

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

Pulmonary system O2 O2 O2 1 acinus ~ 8 generations 10 000 alveoli
Gas exchange units O2 O2

c = 0 D nc = Wc c = c0 Pulmonary system in the bulk on the boundary
Dichotomic branching Densely filling the volume With a large surface area for oxygen transfer to blood O2 1 acinus ~ 8 generations alveoli Gas exchange units in the bulk on the boundary at the entrance c = 0 D nc = Wc c = c0 O2 O2

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 Kitaoka et al. J. Appl. Physiol. 88, 2260 (2000) 3 mm

Idea: to fill densely a given volume with a branching structure
Kitaoka model Idea: to fill densely a given volume with a branching structure

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

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

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

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1 • the previously current box takes the largest index + 1 • the new box takes the largest index + 2

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

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

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

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

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

Kitaoka model 2D labyrinth its skeleton
Felici et al., PRL 92, (2004); Grebenkov et al., PRL 94, (2005)

Kitaoka model Is this model geometry accurate enough?
Since there is no memory, diffusion averages out the effect of the specific geometrical features of the domain DIFFUSION IS NOT SENSITIVE TO LOCAL GEOMETRICAL DETAILS

Oxygen diffusion and lung efficiency Toward lung imaging and understanding

Finite element resolution
c = 0 in the bulk D nc = Wc on the boundary c = c0 at the entrance solving the discretized equations… Felici et al. J. Appl. Physiol. 94, 2010 (2003) Felici et al. Phys. Rev. Lett. 92, (2004) Felici et al. Resp. Physiol. Neurob. 145, 279 (2005)

Diffusion on a skeleton tree
2D labyrinth its skeleton Felici et al., Phys. Rev. Lett. 92, (2004)

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, (2005).

One branch analysis Dnc = Wc c0 a c = 0 c0 al Continuous problem
Discrete problem =(1+a/)-1 =D/W Dnc = Wc ½(ck-1+ck+1)-ck = ck c0 a c = 0 c0 … 1 2 l-1 l l+1 al ext = Wcl+1 Grebenkov et al., PRL 94, (2005)

One branch analysis ext = D cl+1/ ent = D c0/’ ul+a
Continuous problem Discrete problem =(1+a/)-1 =D/W Dnc = Wc ½(ck-1+ck+1)-ck = ck c0 a c = 0 c0 … 1 2 l-1 l l+1 ext = D cl+1/ ent = D c0/’ al ext = Wcl+1 ’ = fl() = a ul+a (u2l- v2l) + aul Grebenkov et al., PRL 94, (2005)

Branch-by-branch computation
… 1 2 l1-1 l1 l1+1 … l-1 l l+1 … 1 2 l2-1 l2 l2+1 Grebenkov et al., PRL 94, (2005)

1ent = D c0/fl () 1 1ext = D cl +1/ 1 … 1 2 l1-1 l1 l1+1 … l-1 l 2ent = D c0/fl () 2 l+1 … 1 2 l2-1 l2 l2+1 2ext = D cl +1/ 2 At branching point: = D/’ ext cl+1 = D[1/fl ()+1/fl ()] 2 1 ext = 1ent+2ent cl+1= c10 = c20 Grebenkov et al., PRL 94, (2005)

1ent = D c0/fl () 1 1ext = D cl +1/ 1 ext = D cl+1/’ … l-1 l l+1 2ent = D c0/fl () 2 2ext = D cl +1/ 2 At branching point: = D/’ ext cl+1 = D[1/fl ()+1/fl ()] 2 1 ext = 1ent+2ent cl+1= c10 = c20 Grebenkov et al., PRL 94, (2005)

Symmetric trees m=2 1 =  = 1 fl () 1= fl ()        
k fl () = =  k=1 m 1= fl ()         Grebenkov et al., PRL 94, (2005)

Symmetric trees m=2 1 =  = 1 fl () 1= fl () 2= fl (1) 1 1
k=1 m 1= fl () 2= fl (1) 1 1 1 1 Grebenkov et al., PRL 94, (2005)

Branching structures are robust against permeability change
Symmetric trees m=2 1 1 fl () = =  k=1 m Branching structures are robust against permeability change 1= fl () 2= fl (1) 2 2 … Total flux at the root: () = D c0/n a(l+1) m-1 n n= fl (fl (fl (…fl ()…))) Grebenkov et al., PRL 94, (2005)

Application to human acini
Haefeli-Bleuer and Weibel, Anat. Rec. 220, 401 (1988)

Human acinus Approximation by a symmetric tree of the same total area
of the same average length of branches of the same branching order (m=2) Grebenkov et al., PRL 94, (2005)

Human acinus Grebenkov et al., PRL 94, (2005)

Oxygen diffusion and lung efficiency Toward lung imaging and understanding

Schematic principle of NMR
Static magnetic field B0 x z y local magnetization 90° rf pulse Grebenkov, Rev. Mod. Phys. 79, 1077 (2007)

Static magnetic field B0 x z y Phase at time T Grebenkov, Rev. Mod. Phys. 79, 1077 (2007)

Static magnetic field B0 Inhomogeneous magnetic field x z y x z y Phase at time T Grebenkov, Rev. Mod. Phys. 79, 1077 (2007)

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)

Healthy acinus Fixed gradient direction
Grebenkov et al., JMR 184, 143 (2007)

Healthy acinus Averaged gradient direction
Grebenkov et al., JMR 184, 143 (2007)

Emphysematous acini How can one model emphysematous acini?
enlargement of the alveolar ducts Emphysema may lead to partial destruction of the alveolar tissue

Emphysematous acini Emphysema may lead to partial destruction of
the internal alveolar tissue

Emphysematous acini ν = 0.6 ν = 0.5 ν = 0.4 ν = 0.3 ν = 0

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

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

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

Perspectives Further theoretical, numerical and experimental study of restricted diffusion in branched or porous structures are important

Thank you for your attention!!!
If you see this slide, the talk is about to end… sorry

What shall we do during “discussion” 4:00 – 4:50?
Please do not leave!!!

Lung imaging with helium-3
Normal volunteer Can one make a reliable diagnosis at earlier stage? Healthy smoker Patient with severe emphysema van Beek et al. JMRI 20, 540 (2004)

Human acinus Haefeli-Bleuer and Weibel, Anat. Rec. 220, 401 (1988)

Denis S. Grebenkov Laboratoire de Physique de la Matière Condensée

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