Transcript otws4 7875
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 E. Weibel H. Kitaoka and co-workers Oxygen diffusion and lung efficiency B. Sapoval and M. Filoche M. Felici (PhD thesis) Toward lung imaging and understanding G. Guillot Pulmonary system O2 O2 O2 1 acinus ~ 8 generations 10 000 alveoli Gas exchange units Dichotomic branching Pulmonary system Densely filling the volume With a large surface area for oxygen transfer to blood O2 in the bulk c = 0 O O D nc = Wc on the boundary 1 acinus ~ 8 generations 10 000 alveoli c = c0 at the entrance Gas exchange units 2 2 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) 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 • 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 • 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 • 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 • 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 21 • 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 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 • 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 • 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 • 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 • 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 21 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 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 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 1 3 2 • 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 1 3 2 • 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 1 3 2 • 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 3 2 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 3 2 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 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 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 2D labyrinth its skeleton Felici et al., PRL 92, 068101 (2004); Grebenkov et al., PRL 94, 050602 (2005) Kitaoka model 2D labyrinth its skeleton Felici et al., PRL 92, 068101 (2004); Grebenkov et al., PRL 94, 050602 (2005) Kitaoka model 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 Is this model geometry accurate enough? Outline of the talk Branched structure of the lung acinus 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, 068101 (2004) Felici et al. Resp. Physiol. Neurob. 145, 279 (2005) Diffusion on a skeleton tree 2D labyrinth Felici et al., Phys. Rev. Lett. 92, 068101 (2004) its skeleton 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). One branch analysis Continuous problem =D/W Dnc = Wc c0 a c = 0 al Discrete problem =(1+a/)-1 ½(ck-1+ck+1)-ck = ck c0 Grebenkov et al., PRL 94, 050602 (2005) 0 1 2 … l-1 l l+1 ext = Wcl+1 One branch analysis Continuous problem =D/W Dnc = Wc c0 a c = 0 al Discrete problem =(1+a/)-1 ½(ck-1+ck+1)-ck = ck c0 0 1 2 … l-1 l ext = Wcl+1 ent = D c0/’ ul+a ’ = fl() = a (u2l- v2l) + aul ext = D cl+1/ Grebenkov et al., PRL 94, 050602 (2005) l+1 Branch-by-branch computation … 0 l-1 l 1 2 l+1 0 1 2 Grebenkov et al., PRL 94, 050602 (2005) … … l1-1 l1 l1+1 l2-1 l2 l2+1 Branch-by-branch computation 1ent = D c0/fl 1() … l-1 l 0 l+1 1 1 2 2 2ent = D c0/fl 2() At branching point: ext = 1ent+2ent cl+1= c10 = c20 1ext = D cl +1/ … … 1 l1-1 l1 l1+1 l2-1 l2 l2+1 2ext = D cl +1/ 2 ext = D[1/f ()+1/f ()] l l cl+1 Grebenkov et al., PRL 94, 050602 (2005) 1 2 = D/’ Branch-by-branch computation 1ent = D c0/fl 1() … l-1 l l+1 ext = 1ent+2ent cl+1= c10 = c20 1 ext = D cl+1/’ 2ent = D c0/fl 2() At branching point: 1ext = D cl +1/ 2ext = D cl +1/ 2 ext = D[1/f ()+1/f ()] l l cl+1 Grebenkov et al., PRL 94, 050602 (2005) 1 2 = D/’ Symmetric trees m 1 m 1 1 = = = 1 k=1 fl () fl () fl () m=2 k 1= fl () Grebenkov et al., PRL 94, 050602 (2005) Symmetric trees m 1 m 1 1 = = = 1 k=1 fl () fl () fl () m=2 1= fl () 2= fl (1) 1 1 1 1 Grebenkov et al., PRL 94, 050602 (2005) Symmetric trees m 1 m 1 1 = = = 1 k=1 fl () fl () fl () m=2 structures 1= Branching fl () against 2= flare (1robust ) 2 2 …permeability change Total flux at the root: () = D c0/n a(l+1) n= fl (fl (fl (…fl ()…))) n m-1 Grebenkov et al., PRL 94, 050602 (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, 050602 (2005) Human acinus 1 10 symmetric acinus ( ) real acinus 0 10 -1 10 -2 10 /L -3 10 -2 10 -1 10 0 10 Grebenkov et al., PRL 94, 050602 (2005) 1 10 2 10 p Outline of the talk Branched structure of the lung acinus Oxygen diffusion and lung efficiency Toward lung imaging and understanding Schematic principle of NMR Static magnetic field B0 z y x local magnetization 90° rf pulse Grebenkov, Rev. Mod. Phys. 79, 1077 (2007) Schematic principle of NMR Static magnetic field B0 z y x Phase at time T Grebenkov, Rev. Mod. Phys. 79, 1077 (2007) Schematic principle of NMR Static magnetic field B0 z Inhomogeneous magnetic field z y x Phase at time T Grebenkov, Rev. Mod. Phys. 79, 1077 (2007) y x 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 1 S 0.9 Fixed gradient direction 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 g 2 4 6 Grebenkov et al., JMR 184, 143 (2007) 8 10 mT/m Healthy acinus 1 S 0.9 Fixed gradient direction 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0 g 2 4 6 Grebenkov et al., JMR 184, 143 (2007) 8 10 mT/m Healthy acinus 1 S 0.9 Averaged gradient direction 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 g 2 4 6 Grebenkov et al., JMR 184, 143 (2007) 8 10 mT/m Emphysematous acini How can one model emphysematous acini? Emphysema may lead to enlargement of the alveolar ducts partial destruction of the alveolar tissue Emphysematous acini Emphysema may lead to partial destruction of the internal alveolar tissue Emphysematous acini 1 S =0.6 =0.5 =0.4 =0.3 =0.2 =0.1 =0.0 0.8 0.6 0.4 0.2 0 0 ν = 0.6 gmT/m 2 ν = 0.5 4 ν = 0.4 6 8 ν = 0.3 10 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!!! [email protected] http://pmc.polytechnique.fr/pagesperso/dg 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 Healthy smoker diagnosis at earlier stage? Patient with severe emphysema van Beek et al. JMRI 20, 540 (2004) Human acinus Haefeli-Bleuer and Weibel, Anat. Rec. 220, 401 (1988)