Journal of Magnetic Resonance 225 (2012) 102–113
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Journal of Magnetic Resonance
journal homepage: www.elsevier.com/locate/jmr
The influence of lung airways branching structure and diffusion time on
measurements and models of short-range 3He gas MR diffusion
Juan Parra-Robles ⇑, Jim M. Wild
Unit of Academic Radiology, University of Sheffield, United Kingdom
a r t i c l e
i n f o
Article history:
Received 8 June 2012
Revised 16 October 2012
Available online 26 October 2012
Keywords:
Hyperpolarized helium
Diffusion MR
Finite element modelling
Lung imaging
a b s t r a c t
Hyperpolarized 3He diffusion experiments have been shown to be sensitive to changes in acinar structure
due to emphysematous lung disease. Extracting quantitative information about lung microstructure from
the diffusion signal is complicated due its dependence on a number of factors including diffusion time
and the complex branching acinar geometry. A theoretical model (cylinder model) has been proposed
as a means of estimating acinar airway dimensions from measured diffusivities. This model assumes that
the effects of acinar branching geometry and finite airway length upon 3He diffusion behaviour are negligible. In this work, we use finite element simulations of diffusion in a model of branching alveolar ducts
to investigate in detail the effects of acinar branching structure and finite airway length on short-range
3
He diffusion measurements. The results show that branching effects have a significant influence upon
3
He diffusivity, even at short diffusion times. The expressions of the cylinder model theory do not account
for significant dependences upon diffusion time, branching geometry and airway length, as a consequence of the oversimplified geometrical model used. The effect of diffusion time on 3He ADC was also
investigated through experiments with healthy human volunteers. The results demonstrate that the cylinder model can produce inaccurate estimates of the airway dimensions as a consequence of incompletely accounting for the diffusion-time dependence in the model equations and confirmed the
predicted limitations of the cylinder model for reliable lung morphometry measurements. The results
and models presented in this work may help in the development of a more realistic theoretical framework for ‘in vivo lung morphometry’ using 3He diffusion MR.
Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction
Hyperpolarized gas (3He and 129Xe) MRI provides information
related to the microstructure and physiology of the lungs. In particular, 3He diffusion experiments have been shown to be sensitive to
changes in acinar structure due to emphysematous lung disease
[1–4]. Most of the in vivo experiments reported to date have measured the apparent diffusion coefficient (ADC) of 3He with pulsed
gradient (PG) methods at diffusion times of a few milliseconds.
The diffusion of 3He gas in lung airways exhibits a non-Gaussian
phase dispersion [5], which results in a non-mono-exponential signal decay, which cannot be accurately described by a single-ADC
model. This signal behaviour originates from effects related to
the complex lung geometry such as: airway anisotropy, varying
airway dimensions and branching structure [6].
Yablonskiy et al. [7] pioneered a technique that uses a theoretical model to obtain estimates of the alveolar duct diameter from
⇑ Corresponding author. Address: Unit of Academic Radiology, Floor C, Royal
Hallamshire Hospital, Glossop Road, Sheffield S10 2JF, United Kingdom.
E-mail address: J.Parra-Robles@sheffield.ac.uk (J. Parra-Robles).
1090-7807/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.
http://dx.doi.org/10.1016/j.jmr.2012.10.008
diffusion weighted 3He MR images of human lung. The duct diameter estimates in patients with emphysema were significantly larger than in normal lungs. These results demonstrated the potential
of the technique as an in vivo quantitative tool to investigate lung
microstructure. In this theoretical analytical model (‘‘cylinder model’’), alveolar ducts were considered as infinitely long non-connected
cylinders with random orientations. This is on the premise that for
short diffusion times, the effects of branching and finite length of
airways are negligible. The model also assumes that the nonmono-exponential diffusion signal decay observed in 3He lung MR
results from the superposition of the mono-exponential signals originating from Gaussian diffusion in each of the individual airways.
In addition to anisotropic diffusion, other effects can influence
the observed signal decay in 3He lung diffusion MR experiments.
When large diffusion gradients are used, localized edge enhancement effects [8] cannot be ignored. Susceptibility-induced field
inhomogeneities [9] have also been shown to influence the lung
3
He diffusion signal behaviour. These effects significantly complicate the determination of accurate relationships between the measured non-Gaussian diffusivities and the lung airway dimensions.
This is compounded by the fact that a wide range of experimental
J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 225 (2012) 102–113
parameters such as diffusion time D, also influence the measured
diffusivity.
The cylinder model was further developed by Sukstanskii et al.
[10], using a more complex geometrical model, first proposed by
Paiva [11], as a basis for Monte Carlo simulations of diffusion. From
the simulation results, new expressions for the relationships between 3He apparent diffusivities and the modelled geometric
parameters were obtained. These new expressions [10] partially
account for non-Gaussian effects by incorporating a first order correction (in b value) to the expressions for the longitudinal and
transverse diffusivities. This updated model also assumes infinite
non-connected airways. It should be noted though, that all computer simulations in [10] were performed for one diffusion time
(D = 1.8 ms) only, although it was stated that the resulting theoretical model is valid over a broad range of diffusion times (up to
10 ms).
In a recent paper [8], we investigated the limits of validity of the
cylinder model through a number of purpose-designed experiments in cylindrical phantoms especially designed to fit the geometrical assumptions of the model. This work demonstrated the
breakdown of the cylinder model equations for large diffusion gradients (>15 mT/m) due to the nature of diffusion in the localization regime and we suggested that effects from acinar branching
and background susceptibility gradients would further limit the
accuracy of the model.
More recently, Sukstanskii et al. [12] published an accuracy
analysis of the cylinder model technique. In this study, through
computer simulations that incorporate branching geometry, results were obtained that led to the conclusion that branching effects are negligible. Unfortunately this paper only reported the
errors that branching effects introduce in the estimated geometrical parameters from the computer simulations; what was not provided were any intermediate results, such as the effect that
branching has on the diffusivity parameters of the theoretical model (i.e. DL0, DT0, bL, bT, see Appendix A for definitions). It was also
stated that the experimental and theoretical evidence of the breakdown of the cylinder model provided in [8] does not apply, this
statement was made without identification of any flaws in experimental design in that work.
In [12], background susceptibility effects were incorporated
into the computer simulations through the use of analytical
expressions describing the localized magnetic field of the distribution of cylindrical alveolar ducts. It was concluded that these effects are negligible at currently used field strengths B0 6 4.7 T
and become only significant (errors in geometrical parameters
16%) at B0 = 7 T. We have recently demonstrated through precise
in vivo experiments at two clinical field strengths (1.5 T and 3 T)
[8], the significant differences (up to 17%) in measured 3He ADC
and estimated airway dimensions that arise as a consequence of
background susceptibility gradients at much lower field strength.
The discrepancy between those experimental results and the
numerical predictions of [12] is due to the oversimplified magnetostatic expressions used in [12] and highlights the importance of solid experimental validation of the foundation assumptions of
theoretical expressions derived from numerical modelling.
In the present work, we go on to demonstrate several limitations of the Sukstanskii cylinder model of lung morphometry
[12] with respect to the effects that the lung branching structure
has on the measured 3He diffusion at different diffusion times.
Evidence of the significant effects that airway branching and finite airway length have upon the measured diffusivities in shortrange diffusion experiments already exists in a number of published articles [13–15] that were not cited in [12]. Fichele et al.
[13] simulated diffusion in several 2D geometric models using finite difference solution of the Bloch–Torrey equation. The results
there indicated that a porous media model was not a good model
103
of lung diffusion. More complex models (i.e. grape and tree-like
structures), which included branching and airway connectivity
provided results that better matched in vivo diffusion experiments.
These results suggested that interconnectivity between airways
does play a significant role in acinar diffusion, even for short-time
range diffusion.
Plotkowiak et al. [15] also used Monte Carlo simulations to
investigate 3He diffusion in a model of a single alveolar sac of finite
length. They found that for D = 1.9 ms and a single alveolar sac
836 lm long, the simulated average (bulk) ADC values were much
smaller than those reported in the literature for healthy lungs.
They found that the length of the alveolar sac model had to be increased by about one order of magnitude to unrealistic values of
5.3–12 mm in order to obtain reasonable ADC values, although
no quantitative investigation of the diffusivity dependence on airway length was presented.
Computer simulations have also been used by other groups to
investigate 3He diffusion in more complex branching structures.
Grebenkov et al. [14,16] used Monte Carlo simulations and experiments in a scaled model of a Kitaoka labyrinth [17]. Their results
showed that the branching structure does affect the measured diffusivities but concluded that short-range diffusion experiments
cannot distinguish the topological structure of the acinus. However, the diffusion time used in this work (D = 10 ms) is significantly longer that those typically used for in vivo 3He diffusion
studies. The Kitaoka labyrinth also may not be the most realistic
model to investigate branching effects for short diffusion times
since its geometry is significantly different from the branching
structure of acinar airways.
A more realistic geometrical model of the acinar branching
structure was used by Perez-Sanchez et al. [18] in Monte Carlo
simulations of 3He diffusion at different inflation stages of the
breathing cycle. However, no quantitative investigation of the effects of branching and finite airway length upon diffusivity or comparison with other theoretical models was reported.
In this paper, the effects of the lung branching structure and finite
airway length in 3He diffusivity in acinar airways are investigated
quantitatively for a range of short diffusion times, using finite element computer simulations. The results of these simulations are
compared to the theoretical predictions of the latest cylinder model
[10,19] and to our previous finite difference simulations [13,20],
which did not include branching effects. The diffusion time dependence of the 3He apparent diffusivity was also further investigated
experimentally, with quantitative results that significantly deviate
from the predictions of the cylinder model. Finally, implications of
these results for ‘‘in vivo MR lung morphometry’’ [10] are discussed.
2. Methods
In a typical pulse gradient (PG) diffusion MR experiment, a
bipolar gradient (Fig. 1) is applied with varying strength and/or
Fig. 1. Diagram of the waveform of the diffusion sensitization gradient used in this
work (s = 0.5 ms, D = 1.8–6 ms).
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duration. The phase dispersion accrued at the end of the bipolar
gradient pulse is proportional to the root mean square (r.m.s) displacement of the spins along the direction of the applied gradient.
For diffusion with an assumed Gaussian phase dispersion, this
r.m.s displacement is the characteristic diffusion length from the
Einstein diffusion equation lD = (2DD)1/2, where D is the diffusion
coefficient. The attenuation of the acquired signal S depends on
the phase dispersion of the spins and hence it is sensitive to the diffusivity of the spins in the media, which is affected by the presence
of restricting boundaries.
The degree of diffusion sensitization depends on the strength
and timing parameters of the diffusion gradient and is commonly
expressed by the b value, which for the waveform of Fig. 1 is given
by [7]: b = (cG)2 [2/3D3 s(D2 + 1/6Ds 8/15s2)]. The b-dependent apparent diffusion coefficient ADC can be calculated as:
ADCðbÞ ¼ 1=b lnðSðbÞ=S0 Þ
ð1Þ
where S(b) is the b-dependent diffusion signal and S0 = S(b = 0).
The MR signal evolution in PG diffusion experiments is described by the Bloch–Torrey equation [13,21]:
h
i
@Mð~
r; tÞ
¼ ic ~
r; tÞ
Gð~
r; tÞ ~
r Mð~
r; tÞ þ Dr2 Mð~
@t
ð2Þ
where M is the complex transverse magnetization (the measured
signal S is proportional to the integral of M over a voxel), G is the
applied field gradient, c is the gyromagnetic ratio and ~
r is the spatial
position vector.
This equation can be solved analytically only for the simplest
confining geometries (e.g. parallel plates, circle) and gradient
waveforms [22]. For more complex geometries, this equation has
to be solved using numerical methods. In this work, Eq. (2) is
solved using the finite element method for the gradient waveform
shown in Fig. 1 and for different combinations of values of G and D.
2.1. Geometrical model and computer simulations
Fig. 2. Geometrical model used in the computer simulations. (a) An alveolar duct
model consists of NS alveolated cylindrical segments of length L, external radius R,
alveolar depth h and inner radius r (r = R h). (b) The branching model consists of a
central duct with NS segments and a branching node at each end, to which two
ducts are connected. The planes formed by the branches at each end (for illustration
purposes, they are shown here in red) are perpendicular. (For interpretation of the
references to colour in this figure legend, the reader is referred to the web version of
this article.)
The geometrical model used in this work is based on the model
of alveolar ducts proposed by Paiva [11], which was adopted by
Sukstanskii et al. [10]. In this model, an alveolar duct consists
(Fig. 2a) of several alveolated cylindrical segments of length L,
external radius R and inner radius r. Alveolar septa of depth
h = R r and thickness, d divide each segment into 8 alveoli. The
new branching model we propose (Fig. 2b) consists of a central
duct with a branching node at each end to which are connected
two branches forming an angle of 90°.
The plane of the branching ducts at one end of the central duct
is rotated by 90° with respect to the plane of the branching ducts at
the other end [18,23,24]. In [12] the importance of the relative orientation of the branches was neglected, arguing that it does not affect the results of the simulations, but no validation of this was
provided.
The duct dimensions were selected to approximate typical
dimensions of alveolar ducts in healthy human lungs: R =
350 lm, L = 240 lm, h = 200 lm and d = 10 lm [19,25]. The
branching nodes consist of two cylinders (radius: 250 lm) connecting the central duct to each of the branches. The length of
these cylinders (210 and 240 lm) where selected such that there
was no overlap between the ducts and are similar to the length
of the alveolar segments. A free diffusion coefficient D0 =
0.88 cm2/s was assumed for the 3He-air mixture in the lungs [26].
The MR diffusion signal in the branching model was calculated
as the volume integral of M over the central duct. Considering only
the signal originating from the central duct, the diffusion signal for
different orientations of the central duct with respect to the applied gradient can be obtained. However, since Eq. (2) is solved
over the whole structure, the presence of branches affects the sig-
nal obtained from the central duct. A similar approach was adopted
for simulations of single ducts (i.e. without branches). Note that
this is similar to the situation found in diffusion imaging experiments at a macroscopic scale: over the time course of the diffusion
experiment a fraction of the spins contributing to a voxel’s signal
have actually sampled regions corresponding to neighbouring voxels. In the cylinder model a voxel contains a large number of infinitely long cylindrical airways with no branching. In our
approach, a voxel contains a large number of finite size ducts with
branches at both ends. These branches are not part of the ‘voxel
volume’ but contribute to the diffusion signal behaviour, since
atoms diffuse between the branches and the ‘voxel airways’.
By only integrating over the central duct, the dependence of DT
and DL on b value and diffusion time can be obtained for our
branching model, allowing direct comparison to the cylinder model ‘single airway’ parameters. [10]. This approach also allows us to
close the peripheral branches and neglect the contribution from
the dead ends and farther airway generations over the diffusion
time range investigated in this work (up to 6 ms).
This approach is different to that adopted by Sukstanskii et al. in
[12], where only atoms whose random walks started within the
analyzed airway were included in the simulation and the net signal
was computed from all of the atoms including those whose final
position is in a different airway. This differs from the actual physical process of MR signal formation: the MR diffusion signal emanating from spins in a given airway at the time of measurement
(or macroscopically from a voxel) does not depend on the path of
the random walk of all the atoms that were inside that airway at
the beginning of the diffusion experiment (i.e. the beginning of
J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 225 (2012) 102–113
the diffusion gradient). In reality, this signal depends on the path of
the random walk of the atoms that were within that airway at the
time of signal acquisition; a fraction of them will have started their
paths inside that airway but another fraction will have originated
in neighbouring airways. Although in certain conditions (i.e.
Gaussian diffusion) both approaches may produce similar results,
this is not always the case when the effects of localized diffusion
and microscopic susceptibility-induced gradients are present.
The choice of the number of segments in the airways of the
branching model and for the integration in the single duct was
based on the fact that the length of a three-segment section
(740 lm) is similar to the average alveolar duct length
(730 ± 258 lm [25]). The number of segments in the single duct
model (9 or 11) was chosen such that its total length is similar
to the longest path length in the branching model.
The computer models were created and solved using Comsol
Multiphysics (COMSOL AB, Stockholm, Sweden) and Matlab (MathWorks, Natick, MA). The boundary conditions assumed impenetrable walls (i.e. zero flow through all external surfaces). Typically 9
or 17 equally spaced b-values between 0.5 and 8.5 s/cm2 were used
in the simulations for each gradient orientation. The gradient
strength was scaled appropriately depending on the diffusion time
which ranged between 1.8 and 6 ms. The number of mesh elements and the time step were optimized such that further decreases in element size and/or time step did not change the
computed solution by more than 1%.
3. Results and discussion
3.1. Transverse diffusivity
To estimate the transverse diffusivity DT (parameters defined in
Appendix A), computer simulations were performed with the gradient oriented perpendicular to the central duct axis (z direction;
Fig. 2b). DT was calculated from the simulated diffusion signal
using Eq. (1).
To account for any dependence on the relative orientation between the gradient direction and the plane formed by the offshooting branches, signals were calculated for 18 equally spaced
angular orientations of the gradient perpendicular to the duct axis
(between 0° and 90° in the xy plane), and their average was used to
calculate DT.
In [12], Sukstanskii et al. assumed that effects due to the relative orientation between branches at either end of the central duct
is negligible and were not considered in the simulations used to assess the influence of branching. The argument given in [12], that
these effects would be insignificant since it is unlikely that atoms
would visit branches at both ends of the duct, is not valid. The signal originating from the central duct would depend on the relative
orientation of the branches if some of the atoms that are in the
duct at the measurement time (signal readout) originated in (or
visited) a branch on one side of the duct and other atoms originated in (or visited) the branch on the other side, even if none have
visited both branches at either end. Fig. 3, clearly visualizes this effect of branching on the spatial distribution of DT. This effect would
be even more significant when branches at both ends of the duct
differ in size and branching angle, which is the case of real lung airways [25], instead of being identical as used in this work for
simplicity.
Fig. 4 shows the b dependence of DT obtained from simulations
with the branching duct (squares) and single duct (circles) models
for two diffusion times (1.8 and 2.5 ms). Although there is a relatively good agreement between the results for a single duct and
the predictions of the cylinder model, the results for the branching
model show major discrepancies with the cylinder model theory.
105
Fig. 3. Spatial distribution of the simulated measured transverse diffusivity (DT) in
the branching model (D = 1.8 ms, G = 27 mT/m). Note the higher diffusivity at the
right hand end of the central duct whose branches show preferential diffusion in
the direction of G.
For the branching model, DT decreases linearly with increasing b
(i.e. bT 6 0), which is the opposite of the linearly increasing behaviour (i.e. bT P 0) predicted by the cylinder model. Furthermore, it
can be seen that the transverse diffusivities obtained with the
branching model are significantly larger (40% at 2.5 ms and larger
at longer diffusion times) than those obtained from the single duct
model and the cylinder model predictions (solid lines).
The increased DT0 values (Fig. 5) and the change in the sign of bT
are due to the presence of branches, since a significant fraction of
the spins contributing to the diffusion signal obtained from the
central duct have sampled regions where less restriction to diffusion along the gradient direction is experienced (i.e. the branches;
see Fig. 3). As a consequence, in the branching model the ‘‘apparent’’ structural length along the direction of the gradient is larger
than R and may become larger than the diffusion length, ld [8]. This
means that the b dependence of DT may be dominated by a behaviour corresponding to a significant fraction of spins sampling a different diffusion regime [8]; such that while for the cylinder model
and single duct simulations, DT increases with increasing b, which
is the behaviour corresponding to the transition from restricted
diffusion to localized diffusion. Whereas for the branching model,
DT actually decreases, which is the behaviour observed in the transition from free diffusion to localized diffusion. This dependence of
the transverse diffusivity DT behaviour on the presence of restriction in the longitudinal direction (i.e. perpendicular to the applied
diffusion gradient) is present in the branching model but is not accounted for in the cylinder model.
As D increases, motional averaging reduces DT0 in the branching
model but not as fast as predicted by the cylinder model because
the increased motional averaging effect is partially compensated
by the increase of the contribution to the duct signal from spins
that have sampled the branched regions.
Results of computer simulations performed with the same L and
with 3 or 4 duct segments (i.e. 740 lm and 990 lm duct length) in
the central duct of the branching model show that DT0 also depends on the duct length, although such dependence is not present
in the cylinder model. bT did not show any dependence on the duct
length within the range of diffusion times investigated in this work
(1.8–6 ms). This dependence of DT0 on duct length is a consequence
of the branching structure and hence was not observed in the results of the simulations with the single duct model.
The relative difference in DT0 between the two duct lengths is
small for short diffusion times, but increases with increasing diffusion time (from 6% for D = 1.8 ms to 13% for D = 6 ms). For the
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Fig. 4. Dependence of DT on b value for single duct (circles) and branching duct (squares) models for two diffusion times (1.8 and 2.5 ms). The theoretical predictions of the
cylinder model (Eq. (4)) are shown as solid lines.
Fig. 5. The diffusion time dependence of DT0 obtained from simulations with the branching phantom is compared to the prediction of the cylinder model (Eq. (A5) in Ref.
[19]).
longer duct, DT0 is lower since a smaller fraction of atoms in the
central duct have sampled less restricted regions (i.e. branches).
This effect would be larger in emphysematous airways (i.e. larger
r) since the helium atoms would experience less restriction and a
larger fraction would visit the branched regions. Note that DT0 depends on the total duct length only and not on L, since the same
diffusion behaviour was obtained for ducts of the same total length
(1 mm), consisting of four segments with L = 240 lm and three
segments and L = 320 lm, respectively.
3.2. Longitudinal diffusivity
To estimate DL, the diffusion gradient was directed along the
axis of the duct (central duct in the branching duct model) and
DL was calculated from the simulated diffusion signal using Eq.
(1). Fig. 6 shows the b-dependence of DL for several diffusion times
obtained from computer simulations using the branching duct and
the single duct (11 segments, total duct length: 2.7 mm) models.
These results highlight two significant effects, which cannot be
explained within the cylinder model theory. First, DL0 (i.e. intercept
at b = 0) significantly decreases with increasing diffusion time. In
the cylinder model, DL0 is independent of the diffusion time. Second, as the diffusion time increases, the b-dependence of DL goes
from the predicted decreasing behaviour to an increasing behaviour. The cylinder model predicts that DL decreases linearly with
b value for all diffusion times and cannot describe the increasing
behaviour. Furthermore, in all cases a significant deviation from
the linear dependence on b predicted by the cylinder model is observed and an additional quadratic term is necessary to fit the
behaviour above moderate b values (b > 3 s/cm2). Hence the argument in [12] that by adding a first order correction to the DL
expression, all non-Gaussian effects are accounted for is not valid
at all diffusion times and quadratic effects may dominate the bdependence of DL (e.g. at 2.5 ms the first order term is nearly negligible compared to the quadratic term).
Fig. 7a shows that DL0 decreases linearly with increasing diffusion time for both the single duct and branching duct models.
The variation is larger for the branching model with DL0 decreasing
by 35% when the diffusion time is increased from 1.8 ms to 6 ms.
For the single duct model the variation of DL0 over this range of diffusion times is still significant (22%). This dependence of DL0 on
diffusion time is not present in the cylinder model [10]. The reason
for the absence of this dependence is not apparent, but may be due
in part to extrapolating results obtained for a single diffusion time
D = 1.8 ms [10] to a wider range of diffusion times in which the
model has not been tested. The omission of this dependence is particularly significant since the cylinder model has been used to draw
important clinical conclusions from the interpretation of diffusion
data obtained at diffusion times up to 10 ms [27] and conclusions
about the nature of structural changes in emphysema were extracted based on the assumption of a diffusion time independence
of DL0.
J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 225 (2012) 102–113
107
Fig. 6. Dependence of the longitudinal diffusivity DL on b-value for (a) the branching duct and (b) the single duct (11 segments) models. Computer simulations were
performed for diffusion times in the range 1.8–6 ms. The predictions of the cylinder model (Eq. (3)) are also included in (a) for comparison.
Furthermore, the values of DL0 predicted by the cylinder model
are also significantly smaller than those obtained with both
the branching duct model and the long duct models for D < 4 ms.
The DL0 values obtained in this work are in good agreement with the
value reported by Plotkoviak et al. [15], 0.32 cm2/s for D = 1.9 ms,
which is also larger than the prediction of the cylinder model.
As discussed above, the b dependence of DL obtained in the simulations (Fig. 6) significantly deviates from the linear behaviour of
the cylinder model for all diffusion times in the investigated range.
For comparison with the cylinder model, Fig. 7b shows the bdependence of the estimated slope (bL) obtained for b 6 3 s/cm2.
The results obtained with both geometrical models differ from
the cylinder model theory. Rather than being always positive, bL
becomes negative for diffusion times above 2.3 ms for the
branching model and 3.5 ms for the single duct. For the branching
model bL reaches a minimum around 3.3 ms and then remains
approximately constant for longer diffusion times.
The absolute values of bL obtained with the branching model are
nearly one order of magnitude smaller than those predicted by the
cylinder model. The values of bL obtained with the single duct
model also deviate from the predictions from the cylinder model,
except at 1.8 ms diffusion time, which by coincidence was the only
diffusion time used in the simulations [10] from which the cylinder
model expressions were derived. Note that the change of sign of bL
for increasing diffusion time is not limited to the branching model
and is also present in the single duct model. This effect might have
been observed in the computer simulations of [10] if longer diffusion times had been studied.
The results presented above show that the values of DL0 and bL
obtained for the branching model differ significantly from those
obtained for a single duct and indicate that branching effects and
the effects of the finite length of alveolar airways on DL cannot
be neglected even for short diffusion times.
3.3. Bulk diffusivity
The diffusion signal obtained from an image voxel originates
from a large number of airways oriented in different directions. In
the cylinder model, the bulk signal is obtained from DL and DT using
Eq. (5) in Appendix A. The results of our computer simulations agree
with earlier simulations [14], and with our previously reported
experimental findings [8] in cylindrical phantoms, in that Eq. (5)
is not valid for large gradients (above 15 mT/m) due to localized
diffusion effects. In our simulations the bulk signal is obtained from
the sum of signals obtained from 36 or 91 uniformly distributed 3D
orientations of the diffusion gradient. Fig. 8 shows the bulk signal
and ADC obtained for two different diffusion times with the branching duct model. There are large discrepancies between the signal
and ADC behaviours predicted by the cylinder model and the results
of our simulations, as a consequence of the significant limitations of
the cylinder model theory identified above.
The difference in ADC between the branching model and cylinder model (Fig. 8) ranges from 18% to 25% in the range of diffusion times investigated here. Even for a single duct, the cylinder
model fails to describe the behaviour at long diffusion times as
expected from the results shown in the previous sections.
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Fig. 7. Comparison of the b-value dependence of (a) DL0 and (b) bL obtained from computer simulations with the branching duct and single duct models alongside the
dependence predicted by the cylinder model (Eqs. (A4) in Ref. [19]).
The consequence of those limitations of the cylinder model
equations can be further appreciated in Fig. 9, where the airway
dimensions estimated from the simulated diffusion data using
the cylinder model are plotted against the diffusion time. The deviation of the estimates of R and r from their known values increases
nearly linearly (slopes 35 and 19 lm/ms for R and r, respectively)
with increasing diffusion time.
Note that to obtain the results of Fig. 9 only two parameters (R
and r) have been fitted to the diffusion signal decay while L has
been fixed to its known value (0.240 mm). It was not possible to
obtain a stable fit of the cylinder model with all three parameters
since the estimated parameters strongly depended on the starting
guess values and varied widely. These results seem to reveal a significant limitation of the cylinder model when seeking to extract
all of the geometrical parameters from the limited number of bvalues used to measure diffusion signals. When comparing the results of the cylinder model to histological measurements, Yablonskiy et al. [19] reduced the number of estimated parameters to
two (R and h = R r), by using the constraint L = 0.765R, however
no physical or histological justification for this reduction in parameter space was given.
This limitation may be even more fundamental and our results
suggest that a standard PG diffusion experiment, as commonly
used for lung diffusion experiments (i.e. a few b values up to
10 cm2/s obtained with a fixed D and varying G), can only reveal
a limited number (i.e. two) of geometrical parameters of the
restricting structures. Further investigation of this subject is required and may help in the design of diffusion experiments from
which more geometrical information can be inferred. This will be
the subject of future research using the numerical modelling tools
developed in this work.
These limitations are due to the complexity of the acinar geometry together with the orientational averaging in the formation of
the bulk pixel signal. As a consequence, different causal changes
in airway geometry can result in similar observed changes in signal
behaviour. Similarly, certain changes in acinar structure may produce opposite effects that cancel each other out and result in no
change in bulk signal behaviour. For example, Fig. 10 shows the
signal behaviour and estimated dimensions for different model
parameters. It can be seen that an increase in the alveolar size L
from 240 lm to 320 lm produces no change in the observed signal
at diffusion time 1.8 ms.
This effect is unique to branching geometries, and can be physically explained within the framework of our model since an increase in L results in an increase in DL which is counteracted by a
decrease in DT, since as the length of the duct increases, the relative
effect of the branches (with less restriction in the transverse direction) is reduced. In models without branches, DL would also increase but DT would remain unchanged, hence an increase in
bulk diffusivity is predicted by the cylinder model.
In this work, different effects are shown to cause diffusion
behaviour that deviates from the cylinder model predictions (e.g.
b and D dependences of DT and DL). Under certain conditions these
could cancel each other out and result in signal behaviour closer to
the theoretical prediction. This would be a coincidental case for
which the combination of complex but incomplete theoretical
expressions would match the actual diffusion behaviour. The limitations of the cylinder model demonstrated in this work become
J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 225 (2012) 102–113
109
Fig. 8. Comparison of bulk ADC obtained from simulations with the branching model, single duct and predicted by the cylinder model (Eq. (6)) at D = 1.8 ms (a) and D = 6 ms
(b).
Fig. 9. Diffusion time dependence of the airway dimensions (R and h) estimated from the simulated diffusion data for the branching model using the cylinder model theory.
The actual airway dimensions are: R = 350 lm and h = 200 lm.
more significant when combined with other sources of error such
as susceptibility effects [9], non-ideal airway geometry (particularly in emphysematous lungs) and low image SNR.
The statistical analysis of the accuracy of the cylinder model in
estimating the airway dimensions [12] are to date incomplete. It is
assumed a minimum SNR of 100 in the 3He NMR diffusion data,
for which the noise follows a Gaussian distribution. Although this
is achievable in ex vivo experiments and in vivo global lung diffusion measurements, in multi-b value in vivo diffusion imaging
experiments, images with lower SNR (and Rician noise distribution) are typically obtained. The influence of several other factors
which cannot be ignored experimentally in vivo (e.g. RF excitation
inhomogeneity, voxel resolution, T2 relaxation) have also not
been considered in those statistical analyses.
110
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Fig. 10. Simulations with the branching phantom showed that changing L from 240 lm to 320 lm does not change the bulk ADC (D = 1.8 ms) as predicted by the cylinder
model (Eqs. (3), (4), and (6)).
4. Experimental validation
Due to the significant differences between the results obtained
from our simulations and the cylinder model theory [10,19], careful assessment of the accuracy of our simulations is necessary. This
is especially needed since our results contradict a previous accuracy analysis of the cylinder model [12], which concluded that
the effects of branching structure and susceptibility gradients
introduce negligible errors in the cylinder model estimates of airway dimensions.
Assessment of the accuracy of a numerical model with further
numerical simulations, as used in [12] is not a reliable approach.
This has already been highlighted experimentally [9] with the
demonstration of the influence of susceptibility effects on measured ADC and estimates of airway dimensions. Another approach
used to validate the cylinder model is the comparison of its airway
size estimates from diffusion experiments to histological measurements in excised lungs [19]. In principle this approach is sound but
has significant limitations due to its ex vivo nature. Firstly the process of fixation and desiccation of the excised lung specimens for
the MR experiments and producing the histological sections may
alter the geometry and size of the airways. Secondly the susceptibility gradients that significantly affect the ADC in vivo [9] are not
the same in the dried lung since water was removed from lung tis-
sue and all the oxygen was eliminated from the airways. Furthermore in [19], an incorrect value of the free diffusion coefficient
was used in the calculations (D0 = 0.88 cm2/s, as typical for He in
air in lungs, instead of the correct D0 = 1.03 cm2/s [26] for the gas
mixture used in those experiments).
To validate the results of our numerical simulations we have
adopted an in vivo experimental approach. Diffusion experiments
in healthy volunteers were conducted at different diffusion times
(see Appendix B for details of the experimental methods). If the
same volunteer is scanned at the same time under the same conditions (i.e. lung inflation, gas mixture, pulse sequence) and only the
diffusion time is changed, the same airway dimensions should be
estimated at all diffusion times using the cylinder model – if its
expressions are correct. Our simulations predicted a bias in size
estimates (R and h) that will increase linearly with increasing diffusion time as shown in Fig. 9. Fig. 11 shows the results of the
in vivo experiments, which confirm the prediction of our numerical
simulations with the estimated airway dimensions increasing linearly with increasing diffusion time. The experimental estimates
deviated from the constant behaviour (horizontal lines) expected
if the cylinder model expressions were correct, with norm of residuals NR = 102 and 70 for R and h, respectively. The linear fits
(R = 238 + 42D and h = 82 + 35D) provided a much better fit with
NR = 5.7 and 9.7 for R and h, respectively. The rate of increase
Fig. 11. Diffusion time dependence of airway dimensions obtained from in vivo experiment in a healthy volunteer. As predicted from our simulations, the estimates of both R
and h deviate from the values predicted by cylinder model (at D = 1.8 ms); the biases of the estimates increase linearly with increasing diffusion time.
J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 225 (2012) 102–113
(slope) of the experimental estimates is higher than in the simulations (Fig. 9), which may be due to susceptibility effects (which
were not included in our simulations) since the effect of background inhomogeneity increase with increasing diffusion time
[28]. These results confirm that the limitations of the cylinder
model identified in this work result in significant inaccuracies
Table 1
Comparison of the dependences of diffusivities on diffusion time and airway
parameters found in this work and the theoretical predictions of the cylinder model.
LT is the total duct length (LT = L NS).
DL0
DL vs b
bL
bT
DL and DT
Cylinder model
FE single duct
FE branching model
Independent of D
Linear dependence
Always P 0
P0
No LT dependence
Depends on D
Nonlinear
60, for D > 3.5 ms
P0
Depends on LT
Depends on D
Nonlinear
60, for D > 2.3 ms
60
Depends on LT
111
when estimating airway geometrical parameters, even in healthy
lungs.
The consequences of using the cylinder model theoretical
expressions with their incorrect diffusion time dependence can
be appreciated when analysing the results recently published by
Narayanan et al. [29]. In that work, the cylinder model was used
to estimate the airway dimensions of children from data acquired
at a diffusion time of 5.2 ms. For children, who have smaller airways (i.e. shorter ducts) we would expect from the results of our
simulations, that branching effects become very significant and
that the airway dimensions would be overestimated. The results
of [29] confirm our predictions, both R and h were grossly overestimated (average values: R = 426 lm and h = 244 lm). It is surprising that the authors and reviewers of that paper did not realize that
those dimensions are larger than average airway dimensions in
adults [21] and unreal in a population of children with median
age 12.8 years.
Fig. 12. Diffusion images obtained from one volunteer for D = 1.8 ms at three different b values: (A) 2.5 cm2/s, (B) 5.0 cm2/s and (C) 7.5 cm2/s. The image intensity has been
corrected for RF depletion and normalized using the first b = 0 image (i.e. image intensity corresponds to S/S0).
112
J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 225 (2012) 102–113
Fig. 13. Fits of the cylinder model (Eqs. (3), (4), and (6)) to diffusion data obtained at 1.8 ms and 4 ms diffusion times.
5. Summary
Table 1 summarizes the main dependences of the diffusivities
(DL, DT) on diffusion time and airway dimensions obtained in this
work for the single duct and branching duct model with finite element (FE) simulations with the theoretical predictions of the cylinder model. The large deviations between the results obtained in
this work and the predictions of the cylinder model demonstrate
that branching effects and finite duct size cannot be ignored in
short range diffusion measurements. Neglecting these effects results in a model that does not describe 3He diffusivity accurately
in acinar airways and consequently predicts incorrect dependences
on airway geometrical parameters and pulse sequence parameters
(e.g. D). Correct determination of the diffusion time dependence of
3
He diffusivity is very important since the cylinder model theory
has been used to interpret data obtained from experiments with
normal, emphysematous and developing lungs that have been performed at different diffusion times [19,29].
In this work, simulations have been performed with a geometrical branching model of dimensions similar to those found in
healthy acinar airways. For emphysematous lungs, where the
geometry is significantly changed by airway wall destruction, airway connectivity would become even more important for gas diffusion and the validity of the cylinder model would be even more
limited. Extending the simulations presented in this work to
emphysematous airways will be the subject of future work and
will require the development of new geometrical models based
on histological data to simulate airway remodelling.
In future work, the influence of several other factors not investigated in this work should also be studied, including: effects from
different branching angles, varying airway sizes and interconnectivity between different branches and/or acini. The experimentally
observed effects of susceptibility-induced field gradients in the
presence of non-Gaussian diffusion [9] also need to be investigated
thoroughly from a sound theoretical basis.
neglected in the cylinder model, have also been identified and
quantified in this work.
The cylinder model represented a seminal approach towards
assessment of lung microstructure but it now needs to be further
assessed and validated before it can be reliably used to extract
accurate microscopic lung morphological information from 3He
diffusion experiments, especially at long diffusion times. This is
particularly the case when attempting to quantify morphological
changes in important clinical questions such as early onset of
emphysema [27] or the alveolarization of developing lungs [29],
where, far-reaching clinical conclusions have been drawn about
the nature of the micro-structural modifications caused by disease
and growth. The results and models presented in this work may
help in the development of a more realistic theoretical framework
for in vivo lung morphometry using 3He diffusion MR.
Appendix A. Cylinder model theory
In the quantitative model of Sukstanskii et al. [10], two bdependent diffusivities were defined with gradients parallel (DL)
and perpendicular (DT) to the alveolar duct axis. The computer simulations indicated that DL and DT depend linearly on b (for b < 10 s/
cm2) according to [10]:
DL ¼ DL0 ð1 bbL DL0 Þ
ð3Þ
DT ¼ DT0 ð1 þ bbT DT0 Þ
ð4Þ
where DL0 and DT0 are, respectively, the longitudinal and transverse
diffusivities when b = 0; and bL and bT are coefficients that are proportional to the second order term (i.e. kurtosis) of the cumulant
expansion of the signal. The expressions for DL0, DT0, bL and bT for
duct models with 4 and 8 alveoli (per duct segment) can be found
in Refs. [10,19], respectively. For an arbitrary angle u between the
airway axis and the gradient, the apparent diffusivity can be obtained
from DL and DT assuming a Gaussian phase distribution as [30]:
2
ADCðuÞ ¼ DL cos2 u þ DT sin u
6. Conclusion
In this work, the influence of interconnectivity of alveolar ducts
upon the diffusion time dependence of the measured 3He diffusion
coefficient has been theoretically and experimentally identified.
Further major limitations of the cylinder model in describing 3He
lung diffusion in lung airways have also been identified. These limitations are a consequence of geometrical assumptions that do not
account for the effects of branching and finite size of alveolar ducts
and results in an oversimplified theoretical model. Significant diffusion time dependences of the measured diffusivities, which were
ð5Þ
In the cylinder model, the bulk signal is calculated as the integral of the signals corresponding to all possible angular directions
with ADCs given by Eq. (5), which can be written as:
SðbÞ ¼ S0 expðbDT Þ
p
1=2
4bðDL DT Þ
U½ðbðDL DT ÞÞ1=2 ð6Þ
where (x) is the error function.
It should be noted that Eq. (6) was first obtained in [30] from
Eq. (5) under the assumption of a Gaussian phase distribution. In
a previous paper [8] we investigated with experiments and
J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 225 (2012) 102–113
computer simulations the validity of this assumption and demonstrated that it is not valid for the gradient strengths (above
15 mT/m) commonly used in short range 3He diffusion experiments in lungs.
Appendix B. Experimental method
Five healthy volunteers aged 24–41 were scanned using a 1.5 T
(GE HDx, Milwaukee, WI) using a quadrature 3He transmit-receive
vest coil (Medical Advances, Milwaukee, WI) with local ethics committee approval. 3He diffusion images were acquired at breathhold (FRC + 1 l) using a gas mixture consisting of 300 ml 3He and
700 ml N2. 3Hewas polarised to 25% using a prototype commercial polariser (Helispin, GE Healthcare) under regulatory licence.
Four scans, corresponding to four different diffusion times (1.4,
1.8, 2.5 and 3.6 ms) were performed for each volunteer consecutively with four separate 1 l doses of the same gas mixture.
A 2D spoiled gradient echo with bipolar diffusion sensitisation
gradients [9] was used (64 64 matrix, TE: 4.8 ms, TR: 8.0 ms,
FOV 35 cm, bandwidth 62.5 kHz). Five coronal slices were acquired
consecutively (i.e. no slice interleaving), with 15 mm slice thickness and 10 mm spacing. Five or six interleaved acquisitions were
obtained for each slice corresponding to 5 or 6 b values. The first
and last acquisitions (with b = 0 s/cm2) were used to obtain flip angle maps for correction of RF depletion effects. The first five interleaves had forward centric phase encoding (i.e. the centre of kspace was acquired first) and b values equally spaced between 0
and 7.5 s/cm2 (except at D = 1.4 ms, where the maximum achievable b was 5.5 s/cm2 due to the gradient strength limitations –
maximum gradient available 33 mT/m). The last interleave was reverse centric phase encoded to accentuate the RF depletion for flip
angle estimation [9].
The diffusion-sensitizing gradient was oriented along the slice
selection axis (i.e. antero-posterior direction) with timing parameters: rise and fall times 0.3 ms and no delay between pulses. The
pulse duration in each scan was selected to achieve the required
diffusion time (D = 1.4, 1.8, 2.5 and 4 ms). The gradient amplitudes
per interleave were varied to achieve equally spaced b values up to
a maximum value of 7.5 s/cm2; except for D = 1.4 ms, where the
maximum achievable b value was 5.5 s/cm2. Fig. 12 shows an
example of the diffusion images and signal decay data obtained
from a healthy volunteer. The average SNR of the first interleave
image (b = 0) was 82 ± 11.
The expressions of the cylinder model were fitted to the acquired diffusion data to estimate the duct dimensions R and h for
each pixel. Whole lung averages were computed after segmentation of the lungs (excluding largest airways). Fig. 13 shows examples of the fits of the cylinder model to the diffusion data at
different diffusion times. The diffusion time dependence of the
estimated average duct dimensions (R and h) were compared to
the cylinder model prediction and the linearly increasing dependence obtained from the simulations performed in this work using
the branching model and goodness of fit was evaluated using the
norm of the residuals NR:
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u N
uX
NR ¼ t ðyi fi Þ2
ð7Þ
i¼1
where i is the index for the D values (N = 4), yi are experimental
estimates of R and h. fi are the predicted values of R and h at each
diffusion time: fi = ax + b for the branching model prediction and
fi = C (where C is a constant) for the cylinder model. Since the cylinder model equations were obtained from simulations at D = 1.8 ms
[10] only, we assumed C equal to the airway dimensions estimated
at this diffusion time.
113
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