Journal of Magnetic Resonance 239 (2014) 143–146

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Journal of Magnetic Resonance

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Response to Commentary on ‘‘The influence of lung airways branchingstructure and diffusion time on measurements and models ofshort-range 3He gas MR diffusion’’

1090-7807/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jmr.2013.10.019

⇑ Corresponding author.E-mail address: j.parra-robles@sheffield.ac.uk (J. Parra-Robles).

Juan Parra-Robles ⇑, Jim M. WildUnit of Academic Radiology, University of Sheffield, United Kingdom

a r t i c l e i n f o

Article history:Received 26 February 2013Revised 25 October 2013Available online 22 November 2013

Keywords:3He Diffusion MRmodel validationLung morphometry

a b s t r a c t

Our extensive investigation of the cylinder model theory through numerical modelling and purpose-designed experiments has demonstrated that it does produce inaccurate estimates of airway dimensionsat all diffusion times currently used. This is due to a variety of effects: incomplete treatment of non-Gaussian effects, finite airway size, branching geometry, background susceptibility gradients and diffu-sion time dependence of the 3He MR diffusion behaviour in acinar airways. The cylinder model is a goodstarting point for the development of a lung morphometry technique from 3He diffusion MR but its lim-itations need to be understood and documented in the interest of reliable clinical interpretation.

� 2013 Elsevier Inc. All rights reserved.

In their Commentary, Yablonskiy et al [1] question the conclu-sion of our recently published paper [2], namely that their‘in vivo lung morphometry technique’ (i.e. the cylinder model)[3,4] ‘‘can produce inaccurate estimates of airway dimensions andhence cannot be used for reliable lung morphometry measurementswithout further development’’. This conclusion is a consequence ofthe results reported in our paper [2], which provided not only de-tailed numerical computer simulations but also experimental dem-onstration of significant inaccuracies in the cylinder model as aconsequence of the incomplete consideration of the effects of: dif-fusion-time dependence, finite airway size and acinar branchingstructure in their current theoretical treatment. In particular, ourpaper demonstrated through numerical simulations that intrinsi-cally account for these factors, that most of the cylinder modelmathematical expressions are incomplete or incorrect.

Instead of addressing the significant limitations of the cylindermodel identified by our simulations and experiments, this Com-mentary [1] provides a historical overview of the development ofthe model and a list of earlier publications, that the Authors useas evidence of validation of the accuracy of their model. The major-ity of this interesting selection of publications, largely arising fromthe Authors group, does not actually assess the accuracy of the cyl-inder model but instead uses it without scrutiny of its validity.

In the only mention of the results of our paper [2] in their Com-mentary, the Authors have totally misinterpreted the experimentalresults shown in Fig. 11 of this paper [2] and incorrectly stated that

there is good agreement between those results and their theoreti-cal model for diffusion times in the range D= 1.6–1.8 ms. Theseexperimental results actually confirm the prediction of our simula-tions, i.e. that the airway sizes estimated with the cylinder modelare inaccurate with an error that increases with increasing diffu-sion time D. To quantify the degree of discrepancy with the cylin-der model we compared the fit of the data to a sloped line (aspredicted by our simulations) with a horizontal line (as predictedif the cylinder model theory is correct). Since the actual dimensionsof the airways are not known (these are in vivo experiments) andthe cylinder model produces different estimates for each diffusiontime, we arbitrarily chose to compare our results to a horizontalline that passes through the estimates corresponding toD = 1.8 ms. That is why both lines cross over at exactly 1.8 ms;there is no agreement between our results and the cylinder model,what that figure actually confirms is that the diffusion time depen-dence in the WU theoretical model is incorrect.

In most of their Commentary [1], the Authors have tried to re-duce the debate to the single issue of the range of validity of thediffusion time dependence of the cylinder model. Their insistenceto now emphasize in this Commentary the importance of the diffu-sion time parameter in the cylinder model theory is at odds withthe casual and inconsistent treatment of this issue in their previouspapers.

At the heart of this issue is the fact that equations that were ob-tained from simulations performed with a single D value, actuallyshow D as a variable parameter [3]. It is not clear how the Authorsarrived at that D dependence, which we have shown in our paperto be incorrect. The presence of this parameter as a variable in themodel is unjustified and misleading.

144 J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 239 (2014) 143–146

It should not be ‘difficult to understand’ for the Authors our opin-ion that their paper [4] does not clarify in any way the range ofvalidity of their theory when they themselves incorrectly usedthe method outside this ‘prescribed range’ at 5 and 10 ms in a pa-per [5] published shortly after, which they do not mention in theirCommentary [1]. In that article, the fact that in the cylinder modeltheory DT0 depends on the diffusion time but DL0 does not (whichwas proven to be incorrect in our paper [2]) was used to draw con-clusions about the nature of the changes in lung microstructureduring emphysema progression. Again, no evidence is reportedanywhere that shows that those equations are valid at diffusiontimes other than the one used to derive them.

In the original theoretical paper [3], the Authors stated thattheir method is valid in the range of ‘several milliseconds’, limitedby the condition that the ‘apparent’ diffusion length ld =

p(2DLD),

where DL is the longitudinal diffusivity [3], is shorter than theaverage length of the acinar airways LA (�730 lm) which resultsin D < �10 ms). In [4] this condition was then changed top

(2D0D) < LA, where D0 is the free diffusion coefficient, which resultsin D < 3 ms. In their most recent review paper [6] this condition hasbeen changed again to

p(2D0 2D) < LA, resulting in D < 1.5 ms.

Why does this condition change without any demonstration ofits validity in the context of the model? What is striking is thataccording to this latest condition, all the experiments that haveever been performed by the WU group in human lungs, even atD = 1.6 ms, now fall outside their new ‘prescribed range’D < 1.5 ms. How can these data be used as evidence of validity ofthe cylinder model when by the Authors’ own definition they wereperformed out of the ‘valid range’?

In the mouse lung adaptation of cylinder model theory, a diffu-sion time range of 0.3–0.6 ms was prescribed and D = 0.44 ms wasactually used in experiments by the WU group [7,8]. If we were toaccept any of the ‘prescribed ranges’ as valid (e.g. the latest versionp

(2D0 2D) < LA [6]), sincep

(2D0 2D) = 400 lm is much larger thanthe average length of mouse acinar airways (�120 lm [9]), then itwould seem that the mouse lung cylinder model was also devel-oped outside the valid range of diffusion times for mouse lungsand the effects of the branching geometry and finite airway sizesshould also have been accounted for in that version of the theoret-ical model.

The use of the cylinder model theory in animal experiments inall of the published WU papers has been performed outside theirown ‘prescribed range’. We reiterate our question raised for humanlungs again for animal lung experiments; how can these data beused as evidence of validity of the cylinder model when by theAuthors’ own definition they were performed out of the ‘validrange’?

In our opinion, the rationale behind the definition of all of theabove conditions is flawed. The spins do not need to travel a dis-tance larger than the average airway length to ‘feel’ the effects ofits finite length, since the farthest a spin can be from a branchingnode or the end of a sac is only half the length of the airway. Soall spins in acinar airways are at a distance equal or smaller than0.5�LA, from a branching node or sac end. Should the conditiontherefore not be

p(2D0�2D) < ½�LA, in which case the prescribed

range becomes D < 0.45 ms?The two papers at the core of the Authors defence of the accu-

racy of the cylinder model [4,10] were already discussed at lengthin our paper [2] and in two earlier articles [11,12] and were provento have significant shortcomings, which must surely raise ques-tions as to them forming a solid basis of validation of the cylindermodel theory. In these two previous papers [11,12], we demon-strated further inaccuracies of the cylinder model theory arisingfrom non-Gaussian phase effects [11] and background susceptibil-ity gradients [12]. Those results have not been challenged by theAuthors in this Commentary or elsewhere.

In [4], Yablonskiy et al. compared estimates of the geometricparameters of alveolar ducts from ex-vivo 3He diffusion experi-ments made in excised lungs to microscopy measurements fromhistological sections obtained from the same lungs, and found anexcellent agreement between the estimates of mean linear inter-cept (Lm) obtained with both methods. In our paper [2], we clearlyexplained the limitations of this validation approach [4]. Mostimportantly, we highlighted that the excellent agreement foundin [4] (and emphasized in Fig. 1 of the Commentary [1]), was ob-tained using an incorrect value for the free diffusivity of 3He(D0 = 0.88 cm2/s), instead of the correct D0 of the gas mixture thatwas actually used in their experiments which may be as high as1.03 cm2/s, depending on the volumes of each lung which werenot reported in [4]. A validation paper, to be trusted, should pro-vide all the parameters used in the experiments and data analysisin order for others to reproduce the work.

In their Commentary [1], the Authors have now argued that theuse of D0 = 0.88 cm2/s for all lungs is justified since the helium con-centration in all lungs was between 15% and 20% and hence D0

never exceeded 0.9 cm2/s. Once more they do not provide anyproof that those were indeed the helium concentrations used intheir experiments. A simple calculation can show that these con-centrations and D0 values are incorrect.

The inflated volumes of lungs M1 and M2 (i.e. lungs with mildemphysema [4]), were 1.1 L and 0.6 L, respectively. What isneeded to properly calculate D0 are the initial gas volumes ofthe partially inflated lungs (i.e. residual volumes) that were mixedwith the 1 L mixture contained in the syringe. Making an edu-cated guess that the residual volume was �50% of the inflatedlung values (a reasonable tidal volume is needed to equilibratethe gas concentration over ‘two or three breaths’), we can esti-mate the D0 values obtained when that amount of nitrogen leftin the lung is mixed with 0.4 L of helium and 0.6 L of nitrogencontained in the syringe. The results are 0.94 cm2/s and0.97 cm2/s for M1 and M2, respectively. These values definitelyexceed the value 0.9 cm2/s conveniently stated by the Authors.As a result, estimates made with the cylinder model using the cor-rect D0 would result in values of Lm and h that would not agreewith histology estimates. It would then appear that the cylindermodel agrees best with histology when fed with incorrect exper-imental parameters. In our opinion, this data should not be shownas supporting evidence for their model without a real assessmentof what impact that error had in their estimates of airwaysdimensions.

In the other key paper used in support of the cylinder model[10], Sukstanskii et al performed an accuracy analysis of the cylin-der model based completely upon computer modelling of diffusionin model airways with branching geometry but did not include anyexperimental validation. Since that paper did not provide interme-diate results of the simulations (e.g. dependence of DL, DT, bL and bT

on experimental parameters b and D), it is hard to assess its meritas a validation of the cylinder model. There are several other issueswith this paper that were also discussed in detail in our paper [2]such as the prediction that the effects of background susceptibilitygradients on the accuracy of airway size estimates are only signif-icant at high field strengths (�7 T), which we have since experi-mentally proven to be wrong in [12]. Until these findings areproperly addressed, this paper [10] cannot be used as reliable proofof the accuracy of their theoretical model.

Other referenced papers (e.g. [13–15]), cited as proof of valida-tion, simply used the original cylinder model theory [16] to fit theMR diffusion data and did not estimate airway dimensions. Thesepapers do not assess the accuracy of the cylinder model, but simplyconfirmed that the model fits the diffusion signal decay. Fitting theold model [16] to the diffusion signal is not evidence that the air-way size estimates obtained with the new equations in the latest

J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 239 (2014) 143–146 145

model [3,4] are accurate, merely that the old equations provide afit to a curve.

With all due respect to the Authors, we are of the opinion thatthe cylinder model theory is not accurate and robust even withinthe ‘prescribed’ 1.6–2 ms range as claimed by the Authors [1]. In[2] we showed that most of their expressions are incorrect orincomplete even in this range. In [2,17], we also showed exampleswhere branching airway geometries with different dimensions re-sult in the same signal behaviour and hence the same cylindermodel estimates at D = 1.8 ms. In this situation, the cylinder modelwould incorrectly predict the same dimensions for the two differ-ent airways. In [12], we showed experimentally that at D = 1.6 msthe airway dimensions estimated by the cylinder model theory aresignificantly affected by the presence of background susceptibilitygradients at the relatively modest clinical field strengths (1.5 and3 T), which the WU theory ignores.

In conclusion, our extensive investigation of the cylinder modeltheory through numerical modelling and purpose-designed exper-iments [2,11,12,18–20] has demonstrated that it does produceinaccurate estimates of airway dimensions at all diffusion timescurrently used. This is due to a variety of effects: incomplete treat-ment of non-Gaussian effects, finite airway size, branching geome-try, background susceptibility gradients and diffusion timedependence of the 3He MR diffusion behaviour in acinar airways.The many inaccuracies discussed above when combined in a com-plex non-linear scenario that contains incomplete and/or incorrectdependences on experimental, lung geometrical and physicalparameters cannot result in a robust and accurate technique.

The cylinder model is a good starting point for the developmentof a lung morphometry technique from 3He diffusion MR but wefeel its limitations need to be understood and documented beforemajor conclusions are drawn in a clinical context. As researcherswith an understanding of the diffusion physics, we and the WUgroup (as well as others in the field) have a responsibility to tacklethese questions, further develop these models and define the re-gimes in which they can be trusted in the interest of reliable clin-ical interpretation. We hope that some of our concerns raised inthis response serve as a basis for constructive discussions withthe Authors in this respect.

Appendix A

Our response to this Commentary [1] has gone through an edi-torial process, with two iterations of revisions. After each revision,the Washington University (WU) group has been given access toour response and have been allowed to make changes to theirCommentary in that knowledge. At the ‘final proof’ stage of the edi-torial process, the Authors (who had not questioned the correct-ness of our numerical and experimental findings in the first twoversions of their Commentary) introduced an Appendix where theylist what they call ‘incorrect statements and assumptions’ in ourpaper. We respond to these comments, which we feel are not sci-entifically substantiated, below.

The Authors try to challenge the results of our simulations byarguing that the parameters DL0, DT0, bL and bT of their model havemeaning only for infinitely long airways. We disagree; it seems thatthe Authors misunderstand both our simulation methods and thephysical and mathematical basis of their own theoretical model.In our paper, the diffusion signal is calculated by integration overthe central duct of our branching geometric model; in this wayapparent diffusion coefficients can be calculated for gradient direc-tions parallel and perpendicular to this central airway (DL and DT,respectively). This is consistent with the theoretical model origi-nally developed by Callaghan et al [21] to describe anisotropic dif-fusion. Nowhere in this theory is there a requirement for the

restricting structures to be infinite. Yablonskiy et al [16] simplyadopted this theoretical formalism and applied it to lung airways,which are finite. Later, a mathematical correction whereby DT andDL are made to depend linearly on b-value, was introduced to maketheir model fit the non-monoexponential behaviour of the simu-lated MR signal. This is simply equivalent to a second order cumu-lant expansion of the MR signal decay, which is applicable to anygeometry, not just to infinite airways.

Hence the parameters DL0, DT0, bL, bT have meaning not only forinfinite airways but for any individual airway, where the directionof the diffusion gradient can be defined parallel and perpendicularto the airway axis (as is the case in or simulations with the signalcalculated from the central duct only). The fact that the results ofour simulations were confirmed experimentally surely validatesthe correctness of our approach and the limitations of the cylindermodel.

The Authors incorrectly argue that their simulation approach‘‘does reflect a real MR-based diffusion measurement process whenthe diffusing spins are ‘‘labelled’’ by diffusion-sensitizing gradientsand the signal is measured at the end of applied diffusion wave-formno matter where their trajectories end’’.

Where the trajectories of the spins end does matter, this is aconsequence of the fact that spatially encoded signal in a DWimaging experiment is acquired after the DW gradients have beenapplied. Our simulation approach [2] reflects the correct physicalmeasurement: only spins whose trajectories end (at the time ofdata acquisition) inside an airway contribute to this airway’s (orvoxel, in the macroscopic equivalent) signal. Spins whose trajecto-ries end outside of the airway do not contribute to the signal mea-sured from that airway (or voxel), no matter if they were inside theairway when they were ‘labelled’ earlier. They do of course contrib-ute to the signal measured from a neighbouring airway (or voxel)and our model of the signal is continuous in this respect. Our paperclearly states that the WU approach would be equivalent to ours incertain cases (‘the detailed balance principle’), but it may not be va-lid in the presence of localized diffusion effects, where large spindephasing occurs due to strong applied diffusion gradients and/or local nonlinear susceptibility-induced inhomogeneity.

The Authors argue because the branches in our model haveclosed ends and are only 0.75 mm long, then it is over-restricteddue to the contribution from atoms which start their diffusionpaths in the closed airways.

The Authors overlook the fact that in our model the signal is cal-culated from the central duct only. Hence for spins that havebounced off the closed ends of the branches to contribute to thecentral duct signal they would have to travel the airway lengthLA = 0.76 mm plus the length of the connecting node �0.3 mm(i.e. �1.06 mm, which is similar to the length of an alveolar sac�1 mm). This distance is larger than the free diffusion path of3He atoms for the longest diffusion time used, D = 6 ms(p

(2D0D) = 1.03 mm).In comparison, the Authors have argued that for D = 1.8 ms, the

fraction of spins that leave a duct is negligible, even though the dis-tance they would need to travel to leave the duct (i.e. ½LA = 0.38 -mm or less) is significantly shorter than the free diffusion lengthp

(2D0D) = 0.56 mm. We find it hard to understand therefore whythe Authors would consider that the closed ends of the brancheswould considerably affect the signal originating from the centralduct; quoting their own words [10]: ’’the signal from the 3He atomsoriginating in one airway (duct or sac) does not depend on whetherthe adjacent airways are ducts or sacs.’’

The Authors go on to state that: ‘The ‘‘bridges’’ between airways(bottlenecks for the diffusing particles) also make the model biasedtoward the motion narrowing regime’.

We would readily compare the bridges in our model to those inthe Authors’ model [10] but this is not possible because they have

146 J. Parra-Robles, J.M. Wild / Journal of Magnetic Resonance 239 (2014) 143–146

never provided any details about those bridges. Our choice (clearlydescribed in [2]) was the simplest possible to prevent overlap be-tween the main duct and the branches and hence allow others toeasily reproduce our geometric model and verify the results ofour simulation if they feel it necessary.

The results of our paper clearly demonstrate that our simula-tions are not biased towards the motion-narrowing regime (dueto over-restriction by the bridges or airway ends), as the Authorsclaim. The DT and DL values (Figs. 4 and 6 in our paper [2]) obtainedwith our model are higher than the cylinder model prediction andnot lower as would be the case if biased towards motion narrowing(unless perhaps their model is more biased towards the motion-narrowing regime than ours?). Furthermore, a choice of links thatacted less as a ‘bottleneck’ than the one used by us would actuallymake the influence of branches even more significant, resulting ineven larger discrepancies with the cylinder model.

Another complaint of the Authors is that our investigation‘‘doesn’t take into account the distribution of airways’ radii and alve-olar sleeve depths (variation of about 17% ...’’. It is not clear why oneshould account for the distribution of airway sizes when testingthe validity of equations designed to be used to estimate mean air-way dimensions from diffusion data, when these equations wereobtained from computer simulations that did not include such avariable distribution? Our simulations proved that these equationsare incorrect or incomplete and their model surely cannot performbetter in the presence of uncertainty, i.e. be better at estimatingthe mean dimensions of a set of airways of different sizes than atdetermining the dimensions of a set of identical airways?

Finally, the Authors state that a shortcoming of our model is’’the fixed (90�) relative orientation of the attached ducts’ planes.This choice (even though used before in several simulation studies)is not validated by any histological data. In our paper (5) these ori-entations were chosen randomly’’.

This statement is scientifically unsubstantiated. Firstly, ourchoice of angle is validated by published measurements with mul-tidetector row X-ray-CT (MDCT) by Tawhai et al [22], which werein agreement with earlier publications [23,24], all of which re-ported values in the range 75–90�. Even if that published evidencecould be ignored, why would their choice of ‘randomly’, which hasnot been used in simulations by anyone else and is not validated byany measurements whatsoever, be superior to any alternativechoice?

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