## Abstract

Functional MRI of the lungs with hyperpolarized helium provides an index of apparent diffusion measured over several seconds (ADC_{sec}) that is only 2% of its free diffusion in air (0.88 cm^{2}/s). The potential of ADC_{sec} to noninvasively assess in vivo lung structure of diseased lungs at the length scales corresponding to several seconds is critically dependent on the exact link between ADC_{sec} and lung peripheral structure. To understand the intruigingly small ADC_{sec}, numerical simulations of gas transport were performed in *1*) a trumpet model, *2*) a symmetrical, and *3*) an asymmetrical multiple-branch-point model of the human acinus. For initial gas boluses in different locations of the acinar models, ADC_{sec} was quantified as follows. At different time intervals, we computed a coefficient of variation (CoV) of the concentration distributions within each acinar model. The slope in the semilog plot of log(CoV) vs. time was proportional to the ADC_{sec} generated by the internal model structure, provided that the outer model boundaries were similar across all models (i.e., similar cumulative cross section vs. average path length). The simulations revealed an ADC_{sec} that amounted to ∼1% of free diffusion in the trumpet model of the acinus, i.e., corresponding to free diffusion within the acinar geometric boundaries. Our simulations show that for initial conditions corresponding to those used in MRI experiments, intra-acinar branching introduces a dramatic diffusion delay, comparable to what is observed experimentally.

- multiple-branch-point model

functional mri of the lungs with hyperpolarized gases, in particular helium-3, has revolutionized the potential for in vivo lung function measurement in health and disease (10). Besides the dynamic imaging of inhaled helium-3 gas distributing over an expanding lung, another interesting application concerns the so-called apparent diffusion coefficient (ADC) (1, 8, 14, 17, 18). Derived on the basis of local signal attenuation owing to the diffusion of the helium-3 molecules after superimposing a specific magnetic field profile, local ADCs are computed. When measuring short-range ADC over milliseconds (ADC_{msec}), as has thus far been most frequently reported in human experiments, the peripheral air spaces that can be covered by molecular diffusion are limited to little more than an alveolated airway or two. Indeed, a typical diffusion distance covered over 1 ms is ∼400 μm (=; the binary diffusion coefficient *D* may be considered 0.88 cm^{2}/s for helium-3 in an air-filled lung). In a normal lung, typical experimental short-range ADC_{msec} is ∼25% of the free diffusion coefficient for hyperpolarized helium-3 in air (7, 18).

When long-range ADC is measured over several seconds (ADC_{sec}) (with a typical diffusion distance in 1 s corresponding to 1.33 cm), diffusive transport may cover an entire gas exchanging unit, an acinus, as it was defined by Haefeli-Bleuer and Weibel (3). Experimental values of ADC_{sec} are typically one order of magnitude smaller than ADC_{msec}, ranging from 1% to 3% of free diffusion, depending on the exact experimental condition (2, 4, 17). In a diseased lung, both local ADC_{msec} and ADC_{sec} have been seen to increase, signaling local widening of air spaces due to destruction of airway walls and/or alveolar septa (14, 17). It has been suggested that long-range ADC_{sec} could be more sensitive than short-range ADC_{msec} because it is an order of magnitude further away from the free diffusion coefficient (thus leaving a greater margin for grading of disease stages) and that ADC_{sec} is much smaller than ADC_{msec} because gas molecules need to negotiate “tortuous” pathways (16).

A potential diagnostic use of ADC measurement will depend on understanding its relation to the complexity of peripheral lung structure in a quantitative way. For this purpose, we simulated diffusion in an anatomically realistic model of the human acinus to assess how diffusional spread is hampered by internal acinar structure. As an alternative to the magnetic resonance technique where concentration differences are not a prerequisite to trace and quantify diffusion of helium-3 in air, we intentionally introduce an initial helium-3 concentration gradient in the model to drive diffusion and quantify effective diffusive spread. Importantly, diffusion is simulated here in a multiple-branch-point model of the human acinus over comparable time intervals to those used to measure ADC_{sec}.

## MATERIALS AND METHODS

#### Model geometries.

Three acinar models have been used to highlight the effect of intra-acinar branching. The three models are presented in an order of decreasing complexity. The asymmetrical multiple-branch-point model (MBPM_{asym}) is an acinar branching structure based on the anatomic data of Haefeli-Bleuer and Weibel (3) that has been extensively used for simulation of respiratory experiments (12). Its branching pattern can be found in Fig. 1*A*. With its acinar entrance at *generation 15*, its shortest pathway ends in *generation 21* and its longest in *generation 26*; the cumulative cross section per generation is shown in Fig. 2 (open circles). The symmetrical multiple-branch-point model (MBPM_{sym}) is a hypothetical acinar branching structure whereby bifurcation in successive generations is purely dichotomous (Fig. 1*B*), with all airways ending in *generation 23* (Fig. 2). All airway lengths and cross sections of the first eight acinar generations in MBPM_{sym} are identical to those of the corresponding generations in MBPM_{asym} (up to 0.72 cm from acinar entrance). Beyond that, minor modifications are implemented to match the average pathway length and total volume of MBPM_{asym}. In this way, MBPM_{asym} and MBPM_{sym} are totally equivalent, except for their branching arrangement. Finally, the trumpet model simply consists of one central channel with nine segments with airway lengths and cumulative cross sections corresponding to those of MBPM_{sym} (solid line in Fig. 2). All three models were discretized, to allow for numerical solution of the gas transport equation: the trumpet model, MBPM_{sym}, and MBPM_{asym} were discretized into respectively 9, 512, and 627 nodes; this discretization is mainly determined by the average pathway distance between two subsequent nodes with respect to typical displacement by diffusion over the course of a numerical time step.

#### Simulation and quantification of diffusion in the acinar models.

With the initial condition of a helium-3 bolus at different locations of the model, diffusive spread was computed in the three models (MBPM_{asym}, MBPM_{sym}, trumpet) with the acinar volume scaled to 3.7 liters and with the model entrance closed. This was done using a one-dimensional (1-D) gas transport equation of diffusion and convection in the lung as given by the partial differential equation (6): (1) where C is gas concentration in each spatial location *z* along each axial pathway; *s* is internal lumen cross section (of the internal channel that excludes the alveoli) and *S* is external cross section (of the envelope cylinder including the alveoli) at each axial distance *z* and time *t*; V is volume; and *D* and V̇ are the binary diffusion coefficient and respiratory flow, respectively. Note that the second term, (*D*/*S*)(∂*s*/∂*z*), has the dimensions of a velocity and results from the assumption of instantaneous radial diffusion in a geometry with increasing lumen cross section. This enables the study of diffusion in a three-dimensional (3-D) structure by means of a 1-D equation (5, 6). Since diffusion is studied during a breath holding phase, the two last terms of *Eq. 1* are discarded.

When *Eq. 1* is solved numerically using fourth-order Runge-Kutta with finite difference equations that can be found elsewhere (13), this results in simulated helium-3 concentration values on each model node at each point in time. The volume corresponding to each node is then used to compute a volume-weighted average and a volume-weighted SD of all concentrations in the model; the ratio of this volume-weighted SD and the volume-weighted average is the volume-weighted coefficient of variation (CoV) used here. The volume-weighting of CoV was necessary for comparison between models because, for instance, in the trumpet model, the first and last node of the model represent, respectively, 0.16% and 65% of the total model volume, while in MBPM_{sym}, the first and one of the last 256 (=2^{8}) nodes of the model represent 0.16% and 0.25% of the total model volume, respectively. The CoV is then plotted in semilog scale as a function of time *t* for the quantification of effective diffusion.

To illustrate how an effective diffusion can be derived from semilog CoV vs. *t* plots, Fig. 3*A* shows CoV curves obtained for a closed cylinder where an initial helium-3 bolus at one of the cylinder ends is allowed to diffuse with *D* = 0.88 cm^{2}/s. In fact, in this simple case, we could verify that the analytical solution, which is a sum of error functions with a *D* dependence, produces identical CoV vs. *t* curves to those obtained numerically. Figure 3*A* shows that the slopes of the semilog CoV vs. *t* plot decrease with increasing cylinder length (ranging between approximately half and double the acinar length). As the cylinder becomes shorter, the rate of diffusive equilibration, or the rate at which CoV tends to zero, is greater. For one given cylinder length (10 mm), Fig. 3*B* shows that varying the initial bolus position between both cylinder ends does not affect the rate of decrease of the CoV curves, i.e., the semilog CoV vs. *t* slope remains the same.

More interestingly, when considering a cylinder of any given length, this semilog CoV slope is directly proportional to the diffusion coefficient. Indeed, when simulating the entire CoV vs. *t* curve over a 0- to 1.0-s time interval for *D* = 0.44 cm^{2}/s and rescaling it to half that time interval, the resulting curve is identical to the one obtained by simulating *D* = 0.88 cm^{2}/s over 0- to 0.5-s time interval. In other words, doubling *D* or halving *t*-axis has the same effect on the CoV curve, i.e., in this CoV vs. *t* representation, changes in *D* and *t* are interchangeable. This applies to all curves in Fig. 3.

## RESULTS

Figure 4 illustrates how the concentration distributions evolve because of diffusion over a 5-s interval (typical of a ADC_{sec} measurement) in the three acinar models, MBPM_{asym}, MBPM_{sym}, and the trumpet model, when an initial helium bolus is placed at the entrance of each model at *t* = 0. The solid lines in Fig. 4 represent the helium-3 concentration obtained for generations up to *generation 23* of MBPM_{sym}, showing complete diffusive equilibration after only 0.5 s. In this perfectly symmetrical model, the concentration in all parallel airways of any given branching generation is the same. Hence, for this particular condition, this model is equivalent to a model with one central pathway, which is attributed one cross section at each point along this pathway that equals the cumulative cross section of all parallel airways with the same distance from acinar entrance, corresponding to the trumpet model. The filled circles in Fig. 4 are the corresponding concentration distributions in the asymmetrical model MBPM_{asym}, showing a parallel heterogeneity in concentration and a much slower diffusive equilibration.

The volume-weighted CoV corresponding to the simulated concentration distributions in Fig. 4 are the dotted lines in Fig. 5, *A–C*. In Fig. 5, *B* and *C*, the very rapid CoV vs. *t* decrease corresponds to the rapid diffusive equilibration of the helium bolus introduced at the model entrance of the MBPM_{sym} and trumpet model. By contrast, Fig. 5*A* (dotted line) reflects the considerable delay of the diffusive equilibration of an initial helium bolus inserted at the model entrance of MBPM_{asym}. All solid lines in Fig. 5, *A–C*, correspond to the diffusive equilibration obtained when the initial helium-3 bolus is not inserted at the model entrance but at a node belonging to successive branching generations of each model. In the trumpet model (Fig. 5*C*), the initial position of the helium-3 bolus in more peripheral acinar generations essentially induces a slight translation on the *t*-axis (Fig. 5*C*). Yet, this does not affect the rate of CoV vs. *t* decrease (i.e., effective diffusive equilibration), which is very rapid in this case. In the case of both MBPM models, however (Fig. 5, *A* and *B*), diffusive equilibration is slowed down to more or less the same extent as soon as the initial helium-3 bolus is placed in a node somewhere within the branching structure. While the level and initial rapid descent of the entire CoV vs. *t* curve depends on the initial distribution of helium-3 boluses over the model, the effective diffusion intrinsic to the structure over most of the measurement interval is estimated from the quasi-linear part of the semilog CoV vs. *t* curves in Fig. 5, i.e., from the chord slope between 0.5 and 5 s.

To assess the effective diffusion induced by intra-acinar branching, we first compute the reference slope corresponding to free diffusion within the acinar confines, i.e., that obtained from the semilog CoV vs. *t* plots corresponding to the trumpet model in Fig. 5*C*. The slopes of the semilog CoV vs. *t* plots obtained in the realistic MBPM_{asym} model are computed for each condition of initial helium-3 bolus location (i.e., using all curves in Fig. 5*A*). Finally, an average effective diffusion for the entire acinus is obtained by assuming that each node of the model has the same probability of being represented in the magnetic resonance ADC measurement (see discussion). Hence, overall effective diffusion was computed as the weighted average of the slopes obtained for each semilog CoV vs. *t* curve in Fig. 5*A*, where the number of nodes in each MBPM_{asym} generation is the weighting factor. For instance, the effective diffusion computed from the simulation of diffusion from a MBPM_{asym} node situated in *generation 16* is 0.36% of free diffusion, while the diffusion from a node situated in *generation 23* is 0.99% of free diffusion. However, because there are only 2 nodes (airways) in *generation 16* and 190 nodes in *generation 23*, the overall estimate of effective diffusion will be much closer to that obtained for *generation 23*. After weighting effective diffusion obtained for each generation by the number of nodes per generation, the resulting effective diffusion leads to 0.97% of free diffusion.

## DISCUSSION

We evaluated effective diffusion in the human acinus by simulating diffusive transport in an anatomy-based model of intra-acinar airway branching (MBPM_{asym}) and by quantifying the rate at which concentration distribution over the model equilibrates. This was done by comparing the semilog CoV vs. *t* plots obtained with MBPM_{asym} (Fig. 5*A*) to those of the acinar trumpet model (Fig. 5*C*), i.e., the envelope of the acinar air space within which free diffusion can take place. This methodology is partly similar to that previously employed to estimate effective diffusion due to airway wall structure (5, 11), where diffusion of a gas bolus was simulated in a cylinder with lateral alveolar walls by random walk and compared with diffusion in a cylinder of similar outer airway cross section, i.e., the envelope of the acinar airway. In that case, it was possible to compare the random walk concentrations at a given location of the alveolar duct model to its analytic solution, which incorporated *D*, to obtain a best-fit *D* or an effective diffusion coefficient. For the problem of effective diffusion in the acinus with all its structural complexity, an analytic solution cannot be obtained. Therefore, effective diffusion was estimated from the rate at which a distribution of concentrations equilibrates in a model with anatomy-based intra-acinar branching (MBPM_{asym}) vs. that in a model without intra-acinar branching (trumpet model).

The reason for the slow diffusion phase induced by airway branching can be understood from the simulation corresponding to an initial helium-3 bolus at the MBPM_{asym} model entrance (Fig. 4; closed symbols). This illustrates how, during the first 0.10 s, the main driving force of diffusive equilibration is the concentration gradient between entrance and end of the MPBM_{asym} (i.e., depending on the initial condition). However, when most of this gradient is abolished, considerable concentration differences remain between parallel airways that take much longer to equilibrate. This is due to the fact that at each individual branch point of the MBPM_{asym}, diffusive flow into both subtended airways is identical (since cross sections of any 2 daughter branches are considered identical in MBPM_{asym}). Consequently, the number of helium-3 molecules entering the volumes subtended from each branch point will be disproportionately high in a smaller subtended volume. This generates a range of concentration differences across parallel airways, and it is the complexity of all possible pathways between parallel units embedded in this MBPM_{asym} network that is determinant of the time it takes for diffusive equilibration. In the MBPM_{sym}, diffusive equilibration of a helium-3 bolus entering the model is much quicker (Fig. 4; solid line) because any incoming gas diffuses toward all branching endings to the same extent and concentration gradients across parallel pathways of the same branching generation are zero. However, as soon as the initial helium-3 bolus is placed in a location that is not symmetrical with respect to the branching structure, the same mechanism as in MBPM_{asym} comes into play. This explains why a slow diffusive equilibration also occurs in the MBPM_{sym} when the initial helium-3 bolus is not located at the model entrance (solid lines in Fig. 5*B*).

While concentration maps in the present model were obtained by solving gas transport equations that were interconnected in a very particular network arrangement, the CoV vs. *t* analysis does eliminate spatial information when reducing concentration maps to a concentration distribution. While rendering the problem more manageable in terms of finding one apparent diffusion value for comparison with experiments, this seemingly simplistic approach is somewhat counterintuitive in the context of diffusion where the basic concept is the mean root-mean-square displacement. While far more complex 3-D analyses or a full computational fluid dynamics solution in a realistic acinar model may be put forward to address ADC_{sec} quantification via computation of 3-D root-mean-square displacements, the resulting concentration distributions would still need to be in agreement with the CoV vs. *t* curves produced here. Our model approach cannot capture any possible anisotropy of ADC_{sec}, which, to the best of our knowledge, has not yet been reported in the literature. However, considering the large number of intra-acinar ducts, it is reasonable to assume a random orientation.

The curves in Fig. 5 suggest that effective diffusion may vary with time, depending in part on the initial condition, whereas both the magnetic resonance experiments and our simulations determine one single effective diffusion coefficient. In fact, the main problem for simulating the experiments is the estimation of the initial condition that best represents the experimental situation. In this study, we have assumed that all nodes of the acinar model contributed equally to overall effective diffusion. However, given the uncertainty regarding the initial condition in the experiments, the first rapid descent of the CoV curve (*t* < 0.5 s) was neglected, and thus our approach may have slightly underestimated effective diffusion obtained experimentally.

The simulated effective diffusion coefficient of the order of 1% of free diffusion in the normal lung may be considered surprisingly similar to the ADC_{sec} that is obtained in magnetic resonance experiments (typically 1–3% of free diffusion in normal humans), given that both methods to estimate one single effective diffusion coefficient are based on totally different principles. In the experiments, an ADC_{sec} value is derived from the MRI signal obtained from any one voxel, which may contain several acini within which diffusion cannot be measured directly. In the acinar model, we actually simulate diffusion in the structure and conversely derive a global number to represent effective diffusion of the acinus as a whole.

Two aspects of our model simulations could have affected the estimate of effective diffusion. First, inter-acinar diffusion was discarded here because, given the dramatic intra-acinar increase in cumulative cross section, diffusive flow out of the acinus via one single airway cross section is much smaller than that within each acinus. Second, collateral ventilation was not considered. While thought to be negligible in normal lungs (9), even a very limited number of collateral ventilatory pathways could have an increasing effect on effective diffusion. To assess the impact of both these effects in a quantitative way, better knowledge on inter-acinar geometry (e.g., branching asymmetry of the airways connecting several acini) and on the number distribution of collateral ventilatory pathways would be required. Finally, the study of effective diffusion as a function of the time interval could also provide useful hints about the functional units that affect estimates of effective diffusion in the lung periphery.

In summary, we simulated diffusion after the introduction of an initial helium-3 concentration gradient in a realistic model of the human acinus. Effective diffusive spread was derived from the time evolution of the CoV of the concentration distribution within the model. For comparable time intervals to those used to measure the long-range apparent diffusion coefficient in functional MRI experiments, we found that the intra-acinar airways represent a restriction to free diffusion of the same order as that obtained experimentally (1%).

## GRANTS

This study was funded by the Fund for Scientific Research-Flanders and the Federal Office for Scientific Affairs (Program PRODEX).

## Acknowledgments

We are grateful to G. K. Prisk for suggestions and comments during the preparation of the manuscript.

## Footnotes

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- Copyright © 2007 the American Physiological Society