Annie Malkus^{1}, Robert V Cadman^{1}, and Sean B Fain^{1,2,3}

We evaluate a morphometric approach for measuring acinar airway and alveolar scales with diffusion weighted MRI of hyperpolarized helium in pediatric asthma subjects.

Forty subjects were scanned with
informed consent under IND#064867. The population included asthmatic and
healthy normal pediatric subjects. Of those, we fit the images from 14 subjects,
aged 14-17 years (mean age 16 ± 1; 8 females
with mean age 16 ± 1). The scan included diffusion
weighting with parameters suited to the morphometric model approach described previously
for subjects with emphysema^{10}.
The model validity extends to $$$R = 400$$$ μm.
Based on work in adults^{6},
we expect the scales present in lungs to be within this regime. Scan parameters
are listed in Table 2.

The image data are used to calculate
the acinar airway scale and the alveolar depth. Specifically, we fit the image
data to the model in equations (A1-A5)^{10}
restated below:

$$S(b)=S_0\exp\left(-bD_T\right)\sqrt{\frac{\pi}{4b\left(D_L-D_T\right)}}Erf\left(\sqrt{b\left(D_L-D_T\right)}\right)\text{(Equation 1)},$$

where $$$Erf()$$$ is the error function, $$$b$$$ is the diffusion weighting parameter

$$b=\left(\gamma G\right)^2\left[\delta^3\left(\Delta -\frac{\delta}{3}\right)+\tau\left(\delta^2-2\delta\Delta+\Delta\tau-\frac{7}{6}\delta\tau+\frac{8}{15}\tau^2\right)\right],$$

$$$\gamma$$$ is the gyromagnetic ratio of the gas, $$$G$$$ is the gradient strength, $$$\delta$$$ is the duration of the diffusion encoding gradient, $$$\Delta$$$ is separation between the beginnings of the lobes of the encoding gradients, and $$$\tau$$$ is the ramp up time.

The $$$D_T$$$ and $$$D_L$$$ are the transverse and longitudinal diffusion coefficients respectively, and $$$S_0$$$ is the signal strength in the absence of diffusion. The diffusion coefficients depend upon $$$R$$$ and $$$h$$$:

$$D_L =D_{L0}\left(1-\beta_LbD_{L0}\right);\\D_T =D_{T0}\left(1+\beta_TbD_{T0}\right),$$

with

$$\frac{D_{L0}}{D_0}=\exp\left[-2.89(h/R)^{1.78}\right];\\ \beta_L=35.6(R/L_1)^{1.5}\exp\left[-4\sqrt{h/R}\right]$$

and

$$\frac{D_{T0}}{D_0}=\exp\left[-0.73(L_2/R)^{1.4}\right]\left(1+\exp\left(-A(h/R)^2\right)\left\{\exp\left[-5(h/R)^2\right] + 5(h/R)^2 - 1\right\}\right);\\\beta_T=0.06;A=1.3+0.25\exp\left[14(R/L_2)^2\right]$$.

The diffusion scales are $$$L_1=\sqrt{2D_0\Delta}$$$ and $$$L_2=\sqrt{4D_0\Delta}$$$ from $$$D_0$$$, the free diffusion coefficient. For the fit, we use a Markov Chain
Monte Carlo (MCMC) Bayesian approach^{11}
implemented in MATLAB (The Mathworks, Natick, MA). To explore the uncertainties
associated with the fits, we employed a MCMC approach using a digital
phantom. The digital phantom was constructed by evaluating the signal model of Equation 1
for values of $$$R$$$ and $$$h$$$ that are consistent with the model’s predicted
valid range. The MCMC fits were conducted by repeating 50 experiments of sampling
the parameter space with a Metropolis Hastings algorithm, keeping the last 1000
steps after a burn in of 1000 steps. We fit the parameters, $$$R$$$, $$$h$$$, and the overall signal strength, $$$S_0$$$.
No noise was added to the phantom signal.

The phantom studies show that the model and MCMC fits are robust for dimensions in the expected physiological range of $$$R = 200-400$$$ μm and $$$h = 100-200$$$ μm. Estimated errors on the order of 50% for $$$R > 200$$$ μm and $$$h > 200$$$ μm are shown in Figure 1. Corresponding image results and estimates of uncertainty in the model fit for a typical normal subject are shown in Figure 2.

Note that the $$$R$$$ values are consistently larger than their corresponding $$$h$$$ values as expected for the model geometry. A summary of $$$R$$$ and $$$h$$$ values for all subjects appears in Table 3. We compared the $$$h/R$$$ values from female subjects to male subjects with an ANOVA.

1. Paiva M. Gaseous diffusion in an alveolar duct simulated by a digital computer. Comput Biomed Res. 1974 Dec;7(6):533-43.

2. O'Halloran RL, Holmes JH et al. The effects of SNR on ADC measurements in diffusion-weighted hyperpolarized He-3 MRI .J Magn Reson. 2007 Mar;185(1):42-9.

3. Wang W, Nguyen NM, et al. Imaging lung microstructure in mice with hyperpolarized 3He diffusion MRI. Magn Reson Med. 2011 Mar;65(3):620-6.

4. Parra-Robles J, Wild JM. The influence of lung airways branching structure and diffusion time on measurements and models of short-range 3He gas MR diffusion. J Magn Reson. 2012 Dec;225:102-13.

5. Kruger SJ, Nagle SK et al. Functional imaging of the lungs with gas agents. J Magn Reson Imaging. 2016 Feb;43(2):295-315.* *

6. Hajari AJ, Yablonskiy DA, et al. Morphometric changes in the human pulmonary acinus during inflation.J Appl Physiol (1985). 2012 Mar;112(6):937-43.

7. Quirk JD, Sukstanskii AL, et al. Experimental evidence of age-related adaptive changes in human acinar airways.J Appl Physiol (1985). 2016 Jan 15;120(2):159-65.

8. Cadman RV1, Lemanske RF Jr, et al. Pulmonary 3He magnetic resonance imaging of childhood asthma. J Allergy Clin Immunol. 2013 Feb;131(2):369-76.e1-5* *

9. Sukstanskii AL, Yablonskiy DA. In vivo lung morphometry with hyperpolarized 3He diffusion MRI: theoretical background. J Magn Reson. 2008 Feb;190(2):200-10.

10. Yablonskiy DA, Sukstanskii AL, et al. Quantification of lung microstructure with hyperpolarized 3He diffusion MRI. J Appl Physiol. 2009 Oct;107(4):1258-65.

11. Sukstanskii AL1, Bretthorst GL, et al. How accurately can the parameters from a model of anisotropic 3He gas diffusion in lung acinar airways be estimated? Bayesian view. J Magn Reson. 2007 Jan;184(1):62-71.Figure
1: Errors from fitting the
model to a digital phantom. We show the normalized error in the alveolar depth,
$$$h$$$, on the left and the acinar airway radius, $$$R$$$, on the right.

Figure 2: Fit
results for one slice in a healthy normal subject. The top row shows histograms
of the fit parameters in the slice. The middle row shows the fit parameters in
each voxel. The bottom plot shows the uncertainty in those fit parameters. The
left column shows plots associated with the $$$R$$$ parameter; the middle with the $$$h$$$
parameter, and the right column with the signal strength.

Table 1: MRI scan parameters, where the notation for diffusion
weighting gradients is the same as in^{10}.

Table 2: Results from fits to all 10 subjects
averaged by sex.