Single Compartment model estimates of acinar duct measurements from inhaled noble gas MRI: Proof of Concept in alpha-1 antitrypsin deficiency emphysema

Eric Lessard^{1}, Alexei Ouriadov^{1}, David G McCormack^{2}, and Grace Parraga^{1}

**Subjects**: Six patients with alpha-1 antitrypsin
deficiency provided written informed consent to an approved study and underwent
spirometry, plethysmography, CT and ^{3}He MRI. Imaging was performed at 3.0 T (MR750, GEHC,
Waukesha, WI) using whole-body gradients (5G/cm maximum) and a rigid linear RF
coil (Rapid Biomedical, Germany). In a
single breath-hold, five interleaved acquisitions (TE = 3 ms, TR = 5 ms,
matrix-size = 128x128, number-of-slices = 7; slice-thickness=30mm, and FOV =
40x40cm) with and without diffusion sensitization were acquired for a given
line of k-space. The diffusion-sensitizing
gradient pulse ramp up/down time = 500 μs, constant time = 460 μs and diffusion
time (Δ) = 1.46 ms resulted in slices acquired at 0, 1.6, 3.2, 4.8 and 6.4 s/cm^{2}.

**Theory**: The short-time diffusivity regime assumes that a thin layer of gas molecules near the spherical walls experience restricted
diffusion while the remaining molecules diffuse freely. The diffusion time of 1.46ms is likely
insufficiently sensitive to longer alveolar length scales (single compartment
approximation) as is the case in extremely severe emphysema that accompanies
AATD. However, by subtracting the
measured apparent diffusion coefficient (ADC) for each b-value from D_{0}
(where D_{0} is the free diffusion coefficient of the gas, D_{0}
≈ 0.84cm^{2}/s) transforms the problem into the short-time diffusion
regime, where the diffusion is restricted near the walls, and unrestricted
within the sphere volume. It follows
that:

$$\frac{D_0 - ADC(t)}{D_0} = 1 - \frac{4}{9\sqrt{\pi}}\frac{S}{V}\sqrt{D_0t}$$

where ADC(t) is
the two b-value ADC(b) and the smallest diffusion time corresponds to the
largest b-value. To generate the diffusion times corresponding to each b-value
and fulfil the condition b=const, we assumed that 1.46ms was the longest
diffusion time and b=1.6s/cm^{2}.

**Image Analysis**: Mean ADC was calculated for all 4 b-values (b = 1.6,
3.2, 4.8, 6.4 s/cm^{2}). The
linear relationship of $$$\frac{D_0 - ADC(t)}{D_0}$$$ and $$$\sqrt{D_0t}$$$ (t = 0.832, 0.94, 1.11,
1.46 ms) was used to generate the slope that approximated S/V. S/V and L_{m}
estimates based on the single compartment model were also corrected using an
empirical coefficient so that subtraction of ADC from D_{0} would not
result in 0 and negative values.

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3 Parra-Robles J etal.
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5 Mitra PP etal. Physical review letters, (1992).

6 Mair RW. etal. Physical review letters, (1999).

7 Jacob RE. etal. Mag Res Med (2005).

8 Miller, GW etal. IEEE TMI, (2007).

AATD,
alpha-1 antitrypsin deficiency; FEV_{1}, forced expiratory volume in 1
second; FVC, forced vital capacity; DL_{CO}, diffusing capability of
the lung for carbon monoxide; RA_{950}, relative area of the CT density
histogram <-950 Hounsfield Units; ADC, apparent diffusion coefficient of
b=1.6s/cm^{2}; Lm, mean-linear-intercept; S/V,
surface-to-volume-ratio; ND, not done.

**Figure 2: Cylindrical and Single
Compartment Model Geometry Schematic and Data**

A) Cylindrical Model schematic

B) Single Compartment Model schematic

C) Cylindrical model theoretical surface-to-volume-ratio (S/V) vs R (r = 300 µm and r/R > 0.4) with individual R and S/V values

D) Single
Compartment Model (D_{0}-ADC)/D_{0} vs sqrt(D_{0}*t)
line-of-best-fit.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

2025