Random walk simulation of the MRI apparent diffusion coefficient in a geometrical model of the acinar tree

Abstract

Apparent diffusion coefficient (ADC) measurement in the lung using gas magnetic resonance imaging is a promising technique with potential for reflecting changes in lung microstructure. Despite some recent impressive human applications, full interpretation of ADC measures remains an elusive goal, due to a lack of detailed knowledge about the structure dependency of ADC. In an attempt to fill this gap we have performed random walk simulations in a three-dimensional geometrical model of the lung acinus, the distal alveolated sections of the lung tree accounting for approximately 90% of the total lung volume. Simulations were carried out adjusting model parameters after published morphological data for the rat peripheral airway system, which predict an ADC behavior as microstructure changes with lung inflation in partial agreement with measured ADCs at different airway pressures. The approach used to relate experimental ADCs to lung microstructural changes does not make any assumption about the cause of the changes, so it could be applied to other scenarios such as chronic obstructive pulmonary disease, lung development, etc. The work presented here predicts numerically for the first time ADC values measured in the lung from independent morphological measures of lung microstructure taken at different inflation stages during the breath cycle.

Figure 1

Three-dimensional representation of the bicylindrical coaxial alveolar duct model. This model requires several independent parameters to be fully characterized: inner and outer radii, length, number of sections in the axial and angular direction, and at least one parameter determining alveolar mouth size. (a) Whole duct. (b) Axial cut allowing us to see the interior of the structure.

Figure 2

Three-dimensional representation of the acinar tree model. Each duct between two nodes corresponds to the bicylindrical coaxial alveolar duct model with inner radius, an outer radius, and length computed for each generation multiplying the values of the previous generation by three constant factors, one for each of the parameters. Bifurcation probability can be constant during a number of initial generations and then decrease geometrically until bifurcation stops spontaneously. The angle between siblings is 80° and the bifurcation plane at each node is rotated 90° with respect to the previous bifurcation plane, where the parent node and its sibling are located.

Figure 3

Scheme of a bifurcation in the acinar tree. Alveolar mouths have not been represented, in order to keep the drawing as clear as possible.

Figure 4

Exponential fit of the signal attenuation values for the simulation of the pure ³He ADC measurement with free diffusion coefficients 1.4 and 1.91 cm²/s, and a bipolar sinusoidal gradient of duration 1.5 ms in the acinar tree model adjusted to reproduce 70% total lung capacity.

Figure 5

Simulation results at different pressures compared to ADC measurements in the rat lung. The points corresponding to the simulations are shown with a bar covering the ADC values obtained from several simulations in the 1.4–1.9 cm²/s range of the free diffusion coefficient. The extremes of this range correspond to 25%Air + ³He (1.4) and pure helium (1.91). The random error of the ADC simulation is described in Convergence of the Simulations. ADC measurements are reported as the average of measures carried out on six different animals, with the standard deviations represented with error bars.