Concise derivation of oscillating-gradient-derived ADC.

The apparent diffusion coefficient (ADC) is analyzed for the case of oscillating diffusion-sensitizing gradients in the high-frequency regime. We provide a concise derivation of the analytical expression for the ADC for an arbitrary number of gradient oscillations N and initial phase φ. It is demonstrated that an ultimate goal - to determine the surface-to-volume ratio (S/V) from MR measurements by using oscillating gradients - can be achieved with cosine-type gradients (φ = 0) for an arbitrary N. However, to determine S/V employing gradients with φ ≠ 0 (including the sine-type gradients) and arbitrary N additionally requires prior knowledge of the time-dependent diffusion coefficient D(t). The latter is rarely known a priori but can be estimated under certain limiting conditions: (i) in the short time regime, when the total diffusion time of the measurements, t, is smaller than the characteristic diffusion time of the microstructural system of interest, an analytical expression for D(t) is available (Mitra's expression) and this allows S/V to be determined in the short time regime with sine-type gradients; (ii) in the important case of purely restricted diffusion, D(t) → 0 at sufficiently long time, the signal becomes independent of φ and behaves as for the cosine-type gradients, thus, allowing determination of S/V.

Exact analytical results for ADC with oscillating diffusion sensitizing gradients.

The apparent diffusion coefficient (ADC) is analyzed for the case of oscillating diffusion sensitizing gradients. Exact analytical expressions are obtained in the high-frequency expansion of the ADC for an arbitrary number of oscillations N. These expressions are universal and valid for arbitrary system geometry. The validity conditions of the high-frequency expansion of ADC are obtained in the framework of a simple 1D model of restricted diffusion. These conditions are shown to be substantially different for cos- and sin-type gradients: for the cos-type gradients, the high-frequency expansion is valid when the period of a single oscillation is smaller than the characteristic diffusion time, the frequency dependence of ADC being practically the same for any N. In contrast, for the sin-type gradients, the high-frequency regime can be achieved only when the total diffusion time is smaller than the characteristic diffusion time, the frequency dependence of ADC being different for different N.

Morphometric changes in the human pulmonary acinus during inflation.

Despite decades of research into the mechanisms of lung inflation and deflation, there is little consensus about whether lung inflation occurs due to the recruitment of new alveoli or by changes in the size and/or shape of alveoli and alveolar ducts. In this study we use in vivo (3)He lung morphometry via MRI to measure the average alveolar depth and alveolar duct radius at three levels of inspiration in five healthy human subjects and calculate the average alveolar volume, surface area, and the total number of alveoli at each level of inflation. Our results indicate that during a 143 ± 18% increase in lung gas volume, the average alveolar depth decreases 21 ±5%, the average alveolar duct radius increases 7 ± 3%, and the total number of alveoli increases by 96 ± 9% (results are means ± SD between subjects; P < 0.001, P < 0.01, and P < 0.00001, respectively, via paired t-tests). Thus our results indicate that in healthy human subjects the lung inflates primarily by alveolar recruitment and, to a lesser extent, by anisotropic expansion of alveolar ducts.

Lung morphometry with hyperpolarized 129Xe: theoretical background.

The (3) He lung morphometry technique, based on MRI measurements of hyperpolarized (3) He gas diffusion in lung airspaces, provides unique information on the lung microstructure at the alveolar level. In vivo 3D tomographic images of standard morphological parameters (airspace chord length, lung parenchyma surface-to-volume ratio, and number of alveoli per unit volume) can be generated from a rather short (several seconds) MRI scan. The technique is based on a theory of gas diffusion in lung acinar airways and experimental measurements of diffusion-attenuated MRI signal. The present work aims at developing the theoretical background of a similar technique based on hyperpolarized (129) Xe gas. As the diffusion coefficient and gyromagnetic ratio of (129) Xe gas are substantially different from those of (3) He gas, the specific details of the theory and experimental measurements with (129) Xe should be amended. We establish phenomenological relationships between acinar airway geometrical parameters and the diffusion-attenuated MR signal for human and small animal lungs, both normal lungs and lungs with mild emphysema. Optimal diffusion times are shown to be about 5 ms for human and 1.3 ms for small animals. The expected uncertainties in measuring main morphometrical parameters of the lungs are estimated in the framework of Bayesian probability theory.

Imaging alveolar-duct geometry during expiration via ³He lung morphometry.

Acinar geometry has been the subject of several morphological and imaging studies in the past; however, surprisingly little is known about how the acinar microstructure changes when the lung inflates or deflates. Lung morphometry with hyperpolarized (3)He diffusion MRI allows non-destructive evaluation of lung microstructure and acinar geometry, which has important applications in understanding basic lung physiology and disease. In this study, we have measured the alveolar and acinar duct sizes at physiologically relevant volumes by (3)He lung morphometry in six normal, excised, and unfixed canine lungs. Our results imply that, during a 37% decrease in lung volume, the acinar duct radius decreases by 19%, whereas the alveolar depth increases by 9% (P < 0.0001 and P < 0.05, respectively via paired t-tests with a Bonferroni correction). A comparison to serial sections under the microscope validates the imaging results and opens the door to in vivo human studies of lung acinar geometry and physiology during respiration using (3)He lung morphometry.

³He lung morphometry technique: accuracy analysis and pulse sequence optimization.

The (3)He lung morphometry technique (Yablonskiy et al., JAP, 2009), based on MRI measurements of hyperpolarized gas diffusion in lung airspaces, provides unique information on the lung microstructure at the alveolar level. 3D tomographic images of standard morphological parameters (mean airspace chord length, lung parenchyma surface-to-volume ratio, and the number of alveoli per unit lung volume) can be created from a rather short (several seconds) MRI scan. These parameters are most commonly used to characterize lung morphometry but were not previously available from in vivo studies. A background of the (3)He lung morphometry technique is based on a previously proposed model of lung acinar airways, treated as cylindrical passages of external radius R covered by alveolar sleeves of depth h, and on a theory of gas diffusion in these airways. The initial works approximated the acinar airways as very long cylinders, all with the same R and h. The present work aims at analyzing effects of realistic acinar airway structures, incorporating airway branching, physiological airway lengths, a physiological ratio of airway ducts and sacs, and distributions of R and h. By means of Monte-Carlo computer simulations, we demonstrate that our technique allows rather accurate measurements of geometrical and morphological parameters of acinar airways. In particular, the accuracy of determining one of the most important physiological parameter of lung parenchyma - surface-to-volume ratio - does not exceed several percent. Second, we analyze the effect of the susceptibility induced inhomogeneous magnetic field on the parameter estimate and demonstrate that this effect is rather negligible at B(0) ≤ 3T and becomes substantial only at higher B(0) Third, we theoretically derive an optimal choice of MR pulse sequence parameters, which should be used to acquire a series of diffusion-attenuated MR signals, allowing a substantial decrease in the acquisition time and improvement in accuracy of the results. It is demonstrated that the optimal choice represents three not equidistant b-values: b(1)=0, b(2)∼2 s/cm(2), b(3)∼8 s/cm(2).

Quantitative assessment of lung microstructure in healthy mice using an MR-based 3He lung morphometry technique.

The recently developed technique of lung morphometry using hyperpolarized (3)He diffusion magnetic resonance (MR) (Yablonskiy DA, Sukstanskii AL, Woods JC, Gierada DS, Quirk JD, Hogg JC, Cooper JD, Conradi MS. J Appl Physiol 107: 1258-1265, 2009) permits in vivo study of lung microstructure at the alveolar level. Originally proposed for human lungs, it also has the potential to study small animals. The technique relies on theoretical developments in the area of gas diffusion in lungs linking the diffusion attenuated MR signal to the lung microstructure. To adapt this technique to small animals, certain modifications in MR protocol and data analysis are required, reflecting the smaller size of mouse alveoli and acinar airways. This is the subject of the present paper. Herein, we established empirical relationships relating diffusion measurements to geometrical parameters of lung acinar airways with dimensions typical for mice and rats by using simulations of diffusion in the airways. We have also adjusted the MR protocol to acquire data with much shorter diffusion times compared with humans to accommodate the substantially smaller acinar airway length. We apply this technique to study mouse lungs ex vivo. Our MR-based measurements yield mean values of lung surface-to-volume ratio of 670 cm(-1), alveolar density of 3,200 per mm(3), alveolar depth of 55 µm, and mean chord length of 62 µm, all consistent with published data obtained histologically in mice by unbiased methods. The proposed technique can be used for in vivo experiments, opening a door for longitudinal studies of lung morphometry in mice and other small animals.

Intracellular water specific MR of microbead-adherent cells: HeLa cell intracellular water diffusion.

The (1)H MR signal arising from flowing extracellular media in a perfused, microbead-adherent cultured cell system can be suppressed with a slice-selective, spin-echo pulse sequence. The signal from intracellular water can, thus, be selectively monitored. Herein, this technique was combined with pulsed field gradients (PFGs) to quantify intracellular water diffusion in HeLa cells. The intracellular water MR diffusion-signal attenuation at various diffusion times was well described by a biophysical model that characterizes the incoherent displacement of intracellular water as a truncated Gaussian distribution of apparent diffusion coefficients (ADCs). At short diffusion times, the water "free" diffusion coefficient and the surface-to-volume ratio of HeLa cells were estimated and were, 2.0 +/- 0.3 microm(2)/ms and 0.48 +/- 0.1 microm(-1) (mean +/- SD), respectively. At long diffusion times, the cell radius of 10.1 +/- 0.4 microm was inferred and was consistent with that measured by optical microscopy. In summary: 1) intracellular water "free" diffusion in HeLa cells was rapid, two-thirds that of pure water; and 2) the cell radius inferred from modeling the incoherent displacement of intracellular water by a truncated Gaussian distribution of ADCs was confirmed by independent optical microscopy measures.

In vivo lung morphometry with hyperpolarized 3He diffusion MRI: theoretical background.

MRI-based study of (3)He gas diffusion in lungs may provide important information on lung microstructure. Lung acinar airways can be described in terms of cylinders covered with alveolar sleeve [Haefeli-Bleuer, Weibel, Anat. Rec. 220 (1988) 401]. For relatively short diffusion times (on the order of a few ms) this geometry allows description of the (3)He diffusion attenuated MR signal in lungs in terms of two diffusion coefficients-longitudinal (D(L)) and transverse (D(T)) with respect to the individual acinar airway axis [Yablonskiy et al., PNAS 99 (2002) 3111]. In this paper, empirical relationships between D(L) and D(T) and the geometrical parameters of airways and alveoli are found by means of computer Monte Carlo simulations. The effects of non-Gaussian signal behavior (dependence of D(L) and D(T) on b-value) are also taken into account. The results obtained are quantitatively valid in the physiologically important range of airway parameters characteristic of healthy lungs and lungs with mild emphysema. In lungs with advanced emphysema, the results provide only "apparent" characteristics but still could potentially be used to evaluate emphysema progression. This creates a basis for in vivo lung morphometry-evaluation of the geometrical parameters of acinar airways from hyperpolarized (3)He diffusion MRI, despite the airways being too small to be resolved by direct imaging. These results also predict a rather substantial dependence of (3)He ADC on the experimentally-controllable diffusion time, Delta. If Delta is decreased from 3 ms to 1 ms, the ADC in normal human lungs may increase by almost 50%. This effect should be taken into account when comparing experimental data obtained with different pulse sequences.

Theoretical limits on brain cooling by external head cooling devices.

Numerous experimental studies have demonstrated that mild hypothermia is a rather promising therapy for acute brain injury in neonates. Because measurement of the resultant cooling of human brain in vivo is beyond current technology, an understanding of physical factors limiting the possible brain cooling would be a substantial achievement. Herein brain cooling by external head cooling devices is studied within the framework of an analytical model of temperature distribution in the brain. Theoretical limits on brain hypothermia induced by such devices are established. Analytical expressions are obtained that allow evaluation of changes in brain temperature under the influence of measurable input parameters. We show that a mild hypothermia can be successfully induced in neonates only if two necessary conditions are fulfilled: sufficiently low cerebral blood flow and sufficiently high value of the heat transfer coefficient describing the heat exchange between the head surface and a cooling device.

Effects of permeable boundaries on the diffusion-attenuated MR signal: insights from a one-dimensional model.

The local magnetization distribution M(x,t) and the net MR signal S arising from a one-dimensional periodic structure with permeable barriers in a Tanner-Stejskal pulsed-field gradient experiment are considered. In the framework of the narrow pulse approximation, the general expressions for M(x,t) and S as functions of diffusion time and the bipolar field gradient strength are obtained and analyzed. In contrast to a system with impermeable boundaries, the signal S as a function of the b-value is modeled well as a bi-exponential decay not only in the short-time regime but also in the long-time regime. At short diffusion times, the local magnetization M(x,t) is strongly spatially inhomogeneous and the two exponential components describing S have a clear physical interpretation as two "population fractions" of the slow- and fast-diffusing quasi-compartments (pools). In the long-diffusion time regime, the two exponential components do not have clear physical meaning but rather serve to approximate a more complex functional signal form. The average diffusion propagator, obtained by means of standard q-space analysis procedures in the long-diffusion time regime is explored; its structure creates the deceiving appearance of a system with multiple compartments of different sizes, while in reality, it reflects the permeable nature of boundaries in a system with multiple compartments all of the same size.

Effects of barrier-induced nuclear spin magnetization inhomogeneities on diffusion-attenuated MR signal.

The spatial distribution of the transverse nuclear spin magnetization, appearing in a single compartment with impermeable boundaries in a Stejskal-Tanner gradient pulse MR experiment, is analyzed in detail. At short diffusion times the presence of diffusion-restrictive barriers (membranes) reduces effective diffusivity near the membranes and leads to an inhomogeneous spin magnetization distribution (the edge-enhancement effect). In this case, the signal reveals a quasi-two-compartment behavior and can be empirically modeled remarkably well by a biexponential function. The current results provide a framework for interpreting experimental MR data on various phenomena, including water diffusion in giant axons, metabolite diffusion in the brain, and hyperpolarized gas diffusion in lung airways.

Theory of FID NMR signal dephasing induced by mesoscopic magnetic field inhomogeneities in biological systems.

A theory of the NMR signal dephasing due to the presence of tissue-specific magnetic field inhomogeneities is developed for a two-compartment model. Randomly distributed magnetized objects of finite size embedded in a given media are modeled by ellipsoids of revolution (prolate and oblate spheroids). The model can be applied for describing blood vessels in a tissue, red blood cells in the blood, marrow within trabecular bones, etc. The time dependence of the dephasing function connected with the spins inside of the objects, s(i), is shown to be expressed by Fresnel functions and creates a powder-type signal in the frequency domain. The short-time regime of the dephasing function for spins outside the objects, s(e), is always characterized by Gaussian time dependence, s(e) approximately exp[-zeta(k)(t/tc)2], with zeta being a volume fraction occupied by the objects, t(c) being a characteristic dephasing time, and the coefficient k depending on the ellipsoid's shape through the aspect ratio of its axes (a/c). The long-time asymptotic behavior of s(e) is always "quasispherical"-linear exponential in time, s(e) approximately exp(-zetaCt/tc), with the same "spherical" decay rate for any ellipsoidal shape. For long prolate spheroids (a/c)<<1, there exists an intermediate characteristic regime with a linear exponential time behavior and an aspect-ratio-dependent decay rate smaller than (zetaC/tc).