Is the “biexponential diffusion” biexponential?

Abstract

Diffusion-weighted signal from the brain, S, deviates from monoexponential dependence on the b-factor. This property is often referred to as biexponential diffusion, since the corresponding model fits data well. The aim of this study is to examine the necessity of using the biexponential model in homogeneous voxels under isotropic diffusion weighting up to b = 2.5 ms/µm 2 . The model is compared to the cumulant expansion of ln S in a power series in b, which takes its origin in fundamental properties of the diffusion-weighted signal, but diverges at large b. Numerous studies [ (1-3) and following papers] indicate that the diffusion-weighted signal from a single voxel can be described as a weighted sum of two exponential functions,where b is the b-value, w 1 and w 2 are the weights, and D 1 and D 2 are the corresponding diffusion coefficients. Equation [1] suggests the presence of two compartments in the brain tissue; however, any attempt to identify them histologically has failed (1,3-6). This supports another interpretation of biexponential diffusion as a manifestation of microscopic restrictions in biological tissue (7-9).In this paper, we scrutinize the description of diffusion in the brain. The aim is to determine whether the biexponential model, Eq. [1], is required by experimental data on a statistically significant level. The good quality of fitting of Eq.[1] to data is not sufficient to prove the relevance of this function. It can be superfluous in the presence of another model with a smaller number of adjustable parameters describing experimental data equally well. METHODS TheoryAs a framework for the present analysis, we use a modelindependent description of the magnitude of the isotropically weighted signal as a power series in b:Medical Physics, Department of Diagnostic Radiology, University Hospital Freiburg, Freiburg, Germany. For the biexponential function, Eq.[1], the convergence radius is determined by the complex-valued solution to the equation S biexp = 0:This value should be compared with the interval of experimental b-factors. If the condition b < b c fulfils, then the series in Eq. [2] can be terminated to a couple of low-order terms. The higher-order terms give a small contribution that cannot be determined from noisy experimental data. This means that the information content of the model, Eq. [1], may be reduced to a smaller number of parameters, which are the first coefficients in Eq. [2]. If so, the model is superfluous and a good accuracy of fitting does not prove its validity.In the opposite case of large b-factors, b > b c , the model represents a specific function that cannot be reduced to a polynomial. In this case, a good quality of fitting gives more credit to the underlying model.The aim of this study can now be formulated as to find out which of the above cases holds for the brain tissue for b-factors up to b = 2.5 ms/µm 2 . MR MeasurementsAll measurements were performed on a 3-T whole-body clinical scanner (Magnetom Trio, Siemens Medical Systems, Erlangen, Germany). The scann...

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

“…Note that the coefficient β L (and a similar coefficient β T in the transverse diffusivity, see Eq. (16) below) reflects non-Gaussian diffusion effects in each individual airway and is proportional to the so called kurtosis K -the second order term in the cumulant expansion of the MR signal (e.g., [33][34][35]…”

Biexponential Signal Attenuation Analysis of Diffusion-weighted Imaging of Breast

“…7,15,17 The decrease in ADC mainly results from increased cellularity, but some tumors show other changes, such as ‰uid viscosity and cell permeability, as well. [23][24][25][26][27][28][29][30] In this respect, biexponential signal analysis will provide more precise information about tissue character. Thus, we performed DWI with high b-values up to 3500 s/mm 2 and examined the character of biexponential signal attenuation using Equation [3], in which fast and slow components are postulated.…”

“…After Niendorf'sˆndings, many researchers have investigated this discrepancy and indicated in‰uence from permeability or restriction of cell membrane [23][24][25] and diŠerence of relaxation time of intra-and extracellular space 26,27 as well as other factors. [28][29][30] However, the detailed cause of biexponential signal attenuation remains unclear. 31 Recently, the parameters f 1 , f 2 and D 1 , D 2 have been used to characterize tissue properties in a clinical context, and the parameters derived from the biexponentialˆtting of signal decay in DWI in human brain have been reported to be diŠerent in ischemia, edema, and tumor and after radiation.…”

Use, misuse, and abuse of apparent diffusion coefficients

“…It is worth stressing that the experimental conditions in Hürlimann's experiment were ordinary. Much larger gradients are now accessible on modern MR scanners so that a non-Gaussian signal attenuation may be relevant for clinical applications (40)(41)(42)(43)(44)(45)(46). For instance, Jacob and coworkers demonstrated a clear deviation from a linear dependence of ln E/E 0 on b for helium-3 gas diffusion in the rat lung (44).…”

“…Moreover, the values of these parameters strongly depend on the range of b-values that are used for fitting. Other limitations of a bi-exponential model are discussed in (45).…”