IVIM and Non-Gaussian DWI of the Breast

Diffusion in Tissue: Fundamentals

The term Gaussian relates to the distribution of the diffusion-driven molecular displacements in a free medium without borders (see Fig. 1.1 ), as in a glass of water or the cystic component of a lesion. This is also known as Brownian motion, “free” or “true” diffusion, in which any water molecule is free to randomly move in any direction without limit. Gaussian relates to the normal shape of the diffusion propagator , a mathematical description of the probability of finding a water molecule at a specific point away from its original position after a given time ( Fig. 8.2 ). In bulk fluids, the Gaussian diffusion coefficient is then linked to the intrinsic properties of the diffusing molecule (mass, size) and the medium (viscosity). This diffusion coefficient is independent of the choice of b values. However, in biological tissues molecular diffusion is hindered by many obstacles (e.g., cell membranes, blood vessels, fibrosis). As a result, diffusion displacements get shorter and the distribution of molecular displacement is no longer Gaussian (see Fig. 1.1 ). Hence diffusion coefficients, though still analyzed with a Gaussian model, are referred to as apparent diffusion coefficients.

Beyond the notion of restricted but Gaussian-appearing diffusion characterized by a single ADC, there are observable deviations from Gaussian behavior both at low b values and high b values. These two regimes are discussed separately later. Within this topic, the distinction between a biophysical model of a system and a purely mathematical representation is important; and we will see that some non-Gaussian descriptions are quite clearly of this latter form, and we do not attempt to explicitly describe or quantify microstructure. Two goals of performing breast DWI predominate: (1) characterization of the underlying physiology/microstructure and (2) generating biomarkers for patient management. These goals often align and are synergistic but must be appropriately prioritized. The microstructure of the breast is quite complex (comprising fibroglandular tissue [FGT] with stroma and lobular/ductal structures, adipose tissue, microvasculature, as well as possible malignant epithelium or benign hyperplasia, cysts, fibrosis, or other growths), and inferring all details of this mixture can be limited by the available data and by intravoxel heterogeneity. It is therefore pragmatic to assume that any successful description of the diffusion decay curve will be at least somewhat empirical. With this in mind, we proceed to discuss IVIM and non-Gaussian DWI contrast in general and for the particular case of breast tissue.

Intravoxel Incoherent Motion (IVIM)

The biophysical model of IVIM is that the DW signal originates from two compartments: molecular diffusion in tissues and microcirculation (perfusion). This is based on the assumption that the flow of blood through capillaries mimics a diffusion process due to the pseudorandom organization of capillaries in tissue. To separate the effects of diffusion and perfusion on the DW signal, a biexponential model can be formulated as follows:

where f is the flowing blood fraction, D * is the pseudodiffusion coefficient associated with blood microcirculation, D _blood is the water diffusion coefficient in blood (a term often omitted or absorbed into D * in practice), and D is the apparent diffusion coefficient in the tissue space. In most cases, the pseudodiffusion coefficient associated with blood microcirculation is much larger (about 10 times larger) than the water diffusion coefficient in tissues. As depicted in Fig. 8.1 , the initial slope of the diffusion decay signal observed at low b values ( b < 200 s/mm ² ) contains a mixture of perfusion and diffusion effects, whereas diffusion effects dominate at higher b values (200 < b < 1000 s/mm ² ). Appropriate b value ranges for IVIM analysis are described in more detail later and depicted in Fig. 8.1 as regimes A and B.

f is sometimes called “ f _IVIM ” or “ f _p ,” D * is sometimes referred to as “ D _p ,” “ ADC _fast ,” or “ ADC _high ,” whereas D is sometimes called “ D _t ,” “ ADC _slow ,” or “ ADC _low ,” not to be confused with the standard ADC , whose calculation includes both perfusion-related and genuine diffusion effects. Regardless of nomenclature, the basic interpretation of the IVIM parameters are that (1) D reflects microstructure of the extravascular space from restricted/hindered diffusion, including cancerous cellularity; (2) f represents a blood volume marker; and (3) D * reflects a combination of blood velocity and vascular architecture. Here we employ the ( D, f, D *) notation when summarizing the literature for consistency.

Although the biexponential model ( Eq. 1 ) is the most commonly used form of IVIM analysis, it should be noted that it reflects only one scenario of the general case. Namely, pseudodiffusive behavior occurs in the long time limit of spins undergoing many directional changes over the echo time. In the opposite, short-time limit (also known as ballistic regime ), where spins largely reside in one capillary segment, the signal behavior is described by a sinc-function expression. In the intermediate case, an admixture of behaviors occurs that can be teased apart with advanced methods of flow compensation and time-dependent IVIM that are beyond the scope of this chapter. But from a fundamental point of view, the approximate nature of the common biexponential description should be kept in mind.

Non-Gaussian Dwi

From an empirical perspective, anything that gives rise to a diffusion signal decay as a function of the diffusion-weighting b value that deviates from a monoexponential form can be defined as non-Gaussian. This definition encompasses a variety of microstructural features, from a single restricted compartment to differently hindered compartments to multiply sized compartments; this empirical point of view leads to the creation and formulation of mathematical descriptions that are able to describe the observed decay curves. In this chapter we will review non-Gaussian MRI techniques that have been applied in breast tissue and breast cancer. Furthermore, we categorize them in two groups: those that sample non-Gaussianity as a function of diffusion weighting strength ( b value) and those that sample non-Gaussianity as a function of diffusion time.

Non-Gaussianity in Spatial Scale: b value Dependence

Acquisition and analysis using DW images at high diffusion weighting ( b ~ 1000–2000 s/mm ² ) allow capturing such diffusion behavior beyond a single Gaussian diffusivity. A detailed description of the mathematics of every non-Gaussian approach is outside the scope of this chapter, and so later sections will focus on an illustrative few as applied to breast DWI, and discussing some of the related literature results. These techniques vary the b value as the primary contrast variable.

Diffusion Kurtosis Imaging

Diffusion kurtosis imaging (DKI), first developed by Jensen and colleagues, is a mathematical representation using an extended b value range that captures the nonmonoexponential (non-Gaussian) nature of the signal decay curve at high b values. DKI includes an additional parameter K , known as the kurtosis (or mean kurtosis MK ). This parameter describes the deviation of the diffusion propagator from a normal shape, and can be thought of as the “peakedness” of the distribution function; a positive value corresponds to a more “peaky” distribution and indicates that the diffusion remains closer to the origin than would be expected for Gaussian diffusion, and so the signal decay appears slower. A K value of zero corresponds to a simple monoexponential decay, whereas a negative K would indicate faster diffusional spread (and is not expected). The kurtosis signal expression is

$S_b=S_0\cdot \exp \left(-b\cdot D_k+\frac{1}{6}{b}^2\cdot D_k{}^2\cdot K\right)$

[2]

where D _k is a diffusivity coefficient. The addition of such successive terms, here in increasing powers of b , is a general strategy of cumulant expansion . Example signal curves are shown in Fig. 8.3 , demonstrating the effect of varying both D _k and K .

As described earlier, the appearance of a non-Gaussian process may arise from microscopic heterogeneity or from larger-scale but subvoxel heterogeneity, and this mathematical representation may not distinguish the two. The excess kurtosis K is therefore empirical, and inferring specific biophysical properties from it usually warrants an accompanying biophysical model. Further, under certain circumstances (large K and beyond a corresponding b value), the kurtosis signal representation begins to increase, which has no physical interpretation. DKI also uses very high b values at which decreasing signal-to-noise is important, and it should be considered carefully.

Multiple Compartments: Finite and Infinite Number of Components

Bi- and Triexponentials

When b values in the higher ranges (e.g., 1000 < b < 3000 s/mm ² ) are acquired, it is possible to apply a biexponential representation that, although mathematically equivalent to the IVIM model, returns two diffusion components and does not attempt to capture perfusion. Perhaps the natural extension of the IVIM model is to extend to three or more finite compartments when including an extended b value range. This approach is no less valid in theory but can fall prey to a limited amount or quality of data given the required computational task. The problem of confidently separating exponential signals that may be not sufficiently represented (from small volume fractions) or distinct enough in decay constants (diffusion coefficients) is challenging.

In general, the equation for representations with n exponential components takes the form

$S_b=S_0\cdot \sum _i^n{C}_i\cdot \exp (-b\cdot D_i)$

[3]

where each component i contributes a certain fraction C _i to the overall signal detected, with a diffusion coefficient D _i (and the sum of the fractions is unity). With useful representation being essentially limited to two or three compartments, it is common to write out each term explicitly. Although there are a growing number of studies that use an explicit triexponential model for DWI in a variety of organs or contexts, moving beyond this may introduce too many variables for the data to support.

Stretched Exponential

Another description of a multicompartmental system uses an assumed distribution of diffusion coefficients, each with their own fractional contribution, relating to a description of the system in terms of anomalous diffusion and fractal geometry. Rather than explicitly model these contributions, it is possible to summarize them using a fewer number of parameters.

In the stretched exponential representation, the monoexponential equation is modified by using a distributed diffusion coefficient ( DDC ), and the exponent itself is raised to the index α:

$S_b=S_0\cdot \text{exp(}-{(b\cdot DDC)}^{\alpha })$

[4]

This results in a single diffusion-coefficient DDC , which represents a summary of the distribution position, and α is a heterogeneity index, which represents the spread. This representation therefore explicitly acknowledges the non-Gaussian nature of the curve, but conceives the system as a collection of Gaussian terms in a specific form, albeit not necessarily modeled on tissue itself, resulting in only one extra variable. Where the index α is 1, the signal curve is exponential; increasing α increases the deviation from monoexponential decay with both faster and slower diffusion features ( Fig. 8.3 ).