Intravoxel incoherent motion (IVIM) is a well-known model used in diffusion-weighted imaging (DWI) to separate the effects of extravascular diffusion and microvascular pseudo-diffusion (cf. perfusion)¹. IVIM modelling requires images to be taken at many different b-values (ideally > 10) and is therefore very time-consuming. Furthermore, the pseudo-diffusion parameters are often strongly corrupted by noise. Relative enhanced diffusivity (RED) is a recently proposed alternative parameter that is strongly weighted by pseudo-diffusion (and inversely by true diffusion), yet it requires images at only 3 different b-values². RED has been shown to provide good discrimination between malignant tumours, benign tumours and healthy tissue. In this work we explore the relationship between RED and the familiar IVIM parameters to promote adoption within the IVIM research community.

IVIM modelling describes the signal attenuation in DWI-MRI using a biexponential model¹:

$$\frac{S_{b}}{S_{0}}=(1-f)\mathrm{e}^{-bD}+f\mathrm{e}^{-b(D+D^{*})}\qquad\qquad(1)$$

where S_b and S₀ are the signals with and without diffusion gradients applied, D is the diffusion rate constant, D* is the pseudo-diffusion rate constant and f is the pseudo-diffusion volume fraction. Fig. 1 displays an illustrative example of the signal attenuation curves (without noise) for two different tissues, one with high D, low D* and low f (denoted B for benign), and the other with low D, high D* and high f (denoted M for malignant). Typically a fitting algorithm would be applied to noisy data to estimate the three parameters (either pixel-wise or, more commonly, after averaging over a region of interest).

In contrast, the parameter RED is obtained directly from data acquired at 3 b-values²:

$$\mathrm{RED}=100\left(\frac{\mathrm{ADC}_{0,1}-\mathrm{ADC}_{1,2}}{\mathrm{ADC}_{1,2}}\right)\qquad\qquad(2)$$

$$\mathrm{ADC}_{i,f}=\frac{\ln(S_{f}/S_{i})}{b_{i}-b_{f}}\qquad\qquad(3)$$

Note that the subtraction in Eq. (2) is necessary to remove, in an approximate sense, the diffusion component from the perfusion component in the numerator. From Fig. 1 and Eq. (2) it is clear that tissue M will have a much higher RED value than tissue B.

Eqs. (1)-(3) can be combined to explore the dependency of RED on the IVIM parameters. However, greater interpretative power may be obtained by applying some approximations to arrive at a simple expression for this dependency. For example, if we assume¹ that ADC_1,2 ≈ D and that in general f << 1, we can apply a Taylor series expansion to obtain the following approximation:

$$\mathrm{RED}\approx\mathrm{RED}_{\mathrm{S}}=\frac{100f}{b_{1}D}\left[1-\mathrm{e}^{-b_{1}D^{*}}\right]\qquad\qquad(4)$$

where b₁ is the intermediate b-value. Eq. (4) suggests that RED is approximately linearly proportional to f, inversely proportional to D, and follows an inverse exponential decay with respect to D*.

To test the accuracy of Eq. (4) we calculated RED and RED_S for several different combinations of D, D* and f, over the domain 0 < b₁ < 700 s/mm² (b₀ = 0; b₂ = 700 s/mm²). The IVIM parameters were taken from ranges appropriate to breast imaging³: D = 1.38 ± 0.65 × 10^-3 mm²/s; D* = 15.9 ± 10.0 × 10 mm²/s; f = 0.1015 ± 0.05. Nine combinations were chosen, one using the Mean of each parameter (denoted MMM) and eight involving the mean ± 1 stdev (High/Low) values (e.g. HLL, etc). Fig. 2 displays the signal attenuation curves for each parameter combination, which represent a reasonable spread of IVIM behaviour.

Fig. 3 compares the values of RED (solid lines) and RED_S (dashed lines) for the nine parameter combinations, as a function of the intermediate b-value, b₁. We observe that RED_S is a reasonable approximation except for those cases with high f and low D*, for which it leads to an overestimation, particularly at low b₁-values. However, it may be argued that this combination is generally not encountered, especially in other more perfuse regions of the body, such as the kidneys, spleen or liver⁴. Note that different parameter combinations can yield very similar RED values (e.g. LHL and HHH), which is actually predicted by Eq. (4). Nevertheless, these tissues could be discriminated simply via their standard ADC values. Furthermore, curves such as those displayed in Fig. 3 could be used to assist the selection of b₁ in the pursuit of optimal tissue discrimination using RED. Lastly, Fig. 4 provides further support that Eq. (4) describes sufficiently the relationship between RED and IVIM, and we note that a lower choice of b₁ results in a greater spread of RED across each parameter range.

RED provides a fast alternative to IVIM modelling that offers valuable information regarding relative changes in properties associated with tissue microstructure and microvasculature. The relationship between RED and IVIM is straightforward and could be used to guide the choice of the intermediate b-value in RED imaging for improving tissue discrimination further.