2.C. IVIM analysis

Parametric maps of diffusion and perfusion with IVIM image analysis were all reconstructed with MATLAB program (Mathworks, Natick, MA, USA). In IVIM model, the signal intensity curves from multiple b‐value DWI experiments were expressed with the following formula:

Where S(b) and S ₀ denote the diffusion‐weighted signal intensities of the pixels with and without diffusion‐encoding gradients (indicated by the b‐value), respectively. D is the apparent diffusion coefficient as reflected by pure molecular diffusion. f denotes the perfusion fraction. D* denotes the pseudo‐diffusion coefficient.

The computation of D used the “traditional” mono‐exponential diffusion model from diffusion‐weighted images at multiple b‐values, with the use of Eq. (2):

To separate diffusion and perfusion in the presence of IVIM effects, a segmented bi‐exponential analysis method was used.²¹, ²², ²³ Since perfusion contribution is negligible in high b‐values DWI measurements, D maps were first computed with the polynomial fitting method using Eq. (2) from the DWI images acquired with higher b‐values, where the lowest b‐value among the higher b‐values DWI for D map computation is named as threshold b‐value.

Secondly, the perfusion fraction (f) is calculated according to Eq. (3),

Where S _int denotes the intercept pixel signal intensity when b‐value extrapolates to 0 from the fitting curves. To calculate the pseudo‐diffusion coefficient (D ^*), a bi‐exponential model of diffusion, as shown in Eq. (1), was used, which was originally described by Le Bihan et al.⁸ D* is calculated using nonlinear least square fitting algorithm by selecting all the b‐values. The trust‐region‐reflective algorithm²⁷ was used in this step.

The optimal threshold b‐values were derived from the following three steps: in the first step, D‐50 was derived from the first‐order poly‐nominal fitting using Eq. (2) with all b‐values between 50–1000 s/mm². Repeating the above steps, D‐100, D‐150, D‐200, D‐300, D‐400, D‐800 were derived with b‐values between 100–1000 s/mm² till to 800–1000 s/mm², here, D‐100, for example, denotes the threshold b‐value equals 100 s/mm². In the second step, seven f values were calculated using Eq. (3) based on different D‐b_value. In the third step, using the D‐b_value and f‐b_value calculated from above two steps, seven D*‐b_value were calculated using Eq. (1). Finally, using Eq. (4), the sum of square residuals (SSR) were calculated to evaluate the difference between the measured data y _i and fitted results f(x _i), so that the threshold b‐values can be determined where the effect of diffusion and perfusion is separated. n denoted the number of measured data.