# Analytical and Numerical Connections between Fractional Fickian and Intravoxel Incoherent Motion Models of Diffusion MRI

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Definitions and Properties

^{2}/s, γ is the gyromagnetic ratio, and Δ$\text{}\ge \delta $ [15]. With the majority of clinical MRI scanners using fixed values of δ and Δ, this expression is routinely simplified:

^{2}. Thus, a common way to adjust diffusion weighting is to vary the gradient amplitude.

^{2}. The most well-known perfusion model—IVIM—extends Equation (2) above by taking a biexponential form [6]. This model allows for the fractional pool size and rate of the more rapidly moving blood pool to be quantified separately from that of the tissue water in the extra-vascular space. A more recent emerging perfusion model—FFD--takes the form of a stretched exponential signal model. This model imposes a fractional time derivative on Fick’s law which allows for the modeling of the non-Gaussian jump length statistics expected from blood perfusion within the multiscale vascular bed [16].

#### 2.1. Biexponential Model

#### 2.2. Stretched Exponential Model

## 3. Materials and Methods

#### 3.1. Biexponential Representation of the Stretched Exponential Model Parameters

#### 3.2. Biexponential Estimates of the Stretched Exponential Model Parameters

#### 3.2.1. Error Analysis

#### 3.2.2. Error Analysis in the Presence of Noise

#### 3.2.3. Model Simulations and Cross-Model Error Analysis

#### 3.3. In Vivo MRI Acquisition

^{2}. DWI acquisition at rest performed 20 signal averages (acquisition time = 16.05 min) in order to increase SNR under low perfusion conditions, while post-exercise acquisition performed 4 signal averages (acquisition time = 3.25 min). All images exhibited SNR > 200, deemed sufficient for robust estimates of parameters [23,24].

## 4. Results

#### 4.1. Fully-Determined Case

^{+}to 1, the pole associated with pseudo-diffusion travels from the center of the unit circle toward the boundary of the unit circle, while the pole associated with diffusion travels from the boundary toward the center of the unit circle. The amplitude of the pole associated with pseudo-diffusion vanishes, whereas the amplitude of the pole associated with diffusion becomes progressively larger. The pole with initially higher amplitude (${f}_{p}$) accounts for the starting values of the stretched exponential, and the pole with a relatively lower initial amplitude ($f$) accounts for the final values of the stretched exponential.

#### 4.2. Over-Determined Case

#### 4.3. Analysis of Fit Error

#### 4.4. In Vivo Results

_{p}shows a strong correspondence between experimental and theoretical values. The pseudo-diffusion coefficient (${D}_{p}$) and diffusion coefficient ($D$) show more scatter around the theoretical curve but largely follow the theoretical prediction. In both muscle groups displayed, these relationships were consistent under physiological conditions at rest and post-exercise hyperemia. Larger deviations of ${D}_{p}$ and $D$ in experimental data appear to occur at lower values of $\widehat{\alpha}$. The greater variance in of ${D}_{p}$ and $D$ can also be explained by Crammer Rao bound analysis of the biexponential model, which shows that the variance of the diffusion rate constant estimates is directly proportional to their cubed value and the quartic of the sum of $D$ and ${D}_{p}$, and inversely related to the quartic of their difference. Thus, the variance in ${D}_{p}$ estimates is expected to be large for large rates (${D}_{p}$) and for small separations between $D$ and ${D}_{p}$ [25].

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Analytical solutions from Equation (22) (Section 3.1) and numerical solutions from Equation (23) are shown for biexponential model parameters (

**a**) f

_{p}, (

**b**) 1 − f

_{p}, (

**c**) D

_{0}, and (

**d**) D, as functions of the stretched exponent $\widehat{\alpha}$, both for $b$= 0, 1, 2, 3 and ${D}_{se}=1/3$.

**Figure 2.**Analytical solutions from Equation (22) (Section 3.1) and numerical solutions from Equation (23) are shown for biexponential model parameters (

**a**) f

_{p}, (

**b**) 1 − f

_{p}, (

**c**) D

_{0}, and (

**d**) D, as functions of the stretched exponent ${D}_{se}$, both for $b$= 0, 1, 2, 3 and $\widehat{\alpha}=0.9$.

**Figure 3.**Analytical results from Equation (22) are shown for biexponential model parameters (

**a**) f

_{p}, (

**b**) 1 − f

_{p}, (

**c**) D

_{0}, and (

**d**) D, as functions of stretched exponential parameters ${D}_{se}$ and $\widehat{\alpha}$, when b = 0, 1, 2, 3 and ${D}_{se}\xb7{b}_{max}\le 1$.

**Figure 4.**Pole behavior of the biexponential model for uniformly spaced b values of b = 0, 1, 2, 3 and ${D}_{se}=1/{b}_{max}$; (

**b**) is the top view of (

**a**). The red trajectory is associated with the pole (${\rho}_{1}={e}^{-{D}_{p}}$) that corresponds to the pseudo-diffusion with amplitude ${f}_{p}$ and the blue trajectory is associated with the pole (${\rho}_{2}={e}^{-D}$) that corresponds to the true diffusion with amplitude $f$. The arrows indicate the direction of the pole trajectory as $\widehat{\alpha}$ increases from 0

^{+}to 1.

**Figure 5.**Biexponential parameter estimates as a function of the stretched exponent $\widehat{\alpha}$. Simulations were implemented by taking uniformly spaced b values with ${b}_{max}=400$ and ${D}_{se}=1/{b}_{max}$. Lines encoded with different colors correspond to the sampling interval of b values, $\tau $, which took discrete values of 1, 5, 10, 20, 30, and 50. The analytical solutions (in black lines) were implemented with $b=0,1,2,3$ and ${D}_{se}=1/400$.

**Figure 6.**Mean squared error calculated from simulations using a b values span between 1 and 20 and uniformly spaced by 1, and ${D}_{se}=1/{b}_{max}$. (

**a**) MSE as a function of $\widehat{\alpha}$; (

**b**) Autocorrelation of fit error at various lags and $\widehat{\alpha}$.

**Figure 7.**Experimental traces of biexponential model parameters obtained from ROI in the rectus femoris at baseline and post-exercise, superimposed by numerical traces. Both numerical and experimental analysis used: $b$ = 0, 3, 7, 10, 15, 20, 25, 30, 40, 50, 70, 100, 200, and 400 $\mathrm{s}/{\mathrm{mm}}^{2}$.

**Figure 8.**Experimental traces of biexponential model parameters obtained from ROI in the vastus lateralis at baseline and post-exercise, superimposed by numerical traces. Both numerical and experimental analysis used: $b$ = 0, 3, 7, 10, 15, 20, 25, 30, 40, 50, 70, 100, 200, and 400 $\mathrm{s}/{\mathrm{mm}}^{2}$.

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Yao, J.; Anjum, M.A.R.; Swain, A.; Reiter, D.A. Analytical and Numerical Connections between Fractional Fickian and Intravoxel Incoherent Motion Models of Diffusion MRI. *Mathematics* **2021**, *9*, 1963.
https://doi.org/10.3390/math9161963

**AMA Style**

Yao J, Anjum MAR, Swain A, Reiter DA. Analytical and Numerical Connections between Fractional Fickian and Intravoxel Incoherent Motion Models of Diffusion MRI. *Mathematics*. 2021; 9(16):1963.
https://doi.org/10.3390/math9161963

**Chicago/Turabian Style**

Yao, Jingting, Muhammad Ali Raza Anjum, Anshuman Swain, and David A. Reiter. 2021. "Analytical and Numerical Connections between Fractional Fickian and Intravoxel Incoherent Motion Models of Diffusion MRI" *Mathematics* 9, no. 16: 1963.
https://doi.org/10.3390/math9161963