Joshua D. Trzasko^{1}, Brent A. Warndahl^{1}, Stephen J. Riederer^{1}, and Adam T. Froemming^{1}

In most diffusion studies, two or more DW images are acquired and an apparent diffusion coefficient (ADC) map is generated, with the goal of providing quantitative diffusion information that is independent of acquisition settings or secondary tissues properties. However, ADC values can vary significantly following protocol changes. In this work, we analytically determine the statistical bias in ADC maps generated from multi-point DWI acquisitions, and show how the derived model rationalizes noise-based error propagation as the source of ADC inconsistencies observed in our own clinical practice.

Purpose

Theory

For brevity, we herein focus on isotropic diffusion. In a DWI exam that generates $$$T$$$ images, the complex-valued signal at any voxel is modeled as

$$f_{t}=me^{-j\phi_{t}}e^{-TE/T2}e^{-b_{t}ADC}+n_{t}~[EQ1]$$

where $$$m$$$ is background signal, $$$\phi_{t}$$$ is image-dependent phase, $$$b_{t}$$$ is the b-value, and $$$n_{t}$$$ is zero-mean Gaussian noise. As DWI often varies the number of excitations (NEX) with b-value, the per-channel variance of is $$$\sigma_{t}=NEX_{t}\sigma_{0}^{2}$$$. ADC values are typically estimated via linear-least-squares fitting of logarithms of magnitudes of DWIs; specifically,

$$\hat{ADC}=\delta_{2}^{T}\left[ \begin{array}{cc}1&-b_{1}\\\vdots&\vdots\\ 1&-b_{T}\end{array}\right]^{\dagger}\left[\begin{array}{c}\ln|f_{1}|\\\vdots\\\ln|f_{T}|\end{array} \right]~[EQ2]$$

where $$$\delta_{2}^{T}=[0;1]^{T}$$$ and $$$\dagger$$$ denotes the pseudo-inverse. It is readily shown that the bias of this method-of-moments estimator is

$$\mathbb{BIAS}\left[\hat{ADC}\right] =\mathbb{E}\left[\hat{ADC}\right]-ADC=\delta_{2}^{T}\left[ \begin{array}{cc}1&-b_{1}\\\vdots&\vdots\\ 1&-b_{T}\end{array}\right]^{\dagger}\left[\begin{array}{c}\mathbb{E}[\ln|f_{1}|]-\ln|\mathbb{E}[f_{1}]|\\\vdots\\\mathbb{E}[\ln|f_{T}|]-\ln|\mathbb{E}[f_{T}]|\end{array} \right]~[EQ3]$$

Denoting $$$SNR=m/(\sqrt{2}\sigma_{0})$$$, the expected value ($$$\mathbb{E}[\cdot]$$$) of the log-Rician distributed quantity $$$\ln|f_{t}|$$$ is:

$$\mathbb{E}[\ln|f_{t}|]-\ln|\mathbb{E}[f_{t}]|=-\frac{1}{2}Ei\left(-NEX_{t}^{-1}\left(SNR~e^{-TE/T2}e^{-b_{t}ADC}\right)^{2}\right)~[EQ4]$$

where $$$Ei(\cdot)$$$ is the exponential integral^{11,9}. Using this identity, and after considerable simplification, it follows that:

$$\mathbb{BIAS}\left[\hat{ADC}\right]=\frac{\sum_{t=1}^{T}Ei\left(-NEX_{t}^{-1}\left(SNR~e^{-TE/T2}e^{-b_{t}ADC}\right)^{2}\right)\left(b_{t}{T}-\sum_{s=1}^{T}b_{s}\right)}{2\left(T\left(\sum_{t=1}^{T}b_{t}^{2}\right)-\left(\sum_{t=1}^{T}b_{t}\right)^{2}\right)}~[EQ5]$$

This generalized model enables prediction of shifts in ADC values under multi-point DWI protocol changes, as well as retrospective corroboration that observed ADC shifts are likely due to statistical artifact. The result derived in [REF9] is a special case of our result in EQ5.

Methods

Results

Discussion

Conclusion

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Fig. 1: Cropped ADC maps of a prostate with suspected PCa generated with two different DWI protocols performed sequentially, identically
windowed/leveled. Note visually apparent intensity differences in both prostatic (1-2) and muscle (3) tissue.

Fig. 2: Plots depicting nominal versus predicted ADC values under each of the tested DWI protocols, for different tissues types in the defined ROIs. Note
that for large ranges of nominal ADC values, the estimated ADC values can significantly differ for the two protocols as well as from the nominal value.

Fig. 3: Table of mean ROI values (in mm/sec x 1e-6) observed in the ADC maps from DWI Protocol 1 (P1) and Protocol 2 (P2), and corresponding values
obtained via Euclidean projection of ROI values onto theoretically predicted curves. The small percentage differences between the measured and
predicted values strongly suggest that the observed ADC differences are strictly due to noise-based error propagation during model fitting.