A data-driven statistical model that estimates measurement uncertainty improves interpretation of ADC reproducibility: a multi-site study of liver metastases

Apparent Diffusion Coefficient (ADC) is a potential quantitative imaging biomarker for tumour cell density and is widely used to detect early treatment changes in cancer therapy. We propose a strategy to improve confidence in the interpretation of measured changes in ADC using a data-driven model that describes sources of measurement error. Observed ADC is then standardised against this estimation of uncertainty for any given measurement. 20 patients were recruited prospectively and equitably across 4 sites, and scanned twice (test-retest) within 7 days. Repeatability measurements of defined regions (ROIs) of tumour and normal tissue were quantified as percentage change in mean ADC (test vs. re-test) and then standardised against an estimation of uncertainty. Multi-site reproducibility, (quantified as width of the 95% confidence bound between the lower confidence interval and higher confidence interval for all repeatability measurements), was compared before and after standardisation to the model. The 95% confidence interval width used to determine a statistically significant change reduced from 21.1 to 2.7% after standardisation. Small tumour volumes and respiratory motion were found to be important contributors to poor reproducibility. A look up chart has been provided for investigators who would like to estimate uncertainty from statistical error on individual ADC measurements.


Appendix 1
The ADC is the decay parameter from an exponential fit of the loss in signal intensity at a given pixel location, between the increasingly diffusion sensitive images (b-100, 500, 900 s/mm 2 ). In clinical data, as the noise distribution is skewed towards the higher b-values, we find that a first order bias correction factor (α ) improves the quality of fit as it removes the SNR dependent bias.
We estimate ADC (referred to as D), through a likelihood-based parameter optimization, log P(I\D, S0) (the probability of the image data given the assumed parameters) b k and the current estimates of ADC, D, and no-diffusion signal, S0 (at b k = 0). This is computed using Where k ∈ 1,2,3 ⎡ ⎣ ⎤ ⎦ refers to the three b-values b k used, e.g. 100, 500 and 900 s/mm 2 .
The signal value for no diffusion (b=0) S0 is the second parameter that is estimated. α is a fixed value defining the amount of bias correction applied and it may be adjusted depending on the amount of image smoothing corresponding to the specific imaging protocol used by the scanner (for our data α was set to the theoretical value of one). An estimate of the standard deviation (SD) of noise in the image is computed from the distribution of second derivatives (for x and y) around zero, in a central rectangular region on the tissue (http://www.tina-vision.net/docs/memos/2008-010.pdf).

Appendix 2
In order to study a change in ADC or reproducibility, we can use the percentage change in mean ADC as given by From a basic statistical level, the wider the distribution of values in the ADC histogram used to determine the mean ADC ( ), the larger the standard error of the mean will be and conversely, the larger the sample size (N) the smaller the standard error of the mean will be.
Standard Error (D) = SD N (2.2) However, we would expect the variable R 12 to be affected by variations in SNR, imaging artifact and motion, to differing degrees between test and retest.
Contributions to errors on R 12 , defined as ε R 12 , can be estimated from error propagation based upon the expected errors ( σ D 1 ,σ D 2 ) on each baseline ADC measurement, D 1 and D 2 . We assume that Where σ D i is a measure of the width of the ADC distribution and N ' is the number of independent measurements in the region.
For mean parameters x and y, with measurement errors in each case defined by (1.5), we wish to find the measurement error on parameter z, defined as Using error propagation, the measurement error on z is given by The derivatives of z with respect to x and y are Hence, we can write ε z = 400 x + y ( ) 2 y 2 σ 2 + x 2 σ y 2 (2.7) Contributions to errors on R 12 , defined as ε R 12 , can therefore be estimated from

Appendix 3
In order to construct likelihood for multiple repeat samples (N), based on the difference between the expected and observed variances, we assume an approximate Gaussian distribution for the difference y 1i − y 2i The last term is normally omitted from fitting routines (on the basis that is a constant), but is needed if the value of σ yi is allowed to vary as part of the fit (as is the case here).
The resulting log likelihood cannot be treated as a χ 2 statistic as the fitting process guarantees that χ 2 = N. However, once the error model is determined we can assess the adequacy of the model for describing sub groups g, with The three parameters we have used to describe observed reproducibility are obtained by fitting the datasets without visible GM, resulting in the error formula ε R 12 2 = β 2 ε R 12 2 σ D 1 ,σ D 2 ( ) + ε R 12 2 σ fix ,σ fix ( ) + ε sys 2