2.1.

Liver Phantoms

Liver specimens were recovered from the University of Arizona’s Department of Pathology for use as MRI phantoms. One- to two-inch thick slices were sectioned during autopsy and fixed in formalin. After the liver was fixed in formalin, biopsies were collected and the phantom placed in an air-tight container. The containers were then placed in the MRI to collect images.

To confirm our observation that MRI of formalin-fixed tissue is comparable to clinical contrast-enhanced MRI, we compared the textures of the liver tissues using a technique introduced by Burgess et al. The method takes the Fourier transform of the data and measures the radially averaged power spectra in frequency space. The results are reported on a log–log scale to determine the slope of the power spectra. Imaging modalities with similar slopes in the spectral density have similar image features.^34,35 We compared healthy and cirrhotic patient data to healthy and cirrhotic phantom images. Patient data were collected at a TR/TE/FA of $4.36\mathrm{ms}/2.3\mathrm{ms}/10\mathrm{deg}$ and phantom data were collected at a TR/TE/FA of $9.79\mathrm{ms}/4.44\mathrm{ms}/10\mathrm{deg}$. All data sets were collected with resolutions of $1.5{\mathrm{mm}}^2$ in-plane resolution at a slice thickness of 3 mm. Results for this experiment are shown in Fig. 1. All of the collected spectra exhibited similar slopes to within experimental error. With this result, we conclude that the formalin-fixed tissue produces images with textures similar to in vivo clinical images of fibrosis. Higher-resolution in vivo experiments are not yet possible due to the limitations of patient respiratory motion.

Fig. 1

(a) Log of the average power spectra $\pm 2\sigma$ of in vivo and (b) ex vivo healthy and diseased liver tissue, plotted against a log frequency scale with $\pm 2\sigma$.

2.2.

Magnetic Resonance Imaging

All images in this optimization study were collected on a Siemens 3T Skyra MRI using the Siemens flex body imaging coil using a 3-D gradient-echo T1-weighted imaging sequence (3-D VIBE, Siemens) with $\mathrm{TR}/\mathrm{TE}=9.79\mathrm{ms}/4.44\mathrm{ms}$, 2 averages, and a range of FAs from $\sim 10$ to $\sim 50\mathrm{deg}$. The FAs available are based on hardware limitations. A field-of-view (FOV) of $26.5\times 26.2\times 3.36\mathrm{cm}$ with a sampling matrix of $768\times 760\times 96$ was selected, resulting in images with isotropic resolution of $0.35{\mathrm{mm}}^3$. All images were collected at room temperature (22°C). The total scan time at one FA was $\sim 25\min$. We collected enough data to fully train the covariance matrices required to calculate the Hotelling observer. This MRI optimization method will eventually be repeated with in vivo data sets once enough patient images are collected.

Before the mathematical observer was trained to perform the task of HF-detection on liver tissue, a basic threshold was implemented to segment out and remove areas of the image that contained blood vessels from the analysis. We found this to be a necessary step in developing the observer technique.

2.3.

Biopsy

Tissue biopsy was used as the gold standard to determine the METAVIR score for the phantoms prior to imaging. Eight scalpel biopsies of $10\times 10\times 1{\mathrm{mm}}^3$ size were collected from each formalin-fixed liver. Large biopsies were possible since the tissue was excised from autopsy. H&E stained slides were prepared for review by a pathologist, who evaluated each sample for the stage of fibrosis. Only phantoms with consistent METAVIR scores were selected to train and test the observers.

2.4.

Local Texture Analysis

We tested texture analyses based on a local, normalized, 2-D discrete autocorrelation (2DAC) and a local, normalized, 2-D discrete circular autocorrelation (2DCC). We found that a Hotelling observer working with 2DCC is capable of distinguishing whether local regions drawn from an $\mathrm{F}0/\mathrm{F}1$ or an F4 liver while requiring a modest amount of training data. The 2DCC is given by

2.5.

Optimal Linear Observer

To perform the classification task between a signal-absent class corresponding to normal liver, and a signal-present class corresponding to fibrotic liver, an optimal linear observer that maximizes detection signal-to-noise ratio (SNR), also known as the Hotelling observer, was trained using local 2DCCs from 2-D slices from MR images of phantoms with METAVIR scores confirmed via biopsy. The set of pixels from the texture analysis were ordered as a $P\times 1$ vector.^36,37 The optimal linear observer template, is also a $P\times 1$ vector and a test statistic is calculated as an inner product between a data vector and the observer vector^36,37

The recovered test statistic $\tau (g)$ is used to make a decision based on a threshold. If $\tau (g)$ is greater than ${\tau }_{\mathrm{th}}$ then it is decided that $H_1$ is true, whereas if $\tau (g)$ is less than ${\tau }_{\mathrm{th}}$, $H_0$ is true.³⁷

2.6.

Ideal Quadratic Observer

The Hotelling observer is the ideal observer only when ${\mathbf{K}}_{\mathbf{0}}\cong {\mathbf{K}}_{\mathbf{1}}$. However, if the statistics are multivariate normal but the covariance matrices are not equal or nearly equal, then the quadratic observer is the ideal observer. It is given by

The ideal quadratic observer can be computed with the same data required to train the linear observer.³⁸ Decisions are made in the same manner as for the linear observer, using a comparison of $\tau (\overrightarrow{g})$ to $t_{\mathrm{th}}$ to decide if $\overrightarrow{g}$ is a member of $H_0$ or $H_1$.

2.7.

Receiver Operator Characteristic Analysis

We recovered the receiver–operator-characteristic (ROC) curves and calculated the AUROC as the figure-of-merit for the observer. To perform ROC analysis, ${\tau }_{\mathrm{th}}$ was varied across the range of possible $\tau$ values spanned by the test statistics ${\tau }_0$ and ${\tau }_1$. ${\tau }_0$ is the test statistics from confirmed signal-absent testing data and ${\tau }_1$ is the test statistics from confirmed signal-present testing data. At each threshold ${\tau }_{\mathrm{th}}$, the false-positive fraction (FPF) and true-positive fraction (TPF) were calculated³⁷ using

For each threshold, the TPF was plotted as a function of the FPF, forming the ROC curve. The figure-of-merit for the ROC curve is the AUROC, which has a possible range from 0.5 to 1.0. An AUROC of 0.5 denotes a situation where the distribution of test statistics fully overlap one another and the observer can do no better than random guessing. An AUROC (or AUC for short) of 1.0 means a complete separation of the test-statistic distributions and perfect observer performance.

2.8.

Curve Fitting

To determine the optimal FA, the AUC was plotted as a function of FA, $A(\theta )$. A smoothing spline was implemented to interpolate the AUC data using the MATLAB^® smoothing toolbox. The optimal FA was selected at the maximum of the spline curve.

3.1.

Liver Phantoms and Biopsy

We collected biopsies from four liver phantoms fixed in formalin for imaging with MRI. Each phantom had biopsies from eight regions assessed by a pathologist. Only phantoms with homogeneous biopsy results were used in this study. Two phantoms were reported as F4, one as F0, and one as F1. Representative images of biopsy slides are provided for each phantom in Fig. 2.

Fig. 2

Biopsy-slide images for all four phantoms. (a) Assessed as an F0 (healthy) liver, (b) F1 (early indications of fibrosis), (c) and (d) both diagnosed as F4 (cirrhotic) livers.

The F0 sample showed no sign of fibrosis, whereas the F1 biopsy showed early fibrosis forming around the portal veins. The two F4 phantoms have a complete ECM and lobules were clearly visible in the biopsy slides. The F0 and F1 livers were used to define the signal-absent class of the linear and quadratic observers and the two F4 livers define the signal-present class to train the model observers. Figure 3 provides a representative MRI of the phantoms at TR/TE $9.79\mathrm{ms}/4.44\mathrm{ms}$ at FA 19 deg associated with each biopsy sample.

Fig. 3

Representative slice images of (a) F0, (B) F1, (c) and (D) F4 phantoms at a 19-deg flip angle. Image resolution is $0.35{\mathrm{mm}}^3$ isotropic.

3.2.

MRI of Liver Phantoms

Each phantom was imaged at five flip angles: 8, 15, 19, 30, and 45 deg in order. Selected slices from each phantom at 19 deg are shown in Fig. 3.

The images from the F0 and F1 phantoms appear relatively untextured and the liver tissue appears uniform in signal throughout a majority of the tissue. The F1 liver in Fig. 3(b) has some features associated with vasculature. This is dependent on the location the slice is removed from during autopsy. The vasculature features, which are dark, are ignored by our analysis. The F4 images suggest that there is visible contrast between the ECM and liver tissue in the cirrhotic livers that appears at the expected length scale associated with fibrosis.

3.3.

Training Model Observers

The set of local 2DCCs from the F0 and F1 phantoms comprised our signal-absent data for training a model observer and the 2DCCs from the two F4 phantoms comprised the signal present data. To avoid bias in the results, only one phantom was used to train the model observer; the other phantom was selected as the testing data. With four phantoms, two in each class, we could derive and test 4 independent observers to check for reproducibility. $7\times 7\text{pixel}$ ROIs were selected with independent gridding to calculate the means and covariance matrices. With this selection method, the liver in Fig. 3(a) had 248,439 ROIs, the liver in Fig. 3(b) had 56,865 ROIs, the liver in Fig. 3(c) had 49,858 ROIs, and the liver in Fig. 3(d) had 127,150 ROIs. The linear observers for each FA are shown in flattened 1-D form of length $P$ in Fig. 4, based notation in Eq. (5). The observer index is the vector component. We find that the templates all detect the same features, regardless of choice of training and testing data and FA—namely the peaks in the 2DCC function associated with the ECM cell size. Figure 5 provides the 2-D representation $M\times N$ of the templates at each FA for one set of training data, based on indexing in Eq. (1). The 2-D templates show a high degree of rotational symmetry, as expected for 2DCCs, which make the results invariant to image rotation.

Fig. 4

Average template $\pm \sigma$ variation between different training and testing data combinations for each acquired flip angle.

Fig. 5

2-D representation of the templates for each acquired flip angle.

The Hotelling observer has a template form $\stackrel{\rightharpoonup }{w}$ that one can visualize, whereas the quadratic observer does not. The sample covariance matrices for each flip angle for a representative signal-absent and signal-present training combination are shown in Fig. 6.

Fig. 6

(a)–(e) The sample covariance matrices for a representative set of training data at each flip angle.

3.4.

ROC Analysis and Curve Fitting

The four phantoms allowed for four different combinations of training and testing data, for which ROC analysis was performed and the AUC for each combination was calculated as a function of flip angle. ROIs were selected with a sliding window to increase sensitivity to local changes in texture; the liver in Fig. 3(a) had 7,686,034 ROIs, the liver in Fig. 3(b) had 2,952,176 ROIs, the liver in Fig. 3(c) had 2,503,899 ROIs, and the liver in Fig. 3(d) had 6,053,726 ROIs. The AUC values are plotted as a function of flip angle in Fig. 7.

Fig. 7

AUC as a function of flip angle for the four independent combinations of training and testing data.

The mean relative AUCs, after a minimal least squares adjustment to remove overall offsets between training and testing combinations, were computed as a function of flip angle, and plotted for the linear and quadratic observers in Figs. 8 and 9, respectively. The optimal flip angle was chosen based on maximizing the AUC and for both the linear and quadratic observers was found to be near 24 deg. The AUC values for the quadratic observer did not improve upon the AUC values of the linear observer.

Fig. 8

Plot of the relative AUC for a linear observer as a function of flip angle.

Fig. 9

Plot of the relative AUC for a quadratic observer as a function of flip angle.