VERDICT‐AMICO: Ultrafast fitting algorithm for non‐invasive prostate microstructure characterization

VERDICT (vascular, extracellular and restricted diffusion for cytometry in tumours) estimates and maps microstructural features of cancerous tissue non‐invasively using diffusion MRI. The main purpose of this study is to address the high computational time of microstructural model fitting for prostate diagnosis, while retaining utility in terms of tumour conspicuity and repeatability. In this work, we adapt the accelerated microstructure imaging via convex optimization (AMICO) framework to linearize the estimation of VERDICT parameters for the prostate gland. We compare the original non‐linear fitting of VERDICT with the linear fitting, quantifying accuracy with synthetic data, and computational time and reliability (performance and precision) in eight patients. We also assess the repeatability (scan‐rescan) of the parameters. Comparison of the original VERDICT fitting versus VERDICT‐AMICO showed that the linearized fitting (1) is more accurate in simulation for a signal‐to‐noise ratio of 20 dB; (2) reduces the processing time by three orders of magnitude, from 6.55 seconds/voxel to 1.78 milliseconds/voxel; (3) estimates parameters more precisely; (4) produces similar parametric maps and (5) produces similar estimated parameters with a high Pearson correlation between implementations, r 2 > 0.7. The VERDICT‐AMICO estimates also show high levels of repeatability. Finally, we demonstrate that VERDICT‐AMICO can estimate an extra diffusivity parameter without losing tumour conspicuity and retains the fitting advantages. VERDICT‐AMICO provides microstructural maps for prostate cancer characterization in seconds.


| INTRODUCTION
Prostate cancer (PCa) is the most frequently diagnosed cancer among men in high-income countries and the second most frequently diagnosed cancer in men worldwide. 1 Currently, digital rectal examination, serum prostate specific antigen (PSA)-a non-specific blood test-and trans-rectal ultrasoundguided biopsy are the primary diagnostic tools. 2 Histological analysis remains the main test for PCa diagnosis and grading, and is the standard procedure used to guide treatment. To obtain histological information a portion of tissue is taken from the prostate and observed under a microscope to detect changes in the tissue architecture (biopsy). However, biopsy is invasive and painful, and can result in complications for the patient. Noninvasive alternatives are highly sought after, as they offer several major benefits: (i) they are safer for the patient; (ii) they do not disrupt tissue, enabling serial examinations and monitoring; (iii) they can examine extended areas of an organ, as opposed to biopsy samples, which are spatially limited. Multiparametric prostate MRI is a non-invasive diagnostic tool, which has been shown to potentially reduce unnecessary primary biopsies in 27% of the patients. However, current non-invasive imaging techniques lack the sensitivity and specificity required for non-invasive cancer grading.

Diffusion-weighted MRI (DW-MRI) is becoming increasingly important in the assessment of malignant tumours. At the 2008 National Cancer
Institute sponsored open consensus conference, experts reached consensus on the use of DW-MRI as a cancer imaging biomarker. 3,4 However, current usage of DW-MRI does not fully exploit its potential. Most cancer DW-MRI studies use the apparent diffusion coefficient (ADC) for tumour assessment. [5][6][7][8] ADC reflects the mobility of water molecules within tissue, which provides useful contrast in cancerous tumours, where ADC values are generally observed to be lower than healthy tissue. However, the ADC values in PCa and benign tissue overlap substantially, confounding the overall specificity. 9,10 ADC is a gross measurement that conflates various physiological parameters, including cell size, shape, permeability, subcellular architecture and vascular perfusion effects. 11 The dependence of ADC on a variety of histological features simultaneously can mask significant alterations in tissue architecture indicative of malignancy. Moreover, ADC lacks biological specificity and fails to associate contrast changes with particular microstructural effects. 10 Advanced model-based imaging techniques can overcome some of the limitations of simplistic diffusion-based indices, such as ADC, by estimating distinct parameters reflecting separate influences on the signal. In particular, in cancer imaging, VERDICT (vascular, extracellular and restricted diffusion for cytometry in tumours) 12,13 is a three compartment model-based DW-MRI technique that is designed to capture the main microstructural properties of cancerous tissue. VERDICT has shown promising results in preclinical studies for characterizing microstructural tissue properties of xenograft tumour models. 12 In a clinical setting for PCa, VERDICT has demonstrated the ability to discriminate normal and malignant prostate tissue 13 and to characterize specific Gleason grades. 14 However, as with many model-based techniques, [15][16][17] VERDICT requires a computationally expensive non-linear fitting procedure to estimate the model parameters. This prevents immediate inspection of VERDICT parameter maps, which is possible with ADC mapping. Non-linear fitting is also vulnerable to local minima, which can cause convergence to a sub-optimal solution, adding to the noise and uncertainty in derived parameter maps. Another limitation of the original VERDICT implementation is that it requires certain parameter constraints for stability, such as fixing the intra-and extra-cellular diffusivities, despite evidence that such parameters may vary. 18,19 Recently, ultrafast-fitting algorithms have been developed to address the high computational cost of model-based microstructure-imaging techniques. [20][21][22] Graphical processing units (GPUs) provide a brute-force solution, using a parallelized approach, to reduce the computational time as in References 20 and 22. Although some overhead lies in GPU-based design and implementation, once adapted for GPU platforms the fitting time can be reduced by several orders of magnitude compared with standard central processing units, and is limited only by the number of cores on the GPU. Alternatively, through linearization and convex optimization, the accelerated microstructure imaging via convex optimization (AMICO) framework 21 addresses the cost limitation for each individual fitting operation and is therefore complementary to speed-ups obtained from GPU computing.
AMICO is guaranteed to find the global minimum, albeit of a reformulated objective function, providing more reliable and stable parameter maps.
Hence, AMICO potentially affords relaxation of overly restricted constraints, such as fixed diffusivity parameters, that were constrained to stabilize non-linear fitting. Both solutions have dramatically reduced the computational time of microstructure-imaging techniques in the brain, e.g. ActiveAx, NODDI and CHARMED. 15,16,23 Whilst GPUs do not reduce the computational cost, only the computational time, AMICO actually reduces the cost via the linearization. AMICO could be implemented in GPUs and further accelerate the fitting. Rapid-fitting algorithms are important to analyse the large volume of data arising from studies and clinical trials and, potentially, to improve patient workflow during the clinical process. For example, the UK Biobank imaging project 24 requires fitting up to 100 000 imaging datasets; only through the use of ultrafast-fitting algorithms NODDI parameters have been included as imaging phenotypes in the project. 25,26 Cohorts in PCa studies are also increasingly large [27][28][29] and thus necessitate similar techniques.
The AMICO framework can be adapted to linearize the estimation of the VERDICT parameters (VERDICT-AMICO), allowing a fast-fitting approach. 30 Here, we present VERDICT-AMICO and demonstrate its benefits over the original VERDICT implementation. We utilize AMICO to fit the VERDICT model for prostate tissue 13 to demonstrate reduction in computation time and robust parameter estimation. We explore the reliability of VERDICT-AMICO using simulated and clinical data. We test whether we can retain the qualitative conspicuity for cancer lesions in parametric maps, and we evaluate the repeatability of the technique within a test-retest experiment. Finally, we use AMICO to demonstrate the possibility of unfixing previously fixed VERDICT parameters.

| THEORY
We first review the original VERDICT model for prostate using non-linear fitting, and the general AMICO framework. 21 Then, we describe VERDICT-AMICO.

| VERDICT for prostate
The VERDICT framework characterizes water diffusion in vascular (VASC), extracellular-extravascular (EES) and intracellular (IC) compartments in tumours. 12 Mathematically, VERDICT is the sum of three parametric models that describe the DW-MRI signal in each separate water population assuming zero exchange between them. The normalized diffusion signal for the VERDICT model is where f i is the proportion of signal with no diffusion weighting (b = 0) from water molecules in population i (IC, EES or VASC), 0 ≤ f i ≤ 1, The model has three different volume fraction parameters: f IC , f EES and f VASC . For prostate, 13 the diffusion signal for the IC compartment (S 1 ) is modelled with impermeable spheres and has d IC (IC diffusivity) and R (cell radius) as parameters. The sphere (terminology from Reference 17) models particles diffusing inside impermeable spherical boundaries with non-zero R using the Gaussian phase approximation. 31 In our implementation, as in the previous works that used the sphere model, 12,13,17 the numerical approximation of the implementation used is 100 roots.
The model for the EES compartment assumes a diffusion tensor (DT) model, in particular an isotropic DT with diffusivity d EES as parameter.
The normalized DT signal is where I is the identity tensor, G is the gradient direction and b = (Δ − δ/3)(γδ|G|) 2 is the b-value for the pulse gradient spin echo (PGSE) sequence, Δ is the time between the onsets of the two pulses, δ is the pulse gradient duration and γ is the gyromagnetic ratio.
The vascular compartment uses the "astrosticks" model (terminology from Reference 17) for isotropically distributed zero-diameter restricting cylinders or "sticks", and has pseudo-diffusivity P as the only parameter. The signal is given by where n is the stick direction with uniform distribution p, p(n) = (4π) −1 .
The model has three independent unknown parameters: f IC , R and f EES . f VASC is calculated as f VASC = 1 − f IC − f EES , and the diffusion and pseudo-diffusion coefficients are fixed, as in Reference 13, to d IC = d EES = 2 × 10 −9 m 2 /s, P = 8 × 10 −9 m 2 /s.

| AMICO
AMICO is a framework that reformulates microstructural imaging techniques as linear systems of equations enabling use of convex optimization techniques (https://github.com/daducci/AMICO/). 21 More concretely, AMICO uses a dictionary of potential parameter combinations, , and convex optimization to find the weight vector b x ∈ R Nk þ for the dictionary elements that best matches the vector containing the N d measurements y ∈ R Nd þ . Hence, the system can be formulated as a convex optimization problem as follows: where ‖·‖ 2 is the standard ℓ 2 -norm in R n , the positivity constraint is explicitly imposed as the coefficients x correspond to volume fractions, Ψ(·) represents a generic regularization function 32 and the parameter λ > 0 controls the trade-off between data and regularization terms. The AMICO dictionary can also be partitioned into different sub-matrices to correspond to different compartments considered in the model of choice. We will use this property to adapt AMICO for the three compartments of the VERDICT model. A pre-defined dictionary for the fitting instead of continuous variables as possible solutions is required to linearize the problem with a fast-computational performance. Nevertheless, the number of solutions is not limited, as the final parameter estimates are a linear combination of each dictionary value according to the real-valued weight vector b x.

| VERDICT-AMICO
We adapt the AMICO framework for prostate VERDICT to estimate the three independent unknown parameters as in Reference 13: f IC , R and f EES .
The dictionary for VERDICT- is partitioned into three sub-matrices, corresponding to the VERDICT compartments: where Φ r ∈ R Nd×Nr , Φ e ∈ R Nd×Ne and Φ v ∈ R Nd×Nv each model the IC, EES and vascular contributions of the diffusion signal in the voxel with The regularization function used is the basic Tikhonov regularization with the same λ value (λ = 0.001) as used in Reference 30.
Here, the full dictionary that mimics the original VERDICT values consists of N k = 19 entries in total combining three sub-dictionaries as follows.
• Each column in Φ r ∈ R N d ×Nr corresponds to the signal attenuation of the water molecules restricted within spheres 17 with a specific radius. We considered N r = 17 radii values linearly spaced from 0.01 μm to 15.1 μm. The corresponding signal profiles are estimated according to the "sphere", assuming an IC diffusion coefficient d IC = 2 × 10 −9 m 2 /s.
• A single compartment, Φ e ∈ R Nd×Ne where N e = 1, describes the EES with the same fixed value for the diffusion coefficient d EES = 2 × 10 −9 m 2 / s. The signal model is free isotropic diffusion.
is considered to account for the vascular volume fraction. Signal is estimated according to the "astrosticks" model and pseudo-diffusivity is fixed, P = 8 × 10 −9 m 2 /s. 13 The estimated coefficients b , and the VERDICT-AMICO estimated parameters are obtained as The original VERDICT only assumes one R per voxel. When there is more than one R per voxel, original VERDICT estimates the average. By averaging the distribution of radii, the two methods are equivalent.
VERDICT-AMICO inherits parameter constraints from the original non-linear implementation of the model. The enhanced robustness of the AMICO fitting may be able to avoid these constraints. We hypothesize that VERDICT-AMICO can estimate a diffusivity parameter without compromising the overall parameter estimation. We previously tested different AMICO dictionaries and observed that different d IC values have a small impact on the other parameter estimates with fixed perfusion. 30 Thus, we unfixed d EES and kept d IC and P fixed with their original values, d IC = 2 × 10 −9 m 2 /s and P= 8 × 10 −9 m 2 /s. 13

| METHODS
We first provide details of the patient population, data acquisition and image processing, region of interest (ROI) selection, and synthetic data. We then describe the experiments comparing the two implementations. Finally, we demonstrate the possibility of estimating an additional previously fixed VERDICT parameter.

| Patient population
This study has been performed with local ethics committee approval as part as of the INNOVATE clinical trial. 33 Between April and July 2016, eight men were prospectively recruited and provided informed written consent. The inclusion criteria were the following: (1) suspected PCa or (2) undergoing active surveillance for known PCa. Exclusion criteria were the following: (1) previous hormonal, radiation therapy or surgical treatment for PCa and (2) biopsy within 6 months prior to the scan. Patient characteristics are shown in Table 1.

| Data acquisition and pre-processing
All patients underwent a standard European Society of Uroradiologists compliant mp-MRI, 2 on a 3 T scanner (Achieva, Philips Healthcare, Best, Netherlands) supplemented by VERDICT DW-MRI. VERDICT DW-MRI was acquired with PGSE and an optimized imaging protocol for VERDICT prostate adapted from Reference 34 with five b-values of 90-3000 s/mm 2 in three orthogonal directions using a pelvic coil. Table 2  To reduce possible artefacts caused by patient movement during scanning, VERDICT DW-MRI data was registered using a rigid registration. 35,36 The transformation matrix was computed using the b = 0 images, and then was applied to the b = 0 and the subsequent DW-MRI.
We normalize the data using the b = 0 images, hence for each voxel the number of normalized measurements is N d = 20 (five b-values with three directions each, and five b = 0 images for each T E ). The AMICO dictionary size is limited by the VERDICT DW-MRI acquisition; we cannot have more dictionary terms than measurements N d as we are using the simple form of regularization. The Likert score is a five-point scoring system used for the interpretation of mp-MRI for PCa. 37 Clinical biopsy information was also used to confirm where to place the ROIs ( Figure 1A).

| ROI selection
We grouped the 28 different ROIs according to Gleason score and prostate zone, as we expect them to have similar underlying microstructures. When the Gleason score was unknown we grouped the ROIs according to the Likert score. This procedure resulted in 10 grouped ROIs (g-ROIs) ( Figure 1B). Table 1 provides details for the ROIs and g-ROIs. We compute the signal-to-noise ratio (SNR) in the VERDICT data using the method of Dikaios et al., 38

| Fitting performance experiments
We test the VERDICT model using both the original non-linear fitting and the linear VERDICT-AMICO. We use synthetic data to compare both implementations for accuracy against ground-truth estimates. Then, to test the fitting performance with clinical data, we compare the variance of the estimated parameters in different known tissue types. We analyse ROIs within the same g-ROIs together. We then contrast the parametric maps in terms of run-time and fitting performance.
Two board-certified radiologists (SP and EJ) examined the VERDICT-AMICO parametric maps for qualitative tumour conspicuity (contrast between tumour and surrounding tissue) to ensure tumour enhancement and clinical relevance. Then, we compared the chi-squared objective function maps ( f obj ), which are sum of square differences adjusted to account for offset Gaussian noise, to evaluate the robustness of the fitting, as in Reference 17. Finally, we test the repeatability of both procedures in three patients (scan-rescan) using Bland-Altman agreement analysis. 41 We emphasize that the different parameter maps for both VERDICT and VERDICT-AMICO should be viewed side by side to aid in their interpretation. For example, the estimated R value is less relevant if the f IC is almost zero in that region, and in areas where f obj is high all estimated parameters are less reliable.

| d EES estimation with VERDICT-AMICO experiment
Here, we use both fitting methods (original non-linear and VERDICT-AMICO) to estimate the extra parameter (d EES ) in two datasets, and we examine the run-time and goodness of fit. The VERDICT-AMICO dictionary with unfixed d EES is • N r = 13 different radii (linearly spaced from 0.01 μm to 15.1 μm) with fixed d IC = 2 × 10 −9 m 2 /s.
• N e = 5 diffusion coefficients for EES: d EES = 1.1 × 10 −9 , 1.6 × 10 −9 , 2.1 × 10 −9 , 2.6 × 10 −9 and 3.1 × 10 −9 m 2 /s.   Figure 2 shows the performance of both fitting methods, VERDICT and VERDICT-AMICO, using synthetic data as a function of SNR. Figure 2A shows the absolute errors for each estimated parameter. We compute the mean absolute errors for the overall combination of the input

| Clinical data
We also evaluate the performance of both methods using clinical data. First, we compare the computational times for parametric maps ( Table 3).
The VERDICT-AMICO formulation reduces the processing time by more than three orders of magnitude, from 6.55 s/voxel to 1.78 ms/voxel. To  board-certified radiologists agreed that tumour conspicuities are similar for the two fitting methods and reveal similar qualitative differences between tumour and normal tissue. For example, in both cases f IC is higher and f EES is lower in tumour compared with normal tissue, as seen in References 13 and 42. In general, the two implementations present similar behaviours. Maps obtained using AMICO appear less noisy than those with the original non-linear fitting. This effect is more evident for the radius map. Figure 5 presents correlations of the estimated parameters from the repeatability experiment for 12 ROIs with both methods. Overall, most parameters are repeatable with high correlation coefficients. The AMICO implementation has the highest correlation coefficients (except for f VASC ). For both methods, R appears the least repeatable. We further test the repeatability of R by recalculating correlation coefficients whilst excluding ROIs with fewer than 10 voxels (ROI 1 from Table 1), and voxels where f IC was too small to produce a reliable signal ( f IC < 0.001).

| DISCUSSION
In this study, we adapted the AMICO framework for the VERDICT technique for PCa characterization to reduce the computational cost of the original fitting. We tested VERDICT-AMICO against the non-linear fitting procedure in terms of (a) accuracy against ground-truth simulation values, (b) computational time, (c) fitting precision in different tissue types, (d) qualitative tumour conspicuity and (e) repeatability. Finally, we demonstrated the estimation of an extra VERDICT parameter: the EES diffusivity (d EES ). The principal benefit of the AMICO framework in prostate is that it provides an acceleration factor of several orders of magnitude compared with non-linear fitting, whilst offering robust parameter estimation, and without significantly affecting the parameter maps.
First, we tested the accuracy of both fitting procedures using synthetic data with an extensive range of parameter values within biophysical limits ( Figure 2). We also checked the accuracy at different SNR levels. For finite SNR, VERDICT-AMICO appears to produce errors that vary less, as a function of parameter values, than the original VERDICT. For infinite SNR, VERDICT-AMICO performance is worse than that of VERDICT; this is an artefact of the regularization term-regularization parameters fail when dealing with perfect data because numerical calculations may cause instabilities and yield an unsatisfactory solution. 43 However, with an SNR of 20, similar to the level in our clinical samples, VERDICT-AMICO is more accurate than VERDICT for most parameters. In the future, to improve the fitting procedure, we could adjust the regularization term ad hoc by estimating the SNR for each region. However, this solution requires a prostate segmentation as input, since the SNR is different in different image regions. The noise model may also influence our results, as AMICO assumes Gaussian noise. Many papers suggest that when the SNR is high enough the noise can be approximated as Gaussian. 39 For SNR = 20, AMICO with Gaussian noise has acceptable results; however, future versions of AMICO will incorporate non-Gaussian noise.
Second, we examined the computational run-time for the two fitting procedures. Results in Table 3 illustrate that AMICO provides an acceleration factor of several orders of magnitude compared with the non-linear fitting, while the parameter estimation remains similar to the original VERDICT fitting. This speed-up makes it practical for researchers to analyse entire volumes, rather than single slices, of prostate data using standard computers. AMICO thus allows the computation of parametric maps without human intervention for precise gland/lesion segmentation.
Additionally, AMICO would allow VERDICT to be computed directly during the scan, similar to the current use of ADC in clinics. Analysing the whole prostate enables the observation of multiple lesions from large datasets, facilitating translation of the method to routine clinical use. VERDICT-AMICO can fit the whole volume of interest (30 976 voxels) in less than 1 minute. Improvement in computation time could also potentially be achieved using large GPUs. 20,22 However, AMICO provides an entirely complementary and direct reduction in computational cost.
We measured the parameter precision in various prostate regions in benign and cancerous tissue (Figure 3). We observed differences in f IC among normal PZ and TZ, with higher f IC for normal TZ. This could potentially be due to the fact that TZ is the main site of origin of prostatic hyperplasia, which is also characterized by a higher volume of epithelial cells, much like tumours. 44 Like VERDICT, VERDICT-AMICO tends to overestimate vascular fraction; we see higher f VASC in the presented results than we would expect from histology. 44 This could be a similar effect to the overestimation of cerebrospinal fluid volume fraction in white matter with NODDI that also arises as a result of fixing diffusivity parameters. 45 We no longer observe vasculature overestimation after unfixing the EES diffusivity.
Visual comparison (Figure 4) by board-certified radiologists showed that the two methods produce similar parametric maps. VERDICT-AMICO retains the qualitative conspicuity in lesions and revealed the same trends observed in Figure 3. Overall, the VERDICT-AMICO formulation produces smoother and more robust maps suitable for radiological inspection, as it enables location of the global minimum more reliably than the original fitting. Greater homogeneity in the minimum objective function maps support this assertion (see the highlighted white square in Figure 4).
We further tested the clinical applicability of VERDICT-AMICO by investigating the repeatability of the parameter estimates ( Figure 5).
Results showed high levels of repeatability for most parameters for both methods. However, the AMICO implementation improves repeatability over the non-linear fitting for f IC and f EES . For R, the repeatability is low for both methods. Future work will investigate this result using histology.  Table 1. C, D, Parametric maps for VERDICT and VERDICT-AMICO, respectively. ROIs are overdrawn on the f IC map for illustrative purposes. The white square highlights a region with greater homogeneity in the objective function for VERDICT-AMICO compared with VERDICT. f IC , intracellular volume fraction; f EES , extracellular-extravascular volume fraction; R, radius [μm]; f VASC , vascular volume fraction. a.u., arbitrary units In our dataset, parameter estimation in tumour regions was less repeatable than in normal tissue (results not shown). This is probably because tumour tissue can be highly heterogeneous, especially when compared with normal tissue, so small errors in the ROI definition can produce greater differences. A more extensive dataset with many tumour types is needed to further test parameter repeatability in cancer lesions.
Next, we examined whether the VERDICT-AMICO improvement in computational time was retained when estimating an additional VERDICT parameter, the diffusivity d EES . 30 We also estimated d EES with the original non-linear fitting ( Figure 6). As expected, the parametric maps presented similar results for both methods. VERDICT-AMICO exhibits the same improvement over the original fitting for computational time and robustness after unfixing d EES . For both fittings, the parameter estimates with the new dictionary are more biophysically plausible and in closer agreement  In Figure 6 we compare VERDICT-AMICO and VERDICT with an extra unfixed parameter, d EES . As in the original VERDICT-AMICO implementation with fixed d EES , the rest of the estimated parameters ( f IC , R, f EES ) do not generally hit the upper boundary constraint compared with the original VERDICT implementation. In Figure 6, R hits the upper boundary in some voxels for VERDICT also after unfixing d EES , which evidences that AMICO improves and stabilizes the fitting. Despite the more realistic estimates after unfixing d EES , both VERDICT and VERDICT-AMICO lose some tumour conspicuity in the f IC maps compared with the original model, which is a clinically important aspect. However, the contrast remains among the full set of maps. For example, we show that the conspicuity can be recovered and perhaps enhanced using the f IC /d EES combination.
One possible explanation for the enhanced conspicuity could be a decreased inter-cell space and increased tortuosity, which is consistent with a higher partial volume of epithelial cells and loss of lumen space. 11 VERDICT with fixed d EES may still have clinical utility for highlighting tumours. In the future, we will run a comprehensive study involving clinicians to further compare the potential clinical use of the two implementations.
The main limitation of the AMICO framework is that the fitting results depend to some extent on the dictionary values, as the regularization has to be set empirically. 21 For this reason, we used histological information to guide the selection of dictionary values. Biopsy results confirm that the estimated maps provide plausible values even with the limited dictionary. However, more samples from larger datasets are required to validate the VERDICT parameters with histology, and this is beyond the scope of this study. Future efforts with larger cohorts will focus on clinical VER-DICT parameter validation, using methods such as those of Reference 52 to further investigate VERDICT as a non-invasive MRI-based cancer biomarker. VERDICT-AMICO also inherits the limitations and assumptions of the VERDICT prostate model, which need to be taken into consideration when interpreting the parameter estimates. For example, the current model does not account for permeability between the different components or for large differences in T 2 within the same voxel (T 2 heterogeneity), 53 which could cause bias in the parameter estimation. In the future, we will study the T 2 and permeability effects and incorporate the findings into the VERDICT model. Nevertheless, the results from this study and previous work provide good evidence that VERDICT-AMICO can provide useful information, improving current methods.
To conclude, VERDICT-AMICO's fast and robust fitting highlights important microstructural differences between tumours and normal prostate tissue in seconds, which is crucial for utilizing VERDICT as a diagnostic tool. Longer-term, fast-fitting algorithms such as AMICO will be essential for the translation of promising techniques like VERDICT to widespread clinical applications. Our results suggest that VERDICT-AMICO maps provide additional value over the original VERDICT prostate implementation, and we hope they will be a valuable source of clinical imaging biomarkers. However, further studies are required to validate this assumption. The VERDICT-AMICO implementation will be freely available online (http://mig.cs.ucl.ac.uk Resources tab). VERDICT-AMICO shows promise as a MRI-based tool for PCa grading and detection, which could be used to enhance the diagnostic potential of current prostate mp-MRI.