Physics-Informed Autoencoder for Prostate Tissue Microstructure Profiling with Hybrid Multidimensional MRI

Abstract

Purpose

To evaluate the performance of Physics-Informed Autoencoder (PIA), a self-supervised deep learning model, in measuring tissue-based biomarkers for prostate cancer (PCa) using hybrid multidimensional MRI.

Materials and Methods

This retrospective study introduces PIA, an emerging self-supervised deep learning model that integrates a three-compartment diffusion-relaxation model with hybrid multidimensional MRI. PIA was trained to encode the biophysical model into a deep neural network to predict measurements of tissue-specific biomarkers for PCa without extensive training data requirements. Comprehensive in silico and in vivo experiments, using histopathology measurements as the reference standard, were conducted to validate the model’s efficacy in comparison to the traditional nonlinear least squares (NLLS) algorithm. PIA’s robustness to noise was tested in in silico experiments with varying signal-to-noise ratio (SNR) conditions, and in vivo performance for estimating volume fractions was evaluated in 21 patients (mean age, 60 years ± 6.6 [SD]; all male) with PCa (71 regions of interest). Evaluation metrics included the intraclass correlation coefficient (ICC) and Pearson correlation coefficient.

Results

PIA predicted the reference standard tissue parameters with high accuracy, outperforming conventional NLLS methods, especially under noisy conditions (r_s = 0.80 vs 0.65, P < .001 for epithelium volume at SNR of 20:1). In in vivo validation, PIA’s noninvasive volume fraction estimates matched quantitative histology (ICC, 0.94, 0.85, and 0.92 for epithelium, stroma, and lumen compartments, respectively; P < .001 for all). PIA’s measurements strongly correlated with PCa aggressiveness (r = 0.75, P < .001). Furthermore, PIA ran 10 000 faster than NLLS (0.18 second vs 40 minutes per image).

Conclusion

PIA provided accurate prostate tissue biomarker measurements from MRI data with better robustness to noise and computational efficiency compared with the NLLS algorithm. The results demonstrate the potential of PIA as an accurate, noninvasive, and explainable artificial intelligence method for PCa detection.

Keywords: Prostate, Stacked Auto-Encoders, Tissue Characterization, MR–Diffusion-weighted Imaging

Supplemental material is available for this article.

©RSNA, 2025

See also commentary by Adams and Bressem in this issue.

Summary

The Physics-Informed Autoencoder model provided accurate and explainable measurements of prostate tissue microstructure using hybrid multidimensional MRI and was more robust to image noise compared with the conventional least-squares algorithm.

Key Points

• ■ A self-supervised deep learning–based method that leverages a biophysical signal model for supervision (named the Physics-Informed Autoencoder, or PIA) provided accurate and explainable prostate tissue microstructure information when compared with histopathologic measurements as the reference standard (intraclass correlation coefficient of 0.94, 0.85, and 0.92 for epithelium, stroma, and lumen compartments, respectively; P < .001 for all), without the need for extensive, annotated training datasets.

• ■ PIA exhibited robust performance against scanner noise, significantly outperforming conventional least-squares methods in both in silico and in vivo experiments (Spearman r = 0.80 vs 0.65, P < .001 for epithelium volume at an SNR of 20:1); this was accomplished with a computational efficiency exceeding conventional methods by a factor of 10 000.

• ■ Tissue-related biomarkers derived from the PIA model significantly correlated with prostate cancer aggressiveness (r = 0.75, P < .001).

Introduction

Prostate cancer (PCa) is alarmingly common, with one in eight male individuals in the United States diagnosed with PCa at some point in their lives (1). Multiparametric MRI with the Prostate Imaging Reporting and Data System (PI-RADS) is considered a standard of care for screening and differential diagnosis of PCa. However, the positive predictive value of PI-RADS version 2.1 is as low as 35%, leading to approximately 1 million unnecessary biopsies each year in the United States alone and causing undue stress to patients (2). In addition, 29% of clinically significant cancers are missed (3). To address the problem of the subjectivity of PI-RADS and to increase diagnostic accuracy, various researchers have turned to biophysiologic compartmental models for noninvasive inference of tissue microstructure (4). The overall approach with these biophysiologic compartmental models involves fitting the MRI data to a predefined function. These functions, typically a sum of decaying exponentials, represent a hypothesis about the underlying signal behavior. Comparative analysis of this approach and other methods for prostate tissue profiling from MRI is presented in Table 1.

Table 1:

Relevant Studies in the Literature for Prostate Tissue Profiling from MRI

A primary challenge associated with multicompartment models is that using the sum of decaying exponentials often leads to ill-posed behavior with nonlinear least squares (NLLS) algorithms, causing difficulties in parameter estimation (15,16). A particular concern is when various tissue compartments exhibit similar MRI decay characteristics. In such cases, the parameter estimation process becomes highly sensitive to initial guesses and noise in the data, leading to a vast solution space. This ambiguity in parameter estimation can substantially degrade the reliability of the model, as small variations in the input data can result in large changes in the estimated parameters, especially in the presence of high levels of noise (17).

There is a growing trend in research exploring the use of supervised deep learning for PCa detection (18,19). However, these models require large amounts of well-labeled training data, and their effectiveness across different MRI vendors remains a concern due to domain discrepancies (20). Physics-informed deep learning (21) aims to integrate physical laws into neural network training, thereby facilitating solution development and avoiding the need for large training datasets. Early applications of this method focused on partial differential equations (22). This approach has been adapted for multiexponential signal models, such as diffusion-relaxation models of white matter microstructure (23) and biexponential intravoxel incoherent motion models (24).

The current work presents an emerging self-supervised deep learning approach that bridges the gap between hypothesis-driven and data-driven methods for MRI signal analysis. Our method leverages the strengths of both paradigms, mitigating their inherent limitations and capitalizing on their complementary advantages. Specifically, the proposed model, Physics-Informed Autoencoder (PIA), encodes the underlying biophysical principles as a prior knowledge constraint within a neural network architecture. This innovation eliminates the need for extensive training on large datasets, a major bottleneck in conventional deep learning approaches. The purpose of this study was to evaluate the performance of PIA in measuring tissue-based biomarkers of PCa using hybrid multidimensional MRI (HM-MRI). The efficacy of our method is comprehensively evaluated through in silico and in vivo experiments, where histopathologic measurements of the true tissue parameters serve as the reference standard for validation.

Materials and Methods

This retrospective study, conducted between June 2022 and July 2024, presents a self-supervised deep learning approach for estimating MRI biomarkers for PCa, with histopathologic confirmation of its measurements. Our framework, PIA, integrates biophysical model–based parameter fitting with deep learning methods. The first set of experiments involves development and analysis of PIA with in silico data, whereas the second set of experiments presents evaluation of PIA’s performance in clinical in vivo prostate MRI scans. This study involved retrospective analysis of prospectively collected data. The study was approved by the institutional review board and complied with the Health Insurance Portability and Accountability Act. Written informed consent was obtained from all included patients.

Biophysical Model: Three-compartment Epithelium, Stroma, and Lumen Diffusion-Relaxation Model

We developed PIA with the histologically verifiable three-compartment diffusion-relaxation model (14) that includes three tissue compartments, consisting of epithelium (ep), stroma (st), and lumen (lu). In this model, signal in each compartment decays as the b value and echo time (TE) increase, at a rate proportional to their volume fractions (ν_n); rates are also related to the individual diffusivities (D_n), and T2 relaxation times (T2_n) such that:

$$\frac{S_{b,\text{TE}}}{S_{0,0}}={{\displaystyle \sum }}_{\text{n}\in \{\text{ep},\text{st},\text{lu}\}}{\nu }_{\text{n}}\times \text{exp}\left(-D_{\text{n}}\times b-\frac{\text{TE}}{\text{T}{2}_{\text{n}}}\right).$$

The current state-of-the-art implementations of this method aim to infer the tissue parameters (ν_n, D_n, T2_n, n ∈ {ep,st,lu}) by fitting HM-MRI data, scanned with various b-value and TE pairs, to the Equation, using the NLLS optimization. Predicted parameters in the Equation are ν_n, D_n, and T2_n. Previous research has demonstrated that tissue volume fraction estimates derived from fitting HM-MRI data to this model using NLLS are valuable biomarkers for cancer detection (25). However, the exploration of diffusivity and T2 measurements within each compartment was not feasible in these studies. This was primarily due to the complexities introduced by the sum of multiexponentials, especially for images with low signal-to-noise ratio (SNR), which hinders accurate estimation of diffusivity and T2 values.

Physics-Informed Autoencoder

Traditional model-based methods treat the tissue-specific biomarkers in the Equation as unknowns in equations, while our proposed solution PIA transforms the problem into a deep learning task and views them as latent variables within an autoencoder. Like all other autoencoders, PIA consists of two parts, an encoder and a decoder. The encoder is a trainable neural network that predicts the underlying tissue-specific biomarkers from the given MRI measurements in its input. On the other hand, the decoder is a nontrainable biophysical model function that reproduces the MRI signals using the output of the encoder (see Fig 1). During training, the encoder learns to emulate the decoder’s physical rules, hence the term physics-informed.

Figure 1:

General flowchart of the proposed model Physics-Informed Autoencoder (PIA) for prostate tissue microstructure analysis. D = diffusivity, DNN = deep neural network, ep = epithelium, Eq. = equation, HM-MRI = hybrid multidimensional MRI, lu = lumen, MSE = mean squared error, st = stroma, TE = echo time, v = volume fraction.

PIA’s encoder is a feed-forward multihead neural network. Multiparametric MRI signals are processed by a six-layer deep neural network with leaky ReLU activations. This shared network extracts the embedding to infer the underlying biomarkers. The embedding is then processed by parameter-specific layers. The volume fraction estimation is a simple classification network with two layers and a softmax activation function. The diffusivity and T2 estimators are modeled with tanh activation functions. This design allows the PIA encoder to predict the diffusivity and T2 of each tissue compartment within their range of realistic values. The outputs of the encoder are fed to the decoder, which is the three-compartment signal model in the Equation, to synthesize an approximation of the input MRI signal.

Training Method

PIA was trained in a self-supervised fashion. The objective of the pretraining phase was to have the encoder learn to emulate the inverse of the biophysical model, especially under adverse noise conditions. The training dataset for the pretraining phase comprised synthetically generated data with various tissue compositions of epithelium, stroma, and lumen compartments. The compartment parameters (ν_n, D_n, T2_n, n ∈ {ep,st,lu}) were sampled uniformly from within biophysically realistic parameter ranges for each compartment, for example, D_st in range 0.7–1.7 μm²/sec (14). It should be noted that the sampled tissue values were never used to supervise PIA; instead, we generated synthetic MRI signals by applying the biophysical model (the Equation).

To establish robustness to noise, the virtual MRI signals were corrupted with normally distributed additive noise at both real and imaginary components, and the magnitude of the noisy signal was used as input to PIA. This procedure made the input magnitude data Rician-distributed, as is the case for in vivo data. The SD of the additive noise was set so that the SNR of the maximum signal amplitude (lowest echo time, lowest b-value signal amplitude) was 20:1. We trained PIA with an SNR of 20:1 because it is reported to be the expected SNR found in prostate tissue (26).

In the forward run, the encoder predicts the underlying tissue parameters from the noise-corrupted signal and the decoder reconstructs the MRI signal based on the encoder’s estimates. The model is trained to minimize the squared error between the reconstructed signal and the noise-free version of the input signal. At every epoch, a new batch of virtual data, each with different random noise and a set of tissue parameters, was generated with random sampling. The training was kept for 50 000 epochs. A learning rate of 0.0003 was used with Adam optimizer. The hyperparameters of the PIA model, including the encoder complexity, learning rate, and length, were set so that the multiheaded encoder could “memorize” the inverse of the Equation under noise-free scenarios. This was established in a hyperparameter tuning phase by measuring the reconstruction error prior to the training with added noise. Details of data synthesis techniques including parameter ranges and the source code are provided in Appendix S1.

In Silico Validation

During inference, the encoder outputs are taken as the tissue parameter estimates of PIA. First, we evaluated PIA’s performance in estimating the underlying parameters using the reference standard volume, diffusivity, and T2 parameters under several conditions to test for robustness to MRI variations: (a) under various noise conditions with SNR levels between 10:1 and 10 000:1, (b) under a different imaging protocol (train with endorectal coil MRI protocol and test with surface coil MRI protocol), and (c) under a different tissue model (train with three-compartment model and test with two-compartment data). Furthermore, we investigated PIA’s performance under conditions in which epithelium and stroma exhibit similar MRI decay characteristics, a situation where NLLS is known to fail. Evaluations were conducted using the following metrics: (a) Spearman correlation coefficient, (b) mean absolute error (MAE), (c) bias, and (d) SD. More detail about the robustness analysis is in Appendix S1.

To assess the speed performance of PIA in comparison to NLLS, we executed both methods on the same set of 20 000 virtual voxels and measured the wall time for their solution on an Intel Xeon Gold 6130 CPU with 2.10 GHz.

Histologic Validation with in Vivo Scans

We validated the performance of PIA’s volume fraction estimations using histologic measurements of patients with PCa. We examined HM-MRI scans from 21 patients with PCa who underwent prostatectomy after imaging (mean age, 60 years ± 6.6 [SD]; all male). This cohort has been previously used in a published work for the histologic validation of HM-MRI using the NLLS method (14); here, we use it to validate the in vivo accuracy of PIA biomarkers and compare them to NLLS and quantitative histology. A total of 71 regions of interest (ROIs), comprising 35 cancerous and 36 healthy tissues from the 21 patients, were evaluated for tissue compartment percentages, using quantitative histology as the benchmark (27,28). Technical detail about the MRI protocol and the histopathologic evaluation is provided in Appendix S1.

Agreement between PIA’s volume fraction estimations and histologic measurements was assessed using the intraclass correlation coefficient (ICC). Performance of PIA’s volume fraction estimations for estimating histologic measurements was evaluated using linear mixed modeling, with PIA volume fraction as the fixed effect and the subjects as random effects; the marginal (unconditional) R² value was used as the metric for prediction performance.

Evaluation of in Vivo Diffusivity and T2 Estimates of PIA

Quantitative histology measurements served as the reference standard for volume fraction estimates and a means to validate PIA’s in vivo performance, as we are not aware of a direct way to validate the performance of PIA in measuring diffusivities and T2 relaxation times of individual tissue compartments in in vivo scans of human prostates (although, in principle, MRI microscopy could provide this information for ex vivo tissues [29]). Correlation of PIA’s diffusivity and T2 measurements of tissue compartments with the Gleason grade, serving as the reference standard, was calculated using the Pearson correlation coefficient. Gleason grades were classified into five categories: healthy (n = 36), 3 + 3 (n = 9), 3 + 4 (n = 14), 4 + 3 (n = 9), and 4 + 4 and above (n = 3).

Diagnostic Utility and Interpretability of PIA’s Measurements

Clinical utility of epithelium and lumen volumes as biomarkers for clinically significant PCa (CSPCa) detection have been previously shown (25). In a receiver operating characteristic curve analysis, including CSPCa (Gleason score 3 + 4 and above) and benign tissues (Gleason score 3 + 3 and below), PIA’s biomarker estimates for the 71 ROIs were compared against the estimates of NLLS and conventional apparent diffusion coefficient (ADC)–based measurements.

The interpretability of PIA biomarkers was analyzed via feature importance in CSPCa detection using the permutation importance method. A random forest model was fit on the in vivo dataset, and the importance of each biomarker was assessed by randomly shuffling its values and observing the resulting decrease in model performance. This process was repeated 10 times to obtain an average importance score for each feature to gauge their respective contributions.

Statistical Analysis

In in silico tests, when comparing PIA’s biomarkers with the estimates from NLLS based on the reference standard, we used Steiger Z test for Spearman r, t test for MAE and bias, and F test for the SD, using a significance P value of .05. When multiple tests were conducted, we applied Bonferroni correction. Consequently, the significance values were reduced to .00139.

In in vivo tests, the metrics obtained using PIA-derived volume fraction estimations were compared with those obtained using NLLS method–derived volume fraction estimations using a one-sided Z test. We used the t test for MAE and absolute bias and the F test for the SD. All standard errors for the differences were determined using the cluster bootstrap method, to account for multiple ROIs defined in each patient. The cluster bootstrap was implemented by resampling, with replacement, the patients to ensure that the correlation in outcome measures between ROIs within the patients was maintained. The cluster bootstrap procedure was iterated B = 9999 times to minimize the simulation error to the extent possible. Analyses were performed in R (version 4.4.1) and Python (version 3.12.5). The significance levels were set to .05.

For diagnostic utility tests, we used DeLong test for area under the receiver operating characteristic curve (AUC) and performed pairwise t tests between each Gleason score group to evaluate the efficacy of PIA’s epithelium diffusivity measurements in detecting PCa aggressiveness.

Results

In Silico Experiments

PIA estimated the imaging biomarkers with superior performance over NLLS with respect to Spearman r, MAE, bias, and SD metrics. At an SNR of 20:1 and in volume of epithelium, for instance, which is the strongest biomarker for PCa detection, PIA’s estimations had significantly higher correlations with the reference standard volume over NLLS (0.80 vs 0.65, P < .001) and lower MAE (0.09 vs 0.12, P < .001). Table 2 presents all results from in silico experiments conducted at an SNR of 20:1.

Table 2:

Comparative Analysis of Simulation Performance: True Biomarkers as Reference Standard versus Estimated Prostate Tissue Parameters Using NLLS and PIA Methods

Figure 2 shows the MAE of PIA and NLLS methods on the parameters (volume, diffusivity, T2) of the epithelium compartment as a function of test set SNR. In an ideal scenario with no or negligible noise levels, NLLS provides the best solution. However, in more realistic operating points of an SNR of 20:1 and worse, the NLLS solution quickly degrades. PIA, on the other hand, shows robustness to noise and keeps reliable estimates. Results of all robustness and sensitivity experiments are presented in Appendix S1.

Figure 2:

Change in mean absolute error (MAE) performance for the two methods, nonlinear least squares (NLLS) (red) versus Physics-Informed Autoencoder (PIA) (blue), on all tissue parameters (volume fraction [Vol.], diffusivity [D.], and T2) of the epithelium (Ep.) compartment, as a function of signal-to-noise ratio (SNR). As expected, NLLS yields accurate measurements under very high SNR levels. However, as noise increases to levels experienced in clinical applications of MRI, the solutions of NLLS quickly degrade (notice the log scale). PIA, however, presents robustness against noise and outperforms NLLS significantly under realistic operating SNR conditions as observed in clinical MRI scans (orange shaded region).

Figure 3 displays scatterplots of true versus predicted measurements for the epithelium compartment, contrasting PIA versus NLLS methods. Areas with a higher scatter point density are depicted with warmer colors to highlight the prediction performance. Scatterplots for the other compartments are provided in Appendix S1.

Figure 3:

Scatterplots compare true versus predicted parameters for the epithelium compartment using nonlinear least squares (NLLS) and Physics-Informed Autoencoder (PIA) methods. D = diffusivity, vol = volume fraction.

Histologic Validation

Figure 4 shows representative images in a patient from the cohort we used in histologic analysis and the accompanying PIA analysis. PIA’s prostate tissue composition measurements demonstrated excellent agreement with quantitative histology, achieving ICC values of 0.94 (epithelium), 0.85 (stroma), and 0.92 (lumen) (P < .001) across the three prostatic compartments. Figure 5 visualizes this performance with scatterplots that directly compare the prostatic tissue volume estimates from PIA against the reference standard provided by quantitative histology. PIA’s prostate tissue composition measurements also demonstrated excellent linear prediction performance with quantitative histology, achieving marginal R² values of 0.88 (epithelium), 0.78 (stroma), and 0.89 (lumen) across the three prostatic compartments.

Figure 4:

Representative images in a 62-year-old male patient with prostate cancer that exhibit two different pathologies on the same section (cancer and cystic atopy). Top row, from left to right: Apparent diffusion coefficient (ADC) map from the axial view (noncontrast), hematoxylin-eosin (H&E)–stained histology slice with ×20 magnification, and image from quantitative histology of the cancer region of interest. Colored overlays on MR images in the second, third, and fourth rows show the Physics-Informed Autoencoder estimates of the volume fraction, ADC, and T2 of the three compartments. Epithelium volume, epithelium ADC, and stroma ADC are great indicators for cancer. Epithelium volume highlights cancer whereas the lumen volume highlights the region with cystic atrophy on the left peripheral zone.

Figure 5:

Scatterplots of tissue composition volume measurements for epithelium, stroma, and lumen. X-axes show the reference standard histology volumes for each region of interest (ROI), and the y-axes show the noninvasive estimates obtained by PIA. Red stars are the cancer ROIs, and green circles are the benign ROIs. PIA = Physics-Informed Autoencoder, Vol = volume.

In addition to yielding more accurate estimates on in silico experiments, PIA outperformed NLLS on in vivo evaluations as well. Table 3 presents a comprehensive histologic validation of PIA, alongside a comparative analysis with the conventional NLLS-based solution across various metrics.

Table 3:

Histologic Validation of Prostate Tissue Volume Estimates in Clinical in Vivo Data Obtained Using NLLS and PIA

Diagnostic Utility and Interpretability of PIA’s Measurements

PIA’s in vivo volume fraction estimates of epithelium present a similar AUC (P = .33) to NLLS (0.99 vs 0.97), surpassing the AUC of conventional ADC of 0.90 (P < .02). More importantly, PIA’s measurements of compartment diffusivities yielded significantly better AUC values for their utility in CSPCa differentiation than the diffusivities of NLLS (0.89 vs 0.62, P < .001 and 0.86 vs 0.60, P < .001 for epithelium and stroma, respectively). The resulting P values of the t tests conducted to assess the statistical significance of differences in epithelium diffusivity measurements between Gleason score groups are presented in Table 4.

Table 4:

Statistical Significance of Differences in PIA-derived Epithelium Diffusivity Measurements between Gleason Score Groups

In silico experiments with various SNR levels demonstrated that PIA is robust in the presence of noise, especially for the exponential terms (ie, the diffusivity and T2 of tissue compartments), whereas the NLLS method struggles with these terms when the noise levels are high (see Figs 2 and 3). In vivo, PIA’s diffusivity estimates for epithelium and stroma demonstrated a strong inverse correlation with cancer aggressiveness, with correlation coefficients of r = −0.75 and r = −0.64, respectively. In contrast, the NLLS-based solution, as observed in in silico experiments, tended to converge to erroneous and less meaningful results for the exponential terms. Figure 6 includes box plots comparing epithelium measurements from both PIA and NLLS-based approaches.

Figure 6:

Box plots display PIA- and NLLS-derived epithelium diffusivity (D_ep) measurements of ROIs in the patient cohort, color coded with respect to each ROI’s cancer aggressiveness, measured with Gleason score. PIA’s measurements of D_ep (left) demonstrate a distinct inverse relationship with cancer aggressiveness, whereas the NLLS-based approach (right) does not. Each box denotes the middle 50% of the data from first quartile to third (IQR), the horizontal line in each box denotes the median, the whiskers denote data within 1.5 × IQR from Q1 and Q3, and outlier points beyond the whiskers were plotted individually. NLLS = nonlinear least squares, PIA = Physics-Informed Autoencoder, ROI = region of interest.

In the feature importance analysis, epithelial volume fraction was a better biomarker than the conventional ADC by a large margin for CSPCa detection. Feature importance and cancer aggressiveness correlation results for other new biomarkers are available in Appendix S1.

Practicality and Speed

On 20 000 virtual voxels, PIA provided a speed improvement of up to a factor of 10 000. NLLS took 2345.39 seconds (about 40 minutes) for 20 000 voxels (almost equivalent to the calculation for one MRI section), where PIA took only 0.18 second.

Discussion

In this article, we presented an emerging method, PIA, that integrates the strengths of physics-based and deep learning–based methods for tissue microstructure profiling using MRI to detect and stage PCa noninvasively. The proposed solution could measure the reference standard prostate tissue volumes as validated with histopathologic measurements (ICC, 0.94, 0.85, and 0.92 for epithelial, stromal, and luminal volume fraction, respectively; P < .001 for all), significantly outperforming NLLS in both in silico and in vivo experiments (r_s, 0.80 vs 0.65; P < .001 for epithelial volume fraction at an SNR of 20:1) and providing an accurate measurement of epithelial diffusivity as a new biomarker for measuring PCa aggressiveness (r = 0.75, P < .001).

We showed that PIA provides noise robustness for the diffusivity and T2 estimates for the tissue compartments in in silico experiments. For in vivo analysis, there is no straightforward way to validate measured diffusivities. Thus, we used correlations of PIA’s measurements with PCa aggressiveness to investigate the diagnostic utility of these measures, not to prove accuracy. Nevertheless, the results showed that measurements of the diffusivity and T2 of each tissue compartment with PIA have potential to produce more effective PCa detection models and improve diagnostic accuracy. We acknowledge that the diffusivity distributions overlap between Gleason scores 3 + 3 and 3 + 4, as well as between 3 + 4 and 4 + 3. This overlap is expected due to the shared Gleason patterns inherent in these scoring categories. Nevertheless, the significant differences observed between benign tissue and Gleason score 3 + 4 indicate that PIA biomarkers can enhance the CSPCa detection. These measurements are important because the existing literature indicates that denser epithelial cells, characterized by a higher nuclear to cytoplasm ratio, are associated with more aggressive cancers. Additionally, an increase in stiffer, diffusion-restricting fibroblasts is noted in the stroma of cancerous tissue (30). Hence, our findings with negative correlation of epithelium diffusivity and moderate positive correlation of epithelium T2 versus the Gleason score were reassuring and consistent with the literature (31,32).

The NLLS-based method had already reported excellent correlation with histology-based tissue volumes in Chatterjee et al (25,33). In addition to superior accuracy over the conventional least squares solution, the speed of the proposed model is critical in clinical contexts. The NLLS model is based on a constrained optimization algorithm, solved de novo for each new voxel. On the other hand, PIA is an analytical function modeled with neural networks, allowing orders of magnitude faster calculations. PIA’s efficiency allows seamless integration with picture archiving and communications systems, enabling real-time estimation of tissue parameters during radiologic examinations. In contrast, the NLLS-based system’s slower processing speed hinders its potential for real-time application.

This work had some limitations and constraints. First, the in vivo validation relied on histopathology as the reference standard, but the study did not account for potential confounding factors such as variations in tissue processing, effect of formalin fixation, interobserver variability among pathologists, or registration errors between MRI and histology. We also acknowledge that the current study focused on data from a single institution and MRI scanner, and the sample size of 21 patients was relatively small. However, it is important to note that our current study provides proof of concept for the PIA model, demonstrating its ability to perform microstructural analysis of prostate tissue, which is a rather uncommon task; therefore, it is susceptible to issues of small sample size and lack of multi-institutional validation. Furthermore, the collection of these unique data (radical prostatectomy specimens, quantitative histology measurements on prostatectomy slices, and pathology to radiology mapping) is very costly. Nevertheless, even with a limited sample size, the statistically significant results observed in this initial study are encouraging.

In conclusion, PIA provides several advantages over conventional methods. First, in contrast to conventional end-to-end deep learning strategies for PCa detection, PIA reduces dependency on extensive training data and learns from physical properties of the tissue structure and imaging. This framework provides a truly explainable artificial intelligence solution. Second, implementation of a denoising autoencoder-inspired training regimen ensures robustness against noise. Finally, the results of our study suggest that PIA can provide new information regarding tissue properties (eg, the diffusivity and T2 of each compartment), and this can potentially improve diagnostic accuracy. As a future direction, prospective studies evaluating the impact of PIA on clinical decision-making and patient management, as well as applying our PIA framework to other microstructural models beyond the HM-MRI context by testing on various models, will help establish the versatility and robustness of our approach.