Tensor-DTI: Deep Learning for DTI and Drug Discovery

• Tensor-DTI is a term denoting methods that use tensor structures to model high-dimensional data in both neuroimaging and drug-target prediction.
• In diffusion MRI, tensor-aware architectures like TW-BAG restore missing DTI slices while preserving anisotropy, achieving up to 78% coefficient error reduction and superior PSNR.
• For drug discovery, contrastive multimodal dual-encoders learn joint molecular-protein representations, outperforming traditional models in accuracy and enrichment metrics.

Tensor-DTI is a contemporary term denoting two major research trajectories: advanced analysis and representation modeling for Diffusion Tensor Imaging (DTI), and a multimodal, deep contrastive architecture for Drug-Target Interaction (DTI) prediction. The unifying concept is the use of tensor structures and tensor-aware deep networks to process, reconstruct, and analyze high-dimensional data—be it 3D neuroimaging or biomolecular interactions—where preserving, inferring, or contrasting tensorial structure is integral to model performance and interpretability. Below, both lines are surveyed with emphasis on model architectures, methodological advances, quantitative evaluation, and implications for clinical and computational biomedicine.

1. Diffusion Tensor Imaging: Tensor Structures and Scalar Metrics

In diffusion MRI, each voxel is associated with a symmetric, positive-definite $3\times3$ diffusion tensor $D(x)$, parameterized by six unique entries: $D_{xx}$, $D_{yy}$, $D_{zz}$, $D_{xy}$, $D_{xz}$, $D_{yz}$, with eigenvalues $\lambda_1 \geq \lambda_2 \geq \lambda_3 > 0$ defining the principal diffusivities. Central derived metrics include:

• Axial Diffusivity (AD): $AD = \lambda_1$
• Mean Diffusivity (MD): $MD = (\lambda_1 + \lambda_2 + \lambda_3)/3$
• Fractional Anisotropy (FA): $$FA = \sqrt{ \frac{3\sum_{k=1}^3 (\lambda_k - \overline{\lambda})^2}{2\sum_{k=1}^3 \lambda_k^2} }$$, where $\overline{\lambda} = MD$

These scalar summaries drive quantification of white matter microstructure, but lose the full tensorial context necessary for tractography and rich multivariate analysis. Preserving and reconstructing the tensor field—in the presence of data loss, noise, or under-sampling—requires models that both respect the positive-definite constraint and enable recovery of scalar and orientation-specific information (Tang et al., 2022).

2. Tensor-DTI for Inpainting and Reconstruction in Neuroimaging

The TW-BAG (Tensor-wise Brain-aware Gate) network addresses the problem of DTI inpainting when DWI slices are missing due to suboptimal clinical acquisition. The architecture is defined as follows (Tang et al., 2022):

• Brain-aware Gate (BAG) Encoder: Four 3D convolutional encoder blocks (channels: 32, 64, 128, 256), each with a dynamic gating branch (produces BAG mask) and a feature branch (LeakyReLU and sigmoid activations), followed by a 512-channel bottleneck.
• BAG Decoder: Symmetric to the encoder, with upsampling and identical gating.
• Tensor-wise Decoders: Six parallel decoders corresponding to each unique tensor component ($D^{c}$, $c\in\{xx,xy,xz,yy,yz,zz\}$), each containing interpolation and two BAG convolutional layers. This design enforces specialization and decouples scaling across tensor components.

The training loss is a masked voxelwise $\ell_1$ reconstruction error within the brain, using ground-truth tensor coefficients. The Human Connectome Project dataset is used for quantitative benchmarking via MAE and PSNR on tensor volumes and FA.

Performance of TW-BAG is exemplified in the table below:

Baseline	MAE (coeff)	PSNR (dB)	FA MAE (in region)	FA MAE (whole)
Cropped	0.0041±0.0004	51.83	0.1888±0.0212	0.0105±0.0025
TW-BAG	0.0009±0.0001	62.41	0.0327±0.0192	0.0018±0.0012

TW-BAG reduces coefficient error by ~78% and FA error by ~83% in disrupted regions. Thus, it achieves whole-brain FA error substantially below clinical group-difference thresholds (e.g., for multiple sclerosis vs. control, $\Delta$FA$\sim$0.03), enabling robust retention of subjects for clinical studies (Tang et al., 2022).

3. Geometric and Statistical Modeling for Tensor DTI Processing

Anisotropy-preserving Metrics: Standard Riemannian distances (affine-invariant, Log-Euclidean) systematically reduce anisotropy during interpolation and averaging, which can undermine microstructural inferences (Collard et al., 2012). The spectral-quaternion metric addresses this by decomposing each tensor as $S=U \Lambda U^T$ and defining a distance

$$d_{SD}^2(S_1, S_2) = k(\Lambda_1, \Lambda_2)\, d_{SO(3)}^2(U_1, U_2) + d_{D^+}^2(\Lambda_1, \Lambda_2)$$

where $k$ is an anisotropy-dependent scalar, $d_{SO(3)}$ is the geodesic distance in rotation space (implemented via quaternions), and $d_{D^+}$ is the log-Euclidean distance on eigenvalues. This construction allows interpolation curves and means that strictly preserve (commute with) the Hilbert anisotropy index $HA = \log(\lambda_1/\lambda_3)$, ensuring biological interpretability during statistical analysis (Collard et al., 2012).

Spatial Bayesian Inference: The spatial Bayesian semiparametric mixture model extends inference beyond aggregate FA/MD, modeling every voxel’s tensor as a mixture of inverse Wishart distributions with neighborhood-correlated labels assigned by a Potts Markov random field. Posterior estimates combine measurement uncertainty with anatomical continuity, increasing sensitivity and specificity for group difference analysis—for example, delineating cocaine-induced FA reductions in corpus callosum splenium (lan et al., 2019).

4. Deep Learning and Data-Driven Optimization in Tensor DTI

Inpainting and Estimation: Recent tensor-DTI frameworks have implemented deep denoising optimization routines, such as DoDTI, which unrolls weighted linear least squares (WLLS) fitting for tensor parameter estimation and incorporates a CNN denoiser acting directly on the tensor map rather than the raw DWI images (Li et al., 2024). The optimization is solved by unrolled ADMM with fixed or learnable parameters, yielding superior performance in NRMSE and SSIM of FA/MD/AD/RD over traditional model-based estimators, even under high noise, small numbers of gradient directions, and varied acquisition schemas.

Transformer-based Estimation: Transformer-based DTI estimation, given only six DWI directions, leverages both local DWI intensity and spatial context via stacked transformer blocks. Two stages (signal-only, then signal+tensor refinement) yield errors on FA, MD, and orientation angle comparable to reference solutions based on 30–88 DWIs, enabling drastically reduced scan times for neonates and infants (Karimi et al., 2022).

Method	Tensor MAE	MD MAE	FA MAE	Angle Error
CWLLS	0.450±0.040	0.413±0.042	0.155±0.017	20.2°±2.45°
Transformer	0.118±0.019	0.042±0.029	0.071±0.003	11.6°±2.07°

5. Tensor-DTI in Drug-Target Interaction Prediction

Contrastive Multimodal Dual-Encoder: In the context of computational drug discovery, Tensor-DTI denotes a dual-encoder architecture that jointly learns molecular and protein latent spaces via contrastive separation. Its salient components are (Gil-Sorribes et al., 9 Jan 2026):

• Drug Encoder: Graph Convolutional Network (pretrained), outputs $h_G$.
• Protein Encoder: Transformer-based LM (SaProt or ESM-2), outputs $h_P$; additionally a pocket encoder (GearNet) based on predicted binding site graphs.
• Projection/Classifier Heads: Project embeddings to shared latent space ($D=256$), use concatenated embeddings for classification/regression.
• Training Objective: Combined binary cross-entropy for DTI, mean-squared error for DTA, and a contrastive margin loss:

$$L_\text{contrast} = \sum_{(d,p)} \max\left(0, \alpha + \|z_d - z_p\|_2 - \|z_d - z_{p^-}\|_2\right)$$

with $\alpha=1.0$, negatives $p^-$ randomly drawn in-batch.

Performance on benchmark datasets demonstrates superiority or competitiveness over ConPLex, MolTrans, and EnzPred-CPI in AUPR, both in-distribution and under stringent drug/target family holdout (e.g., Unseen Targets: 0.839±0.003) (Gil-Sorribes et al., 9 Jan 2026).

6. Evaluation, Applications, and Interpretability

Neuroimaging Inpainting: Tensor-DTI approaches, particularly TW-BAG and related architectures, restore DTI volumes with minimal residual error, correcting FA/MD/AD to within subgroup difference levels—crucial for clinical trial inclusion and accurate white matter quantification.

Representation Learning: Tensor-DTI also encompasses interpretable latent representations for tractography, for example, arranging tract-level FA values in a 2D spatial grid and using a $\beta$-TC-VAE with spatial broadcast decoder. This approach achieves classification F1 improvements of $\sim$15.7% over baseline 1D DNNs and higher Mutual Information Gap for disentanglement (Singh et al., 25 May 2025).

Drug Discovery and Screening: In large-scale chemical screening, Tensor-DTI achieves high enrichment and reliability for kinase and non-kinase families, remaining robust for targets withheld from training and efficient in computational resources. The introduction of auxiliary confidence and unfamiliarity models further refines prediction reliability and out-of-distribution awareness (Gil-Sorribes et al., 9 Jan 2026).

7. Future Directions and Open Issues

• Extension of spatial Bayesian and geometric frameworks to non-Gaussian diffusion, higher-order tensor/compartment models, and general positive semi-definite fields.
• Integration of tensorial uncertainty estimates into deep learning pipelines for both structural neuroimaging and DTI-based drug discovery.
• Further systematic comparison between explicit tensor-space regularization (DoDTI) and implicit deep representation learners (transformer/CNN/contrastive architectures).
• Broader adoption and clinical translation contingent on detailed reporting of acquisition heterogeneity, synthetic-to-real generalization, and open-source dissemination.

References:

• (Tang et al., 2022): Introduction of TW-BAG for DTI inpainting with detailed quantification.
• (Collard et al., 2012): Anisotropy-preserving Riemannian metric for DTI.
• (lan et al., 2019): Spatial Bayesian mixture modeling of DTI tensors.
• (Li et al., 2024): Optimization-based denoising and estimation with deep ADMM for DTI.
• (Singh et al., 25 May 2025): Interpretable β-TC-VAE latent representations of tract-level FA.
• (Karimi et al., 2022): Transformer-based DTI estimation from minimal DWIs.
• (Gil-Sorribes et al., 9 Jan 2026): Multimodal contrastive dual-encoder for drug-target and macromolecule interaction prediction.