Med-DDPM: Diffusion Models in Medical Imaging

• Med-DDPM is a framework that adapts denoising diffusion probabilistic models for synthesizing, denoising, and reconstructing clinically relevant images.
• It leverages conditional generation and hybrid data-consistency techniques with PET and MR inputs to enhance anatomical fidelity and reduce noise.
• Empirical evaluations show improved PSNR and SSIM metrics over traditional methods, with robust uncertainty quantification for clinical reliability.

Med-DDPM designates a class of Denoising Diffusion Probabilistic Model (DDPM) frameworks and variants adapted for medical imaging and related biomedical generation and analysis tasks. These models apply the iterative, probabilistic denoising and distribution-matching principles of DDPMs to synthesize, denoise, or reconstruct clinically relevant data, offering flexibility in leveraging anatomical priors and handling complex noise characteristics.

1. Principles of DDPM in Medical Imaging

DDPMs learn mappings from a simple distribution (e.g., Gaussian noise) to a desired data distribution (such as medical images) through a two-step Markov process. The forward process incrementally corrupts a clean image $x_0$ with Gaussian noise:

$$q(x_{1:T} | x_0) = \prod_{t=1}^T q(x_t | x_{t-1}), \quad q(x_t | x_{t-1}) = \mathcal{N}(x_t; \sqrt{1-\beta_t} x_{t-1}, \beta_t I)$$

At each time step $t$, $$q(x_t|x_0) = \mathcal{N}(x_t; \sqrt{\overline{\alpha}_t} x_0, (1-\overline{\alpha}_t)I)$$ with $$\overline{\alpha}_t = \prod_{l=1}^t \alpha_l,~\alpha_l=1-\beta_l$$.

The reverse (generation) process is learned by a neural network that predicts the denoising direction:

$$x_{t-1} = \frac{1}{\sqrt{\alpha_t}}[x_t - \frac{\beta_t}{\sqrt{1-\overline{\alpha}_t}} \epsilon_\theta(x_t, t)] + \sigma_t z$$

where $\epsilon_\theta$ predicts the noise and $z \sim \mathcal{N}(0, I)$.

In medical imaging, this machinery is used to remove statistical noise (e.g., low-dose PET) or to sample from distributions of plausible high-quality images that match clinical data (Gong et al., 2022).

2. Model Variants and Methodologies

Med-DDPM frameworks are configured to exploit domain knowledge and data modality priors in various ways:

• Conditional Generation: Inputs may include the noisy medical scan and/or auxiliary anatomical priors (such as MRI alongside PET). The noise predictor network thus takes a multi-channel input, e.g., $$\epsilon_\theta(x_t, t, x_\text{noisy}, x_\text{prior})$$.
  • Example: For PET denoising, PET and MR images are provided together to enable better anatomical fidelity (Gong et al., 2022).
• Hybrid Conditioning with Data Consistency: The network is trained with only the prior image; during inference, the noisy measurement itself is enforced as a data-consistency constraint in the iterative reverse process. This is operationalized via an adjusted denoising update:

$$x_{t-1} = \frac{1}{\sqrt{\alpha_t}} [ x_t - \frac{\beta_t}{\sqrt{1-\overline{\alpha}_t}} \epsilon_\theta(x_t, t, x_\text{prior}) ] - \frac{\sigma_t^2}{\sigma_d^2}(x_\text{noisy} - x_t) + \sigma_t z$$

Four core strategies emerge:

Method	Network Input	PET Role in Inference
DDPM-PET	PET only	as input
DDPM-MR	MR only	as input
DDPM-PETMR	PET + MR	both as input
DDPM-MR-PETCon	MR only	PET as data-consistency

This design enables operation across varying noise levels and anatomical contexts.

3. Empirical Evaluation and Performance

Performance is assessed primarily via global and regional quantitative metrics including PSNR (Peak Signal-to-Noise Ratio) and SSIM (Structural Similarity). The paper reports experiments on 120 brain $^{18}$F-FDG and 140 brain $^{18}$F-MK-6240 PET datasets, with MR priors for a subset (Gong et al., 2022).

Key findings:

• Global Performance: Approaches incorporating PET data (DDPM-PET, DDPM-PETMR) achieve superior metrics compared to both nonlocal mean (NLM) filtering and Unet-based methods.
• Impact of MR Priors: Adding MR prior as an input reduces noise and uncertainty in reconstructed images. However, use of MR alone (DDPM-MR) can result in bias where PET-specific intensity matters.
• Hybrid Data Consistency: Employing DDPM-MR-PETCon, which combines MR priors with PET-driven data consistency during inference, yields the best local quantification and lowest uncertainty—especially reflected in region-level evaluation.

Uncertainty quantification is facilitated by running multiple reverse processes, illustrating reduced uncertainty with MR priors and further constriction with PET constraints.

4. Architecture Design and Implementation Considerations

The denoising network is typically a UNet-variant, adapted for the input dimensionality and channel configuration dictated by the chosen conditioning paradigm. Inputs are multi-modality images (e.g., PET, MR), and data-consistency constraints are realized via explicit terms in the denoising formula, imposing fidelity to observed degradation.

The approach requires paired datasets for supervised training (e.g., low-dose/high-dose PET, MR), and is amenable to extension for 3D volumetric architectures. The model is agnostic to precise noise levels in certain configurations (notably DDPM-MR-PETCon).

Computational demands are nontrivial but manageable, benefiting from parallelism in UNet architectures and the possibility of optimization in the reverse process (e.g., fewer reverse steps or acceleration schemes).

5. Advantages, Limitations, and Uncertainty Quantification

Advantages:

• Flexible Utilization of Priors: Both PET and MR data can be leveraged to maximize image fidelity and anatomical coherence.
• Noise-Level Agnosticism: The hybrid approaches enable robustness across variable acquisition quality.
• Uncertainty Estimation: The stochastics of the reverse process support the generation of uncertainty maps, valuable for downstream clinical interpretability and risk assessment.

Limitations and caveats:

• Reliance solely on MR priors may induce anatomical bias in regions where PET intensity is uniquely informative; MR as a sole prior is discouraged for intensity-sensitive applications.
• Inference time and scalability to full 3D or higher-resolution volumes remain open for further optimization.

6. Future Directions and Clinical Relevance

Planned advancements include development of genuine 3D architectures and reduction of inference latency. The generality of the framework suggests adaptability for whole-body reconstruction and integration into existing PET image reconstruction pipelines.

Med-DDPM forms a flexible tool for denoising and uncertainty-aware synthesis in medical imaging, demonstrating robust improvement over classical and CNN-based methods, particularly when exploiting multi-modal priors and data-consistency constraints. The methodology directly addresses clinical needs for image quality enhancement under dose and acquisition constraints, with implications for diagnostic reliability and workflow efficiency (Gong et al., 2022).