TorchDecon implements a deep learning-based approach for cell type deconvolution, originally proposed by Menden et al. (2020) in their Scaden algorithm. This vignette provides a comprehensive overview of the mathematical framework and algorithmic principles underlying TorchDecon.
Author: Zaoqu Liu ([email protected])
Cell type deconvolution aims to estimate the cellular composition of bulk tissue samples. Given a bulk expression profile \(\mathbf{x} \in \mathbb{R}^G\) (where \(G\) is the number of genes), we seek to estimate the cell type fraction vector \(\mathbf{f} \in \mathbb{R}^K\) (where \(K\) is the number of cell types) such that:
\[\sum_{k=1}^{K} f_k = 1, \quad f_k \geq 0 \quad \forall k\]
Traditional deconvolution methods (e.g., CIBERSORT, MuSiC) rely on:
\[\mathbf{x} = \mathbf{S} \cdot \mathbf{f} + \boldsymbol{\epsilon}\]
where \(\mathbf{S} \in \mathbb{R}^{G \times K}\) is the signature matrix.
TorchDecon’s deep learning approach instead learns a non-linear mapping directly from bulk expression to cell type fractions:
\[\mathbf{f} = \mathcal{F}_\theta(\mathbf{x})\]
where \(\mathcal{F}_\theta\) is a neural network with parameters \(\theta\).
┌─────────────────────────────────────────────────────────────────┐
│ TorchDecon Pipeline │
├─────────────────────────────────────────────────────────────────┤
│ │
│ ┌──────────────┐ ┌──────────────┐ ┌──────────────┐ │
│ │ scRNA-seq │───▶│ Bulk │───▶│ Training │ │
│ │ Reference │ │ Simulation │ │ Data │ │
│ └──────────────┘ └──────────────┘ └──────────────┘ │
│ │ │
│ ▼ │
│ ┌──────────────┐ ┌──────────────┐ ┌──────────────┐ │
│ │ Cell Type │◀───│ Trained │◀───│ Training │ │
│ │ Fractions │ │ Ensemble │ │ Loop │ │
│ └──────────────┘ └──────────────┘ └──────────────┘ │
│ ▲ │
│ │ │
│ ┌──────────────┐ │
│ │ Real Bulk │ │
│ │ Data │ │
│ └──────────────┘ │
│ │
└─────────────────────────────────────────────────────────────────┘For each simulated sample \(s\):
Generate random fractions: Sample \(\mathbf{f}^{(s)} \sim \text{Dirichlet}(\boldsymbol{\alpha})\) or uniform distribution, then normalize:
\[f_k^{(s)} = \frac{u_k}{\sum_{j=1}^K u_j}, \quad u_k \sim \text{Uniform}(0, 1)\]
Sample cells: For each cell type \(k\), sample \(n_k = \lfloor f_k^{(s)} \cdot N \rfloor\) cells (with replacement), where \(N\) is the total cells per sample.
Aggregate expression: Sum the expression across sampled cells:
\[x_g^{(s)} = \sum_{k=1}^{K} \sum_{i \in C_k^{(s)}} c_{gi}\]
where \(C_k^{(s)}\) is the set of sampled cells from type \(k\), and \(c_{gi}\) is the count for gene \(g\) in cell \(i\).
To improve model generalization, TorchDecon generates “sparse” samples where some cell types are absent:
\[f_k^{(s)} = \begin{cases} \tilde{f}_k / \sum_{j \in \mathcal{A}} \tilde{f}_j & \text{if } k \in \mathcal{A} \\ 0 & \text{otherwise} \end{cases}\]
where \(\mathcal{A} \subset \{1, ..., K\}\) is a randomly selected subset of cell types.
\[\tilde{x}_g = \log_2(x_g + 1)\]
This transformation: - Reduces the dynamic range of expression values - Stabilizes variance - Makes the data more normally distributed
For each sample \(s\):
\[\hat{x}_g^{(s)} = \frac{\tilde{x}_g^{(s)} - \min_j(\tilde{x}_j^{(s)})}{\max_j(\tilde{x}_j^{(s)}) - \min_j(\tilde{x}_j^{(s)})}\]
This ensures: - All features are in \([0, 1]\) - Sample-specific technical variations are mitigated - Neural network training stability is improved
Genes are filtered based on variance across samples:
\[\text{Var}(x_g) = \frac{1}{n-1} \sum_{s=1}^{n} (x_g^{(s)} - \bar{x}_g)^2 > \tau\]
where \(\tau\) is the variance threshold (default: 0.1).
Each model in the ensemble is a fully connected neural network:
\[\mathbf{h}^{(l)} = \sigma(\mathbf{W}^{(l)} \mathbf{h}^{(l-1)} + \mathbf{b}^{(l)})\]
where: - \(\mathbf{h}^{(l)}\) is the hidden representation at layer \(l\) - \(\mathbf{W}^{(l)}, \mathbf{b}^{(l)}\) are learnable weights and biases - \(\sigma(\cdot)\) is the ReLU activation: \(\sigma(z) = \max(0, z)\)
The final layer uses softmax to ensure valid probability distribution:
\[f_k = \frac{\exp(z_k)}{\sum_{j=1}^K \exp(z_j)}\]
This guarantees: - \(f_k \in (0, 1)\) for all \(k\) - \(\sum_k f_k = 1\)
During training, dropout randomly zeroes elements:
\[\tilde{h}_i = \begin{cases} h_i / (1-p) & \text{with probability } 1-p \\ 0 & \text{with probability } p \end{cases}\]
Mean Squared Error (MSE) between predicted and true fractions:
\[\mathcal{L}(\theta) = \frac{1}{n \cdot K} \sum_{s=1}^{n} \sum_{k=1}^{K} (\hat{f}_k^{(s)} - f_k^{(s)})^2\]
Parameter updates using Adam:
\[\begin{aligned} m_t &= \beta_1 m_{t-1} + (1 - \beta_1) g_t \\ v_t &= \beta_2 v_{t-1} + (1 - \beta_2) g_t^2 \\ \hat{m}_t &= m_t / (1 - \beta_1^t) \\ \hat{v}_t &= v_t / (1 - \beta_2^t) \\ \theta_{t+1} &= \theta_t - \eta \cdot \hat{m}_t / (\sqrt{\hat{v}_t} + \epsilon) \end{aligned}\]
Default hyperparameters: \(\beta_1 = 0.9\), \(\beta_2 = 0.999\), \(\eta = 10^{-4}\)
Final prediction is the arithmetic mean across three models:
\[\hat{\mathbf{f}} = \frac{1}{3} \sum_{m \in \{256, 512, 1024\}} \mathcal{F}_{\theta_m}(\mathbf{x})\]
Ensemble benefits: - Reduced variance in predictions - Improved robustness to initialization - Better generalization
| Model | Layer Dimensions | Dropout Rates | Total Parameters* |
|---|---|---|---|
| M256 | G → 256 → 128 → 64 → 32 → K | 0, 0, 0, 0 | ~G×256 + 50K |
| M512 | G → 512 → 256 → 128 → 64 → K | 0, 0.3, 0.2, 0.1 | ~G×512 + 200K |
| M1024 | G → 1024 → 512 → 256 → 128 → K | 0, 0.6, 0.3, 0.1 | ~G×1024 + 800K |
*Approximate; depends on number of genes (G) and cell types (K)
Deep neural networks are universal function approximators (Hornik, 1991). Given sufficient capacity, TorchDecon can theoretically approximate any continuous mapping from expression space to fraction space.
Menden, K., et al. (2020). Deep learning-based cell composition analysis from tissue expression profiles. Science Advances, 6(30), eaba2619.
Hornik, K. (1991). Approximation capabilities of multilayer feedforward networks. Neural Networks, 4(2), 251-257.
Kingma, D. P., & Ba, J. (2015). Adam: A method for stochastic optimization. ICLR.
Package Author: Zaoqu Liu
Contact: [email protected]
GitHub: https://github.com/Zaoqu-Liu/TorchDecon