Synopsis

In this study we propose a method to quantify the distribution of Super Paramagnetic Iron Oxide (SPIO) particles with MRI. This task is particularly challenging due to the extreme distortion that these particles produce in the image. Our method is based on a supervised feed-forward deep learning model. The estimation of total quantity of SPIO was in the order of 9% error. This is potentially useful for detecting breast cancer metastasis by identifying residual particles in the breast and eventually other organs.

Introduction

Methods

The main idea is to directly visualize the field distortion, and from there the SPIO distribution. To measure the field map produced by the SPIO particles we acquire 25 slices perpendicular to the excitation plane to create an image of two channels where the odd numbered slices go into the first channel and the even numbered go into second channel. This is a descriptor of the distortion produced by the particles. We know that In MRI, the local magnetic field is the convolution of the dipole kernel with the susceptibility, which is proportional to the distribution of SPIO³ $$$C(\mathbf{r})$$$. This is

$$\Delta B_z (\mathbf{r})\propto C(\mathbf{r})*D(\mathbf{r}), \quad\text{with}\quad D(r, \theta) = \frac{3\cos^2(\theta)-1}{4\pi r^3} .$$

We use this to simulate the distortion produced by a given distribution of particles. With this we created the required amount of data to train a deep learning model. We use U-Net⁴, inspired in the success of DeepQSM⁵ and QSMnet⁶, as the base model. We extend the bottleneck of the neural network (central section) using residual blocks⁷ to increase the convergence rate of the model and to remove the depth concatenation operations. We call our method DeepSPIO. To train the network we used a modified MSE loss function to account for the unnormalized training set and give every sample the same weight.

$$L = \frac{1}{2N}\sum_{i=1}^N \Bigg | \Bigg | P \odot \Bigg(\frac{\alpha_i}{S_i} - \frac{\hat{\alpha_i}(w)}{S_i}\Bigg)\Bigg | \Bigg |_F^2 + \lambda \sum_{l=1}^L||w_l||_2^2.$$

Where

$$S_i = ||C_i||_1\quad\text{and}\quad C_i = \Psi^{-1}\alpha_i.$$

$$$\alpha_i$$$ corresponds to the wavelet transform (Daubechies 4 of 5 levels) of the sample $$$i$$$ of SPIO distribution. $$$P$$$ is a preconditioner. $$$\Psi$$$ corresponds to the wavelet transform. $$$L$$$ is the number of layers. $$$N$$$ is the number of training samples. Figure 1 shows a summary of the problem.

The method is robust enough such that other anatomies can be used as training. Therefore, we simulated SPIO distortions with brain and water proton density images, but other contrasts can also be used. We generated a uniform concentration of SPIO in two different shapes: metaballs and toruses. We sampled exponentially between 0.2734 and 17.5 $$$\frac{\mu\text{g}}{\text{mm}^3}$$$ with 400,000 samples. We generated a SPIO distribution mixing the geometric figures with overlapping (in case of overlap the concentrations are added) with random positions, rotations and sizes. The output is a wavelet transform of the middle slice ($$$256 \times 256$$$ coefficients). We generated 2 million samples in this way.

We used the following metrics to measure the quality of the estimation:

$$\text{NMAE}:= \frac{||C - \hat{C}||_1}{||C||_1} \quad\text{and}\quad \text{Integral Error}:= \frac{| \text{ } ||C||_1 - ||\hat{C}||_1 \text{ }|}{||C||_1}.$$

We also used the dice’s similarity coefficient with binarized $$$C$$$ and $$$\hat{C}$$$:

$$\text{Dice}:= \frac{2|B_C \cap B_{\hat{C}}|}{|B_C| + |B_{\hat{C}}|}.$$

Results

We constructed a test dataset comprising only of unseen simulations from brain images, with a linear distribution of SPIO particles concentration in the same range as the training set. And we obtained the following result:

• NMAE: $$$26.02\% \pm 9.82\%.$$$
• Integral Error: $$$8.83\% \pm 8.51\%.$$$
• Dice: $$$91.28\% \pm 4.82\%.$$$
  

Furthermore, to show the ability of the algorithm to perform in images with different intensities, textures and shapes we quantified the SPIO distribution in simulated breast and brain images (Figures 3 and 4). We also scanned an agar phantom with SPIO particles in it to test the method as is seen in Figure 5.

Discussion and Conclusion

Acknowledgements

References

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