Introduction

In quantitative MRI, tissue properties are estimated from MRI data using bio-physical models that relate the MRI signal to the underlying tissue properties via model parameters. Conventionally, such parameter estimation is done using least squares fitting to images with different contrast weightings. However, least squares fitting is strongly affected by imaging noise, either overfitting to the noise, or ending up in local minima. Moreover, parameters with low sensitivity are estimated with poor accuracy. Deep learning for estimating quantitative MRI parameter maps has the potential to be more robust to noise and thus improve the accuracy^1-4. These conventional deep learning implementations are not robust to acquisition setting changes, as they are dependent on the input being either a fixed set of input signals (fully connected networks) or a series of regularly sampled signals (recurrent neural networks). Our group was the first to overcome this by using Neural Controlled Differential Equations^5,6 for intravoxel incoherent motion (IVIM) MRI⁷. Here, we implement NCDEs as a generic tool for quantitative MRI parameter estimation. We show its performance on simulated data for IVIM MRI, extended Tofts-Kety dynamic contrast enhanced (DCE) MRI and variable flip angle (VFA) T1-mapping. We further show its performance on in-vivo data for IVIM MRI.

Methods

1. Quantitative MRI
Quantitative MRI models allow us to assess tissue properties by using biophysical models to describe the MRI signal intensity as a function of a changing variable, e.g. diffusion weighting (b) in IVIM, flip angle (FA) in VFA and time (t) in DCE.
In IVIM MRI, the signal intensity at diffusion weighting $$$b$$$ depends on the diffusion coefficient ($$$D$$$), pseudo diffusion coefficient ($$$D^{*}$$$), perfusion fraction ($$$f$$$) and baseline signal intensity ($$$S0$$$):
$$S(b)=S0((1-f)e^{-bD}+fe^{-bD*})\hspace{10mm}(1)$$
In VFA T1 mapping, the signal intensity at flip angle $$$FA$$$ depends on the longitudinal relaxation time ($$$T1$$$) and signal at maximal longitudinal magnetization ($$$S0$$$):
$$S(FA)=S_{0}\frac{1-e^{\frac{-TR}{T1}}}{1-cos(FA)e^{\frac{-TR}{T1}}}sin(FA)\hspace{10mm}(2)$$
In DCE MRI, the concentration of gadolinium in tissue ($$$C_{t}$$$), at timepoint $$$t$$$ depends on the gadolinium reflux rate ($$$k_{e}$$$), fractional volume of extravascular extracellular space ($$$v_{e}$$$), fractional volume of plasma ($$$v_{p}$$$) and bolus arrival time $$$d\tau$$$:
$$C_{t}(t)=v_{p}C_{p}(t)+k_{e}v_{e}\int{}C_{p}(\tau)e^{-k_{e}(t-\tau)}d\tau\hspace{10mm}(3)$$
2. NCDE model
We trained an NCDE (figure 1) to map the input sequence to an output vector $$$y$$$ [6]. $$$y$$$ contains the coefficients to compute parameters, i.e. for $$$D$$$:
$$D=D_{min}+sigmoid(y[1])(D_{max}-D_{min})\hspace{10mm}(4)$$
Based on the predicted model parameters together with the set of measured values on the x-axis, a signal decay curve $$$S_{pred}$$$ is predicted according to (1-3).
For each quantitative MRI model, a dedicated NCDE is trained by minimizing a physics-informed loss², comparing the input signal ($$$S_{input}$$$) intensities to ($$$S_{pred}$$$), following:
$$L=\frac{1}{len(S_{input})}\Sigma(S_{input}-S_{pred})^{2}\hspace{10mm}(5)$$

3. Simulations
Per model training, 1,000,000 signal decay curves were simulated based on the respective IVIM, VFA and DCE model as in (1-3) from uniformly sampled combinations of parameters, the ranges of parameter values are described in table 1. Rician noise was added to the generated signal decay curves to make the signal-to-noise ratio (SNR) uniformly distributed between 5 and 50. The signal decay curves were used to train the NCDE model using the loss in (5).
Similarly, 100,000 signal decay curves were generated for evaluation of the NCDE model and non-linear least squares fitting. Performance was measured using parameter-specific normalized root mean square error (nRMSE) as a function of SNR.

4. In-vivo data
In-vivo IVIM data (Table 2) from 7 healthy volunteers are used for training of the NCDE and evaluation of the NCDE and least squares fitting. In 10 healthy volunteers, the NCDE estimation was visually compared to least squares.

Results

1. Simulations
For high-sensitivity parameters (e.g. $$$D,f,k_{e},v_{e}$$$), the NCDE could infer QMRI parameters from variable length signal decay curves with similar accuracy as least squares minimization. For low-sensitivity parameters (e.g. $$$D^{*},d\tau,v_{p}$$$) and when SNR is low, the NCDEs improved over least squares minimization (Figure 2).

2. In-vivo data
For high-sensitivity parameters (e.g. $$$D$$$), the NCDE estimates parameter maps which are very similar to the LSQ. For low-sensitivity parameters, the NCDE estimates less polarized parameter maps than least squares estimation (figure 3).

Discussion

We are the first to demonstrate the applicability of NCDEs as a generic tool for quantitative MRI parameter estimation by showing that NCDEs have comparable or improved performance to least squares minimization. NCDEs can improve parameter estimation under conditions of higher uncertainty (e.g. low SNR, low-sensitivity parameters).

Conclusion

NCDEs can be used as a generic tool for quantitative MRI parameter estimation. NCDEs outperform least squares fitting when SNR is low and in low-sensitivity parameters.