Modelling Non-stationary and Non-separable Spatio-Temporal Changes in Neurodegeneration via Gaussian Process Convolution

Abstract

Modelling longitudinal changes in organs is fundamental for the understanding of biological and pathological processes. Most of the previous works on spatio-temporal modelling of image time series relies on the assumption of stationarity of the local spatial correlation, and on the separability between spatial and temporal processes. These assumptions are often made in order to lead to computationally tractable approaches to longitudinal modelling, but inevitably lead to an oversimplification of the complex spatial and temporal dynamics underlying the biological processes. In this work we propose a novel spatio-temporal generative model of time series of images based on kernel convolutions of a white noise Gaussian process. The proposed model is parameterised by a sparse set of control points independently identified by specific spatial and temporal parameters. This formulation is highly flexible and can naturally account for spatially and temporally varying dynamics of changes. We demonstrate a preliminary application of our non-parametric method on the modelling of within-subject structural changes in the context of longitudinal analysis in Alzheimer’s disease. In particular we show that our method provides an accurate description of the pathological evolution of the brain, while showing high flexibility in modelling and predicting region-specific non-linearity due to accelerated structural decline in dementia.

S. Ourselin—Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (www.loni.ucla.edu/ADNI). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: www.loni.ucla.edu/ADNI/Collaboration/ADNI_Authorship_list.pdf.

Appendix

We explicit here the derivatives of the log-marginal likelihood (5) with respect to the model parameters $\varvec{\theta }$:

$$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}\varvec{\theta }}\log \mathcal {L} =&-\frac{1}{2} Tr\left( \varSigma (\varvec{\theta })^{-1} \frac{\mathrm {d} \varSigma (\varvec{\theta })}{\mathrm {d}\varvec{\theta }} \right) - y^T \varSigma (\varvec{\theta })^{-1} \frac{\mathrm {d} \varSigma (\varvec{\theta })}{\mathrm {d}\varvec{\theta }} \varSigma (\varvec{\theta })^{-1} y. \end{aligned}$$

(14)

With the following simplification

$$\begin{aligned} A = \left( \frac{1}{\sigma _x^2} Id_{N_w} + \frac{1}{\sigma _{\epsilon }^2}K^TK \right) ^{-1}. \end{aligned}$$

the derivative are as follows:

• Noise parameter.

  $$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}\sigma _{\epsilon }^2}\log \mathcal {L}&= \frac{1}{2} [- N + \frac{1}{\sigma _{\epsilon }^2} Tr(K^T K A) \end{aligned}$$

  (15)
  $$\begin{aligned}&\quad \,+ y^T \left( \frac{1}{\sigma _{\epsilon }^2} Id_{N} - \frac{2}{\sigma _{\epsilon }^4} K A K^T + \frac{1}{\sigma _{\epsilon }^6} K A K^T K A K^T \right) y ]. \end{aligned}$$

  (16)

• Amplitude parameter.

  $$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}\sigma _x^2}\log \mathcal {L}&= -\frac{1}{2} Tr(\frac{\sigma _{\epsilon }^2}{\sigma _x^2} K^T K) - Tr(\frac{\sigma _x^2}{\sigma _{\epsilon }^4} K^T K A K^T K) \end{aligned}$$

  (17)
  $$\begin{aligned}&\quad \,+ \!\frac{\sigma _x^2}{2} \left( \frac{1}{\sigma _{\epsilon }^2} y^T K - \frac{1}{\sigma _{\epsilon }^4}y^T K A K^T K \!\right) \left( \frac{1}{\sigma _{\epsilon }^2} K^T y - \frac{1}{\sigma _{\epsilon }^4} K^T K A K^T y \right) \end{aligned}$$

  (18)

• Control points parameters.

  $$\begin{aligned} \frac{\mathrm {d}}{\mathrm {d}\mathcal {\theta }_j}\log \mathcal {L}&= -\frac{\sigma _x^2}{\sigma _{\epsilon }^2} Tr(\frac{\mathrm {d}K}{\mathrm {d}\mathcal {\theta }_j} K^T) + \frac{\sigma _x^2}{\sigma _{\epsilon }^4} Tr( K^T K A K^T \frac{\mathrm {d}K}{\mathrm {d}\mathcal {\theta }_j}) \end{aligned}$$

  (19)
  $$\begin{aligned}&\quad \, -2 \frac{\sigma _x^2}{\sigma _{\epsilon }^4} (y^T K \frac{\mathrm {d}K}{\mathrm {d}\mathcal {\theta }_j} y )\end{aligned}$$

  (20)
  $$\begin{aligned}&\quad \, +2 \frac{\sigma _x^2}{\sigma _{\epsilon }^6} ( y^T K A K^T \frac{\mathrm {d}K}{\mathrm {d}\mathcal {\theta }_j} K^T y + y^T K A K^T K \frac{\mathrm {d}K}{\mathrm {d}\mathcal {\theta }_j} y )\end{aligned}$$

  (21)
  $$\begin{aligned}&\quad \, -2 \frac{\sigma _x^2}{\sigma _{\epsilon }^8} (y^T K A K^T \frac{\mathrm {d}K}{\mathrm {d}\mathcal {\theta }_j} K^T K A K^T y ). \end{aligned}$$

  (22)