Analysis of MRI stomach scans using differential equations

Differential equations (DE) are commonly used to describe dynamic systems evolving in one
(e.g. time) or more dimensions (e.g. space and time). In real applications the parameters defining the theoretical model describing the phenomenon under consideration are often unknown and need to be estimated from the available measurements. This estimation task has been extensively discussed in the statistical literature and probably the most popular procedures are those relying on nonlinear least squares (Bielger et al, 1986). These approaches are computationally intensive and often poorly suited for statistical inference. An attractive alternative is represented by the penalized smoothing procedure introduced by Ramsay et al. (2007). This approach can be viewed as a generalization of the L-spline framework (see Welham et al., 2006 among others) where the flexibility of a high dimensional B-spline expansion of the state function is counterbalanced by a penalty term defining the (set of) differential equation(s) synthesizing the dynamics under investigation. The fidelity of the extracted signal to the hypothesized model is then tuned by a “DE-compliance” parameter to be extracted form the data
too. This approach works reasonably well both with linear and nonlinear differential models but, in the latter case, due to the implicit link between the vector of unknown DE parameters and the spline coefficients, the computational burden tends to increase and the optimization of the compliance
parameter can be demanding. To overcome these drawbacks we adopt the quasilinearized (QL)
ODE-P-spline approach proposed by Frasso et al. (2014). The quasilinearization (Bellman and
Kalaba, 1965) step greatly reduces the computational cost of the estimation procedure making the
penalty term a second order polynomial function of the DE parameters.
As motivating example we present the results of a QL-ODE-P-spline analysis of a set of MRI
scans describing stomach contractions during digestion. For illustrative purposes we analyze the
measurements related to a single slice of the stomach. Finally, we conclude with a discussion of the possible further extensions of the presented methodology.