  Reliable analysis of comprehensive two-dimensional gas chromatography -
time-of-flight mass spectrometry (GC$\times$GC-TOFMS) data is considered to be
a major bottleneck for its widespread application. For multiple samples,
GC$\times$GC-TOFMS data for specific chromatographic regions manifests as a 4th
order tensor of I mass spectral acquisitions, J mass channels, K modulations,
and L samples. Chromatographic drift is common along both the first-dimension
(modulations), and along the second-dimension (mass spectral acquisitions),
while drift along the mass channel and sample dimensions is for all practical
purposes nonexistent. A number of solutions to handling GC$\times$GC-TOFMS data
have been proposed: these involve reshaping the data to make it amenable to
either 2nd order decomposition techniques based on Multivariate Curve
Resolution (MCR), or 3rd order decomposition techniques such as Parallel Factor
Analysis 2 (PARAFAC2). PARAFAC2 has been utilised to model chromatographic
drift along one mode, which has enabled its use for robust decomposition of
multiple GC-MS experiments. Although extensible, it is not straightforward to
implement a PARAFAC2 model that accounts for drift along multiple modes. In
this submission, we demonstrate a new approach and a general theory for
modelling data with drift along multiple modes, for applications in
multidimensional chromatography with multivariate detection.
