Recent advances in chemical multi-way calibration with second-order or higher-order advantages: Multilinear models, algorithms, related issues and applications

Highlights

• The theories and applications of multi-way calibration have been systematically introduced.

• The theory part focused on multilinear models, algorithms and some fundamental issues.

• Different high-order instrumental data acquisition procedures have been described.

• Representative applications published from 2015 to 2020 have been reviewed.

• The advantages and contributions of multi-way calibration methods have been emphasized.

Abstract

This review discusses the recent advances in both theories and analytical applications of multi-way calibrations based on various high-order analytical data. In the theory part, we focus on some aspects of multi-way calibration, such as multilinear models and their extensions, multi-way calibration algorithms with second-order or higher-order advantages, and other fundamental issues. According to different types of high-order instrument signals, recent applications of second-, third-, and fourth-order calibrations are then discussed, and their contributions to green analytical chemistry are highlighted.

Introduction

With the increasing number of available second- and higher-order analytical instruments, the multidimensional experimental data array for each sample can be obtained more and more easily and quickly. It is a challenge for analysts to process and make full use of this kind of data, and is also a major opportunity for the revolution in analytical chemistry. Multi-way calibration is a powerful analytical method based on high-order instrument data for the quantification of analytes of interest in complex systems. The work of multi-way calibration first began in 1978 [1], and in the past two decades, it has made tremendous progress and has become a hotspot of theoretical interest and intensive experimental research [[2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]], which benefits from the emergence of more high-order analytical instruments and efficient algorithms, as well as their revolutionary “second-order advantage” [24], that is, one can directly quantify several components of interest even in the presence of uncalibrated components in predicted samples. It realizes the dreams of many analysts in a simple and special way, and a large number of applications in various scientific fields have confirmed its derived benefits: (1) clean-up and pre-separation steps may no longer be needed; (2) in combination with spectroscopic technology, real-time reaction process monitoring of target analytes in complex systems can be achieved; (3) in combination with chromatographic technology, full chromatographic separation of the analytes is not required and the experiment process is much simpler; (4) baseline effects in chromatographic analysis can be easily modeled and removed from the scene; (5) the same calibration set can be applied to multiple scenes with different unknown background interferences; (6) the sensitivity, selectivity and anti-collinearity increase as the number of instrumental modes increases, which can be regard as the “higher-order advantages” [12].

Non-pollution or little pollution during chemical experiment will be the trend of analytical chemistry, while the advantages of the multi-way calibration are highly consistent with the concept of green analytical chemistry. The use of multi-way calibration always makes the analysis process simpler and greener, and “mathematical separation” will completely or partially replace the “physical/chemical separation”, which saves time, cost, solvent, manpower and energy. Although some recent reviews have covered multi-way calibration in different fields based on different focuses [[15], [16], [17], [18],[21], [22], [23]], a more comprehensive review is needed. In this context, this paper systematically reviews models (focus on multilinear models), algorithms with second-order or higher-order advantages, general considerations and recent representative applications for multi-way calibration along the basic logic from theory to practical application, aiming to provide a guide for analysts. In addition, the contribution of multi-way calibration to green analytical chemistry is highlighted.