In this work, the problem of metabolic flux estimation is formulated as a problem of parameter estimation from incomplete labelling data. The expectation/conditional maximisation (ECM) algorithm is used to determined a maximum-likelihood (ML) estimate because of its simplicity and stable convergence. We propose to simplify a nonlinear inverse problem, generally numerically solved by an iterative optimisation algorithm, to a linear regression problem which is arrived at from a linear-in-the-parameter formulation during a partial optimisation process of the ECM algorithm. Three linear least square algorithms, the ordinary least squares (LS), the total least squares (TLS) and the constrained least squares (CLS), have been tested to solve the linear regression in this step. Using simulations, resulting parameter estimates and errors in flux estimation are compared and evaluated. The performance of the algorithms are investigated under two scenarios; when the labelling data are corrupted by a wide range of noise and when the labelling data are incompletely observed. Results suggest that the estimates from the ECM algorithm using CLS produce results superior to other combinations and have potential to be refined to improve its performance in metabolic flux estimation