The use of curve-fitting and compartmental modelling for calculating physiological parameters from measured data has increased in popularity in recent years. Finding the ‘best fit’ of a model to data involves the minimization of a merit function. An example of a merit function is the sum of the squares of the differences between the data points and the model estimated points. This is facilitated by curve-fitting algorithms. Two curve-fitting methods, Levenberg-Marquardt and MINPACK-1, are investigated with respect to the search start points that they require and the accuracy of the returned fits. We have simulated one million dynamic contrast enhanced MRI curves using a range of parameters and investigated the use of single and multiple search starting points. We found that both algorithms, when used with a single starting point, return unreliable fits. When multiple start points are used, we found that both algorithms returned reliable parameters. However the MINPACK-1 method generally outperformed the Levenberg-Marquardt method. We conclude that the use of a single starting point when fitting compartmental modelling data such as this produces unsafe results and we recommend the use of multiple start points in order to find the global minima.