# Global and Three-Dimensional Fitting

Global fitting is a relatively new mathematical technique that builds on the principles of standard non-linear Least Squares Fitting (LSF). Used to fit multiple fits simultaneously, global fitting is built on the concept of sharing parameters across fits and can be applied across different models.

By sharing parameters across a number of fits, the fitting process finds the optimum global parameter value for each of the fits. Essentially this returns the same fitted result for each of the shared parameters across all the fits selected. A global fitting scenario could involve, for example, the fitting of five different dose response curves together, with all the fits sharing the same upper asymptote, but the researcher wants to avoid locking them all at the same value.

Traditionally the only way to do this would be to lock the upper asymptote parameters at the same value for all the fits. Using global fitting, however, the computer is asked to determine what the best common value would be for all five curves.

### Global fitting techniques

Global fitting allows users to share parameters across fits that are not using the same model. The example below illustrates how a linear fit can be combined with an extended Michaelis-Menten fit to create a better estimate for the parameters.

*Fig 1: Combining fits across models using global fitting*

Global fitting is becoming more popular across a number of different areas, as well as being applied to multiple dose response curve fitting. Another technique is to fit high and low control values and use the results of those high and low control fits as the constraints and locking mechanism for dose response fits.

The example in Fig 2 below shows how global fitting can be used in Ki determination, demonstrating how the same dataset can be globally fitted using different models. Checking the results clearly shows that the competitive model is a better fit to the data.

*Fig 2: Applying global fitting to Kidetermination*

### 3D Enzyme Kinetics

3D fitting is a technique that is becoming more widely used for example in Schild analysis and enzyme kinetic data analysis. Used in a similar way to global fitting, 3D fitting employs similar methods and principles but produces a three-dimensional plot, whereas global fitting is normally used on two-dimensional data.

In the example below, the results of a 3D fit are used to determine whether data is competitive or non-competitive.

*Fig 3: Competitive and non-competitive data*

3D fitting is also used quite heavily in Schild analysis, massively improving the accuracy of the results that are generated through a series of data transformations.

*Fig 4: Schild analysis using 3D fitting*

Because it minimizes the number of calculations and data transformations that need to be processed to perform a Schild analysis, 3D fitting reduces error values and provides a simpler interpretation of results.

However, it can be difficult to interpret graphs and extract results from a 3D ‘surface’. For example, when viewing two 3D curves, it may not be straightforward to interpret the difference between the two and determine whether the data set is non-competitive or competitive in nature. To accurately determine the competitive nature of an analysis, researchers need to look at the final results.

### Automate for best results – a best practice process

Fig 5 below shows an example automated process.

*Fig 5: An example automated process*

In the initial pre-analysis check phase, the researcher checks the concentration ranges in the data set to ensure there are sufficient data points for a good quality fit. Before fitting the data set, robust fitting can be performed to smooth or reject data points based on outlier status.

In the next stage of the process, the researcher applies data normalization to reduce noise in the data and bring all measurements into a defined range.

Once normalized, data can be analyzed and an appropriate fitting model applied. Examples include fitting a single curve using a standard dose response model, or bell-shaped dose response curve, or a five- or three- parameter curve. Multiple curves can also be analyzed, enabling the researcher to build a library of curves that are related. For example, the researcher can fit more than one model at the same time, such as a two-site and single-site dose response model, and then perform a check such as a T test or AUC, or other fitting analysis to see which one of those two models has fitted the data best.

Additional QC tests can then be performed to check all the fitting statistics and all the parameter values produced at the end of the fitting process. Always check the error values that are associated with those parameters. Check if your IC_{50} value is greater than the highest concentration tested, which indicates that you are trying to extract data from the data set which is in a non-defined area of the curve.

Researchers can also perform other simple checks, for example:

- Check if the minimum response that was produced is greater than 50 percent, in which case all the data points are too high
- Check if the maximum response is less than 50 percent, in which case it may be non active.

Goodness of fit can be checked using residuals, F test, T test and Chi^{2} techniques as discussed in part 2 of this best practice series, Resolving Fitting Issues. If the fit failed, changes may need to be made to the parameters to produce more meaningful results, and the fit performed again.

At this stage, manual QC intervention may be necessary to reanalyze the data, and check for errors before refitting, such as the rejection of too many data points by the end.

When the fit is successful, researchers generate a report on that particular fit. The report could include:

- All parameter values including error estimates
- Fitting statistics, such as Chi
^{2}values, F test and T test - Number of knocked out points
- Graphical results
- Intersect values, such as EC
_{20}s, EC_{80}s, IC_{20}s and IC_{80}s, and their error values

## Summary - how to achieve maximum fitting results

Using the best practice guidelines above, plus automation wherever possible, you can improve consistency, reduce errors and optimize the fitting process to be faster and more efficient. From a mathematical perspective, it makes sense to automate as much of the fitting process as possible. Automated fitting consistently analyzes data from start to finish of the process and eliminates curves as soon as poor quality is detected. Improving consistency means that researchers can directly compare curves knowing that the data sets have been produced in the same way with the same techniques and analysis interpretation.

Being faster and more efficient than manual processes, automation speeds up fitting, removing systematic or human error. Using best practices with automation increases throughput and saves time, reducing the need for manual QC intervention and accelerating decision making.