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Linear Equation

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Mitchell Buckley

A linear fit is one of the most prevalent ‘curve’ fit types used across every industry given its applicability to diverse data sets. A linear equation is an algebraic equation in which the highest power of either variable is one and represents a straight line when graphed on an X, Y coordinate plane. 

Within the world of drug discovery, the linear fit has an enormous number of applications including processes such as understanding the effect of a drug at saturating exposures on tumor size or quantifying the elimination rate in a hepatocyte stability assay.

To implement the linear fit, users can follow these steps:

  1. Creating readouts for X and Y input data such as timepoint and tumor size, respectively
  2. Creating a plot type readout to generate the curve fit
  3. Data calculations for different intercepts (optional)
  4. Editing and saving the curve

Let’s work through an example where we will implement this curve fit within our “PD-1 Tumor Mouse Model - Saturating Dose” protocol.

Create a numerical readout for the X-axis values:

Create a numerical readout for the Y-axis values:

Create a readout to plot the curve and calculate any optional intercept values:

Pro tip: Create conditional readouts for parameters such as mouse strain, route of administration, or concentration of drug, for example.

Our protocol now captures our input data and automatically generates a linear fit plot for each entity per run of the protocol. As shown below, be sure to customize your report to include the relevant auto calculated parameters such as area under the curve and slope. See below for a full list of auto-calculated parameters.

Fit Parameters:

The linear fit is performed using the following equation:

Y=mx+b

This is the slope intercept form of a linear equation describing a straight line on a two dimensional graph. 

Where

Y=the dependent variable which changes based on the value of x

X= the independent variable 

m= the slope of the line which describes how much the Y variable changes for every one unit change of the X variable

b= The y-intercept which is the point in which the line crosses the Y-axis (x=0)

The following parameters will automatically be calculated by CDD Vault for this plot type:

  • Area under the curve - observed 
  • Maximum measured
  • Minimum measured
  • N
  • R-squared
  • Slope
  • Y-intercept

Area under the curve:

  • The CDD Vault AUC is calculated as the (positive area - negative area) under the curve where:
    • The curve is defined as straight line connections between responses, not the fitted curve (Linear Trapezoidal Non-uniform grid method)
    • Units are response units * dose_units
    • The baseline is set at the negative control mean or 0 if control data is not available

Additional calculations:

Please note that all auto-calculated parameters listed above can be used in custom calculations as separate readouts. These parameters can be accessed within our robust formula editor. 

For example:

Tumor size difference: Maximum measured value - minimum measured value

[PD1 Tumor Mouse Model (EGFR Ex20Ins mutant) -> Timepoint - Tumor Size :  -> Maximum measured (mm^2)]-[PD1 Tumor Mouse Model (EGFR Ex20Ins mutant) -> Timepoint - Tumor Size :  -> Minimum measured (mm^2)]

Predicting tumor kill time: Y-tinercept/ slope

[PD1 Tumor Mouse Model (EGFR Ex20Ins mutant) -> Timepoint - Tumor Size :  -> Y-intercept (mm^2)]-[PD1 Tumor Mouse Model (EGFR Ex20Ins mutant) -> Timepoint - Tumor Size :  -> Slope (mm^2)]