The exponential accumulation and decay function is useful for modeling many different chemical and biological processes including receptor binding, signal transduction pathways, or cellular proliferation and apoptosis.
This particular curve fit aims to model a process comprising two phases including the accumulation and decay phase each with its own set of calculated readouts and fit parameters. If you are interested in modeling exponential accumulation and decay data sets within the context of pharmacokinetics (oral dosing), please see this article.
To implement the accumulation and exponential decay curve fit, users can follow these steps:
- Creating readouts for input data such as optical density (Y) and time (X)
- Creating a plot type readout to generate the curve fit
- Data calculations for different intercepts (optional)
- Editing and saving the curve
Let’s work through an example where we will implement this curve fit within our “Bacterial proliferation and apoptosis” 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 cell line or concentration of drug.
Our protocol now captures our input data and automatically generates an exponential accumulation and decay curve for each entity per run of the protocol. As shown below, be sure to customize your report to include the relevant calculations such as K1 and K2, the accumulation and decay rate constants. See below for a full list of auto-calculated parameters.
Fit Parameters:
The Exponential Accumulation and Decay curve fit is performed using the following equation:
This equation is a bi-exponential model often used to describe processes where two different rates of exponential accumulation and decay are involved.
Where
Y= The dependent variable readout at time, t
C= The scaling constant related to the maximum amplitude of the process
K1= The accumulation rate constant
t= The time variable
K2= The decay rate constant
The accumulation rate constant K1 is the rate at which the dependent variable increases during the accumulation phase before reaching the peak expressed in units of independent variable-1 (e.g., hour-1). The decay rate constant K2 represents the rate at which the dependent variable decreases during the decay phase, also expressed in units of dependent variable-1 (e.g., hour-1).
The following parameters will automatically be calculated by CDD Vault for this plot type:
- Amplitude
- Area under the curve - observed
- Maximum measured
- Minimum measured
- N
- Peak
- R-squared
- K1 rate constant 1
- K2 rate constant 2
- R-squared
While our pharmacokinetic specific curve fits will auto-calculate the half life based on the decay rate constant, users may find value in defining custom calculations to yield the half life for the accumulation and decay phase in this plot type. Given the biphasic nature of this curve type (accumulation and decay phases) it is important to note that two half lives can be calculated, one for each phase.
Half life is calculated by T ½ = ln(2)/K where k is either the accumulation rate constant or the decay rate constant.
In the context of bacterial growth, it may be of value to quantify the bacterial doubling time using the K1 accumulation rate constant.
- Doubling time or T1/2 accumulation = ln(2)/ K1
Similarly, it may be of value to quantify the bacterial apoptosis half life using the K2 decay rate constant:
- Apoptosis half life T1/2= ln(2)/K2
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.