Delays are ubiquitous in pharmacokinetics (PK) and pharmacodynamics (PD) studies. Transit compartment models, described by systems of ordinary differential equations, have been widely used to describe delayed outcomes in PK and PD studies. The obvious disadvantage for this type of model is it requires manually finding proper values for the number of compartments. In addition, it may require many differential equations to fit the data and may not adequately describe some complex features.
Delay differential equations (DDEs) provide an alternative way to model delayed outcomes that does not suffer these disadvantages. In this blog post, I will introduce DDEs and demonstrate their relationship with traditional models such as transit compartment models, typical absorption models, and models for describing atypical absorption profiles.
Why is modeling delays important?
Delays commonly occur in pharmacology. For example, have you ever had a headache and noticed that there is a delay between the time you swallow some ibuprofen tablets and the time when you start feeling better? This is due partly to an absorption delay arising from the time it takes for the drug to be transported from the depot site to the central compartment after drug administration. Some drugs are administered as pro-drugs that must be metabolized to the active drug. In these cases, the drug effect may be delayed with respect to the parent drug concentration in the central compartment.
Introducing DDEs
For ordinary differential equations, the future state of the system is totally determined by its present value. While for delay differential equations, the future state of the system is determined by both its present AND past values. This means that for DDEs, one has to specify the history function, which gives the behavior of the system prior to time 0 (assuming that we start at t = 0).
Delay differential equations have been widely used in the biological sciences and engineering to model delayed outcomes. The first biological application of DDEs for investigating predator-prey interaction dynamics dates to the 1920s. However, this approach did not become widely adopted until the 1970s. In recent years, we have seen DDEs starting to be used for pharmacological applications.
The discrete delay approach
Differential equations that only involve discrete delays are called discrete delay differential equations. Viral dynamics have been modeled using discrete DDEs. For example, the human immunodeficiency virus infects CD4+ T cells. A temporal delay exists between the time of the initial T-cell infection and the onset of viral production. The HIV kinetics model that uses discrete DDEs assumes that the length of time from the initial infection to viral production is the same for all acutely infected T-cells. But, this is not true in reality. How do we accommodate the fact that the delay time varies between infected cells?
The distributed delay approach
If you use a weighted average for all possible values for the delay time, you get the convolution of the signal to be delayed and the probability density function (PDF) of the delay time. This type of delay is referred to as a “distributed delay.” The PDF of the delay time is often called the “delay kernel.”
The distributed delay approach includes the discrete delay approach as a special case. It also incorporates a wide array of pharmacokinetics and pharmacodynamics models as special cases including transit compartment models, typical absorption models, and a number of atypical (or irregular) absorption models. This is done through assuming a specific distribution form for the delay time.
Bringing the power of DDEs to pharmacometrics
Phoenix NLME offers integration of a model delay (discrete or distributed) function eliminating the need to add complex lines of code for each delay differential equation. The new delay function greatly simplifies modeling delayed outcomes, an important function in several therapeutic areas such as oncology, diabetes and arthritis. In Phoenix 7.0 you can add a delay function with a single Phoenix Modeling Language (PML) command avoiding inefficient workarounds and approximations.
To learn how DDEs are implemented in Phoenix NLME, please watch this webinar that I gave on this topic.
Author’s note: This blog post received editorial support from Suzanne Minton.