Estimating the Number of Transit Compartments Using a Distributed Delay Model
Delays are ubiquitous in pharmacokinetic (PK) and pharmacodynamic (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. This type of model has the disadvantage of requiring manually finding proper values for the number of compartments. In addition, transit compartment models may require many differential equations to fit the data and may not adequately describe some complex features.
Delay differential equations have been widely used in the biological sciences and engineering to model delayed outcomes. This approach that does not suffer the disadvantages incurred by using transit compartment models. Differential equations that only involve discrete delays are called discrete delay differential equations. The distributed delay approach includes the discrete delay approach as a special case. This is done through assuming a specific distribution form for the delay time.
The maturation of blood cells from the early stage precursors in the bone marrow to the mature cells observed in the circulation is an example of a system that exhibits delays. For example, red blood cells develop from precursors in the bone marrow. Upon the stimulation with hematopoietic growth factors, they mature and are released to the circulation where they carry hemoglobin. Likewise, white blood cells originate from myeloid stem cells. Upon stimulation by cytokines, they mature and are released from the bone marrow to the circulation. In drug development, a common application of models incorporating delays is to evaluate the effects of drugs on hematopoietic cells. This application has particular use in oncology drug development as many chemotherapeutic agents are toxic to hematopoietic cells.
The gamma-distributed delay model has been introduced to extend the classic transit compartment model of chemotherapy-induced myelosuppression. The gamma distribution provides an additional shape parameter that is equal to the number of transit compartments if it assumes an integer value.
The objective of this presentation was to demonstrate deterministic identifiability of the distributed delay model given typical data showing the effect of chemotherapy on white blood cells. The analysis has been performed using the delay operator implemented in Phoenix 8. By watching this webinar, you can learn how the Phoenix delay operator can provide the following benefits:
- Eliminate the need to add complex lines of code for each delay differential equation
- Simplify modeling delayed outcomes
- Avoid inefficient workarounds and approximations