Many biological systems exhibit time delays. For instance, there is a delay between the time that a virus infects a cell and the onset of viral production. Mathematical models that incorporate these temporal delays are often used to characterize the pharmacokinetics/pharmacodynamics (PK/PD) of drugs. In this blog post, I’ll discuss some examples of biological systems that exhibit delays and some considerations for building models with distributed delays to study these systems.
Biological systems that exhibit delays
In drug development, two major applications of models incorporating delays are to evaluate orally dosed drugs and the effects of drugs on hematopoietic cells. The latter has particular use in oncology drug development as many chemotherapeutic agents are toxic to hematopoietic cells.
The transport of the drug molecules from the gut to the systemic circulation takes time that varies from minutes to hours. Orally administered drugs first disintegrate and dissolve in the GI tract. Then, they permeate through the intestinal membrane, enter the portal vein, and the hepatic veins in the liver. Finally, they enter the circulation. These steps introduce a delay in the drug plasma concentration time course data.
Another example of a system that exhibits delays is the maturation of blood cells from the early stage precursors in the bone marrow to the mature cells observed in the circulation. 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.
The concept of the “distributed delay”
Biological systems exhibiting delays consist of a population of mediators (eg, cells or drug molecules). Delays for the mediators can be “discrete” or “distributed.” A model that incorporates a discrete delay assumes that the length of delay is the same for all mediators.
By contrast, in systems with distributed delays, each mediator is assigned with a residence time (delay). For example, red blood cells live in the circulation for about 100 to 120 days. They are continuously being produced by the bone marrow, entering circulation, dying, and then being replaced by new red blood cells. Thus, the population of red blood cells in the circulation at any given time has a distribution of ages depending on when they entered circulation.
How do we describe the distribution of ages in the population mathematically? A random variable assigns each cell with its residence time. The probability density function describes the distribution of residence times for the population. Now we need to describe the relationship between the input of new cells to the circulation and the output (cell death). That relationship is described by the convolution integral.
To figure out the rate of the output of the mediators (cells), we need to know the distribution of the cells’ ages and the rate of the input (progenitor maturation). The input rate must be known well beyond the times that we need to predict. Keeping with our hematopoietic cell population example, the output of the system is the convolution of the input with the distribution of cell ages.
Modeling the past
To predict the output of a system in the future, we must know the input to the system in the past. These systems can be subdivided into those with a constant past and those with a variant past. If the system was at a constant, zero, steady state, there is no need to model the past. For example, this is the case in a clinical drug trial when the patients have never received any drug doses prior to the start of the trial.
But when the system has a non-zero baseline, we have to account for it. For instance, the number of hematopoietic cells in circulation is typically at some steady state which is a non-zero, constant value.
Sometimes, a system’s baseline varies with time. For example, if a drug is given repeatedly and achieves steady state, the mean concentration of the drug is fairly constant, but concentration of the drug at any given time varies. Thus, if you started a clinical trial where the patients were already dosed to steady state for a drug, you would have to account for this prior history.
Delay differential equations—a new pharmacometrics tool
PK/PD models typically model these delay-time distributions by means of compartments and inter-compartment flows such as simple absorption models. These models automatically perform the necessary convolution by means of differential equations or their closed-form equivalents. However, a single intermediate compartment or a fixed chain of intermediate compartments may not give enough freedom in modeling the desired delay. This gives rise to the need for more flexible means of modeling distributed delays.
The distributed delay approach is an alternative way to model delayed outcomes that does not suffer these disadvantages. To learn more about mathematical tools to model pharmacokinetic/pharmacodynamic (PK/PD) systems with distributed delays, please watch this webinar by Dr. Wojciech Krzyzanski from SUNY-Buffalo.