Sometimes we want to see how people change over time, as in studies of human development and lifespan. When we test the same group of individuals repeatedly over an extended period of time, we are conducting longitudinal research. Longitudinal research is a research design in which data-gathering is administered repeatedly over an extended period of time. For example, we may survey a group of individuals about their dietary habits at age 20, retest them a decade later at age 30, and then again...

Time differentiation, convolution, integration, and periodicity are fundamental concepts in analyzing functions and signals over time. Each concept provides a unique perspective on how functions evolve, interact, and repeat, offering essential tools for various scientific and engineering applications.
Time differentiation involves analyzing the rate of change of a function over time. Mathematically, it is the derivative of a function with respect to time. This concept can be likened to tracking...

According to statistical moment theory, mean residence time (MRT) is an important measure in pharmacokinetics. MRT can be defined as the expected mean of a probability density function distribution. It provides valuable insights into drug disposition in the body.
After the administration of a drug through intravenous bolus injection, the drug molecules are distributed throughout the body and remain there for varying periods. The MRT represents the average time these drug molecules stay in the...

The long-term stability of drug products is critical to ensuring their quality, safety, and effectiveness over time. Stability directly influences a product's ability to maintain its intended characteristics, ensuring it performs as expected during its intended shelf life. Key attributes such as drug potency, impurities, dissolution, and other physicochemical measures of performance are tested to assess stability. These parameters indicate how well the product retains its quality over time and...

Drug elimination from the body primarily occurs through metabolic and excretion pathways. Hepatic metabolism transforms lipophilic drugs into hydrophilic forms for excretion, typically via enzymatic processes classified as phase I (modification) and phase II (conjugation). Renal excretion eliminates drugs and metabolites through filtration and secretion in the kidneys. Impairment in liver or kidney function can hinder these processes, delaying drug clearance and extending the drug’s half-life.

A function is continuous at a point a if three conditions are met: the function is defined at a, the limit of the function as x approaches a exists, and this limit equals the function’s value. Mathematically, this is written asThis definition ensures the graph of the function does not exhibit any breaks, holes, or jumps at that point. Discontinuities occur when any of these conditions fail. A removable discontinuity exists when the two-sided limit exists but the function is either undefined or...