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Related Concept Videos

Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...
Linear Differential Equations01:27

Linear Differential Equations

The integrating factor method provides a systematic way to solve first-order linear differential equations, especially those that cannot be handled by separation of variables. This method is particularly useful in modeling time-dependent physical systems influenced by both constant inputs and resistive forces. A common example is the motion of a car subjected to a constant engine force while experiencing air resistance proportional to its velocity.In such scenarios, Newton’s second law yields a...
Differential Equations: Problem Solving01:21

Differential Equations: Problem Solving

When analyzing the motion of falling objects, it is essential to consider not only the force of gravity but also the opposing force of air resistance. A practical example involves releasing a heavy test weight during a safety check on a ship. As the weight falls from rest, gravity accelerates it downward while air resistance exerts an upward force that increases with velocity. This dynamic interplay of forces is well described by differential equations, which provide a mathematical framework...
Difference Equation Solution using z-Transform01:24

Difference Equation Solution using z-Transform

The z-transform is a powerful tool for analyzing practical discrete-time systems, often represented by linear difference equations. Solving a higher-order difference equation requires knowledge of the input signal and the initial conditions up to one term less than the order of the equation.
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Reaction Mechanisms: Rate-limiting Step Approximation01:29

Reaction Mechanisms: Rate-limiting Step Approximation

The rate-determining step, or RDS, in a chemical reaction is the slowest step that determines the overall reaction rate. It is identified by using the observed rate law and typically involves approximation methods like the RDS approximation or the steady-state approximation.In the RDS approximation, also known as the rate-limiting-step or equilibrium approximation, the reaction mechanism consists of one or more reversible reactions near equilibrium, followed by a slower RDS, and then one or...
Transmission-Line Differential Equations01:26

Transmission-Line Differential Equations

Transmission lines are essential components of electrical power systems. They are characterized by the distributed nature of resistance (R), inductance (L), and capacitance (C) per unit length. To analyze these lines, differential equations are employed to model the variations in voltage and current along the line.
Line Section Model
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Related Experiment Video

Updated: May 10, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Solving delay differential equations in S-ADAPT by method of steps.

Robert J Bauer1, Gary Mo, Wojciech Krzyzanski

  • 1ICON Development Solutions, Hanover, MD 21076, USA.

Computer Methods and Programs in Biomedicine
|July 2, 2013
PubMed
Summary

This study implemented a delay differential equation (DDE) solver in S-ADAPT, enhancing its capabilities for pharmacokinetic and pharmacodynamic modeling. The new solver accurately simulates DDEs, improving population analysis.

Keywords:
Delay differential equationsMethod of stepsPharmacodynamicsPharmacokinetics

Related Experiment Videos

Last Updated: May 10, 2026

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

Area of Science:

  • Pharmacometrics
  • Computational Biology
  • Mathematical Modeling

Background:

  • S-ADAPT, a pharmacokinetic/pharmacodynamic (PK/PD) modeling software, currently lacks a solver for delay differential equations (DDEs).
  • Existing ordinary differential equation (ODE) solvers in S-ADAPT, like LSODA, are not suitable for DDEs.
  • Accurate simulation of DDEs is crucial for advanced PK/PD analyses, including population modeling.

Purpose of the Study:

  • To implement a novel DDE solver within the S-ADAPT software.
  • To extend S-ADAPT's capabilities for analyzing complex biological systems described by DDEs.
  • To validate the accuracy and performance of the newly implemented DDE solver.

Main Methods:

  • Implementation of a DDE solver in S-ADAPT utilizing the method of steps.
  • Transformation of DDE systems into equivalent ODE systems for numerical solution.
  • Validation against analytically derived solutions for linear DDEs and comparison with MATLAB's dde23 for nonlinear DDEs.
  • Testing parameter estimation using importance sampling expectation-maximization on simulated population PK/PD data.

Main Results:

  • The implemented DDE solver in S-ADAPT demonstrated high accuracy, with solutions agreeing to at least 7 significant digits compared to analytical and MATLAB solutions.
  • The solver successfully handled scalar linear DDEs with various input types (bolus, infusion).
  • Population parameter estimates obtained using S-ADAPT's new solver closely matched the true parameters in simulated PK/PD data.

Conclusions:

  • The integration of a DDE solver significantly enhances S-ADAPT's utility for complex PK/PD modeling and simulation.
  • The method of steps provides a robust approach for solving DDEs within computational modeling software.
  • S-ADAPT is now better equipped for advanced population analyses involving time-delayed biological processes.