Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Modeling with Differential Equations01:25

Modeling with Differential Equations

152
Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
152
Exponential Equations for Modeling Growth01:26

Exponential Equations for Modeling Growth

409
Exponential models are essential for describing rapid, multiplicative changes in natural systems, such as population growth. When a population doubles at regular intervals, the process can be modeled using a suitable base. For instance, a bacterial culture that doubles every three hours follows the model n(t)=n0⋅2t/3, where n(t) is the population at the time t.A more general model uses the natural base e, especially for continuous growth. This takes the form n(t)=n0⋅ert, where r is...
409
Exponential Growth01:29

Exponential Growth

138
Bacterial populations exhibit exponential growth when conditions such as nutrient availability and temperature are favorable. In this phase, cells reproduce through binary fission, where each cell divides into two identical daughter cells. This process causes the population to double at regular intervals, resulting in a growth rate that is directly proportional to the current number of cells. As the population increases, the number of new cells formed during each generation also grows, creating...
138
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

387
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
387
Linear Differential Equations01:27

Linear Differential Equations

137
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...
137
Population Growth00:57

Population Growth

29.4K
Population size is dynamic, increasing with birth rates and immigration, and decreasing with death rates and emigration. In ideal conditions with unlimited resources, populations can increase exponentially, which plots as a J-shaped growth rate curve of population size against time. This type of curve is characteristic of newly-introduced invasive species, or populations that have suffered catastrophic declines and are rebounding.
29.4K

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Text mining, a race against time? An attempt to quantify possible variations in text corpora of medical publications throughout the years.

Computers in biology and medicine·2016
Same author

Matched Backprojection Operator for Combined Scanning Transmission Electron Microscopy Tilt- and Focal Series.

Microscopy and microanalysis : the official journal of Microscopy Society of America, Microbeam Analysis Society, Microscopical Society of Canada·2015
Same author

Principal Component Analysis of gait in Parkinson's disease: relevance of gait velocity.

Gait & posture·2013
Same author

Stochastic modelling of biased cell migration and collagen matrix modification.

Journal of mathematical biology·2009
Same author

Computing reconstruction kernels for circular 3-D cone beam tomography.

IEEE transactions on medical imaging·2008
Same author

Mathematics in biomedical imaging.

International journal of biomedical imaging·2008
Same journal

Phenotypic plasticity trade-offs in an age-structured model of bacterial growth under stress.

Journal of mathematical biology·2026
Same journal

Intraspecific interactions facilitate mutualism across multilayer networks under weak selection.

Journal of mathematical biology·2026
Same journal

A two-species competition model on a compact metric graph for the invasion and competition of Aedes Aegypti and Aedes Albopictus mosquitoes in Florida.

Journal of mathematical biology·2026
Same journal

Superinfection and the hypnozoite reservoir for Plasmodium vivax: a multitype branching process approximation.

Journal of mathematical biology·2026
Same journal

Correction to: Superinfection and the hypnozoite reservoir for Plasmodium vivax: a general framework.

Journal of mathematical biology·2026
Same journal

Stoichiometric balance and sustained rhythms.

Journal of mathematical biology·2026
See all related articles

Related Experiment Video

Updated: Mar 19, 2026

The Use of Chemostats in Microbial Systems Biology
13:19

The Use of Chemostats in Microbial Systems Biology

Published on: October 14, 2013

31.9K

Numerical rate function determination in partial differential equations modeling cell population dynamics.

Andreas Groh1, Holger Kohr2, Alfred K Louis3

  • 1Hexagon Metrology PTS, Walter-Zapp-Strasse 4, 35578, Wetzlar, Germany. andreas.groh@hexagonmetrology.com.

Journal of Mathematical Biology
|June 14, 2016
PubMed
Summary
This summary is machine-generated.

This study presents a novel method to determine unknown rate functions in partial differential equations (PDEs) using discrete measurements. The approach effectively solves ill-posed inverse problems in population balance equations (PBEs) with regularization.

Keywords:
Cell population dynamicsInverse problemParameter estimationPartial differential equationPopulation balance equation

More Related Videos

Author Spotlight: Unveiling the Polyfunctionality and Heterogeneity in Immune Responses
09:43

Author Spotlight: Unveiling the Polyfunctionality and Heterogeneity in Immune Responses

Published on: March 8, 2024

2.6K
Isolation and Time-Lapse Imaging of Primary Mouse Embryonic Palatal Mesenchyme Cells to Analyze Collective Movement Attributes
07:13

Isolation and Time-Lapse Imaging of Primary Mouse Embryonic Palatal Mesenchyme Cells to Analyze Collective Movement Attributes

Published on: February 13, 2021

2.7K

Related Experiment Videos

Last Updated: Mar 19, 2026

The Use of Chemostats in Microbial Systems Biology
13:19

The Use of Chemostats in Microbial Systems Biology

Published on: October 14, 2013

31.9K
Author Spotlight: Unveiling the Polyfunctionality and Heterogeneity in Immune Responses
09:43

Author Spotlight: Unveiling the Polyfunctionality and Heterogeneity in Immune Responses

Published on: March 8, 2024

2.6K
Isolation and Time-Lapse Imaging of Primary Mouse Embryonic Palatal Mesenchyme Cells to Analyze Collective Movement Attributes
07:13

Isolation and Time-Lapse Imaging of Primary Mouse Embryonic Palatal Mesenchyme Cells to Analyze Collective Movement Attributes

Published on: February 13, 2021

2.7K

Area of Science:

  • Applied Mathematics
  • Computational Biology
  • Chemical Engineering

Background:

  • Inverse problems in partial differential equations (PDEs) are often ill-posed, requiring regularization for stable solutions.
  • Population balance equations (PBEs) model the evolution of cell populations based on size, crucial in biological and chemical systems.
  • Accurate determination of rate functions within PDEs is essential for predictive modeling.

Purpose of the Study:

  • To introduce a novel regularization method for solving inverse problems in size-structured population balance equations (PBEs).
  • To determine an unknown rate function within a PBE using discrete measurements.
  • To address the ill-posed nature of inverse problems by mitigating measurement error amplification.

Main Methods:

  • The approximate inverse method, a pointwise regularization scheme, is employed.
  • Separation of mollification in time and size variables is a key technique.
  • Numerical instability is avoided by shifting differentiation to an analytically defined function.

Main Results:

  • The developed method successfully determines unknown rate functions in size-structured PBEs.
  • Numerical experiments with simulated data disturbances validate the scheme's performance.
  • The regularization approach effectively handles measurement and model errors.

Conclusions:

  • The proposed approximate inverse method offers a robust solution for inverse problems in PBEs.
  • The technique provides a reliable way to estimate rate functions from discrete population data.
  • This work contributes to accurate modeling of cell population dynamics through improved parameter estimation.