page_title: Hyperbolastic functions
section_title: Applications
hierarchical_section_title: Hyperbolastic functions / Applications
caption_attribution_description: English: 3D Hyperbolastic graph of phytoplankton biomass as a function of nutrient concentration and time
context_page_description: The hyperbolastic functions, also known as hyperbolastic growth models, are mathematical functions that are used in medical statistical modeling. These models were originally developed to capture the growth dynamics of multicellular tumor spheres, and were introduced in 2005 by Mohammad Tabatabai, David Williams, and Zoran Bursac. The precision of hyperbolastic functions in modeling real world problems is somewhat due to their flexibility in their point of inflection. These functions can be used in a wide variety of modeling problems such as tumor growth, stem cell proliferation, pharma kinetics, cancer growth, sigmoid activation function in neural networks, and epidemiological disease progression or regression.
  The hyperbolastic functions can model both growth and decay curves until it reaches carrying capacity. Due to their flexibility,
  these models have diverse applications in the medical field, with the ability to capture disease progression with an intervening
  treatment. As the figures indicate, the hyperbolastic functions can fit a sigmoidal curve indicating that the slowest rate occurs at
  the early and late stages.
context_section_description: According to stem cell researchers at McGowan Institute for Regenerative Medicine at the University of Pittsburgh, "a newer model [called the hyperbolastic type III or] H3 is a differential equation that also describes the cell growth. This model allows for much more variation and has been proven to better predict growth."
  The hyperbolastic growth models H1, H2, and H3 have been applied to analyze the growth of solid Ehrlich carcinoma using a variety of treatments.
  In animal science, the hyperbolastic functions have been used for modeling broiler chicken growth. The hyperbolastic model of type III was used to determine the size of the recovering wound.
  In the area of wound healing, the hyperbolastic models accurately representing the time course of healing. Such functions have been used to investigate variations in the healing velocity among different kinds of wounds and at different stages in the healing process taking into consideration the areas of trace elements, growth factors, diabetic wounds, and nutrition.
  Another application of hyperbolastic functions is in the area of the stochastic diffusion process, whose mean function is a hyperbolastic curve of type I. The main characteristics of the process are studied and the maximum likelihood estimation for the parameters of the process is considered.
  To this end, the firefly metaheuristic optimization algorithm is applied after bounding the parametric space by a stage wise procedure. Some examples based on simulated sample paths and real data illustrate this development. A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called Brownian motion. The hyperbolastic function of type III was used to model the proliferation of both adult mesenchymal and embryonic stem cells; and, the hyperbolastic mixed model of type II has been used in modeling cervical cancer data. Hyperbolastic curves can be an important tool in analyzing cellular growth, the fitting of biological curves, and the growth of phytoplankton.
  In forest ecology and management, the hyperbolastic models have been applied to model the relationship between DBH and height.
  The multivariable hyperbolastic model type III has been used to analyze the growth dynamics of phytoplankton taking into consideration the concentration of nutrients.
caption: 3D Hyperbolastic graph of phytoplankton biomass as a function of nutrient concentration and time
exif: {}