Hyperbolastic functions


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. In addition to the presenting sigmoidal shapes, it can also accommodate biphasic situations where medical
interventions slow or reverse disease progression; but, when the effect of the treatment vanishes, the disease will begin the
second phase of its progression until it reaches its horizontal asymptote.
One of the main characteristics these functions have is that they cannot only fit sigmoidal shapes, but can also model biphasic growth patterns that other classical sigmoidal curves cannot adequately model. This distinguishing feature has advantageous applications in various fields including medicine, biology, economics, engineering, and computer aided system theory.

Function H1

The hyperbolastic rate equation of type I, denoted H1, is given by:
where is any real number and
is the population size at. The parameter represents carrying capacity, and parameters and jointly represent growth rate. The parameter gives the distance from a symmetric sigmoidal curve. Solving the hyperbolastic rate equation of type I for gives:
where is the inverse hyperbolic sine function. If one desires to use the initial condition, then can be expressed as:
If, then reduces to:
The hyperbolastic function of type I generalizes the logistic function. If the parameters, then it would become a logistic function. This function is a hyperbolastic function of type I. The standard hyperbolastic function of type I is

Function H2

The hyperbolastic rate equation of type II, denoted by H2, is defined as:
where is the hyperbolic tangent function, is the carrying capacity, and both and jointly determine the growth rate. In addition, the parameter represents acceleration in the time course. Solving the hyperbolastic rate function of type II for gives:
If one desires to use initial condition then can be expressed as:
If, then reduces to:
The standard hyperbolastic function of type II is defined as:

Function H3

The hyperbolastic rate equation of type III is denoted by H3 and has the form:
where > 0. The parameter represents the carrying capacity, and the parameters and jointly determine the growth rate. The parameter represents acceleration of the time scale, while the size of represents distance from a symmetric sigmoidal curve. The solution to the differential equation of type III is:
with the initial condition we can express as:
The hyperbolastic distribution of type III is a three-parameter family of continuous probability distributions with scale parameters > 0, and ≥ 0 and parameter as the shape parameter. When the parameter = 0, the hyperbolastic distribution of type III is reduced to the weibull distribution. The hyperbolastic cumulative distribution function of type III is given by:
and its corresponding probability density function is:
The hazard function is given by:
The survival function is given by:
The standard hyperbolastic cumulative distribution function of type III is defined as:
and its corresponding probability density function is:

Properties

If one desires to calculate the point where the population reaches a percentage of its carrying capacity, then
one can solve the equation:
for, where. For instance, the half point can be found by setting.

Applications

According to stem cell researchers at McGowan Institute for Regenerative Medicine at the University of Pittsburgh, "a newer model 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.

Hyperbolastic regressions

Hyperbolastic regressions are statistical models that utilize standard hyperbolatic functions to model a dichotomous outcome variable. The purpose of binary regression is to predict a binary outcome variable using a set of explanatory variables. Binary regression is routinely used in many areas including medical, public health, dental, and biomedical sciences. Binary regression analysis was used to predict endoscopic lesions in iron deficiency anemia. In addition, binary regression was applied to differentiate between malignant and benign adnexal mass prior to surgery.

The hyperbolastic regression of type I

Let be a binary outcome variable which can assume one of two mutually exclusive values, success or failure. If we code success as and failure as, the hyperbolastic success probability of type I as a function of explanatory variables is given by:

where are model parameters. The odds of success is the ratio of probability of success to the probability of failure. For hyperbolastic regression of type I, the odds of success is denoted by and expressed by the equation:
The logarithm of is called the logit of Hyperbolastic of type I. The logit transformation is denoted by and can be written as:

The hyperbolastic regression of type II

For the binary outcome variable, the hyperbolastic success probability of type II as a function of explanatory variables is:
For the hyperbolastic regression of type II, the odds of success is denoted by and is given by:
The logit transformation is denoted by and is given by: