Hyperbolastic functions

Graphic describing the Hyperbolastic Type I function with varying parameter values.

Graphic describing the Hyperbolastic Type I function with varying parameter values.

Graphic describing the Hyperbolastic Type II function with varying parameter values.

Graphic describing the Hyperbolastic Type II function with varying parameter values.

Graphic describing the Hyperbolastic Type III function with varying parameter values.

Graphic describing the Hyperbolastic Type III function with varying parameter values.

Graphic describing the Hyperbolastic Type III function with varying parameter values.

Hyperbolastic Functions

The Hyperbolastic functions are types of mathematical functions introduced in 2005 by Tabatabai et. al.. These functions can be used in a wide variety of modeling problems including, but not limited to, tumor growth; stem cell proliferation; wound healing; dose response analysis in toxicology, pharmacology, and pharma kinetics; cancer growth; growth of humans, animals, and plants; growth of healthcare cost; artificial neural networks in machine learning; dissociation of double-stranded DNA as a function of temperature; cognitive decline in dementia patients using Mini-Mental State Examination (MMSE) score as a function of age; and epidemiological disease progression or regression.

The Hyperbolastic Function H1

The Hyperbolastic rate equation of type I is denoted by H1 and is given by:

$${\frac {dP(x)}{dx}}={\frac {P(x)}{M}}\left(M-P\left(x\right)\right){\frac {\delta +\theta }{\sqrt {1+x^{2}}}},$$

where x is any real number and $P\left(x\right)$ is the population size at x. The parameter M represents carrying capacity, and parameters $\delta$ and $\theta$ jointly represent growth rate. The parameter $\theta$ gives the distance from a symmetric sigmoidal curve. Solving the Hyperbolastic rate equation of type I for $P\left(x\right)$ gives:

$$P(x)={\frac {M}{1+\alpha e^{-\delta x_{0}-\theta \times arcsinh\left(x\right)}}},$$

where $arcsinh\left(x\right)$ is the inverse hyperbolic sine function of x If one desires to use the initial condition $P\left(x_{0}\right)=P_{0}$, then $\alpha$ can be expressed as:

$\alpha ={\frac {M-P_{0}}{P_{0}}}e^{\delta x_{0}+\theta \times arcsinh\left(x_{0}\right)}$.

If $x_{0}=0$, then $\alpha$ reduces to:

$\alpha ={\frac {M-P_{0}}{P_{0}}}$.

We call the function $P\left(x\right)$ the Hyperbolastic function of type I. The standard Hyperbolastic function of type I is defined as:

$P(x)={\frac {1}{1+e^{-x-arcsinh\left(x\right)}}}$.

The Hyperbolastic Function H2

The Hyperbolastic rate equation of type II is denoted by H2 and is defined as:

${\frac {dP(x)}{dx}}=\alpha \delta \gamma P^{2}\left(x\right)x^{\gamma -1}tanh\left({\frac {M-P\left(x\right)}{\alpha \times P\left(x\right)}}\right)/M,$

where x is any real number and parameter $\gamma$ > 0. The symbol $tanh\left(\cdot \right)$ denotes the hyperbolic tangent function, M is the carrying capacity, and both $\delta$ and $\gamma$ jointly determine the growth rate. In addition, the parameter $\gamma$ represents acceleration in the time course. Solving the Hyperbolastic rate function of type II for $P\left(x\right)$ gives:

$P(x)={\frac {M}{1+\alpha \times arcsinh\left(e^{-\delta x^{\gamma }}\right)}}$.

If one desires to use initial condition $P\left(x_{0}\right)=P_{0}$, then $\alpha$ can be expressed as:

$\alpha ={\frac {M-P_{0}}{P_{0}\times arcsinh\left(e^{-\delta x_{0}^{\gamma }}\right)}}$.

If $x_{0}=0$, then $\alpha$ reduces to:

$\alpha ={\frac {M-P_{0}}{P_{0}\times arcsinh\left(1\right)}}$.

The standard Hyperbolastic function of type II is defined as:

$P(x)={\frac {1}{1+arcsinh\left(e^{-x}\right)}}$.

The Hyperbolastic Function H3

The Hyperbolastic rate equation of type III is denoted by H3 and has the form:

${\frac {dP(t)}{dt}}=\left(M-P\left(t\right)\right)\left(\delta \gamma t^{\gamma -1}+{\frac {\theta }{\sqrt {1+\theta ^{2}t^{2}}}}\right)$,

where t > 0. The parameter M represents the carrying capacity, and the parameters $\delta ,$ $\gamma ,$ and $\theta$ jointly determine the growth rate. The parameter $\gamma ,$ represents acceleration of the time scale, while the size of $\theta$ represents distance from a symmetric sigmoidal curve. The solution to the differential equation of type III is:

$P(t)=M-\alpha e^{-\delta t^{\gamma }-arcsinh\left(\theta t\right)}$,

with the initial condition $P\left(t_{0}\right)=P_{0}$ we can express $\alpha$ as:

$\alpha =\left(M-P_{0}\right)e^{\delta t_{0}^{\gamma }+arcsinh\left(\theta t_{0}\right)}$.

The standard Hyperbolastic of type III is defined as:

$P\left(t\right)=1-e^{-t-arcsinh\left(t\right)}$.

The Hyperbolastic distribution of type III is a three-parameter family of continuous probability distributions with scale parameters $\delta$ > 0, and $\theta$ ≥ 0 and parameter $\gamma$ as the shape parameter. When the parameter $\theta$ = 0, the Hyperbolastic distribution of type III is reduced to the Weibull distribution. The Hyperbolastic cumulative distribution function is given by:

$F(x;\delta ,\gamma ,\theta )={\begin{cases}1-e^{-\delta x^{\gamma }-arcsinh\left(\theta x\right)}&x\geq 0,\\0&x<0\end{cases}}$,

and its corresponding probability density function is:

$f(x;\delta ,\gamma ,\theta )={\begin{cases}e^{-\delta x^{\gamma }-arcsinh\left(\theta x\right)}\left(\delta \gamma x^{\gamma -1}+{\frac {\theta }{\sqrt {1+\theta ^{2}x^{2}}}}\right)&x\geq 0,\\0&x<0\end{cases}}$.

The hazard function h (or failure rate) is given by:

$h\left(x;\delta ,\gamma ,\theta \right)=e^{-\delta x^{\gamma }-arcsinh\left(\left(\theta x\right)\right)}.$

The survival function S is given by:

$S(x;\delta ,\gamma ,\theta )=e^{-\delta x^{\gamma }-arcsinh\left(\theta x\right)}.$

Applications

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 are often considered as a useful tool 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 produced explicit functions 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 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.