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, 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, agronomy, and computer aided system theory.
Function H1 The hyperbolastic rate equation of type I, denoted H1, is given by
d P ( x ) d x = P ( x ) M ( M − P ( x ) ) ( δ + θ 1 + x 2 ) , {\displaystyle {\frac {dP(x)}{dx}}={\frac {P(x)}{M}}\left(M-P\left(x\right)\right)\left(\delta +{\frac {\theta }{\sqrt {1+x^{2}}}}\right),}
where x {\displaystyle x} is any real number and
P ( x ) {\displaystyle P\left(x\right)} is the population size at x {\displaystyle x} . The parameter M {\displaystyle M} represents carrying capacity, and parameters δ {\displaystyle \delta } and θ {\displaystyle \theta } jointly represent growth rate. The parameter θ {\displaystyle \theta } gives the distance from a symmetric sigmoidal curve. Solving the hyperbolastic rate equation of type I for P ( x ) {\displaystyle P\left(x\right)} gives
P ( x ) = M 1 + α e − δ x − θ arsinh ( x ) , {\displaystyle P(x)={\frac {M}{1+\alpha e^{-\delta x-\theta \operatorname {arsinh} (x)}}},}
where arsinh {\displaystyle \operatorname {arsinh} } is the inverse hyperbolic sine function. If one desires to use the initial condition P ( x 0 ) = P 0 {\displaystyle P\left(x_{0}\right)=P_{0}} , then α {\displaystyle \alpha } can be expressed as
α = M − P 0 P 0 e δ x 0 + θ arsinh ( x 0 ) {\displaystyle \alpha ={\frac {M-P_{0}}{P_{0}}}e^{\delta x_{0}+\theta \operatorname {arsinh} (x_{0})}} . If x 0 = 0 {\displaystyle x_{0}=0} , then α {\displaystyle \alpha } reduces to
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