Table 1 The obtained models.

From: Fractional calculus in mathematical oncology

 

Treated case

Untreated case

Integer order exponential model

\(\frac{dv}{dt}=0.046\cdot v\left(t\right)\)

\(\frac{dv}{dt}=0.109\cdot v\left(t\right)\)

Fractional order exponential model

\(\frac{{dv}^{0.97}}{dt}=0.046\cdot v\left(t\right)\)

\(\frac{{dv}^{0.98}}{dt}=0.109\cdot v\left(t\right)\)

Integer order logistic model

\(\frac{dv}{dt}=0.0735\cdot v\left(t\right)\cdot \left(1-{\left(\frac{v\left(t\right)}{2.81}\right)}^{3.35}\right)\)

\(\frac{dv\left(t\right)}{dt}=0.565\cdot v\left(t\right)\cdot \left(1-{\left(\frac{v\left(t\right)}{35.2}\right)}^{0.134}\right)\)

Fractional order logistic model

\(\frac{d{v}^{0.94}\left(t\right)}{dt}=0.0735\cdot v\left(t\right)\cdot \left(1-{\left(\frac{v\left(t\right)}{2.81}\right)}^{3.35}\right)\)

\(\frac{d{v}^{0.88}\left(t\right)}{dt}=0.565\cdot v\left(t\right)\cdot \left(1-{\left(\frac{v\left(t\right)}{35.2}\right)}^{0.134}\right)\)

Integer order Gompertz model

\(\frac{dv\left(t\right)}{dt}=0.204\mathit{ln}\left(\frac{8.81}{v\left(t\right)+5.04}\right)\)

\(\frac{dv\left(t\right)}{dt}=0.106\mathit{ln}\left(\frac{26.09}{v\left(t\right)+2.24}\right)\)

Fractional order Gompertz model

\(\frac{d{v}^{0.95}\left(t\right)}{dt}=0.204\mathit{ln}\left(\frac{8.81}{v\left(t\right)+5.04}\right)\)

\(\frac{d{v}^{0.86}\left(t\right)}{dt}=0.106\mathit{ln}\left(\frac{26.09}{v\left(t\right)+2.24}\right)\)

Integer order generalized Bertalanffy–Pütter model

\(\frac{dv}{dt}=0.168\cdot {v}^{2.18}-0.1\cdot {v}^{2.66}\)

\(\frac{dv}{dt}=0.894\cdot {v}^{1.38}-0.741\cdot {v}^{1.44}\)

Fractional order generalized Bertalanffy–Pütter model

\(\frac{d{v}^{0.96}}{dt}=0.168\cdot {v}^{2.18}-0.1\cdot {v}^{2.66}\)

\(\frac{d{v}^{0.96}}{dt}=0.894\cdot {v}^{1.38}-0.741\cdot {v}^{1.44}\)

Integer order particularized Bertalanffy–Pütter model

\(\frac{dv}{dt}=0.064{v}^{2.41}-\mathit{ln}\left(v\right)\cdot 0.063\cdot {v}^{2.41}\)

\(\frac{dv}{dt}= 0.203{v}^{0.74}-\mathit{ln}\left(v\right)\cdot 0.011\cdot {v}^{0.74}\)

Fractional order particularized Bertalanffy–Pütter models model

\(\frac{d{v}^{0.96}}{dt}=0.064{v}^{2.41}-\mathit{ln}\left(v\right)\cdot 0.063\cdot {v}^{2.41}\)

\(\frac{d{v}^{0.95}}{dt}= 0.203{v}^{0.74}-\mathit{ln}\left(v\right)\cdot 0.011\cdot {v}^{0.74}\)

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