Curve fitting is dependent on the package minpack.lm
(http://cran.r-project.org/web/packages/minpack.lm/index.html).
Monotonic(sigmoidal) equations are as follows:
Hill: $$E = 1/\left( {1 + {{\left( {\alpha /c} \right)}^\beta }} \right)$$
Hill_two: $$E = \beta c/\left( {\alpha + c} \right)$$
Hill_three: $$E = \gamma /\left( {1 + {{\left( {\alpha /c} \right)}^\beta }} \right)$$
Hill_four: $$E = \delta + \left( {\gamma - \delta } \right)/\left( {1 +
{{\left( {\alpha /c} \right)}^\beta }} \right)$$
where \(\alpha\) = EC50, \(\beta\) = H (Hill coefficient), \(\gamma\) = Top,
and \(\delta\) = Bottom
Weibull: $$E = 1 - \exp ( - \exp (\alpha + \beta \log (c)))$$
Weibull_three: $$E = \gamma \left( {1 - \exp \left( { - \exp \left( {\alpha +
\beta \log \left( c \right)} \right)} \right)} \right)$$
Weibull_four: $$E = \gamma + \left( {\delta - \gamma } \right)\exp \left(
{ - \exp \left({\alpha + \beta \log \left( c \right)} \right)} \right)$$
Logit: $$E = {(1 + \exp ( - \alpha - \beta \log (c)))^{ - 1}}$$
Logit_three: $$E = \gamma /\left( {1 + \exp \left( {\left( { - \alpha } \right)
- \beta \log \left( c \right)} \right)} \right)$$
Logit_four: $$E = \delta + \left( {\gamma - \delta } \right)/\left( {1 + \exp \left
( {\left( { - \alpha } \right) - \beta \log \left( c \right)} \right)} \right)$$
where \(\alpha\) is the location parameter and \(\beta\) slope parameter.
\(\gamma\) = Top, and \(\delta\) = Bottom
BCW: $$E = 1 - \exp \left( { - \exp \left( {\alpha + \beta \left(
{\frac{{{c^\gamma } - 1}}{\gamma }} \right)} \right)} \right)$$
BCL: $$E = {(1 + \exp ( - \alpha - \beta (({c^\gamma } - 1)/\gamma )))^{ - 1}}$$
GL: $$E = 1/{(1 + \exp ( - \alpha - \beta \log (c)))^\gamma }$$
Non-monotonic(J-shaped) models:
Hill_five: $$E = 1 - \left( {1 + \left( {\gamma - 1} \right)/\left( {1 + {{\left( {\alpha /c}
\right)}^\beta }} \right)} \right)\left( {1 - 1/\left( {1 + {{\left( {\delta /c} \right)}
^\varepsilon }} \right)} \right)$$
Brain_Consens: $$E = 1 - \left( {1 + \alpha c} \right)/\left( {1 + \exp \left(
{\beta \gamma } \right){c^\beta }} \right)$$
where \(\alpha\) is the initial rate of increase at low concentration, \(\beta\)
the way in which
response decreases with concentration, and \(\gamma\) no simple interpretation.
BCV: $$E = 1 - \alpha \left( {1 + \beta c} \right)/\left( {1 + \left( {1 + 2\beta
\gamma } \right){{\left( {c/\gamma } \right)}^\delta }} \right)$$
where \(\alpha\) is untreated control, \(\beta\) the initial rate of
increase at low concentration, \(\gamma\) the concentration cause 50% inhibition,
and \(\delta\) no simple interpretation.
Cedergreen: $$E = 1 - \left( {1 + \alpha \exp \left( { - 1/\left( {{c^\beta }}
\right)} \right)} \right)/\left( {1 + \exp \left( {\gamma \left({\ln
\left( c \right) - \ln \left( \delta \right)} \right)} \right)} \right)$$
where \(\alpha\) the initial rate of increase at low concentration, \(\beta\)
the rate of hormetic effect manifests itself,
\(\gamma\) the steepness of the curve after
maximum hormetic effect, and \(\delta\) the lower bound on the EC50 level.
Beckon: $$E = \left( {\alpha + \left( {1 - \alpha } \right)/\left( {1 + {{\left(
{\beta /c} \right)}^\gamma }} \right)} \right)/\left( {1 + {{\left(
{c/\delta } \right)}^\varepsilon }} \right)$$
where \(\alpha\) is the minimum effect that would be approached by
the downslope in the absence
of the upslope, \(\beta\) the concentration at the midpoint of the falling slope,
\(\gamma\) the steepness of the rising(positive) slope, \(\delta\) the concentration
at the midpoint of the rising slope, and \(\epsilon\) the steepness of the
falling(negative) slope.
Biphasic: $$E = \alpha - \alpha /\left( {1 + {{10}^{\left( {\left( {c - \beta }
\right)\gamma } \right)}}} \right) + \left( {1 - \alpha } \right)/\left
( {1 + {{10}^{\left( {\left( {\delta - c} \right)\varepsilon } \right)}}}
\right)$$
where \(\alpha\) is the minimum effect that would be approached by the
downslope in the absence
of the upslope, \(\beta\) the concentration at the midpoint of the falling slope,
\(\gamma\) the steepness of the rising(positive) slope, \(\delta\) the concentration
at the midpoint of the
rising slope, and \(\epsilon\) the steepness of the falling(negative) slope.
In all, \(E\) represents effect and \(c\) represents concentration.