This document discusses various mathematical models that can be used to describe the relationship between drug concentration and pharmacological effect. It describes linear, non-linear, E max, and Hill models. The linear model shows a direct proportional relationship between effect and concentration. The non-linear model uses logarithmic transformation. The E max and Hill models describe the non-linear saturation of effect at high concentrations, with the Hill model allowing for a steeper or shallower concentration-response curve. The therapeutic concentration range is where pharmacological effects are most useful and toxicity is avoided.

1. S.PRIYA
PHARM.D IV YR

2. MATHEMATICAL MODELS
 A mathematical model is a description of
a system using mathematical concepts and language.
The process of developing a mathematical model is
termed mathematical modeling.
 Mathematical models that relate pharmacological
effect to a measured drug concentration in plasma or
at the effector site can be used to develop quantitative
relationships.

3. Some of the commonly used
relationships or models are:
 Linear model
 Non-linear or Logarithmic model
 E max model/ Hyperbolic model
 Hill model/ sigmoid- Emax model

4. LINEAR MODEL
When the pharmacological effect (E) is directly
proportional to the drug concentration (C), the
relationship may be written as:
E = PC+ E0
Where P is the slope of the line obtained from the plot of
E versus C and E0 is the extrapolated y-intercept called
as baseline effect in the absence of drug.

6. NON-LINEAR/ LOGARITHMIC
MODEL
If the concentration- effect relationship doesnot confirm
to a simple linear function logarithmic transformation
of the data is needed.
E= P log C+ I
Where I is empirical constant, this transformation is
popular because it expands the initial part of the curve
where response is changing markedely with a small
change in the concentration and contracts the latter
part where a large change in concentration produces
only a slight change in response.

8. An important feature of this transformation is the linear
relationships between drug concentration and
response at concentrations producing effects of
between 20 to 80% of the maximum effect. Beyond
this range, a large dose produces a larger concentration
of drug in the body.

9. E MAX MODEL/ HYPERBOLIC
MODEL
Unlike earlier models, these models describe the non-
linear concentration –effect relationship i.e., the
response increases with an increase in drugs
concentration at a low concentration and tends to
approach maximum( asymptote) at high
concentration. Such a plot is characteristics of most
concentration-response curve. None of the preceding
models can account for the maximal drug effect as the
Emax models.

11. Michaelis- Menten equation for a saturable
process(saturation of receptor sites by the drug
molecules) is used to describe such a model.
E = Emax C/ C50+C
Where, Emax is maximum effect, and
C50 is the concentration at which 50% of the
effect is produced.
When C˂˂C50, the equation reduces to a linear
relationship.

12. HILL MODEL/ SIGMOID-EMAX
MODEL
In certain cases, the concentration- response
relationship is steeper or shallower than that
predicted. A better fit may otherwise be obtained by
considering the shape factor ‘h’ also called as hill
coefficient, to account for deviations from a perfect
hyperbola, and the equation so obtained is called hill
equation.

14. E = EmaxCh/Ch50+Ch
If h=1, a normal hyperbolic plot is obtained and the
model is called emax model. Larger the value of h,
steeper the linear portion of the curve and greater
its slope. Such a plot is often sigmoidal and thus,
the Hill model may also be called as Sigmoidal-
Emax model.

15. DRUG LEVELS IN
BLOOD

16. THERAPEUTIC CONCENTRATION
RANGE
The therapeutic effectiveness of a drug depends upon its
plasma concentration.
There is an optimum concentration range in which
therapeutic success is most likely and concentrations
both above and below it are more harmful than useful.
Such a range is thus based on the difference between
pharmacological effectiveness and toxicity.

23. REFERENCES
 Biopharmaceutics and pharmacokinetics
Author: D M Brahmankar and Sunil B Jaiswal
Pg no :392-395
 https://www.slideshare.net/coolarun86/therapeutic-
drug-monitoring-43196669