With most drugs, the response produced is reversible i.e. a reduction in concentration at the site of action reverses the effect.

With most drugs, the response produced is reversible i.e. a reduction in concentration at the site of action reverses the effect. The response produced by a drug can be classified into two categories –

__1. Graded response__ - *is the one where intensity of effect increases with the dose or concentration of drug*. A majority of drugs produce* *graded response. The* *response can be measured on a continual basis in such cases and establishing a linear relationship between drug concentration and intensity of response is easy.

__2. Quantal Response__ *– is the one where the drugs may either show their effect or not at all i.e. *the responses are not observed on a continuous basis, for example,* *prevention of seizures by phenytoin. Such responses are also called as *all-or-none responses*. Thus, establishing a concentration-response relationship in such circumstances is difficult but can be developed in terms of the *frequency* with which a particular event occurs at a given drug concentration.

* *

*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. Such models are often called as *pharmacodynamic models. Some of the* *commonly used relationships or models are discussed below.

__1. Linear Model:__ When the pharmacological effect (E) is directly proportional to the drug concentration (C), the relationship may be written as:

E = PC + E0 (13.1)

where P is the slope of the line obtained from a plot of E versus C and Eo is the extrapolated y-intercept called as baseline effect in the absence of drug.

__2. Non-linear/Logarithmic Model:__ If the concentration-effect relationship does not conform to a simple linear function, a logarithmic transformation of the data is needed.

E = PlogC + 1 (13.2)

where *I* is empirical constant. This transformation is popular because it expands the initial part of the curve where response is changing markedly with a small change in concentration and contracts the latter part where a large change in concentration produces only a slight change in response. 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 (Fig. 13.1). Beyond this range, a larger dose produces a larger concentration of drug in the body.

Fig. 13.1 *A typical sigmoidal shape log drug concentration-effect relationship*

__3. Emax Model/Hyperbolic Model:__ Unlike earlier models, these models describe non-linear concentration-effect relationships i.e. the response increases with an increase in drug concentration at low concentrations and tends to approach

Fig. 13.2 *A hyperbolic concentration-response relationship based on Emax model*

maximum (asymptote) at high concentrations (Fig. 13.2). Such a plot is characteristic of most concentration-response curves. None of the preceding models can account for the maximal drug effect as the *Emax* *models*.

*Michaelis-Menten equation *for a saturable process (saturation of receptor sites by the* *drug molecules) is used to describe such a model.

where

Emax = maximum effect, and

C50 = the concentration at which 50% of the effect is produced.

When C << C50, the equation reduces to a linear relationship. In the range 20 to 80%, the Emax model approximates equation 13.2.

__4. Hill Model/Sigmoid-Emax Model:__ In certain cases, the concentration-response relationship is steeper or shallower than that predicted from equation 13.3. 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* (equation 13.4).

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 sigmoid-Emax model (Fig. 13.3).

Fig. 13.3* **Effect of shape factor n on the concentration-response curves*

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