### Basic functions: Power functions

### Context 3: Equivalent doses of antipsychotics

Antipsychotics are drugs that help to get out of a psychosis. Atypical antipsychotics are used in the treatment of mental illnesses, such as schizophrenia, acute mania, maintenance treatment of bipolar disorder, and acute psychosis. Examples of atypical antipsychotics are clozapine, risperidone, olanzapine, quetiapine and aripiprazole. Patients sometimes take several antipsychotics at the same time. To get an idea of the total strength of the drug use, the different doses are converted to the dose of one type of drug, such as chlorpromazine (CPZ). Andreasen and colleagues reported in the article "Antipsychotic dose equivalents and dose-years: a standardized method for comparing exposure to different drugs." ( *Biol. Psychiatr.* 67, 255–262, 2010) power relationships of the form \[y=c\cdot x^b\] where \(y\) is the dose of a given antipsychotic, \(x\) is the equivalent dose of CPZ, and \(b\) and \(c\) are drug-specific, experimentally determined numbers. The table below lists the power relationships found for the aforementioned antipsychotics by Andreasen and colleagues.

Antipsychotic |
Relation ( \(x={}\) CPZ dose ) |
Equivalent to 100mg CPZ |

clozapine | \(\quad y=2.027\,x^{0.864}\) | \(\quad 108\) |

risperidone | \(\quad y=0.019\,x^{0.924}\) | \(\quad 1.32\) |

olanzapine | \(\quad y=0.086\,x^{0.870}\) | \(\quad 4.75\) |

quetiapine | \(\quad y=2.806\,x^{0.852}\) | \(\quad 142\) |

aripiprazole | \(\quad y=0.255\,x^{0.700}\) | \(\quad 6.42\) |

chlorpromazine (CPZ) | \(\quad y=x\) | \(\quad 100\) |

Suppose a patient uses \(5\,\mathrm{mg}\) olanzapine and \(50\,\mathrm{mg}\) quetiapine daily. What is the equivalent dose of chlorpromazine?

In the table we read that \(x\,\mathrm{mg}\) CPZ is equivalent to \(y_\mathrm{olanzapine}=0.086\,x^{0.870}\,\mathrm{mg}\) olanzapine and is also equivalent to \(y_\mathrm{quetiapine}=2.806\,x^{0.852}\,\mathrm{mg}\) quetiapine. Given the daily doses of olanzapine and quetiapine, i.e. given the \(y\) values, this allows us to calculate the equivalent CPZ doses. \[\begin{array}{ccccccccc} x_\mathrm{olanzapine} &=& \left(\frac{y_\mathrm{olanzapine}}{0.086}\right)^{\frac{1}{0.870}}&=&\left(\frac{5}{0.086}\right)^{\frac{1}{0.870}}&\approx& 106.69\\[0.25cm] x_\mathrm{quetiapine} &=& \left(\frac{y_\mathrm{quetiapine}}{2.806}\right)^{\frac{1}{0.852}}&=&\left(\frac{50}{2.806}\right)^{\frac{1}{0.852}}&\approx& 29.39\end{array}\] The total daily use of antipsychotics by this patient is thus equivalent to \(x_\mathrm{olanzapine}+x_\mathrm{quetiapine}=106.69+29.39=136.08\,\mathrm{mg\;CPZ}\).