Medicine and Math

                         Medicine and Math

Prescriptions and Medication

Regularly, doctors write prescriptions to their patients for various ailments.  Prescriptions indicate a specific medication and dosage amount.  Most medications have guidelines for dosage amounts in milligrams (mg) per kilogram (kg).  Doctors need to figure out how many milligrams of medication each patient will need, depending on their weight.  If the weight of a patient is only known in pounds, doctors need to convert that measurement to kilograms and then find the amount of milligrams for the prescription.  There is a very big difference between mg/kg and mg/lbs, so it is imperative that doctors understand how to accurately convert weight measurements.  Doctors must also determine how long a prescription will last.  For example, if a patient needs to take their medication, say one pill, three times a day.  Then one month of pills is approximately 90 pills.  However, most patients prefer two or three month prescriptions for convenience and insurance purposes.  Doctors must be able to do these calculations mentally with speed and accuracy.

Doctors must also consider how long the medicine will stay in the patient’s body.  This will determine how often the patient needs to take their medication in order to keep a sufficient amount of the medicine in the body.  For example, a patient takes a pill in the morning that has 50mg of a particular medicine.  When the patient wakes up the next day, their body has washed out 40% of the medication.  This means that 20mg have been washed out and only 30mg remain in the body.  The patient continues to take their 50mg pill each morning.  This means that on the morning of day two, the patient has the 30mg left over from day one, as well as another 50mg from the morning of day two, which is a total of 80mg.  As this continues, doctors must determine how often a patient needs to take their medication, and for how long, in order to keep enough medicine in the patient’s body to work effectively, but without overdosing.

The amount of medicine in the body after taking a medication decreases by a certain percentage in a certain time (perhaps 10% each hour, for example). This percentage decrease can be expressed as a rational number, 1/10. Hence in each hour, if the amount at the end of the hour decreases by 1/10 then the amount remaining is 9/10 of the amount at the beginning of the hour. This constant rational decrease creates a geometric sequence.  So, if a patient takes a pill that has 200mg of a certain drug, the decrease of medication in their body each hour can be seen in the folowing table. The Start column contains the number of mg of the drug remaining in the system at the start of the hour and the End column contains the number of mg of the drug remaining in the system at the end of the hour.

Hour	Start	End
1	200	9/10 x 200 = 180
2	180	9/10 x 180 = 162
3	162	9/10 x 162 = 145.8
.	.	.

The sequence of numbers shown above is geometric because there is a common ratio between terms, in this case 9/10. Doctors can use this idea to quickly decide how often a patient needs to take their prescribed medication.