# DRUG EXCRETION

### CLEARANCE

• Let’s imagine that we have a tube and it contains one liter of plasma. let’s imagine that we put five grams of a drug into this liter of plasma. You can easily imagine that the concentration of the drug inside this litre must be 5g/L. Now I want you to imagine that the kidneys excrete 5 grams from the body, so kidenys take five grams out of this liter of plasma and remove it from the body. Well it’s very easy to imagine 5g of the drug being removed but what’s harder to conceptualize in your head is the fact that the kidneys have therefore cleared 1 litre of the drug by removing those 5 grams.
• Now we know it’s one liter because I told you as one leader at the beginning of this exercise but if we didn’t know was one liter we could easily calculate that. We would take the 5 grams which were removed from the body and divide by the concentration of 5g/L and this would give us a clearance of one liter.
• So the amount removed by the kidneys divided by the concentration in the plasma will give you the clearance and that represents the volume that held the amount of the drug that was cleared by the body.
• Clearance represents the volume of blood cleared of a drug.
• It’s the volume of blood that contained the amount of drug that the kidneys removed.
• It’s a number in liters per minute or volume flow that the kidneys are clearing and we can calculate it by taking the excretion rate for example 5g/min and then dividing it by the plasma concentration, for example if the plasma concentration were 5g/L we could put that on the bottom and then we divide these two numbers and we would get a clearance rate of 1L/min for drug X by the kidneys.
• Clearance of drugs mostly occurs via the liver or the kidneys.
• The liver clears drugs in two ways: It can bio transform the drug into metabolites and It also can excrete drugs directly into the bile and then they go out in the GI tract into the stool.
• The kidneys can clear a drug by secreting it or filtering it and excreting the drug into the urine
• If you know that the liver and kidneys clear drugs from the body then it’s easy to imagine that in liver or kidney disease the clearance rate may fall, this means that the concentration of the drug in the plasma may rise and sometimes toxicity can occur and many drugs need to have their dosages adjusted in patients who have kidney or liver disease.
• You can also calculate the clearance if you know the volume of distribution but in order to do this you need to know something about the drug called the elimination constant (Ke)
• The Elimination constant represents how quickly the kidneys clear a volume of plasma of the drug. So if you know that constant then the clearance of drug X will be equal to the volume of distribution times the elimination constant.
• This equation sometimes confuses, because it implies that the larger your volume of distribution, the more the kidneys clear the drug and that should make sense to you if you think about what the volume of distribution means.
• So let’s imagine that we’ve got our tube again with 1 liter of plasma inside it and let’s imagine that that one liter represents the volume of distribution of drug A. We put 5 grams of drug a into that 1 liter and it distributes throughout the 1 liter because that’s its volume of distribution. Now I want you to imagine that the kidneys clear 5 grams a minute by excreting the drug in the urine and the clearance of drug A will be one litre.
• Now let’s imagine we have drug B. That drug B has a volume of distribution of 10 liters instead of 1 liter, so I’m going to fill this tube with 10 liters of plasma. Now I’m going to put 5 grams of drug B inside. Now the kidneys excrete drug B the same way they do drug A, so they’re gonna remove these five grams but the clearance for drug B is going to be much greater by removing five grams of drug B you’ve just cleared ten liters of fluid because it’s volume of distribution is so much higher.
• Going back, When the volume of distribution goes up the clearance goes up. Remember the clearance is not the amount of the drug being removed by the kidneys, it’s the volume of plasma that has been cleared of the drug. So the higher the volume of distribution the more plasma that gets cleared.
• So I’ve summarized that for you up here. The higher the volume of distribution the higher of clearance. Suppose 10 grams an hour is removed from the body. If you have a higher volume of distribution for the drug there’s a higher volume holding that 10 grams and therefore you have a higher clearance per unit time.
• One of the reasons this equation can be confusing is because it’s often illustrated as an equation to calculate the clearance from the volume of distribution and the elimination constant but that’s really not how it’s usually used. It’s usually used to calculate an elimination constant Ke for different drugs. So if you think about it that way then the elimination constant is equal to the clearance divided by the volume of distribution and this makes a lot more sense intuitively.
• The higher your clearance the higher your elimination constant. Conversely the lower your clearance the lower your elimination constant. If you’ve got a large volume of distribution this is going to shrink or decrease the magnitude of the elimination constant. This is the way it’s usually used in the pharmacology literature to calculate elimination constants for different drugs, so that you can compare one to another.
• You can also determine the clearance of a drug by monitoring its plasma concentration over time as its excreted from the body. let’s look at this figure here. Imagine that we administer a drug at time zero. what we would see is that the plasma concentration would begin to rise then it would peak and then it would begin to fall as the drug is excreted from the body and eventually it would fall to zero.
• You can use the area under the curve which can be determined mathematically to calculate the clearance. If you take the dose that was initially administered and divide by the area under the curve you will get an answer that has units of liters per minute and that answer will be the clearance of that drug from the body.

### HALF-LIFE

• The half-life of a drug is the time required to change the amount of drug in the body by one half
• It’s usually determined by calculating the time required for the concentration of the drug to fall by 50%.
• The half-life depends on the volume of distribution and the clearance of the drug.
• You can imagine that, drugs with a very large volume of distribution take a long time for their levels to fall. You can also imagine that drugs that have a very high clearance rate fall more quickly than drugs that have a slow clearance rate. So the equation for half-life is that the half-life is equal to 0.7 times the volume of distribution divided by the clearance.
• So let’s talk about what’s going on inside the body after each half-life. Let’s imagine that at time 0 we have 100% of the drug inside the body. well once one half-life goes by the amount of the drug in the body will have fallen by 50%. After two half-lives were down to 25%. After three we’re down to 12.5%, after four we’re down to 6.25% and then starts to get even lower.
• Now it turns out that once the amount of the drug remaining in the body falls below 6.25% then most drugs don’t have anymore therapeutic effect. So we generally say it takes four half-lives get the drug out of the body, which means it takes four half lives for it to fall to such a low level that it’s not going to have any clinical impact for the patient.
• So the half-life has a very important role to play in the amount of time it takes for a drug concentration to reach steady state inside a patient.
• Let’s review this chart here. In this chart we have the number of half-lives of the drug along the x-axis and we have the amount of drug or the dose that we’ve administered to the patient on the y-axis. So let’s imagine that at time zero we give the patient a dose of 10mg and then we let one half-life go by, well the amount that’s inside the patient is going to fall from ten to five after one half-life that’s a 50% drop.
• But now let’s imagine we give another 10mg dose, well this is going to pump the amount of drug in the patient up to 50mg. When another half-life goes by, we’re going to fall down to 7.5mg right here. But then we give another 10mg dose and we go up to 17.5mg right here. After another half-life we fall to half of 17.5 which is 8.75. Now we give another ten milligrams and we go up to 18.75. After one more half life we now fall to 9.4 which is half of 18.75. Now we give another dose and we go up almost to 20.
• What’s going to happen after we pass four half-life. So we’re basically going to go up and down from 20 down to 10 up to 20 down to 10 and up to 20. So what you can see is that somewhere between four and five half-lives we reach steady state such that the dose is going up and down between the same two numbers.
• So it’s very important you understand this chart and it’s very important that you understand that it takes four to five half-life’s in order to reach steady state when you’re giving a repeated dose of the same amount to a patient over time.