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Important parameters in the Fish Tank 1: Volume, Clearance, and k

Important parameters in the Fish tank:
Volume of Distribution, Clearance, and Elimination Rate Constant
Now, let’s take a closer look at one of the elements that we just mentioned, the volume of distribution. This is the volume of the tank, and again, if we’re talking about a human being that red human there is our tank, even though it’s shaped like a human, we’re going to treat it as a tank. Now, let’s assume what what happens when we add drug to a patient. It’s like adding drug to a tank or a beaker that we can’t see. So I’ve put the beakers behind a screen. This is much like when we give drug to a patient we add the drug and we don’t know how much drug is actually in the patient.
We don’t know what the concentration is until we take a sample of blood from the patient. So, in this case, what we’re gonna do is add 100 milligrams of drug on the A side, we add 100 milligrams. On the B side, we had 100 milligrams. but on the A side the concentration that results from that 100 milligrams is 50 milligrams per liter. The concentration that results on the B side is only 10 milligrams per liter. So intuitive intuitively which side was the drug placed into a larger volume? Well, we can remove the screen and see when we add 100 milligrams to a much larger tank, the concentration is only 10 milligrams per liter.
If the tank is much smaller, the concentration of fish in that tank will be much larger. And so A gives us a concentration of 50 milligrams per liter. We can take this a step further and say that the volume of the tank is actually represented by the ratio of the dose that we gave to the concentration that resulted. In the larger tank, that we have a volume of 10 liters because 100 milligrams only produced a concentration of 10 milligrams per liter. With a smaller tank of a volume of only 2 liters, the same 100 milligram dose produced a concentration of 50 milligrams per liter.
So, we can relate this to the application of volume of distribution because it enables us to determine a dose that would be required to achieve a certain concentration or if we give a certain dose to a patient, if we know what the volume of the drug is in that patient, we can predict what the serum concentration. Dose equals concentration times volume or concentration equals the dose divided by the volume. Volume is simply a ratio of dose to the concentration that results. But we know that the patient is not a large tank filled with nothing but blood. So what actually happens that determines this ratio of dose to concentration?
Well, this schematic I think illustrates it well from winters textbook a basic clinical pharmacokinetics. It shows that in case A, we have administered 12 milligrams of drug. Now, two of those milligrams remain in the blood, ten of those milligrams distribute from the blood into the tissue. So when we measure a serum concentration, remember our frame of reference is always serum or plasma when we measure a serum concentration, we see that the concentration is only two milligrams per liter. So, since we have 12 milligrams that we’ve given the patient and we have a concentration of only two. It appears that we’ve placed that drug into six liters of volume.
Now take a look at case B Case B shows a higher state of tissue distribution. Now, 11 milligrams have distributed from the of that dose have distributed from the plasma into the tissues and only one milligram per liter remains as the concentration. So, it appears as though, we now have a value of 12 liters, only with a concentration of one milligram per liter. So, it appears that that the volume is much larger but the patient has not changed It’s the distribution of drug out of the blood that gives us a much lower concentration that makes it appear as though the volume is larger.
This is why we refer to it as the volume of distribution because the ratio of drug, given the dose, given to the serum concentration is not dependent on patient size. It’s dependent on the distribution of drug from the blood into the tissues. and example C shows high plasma protein binding in this case, the drug tends to remain in the plasma. So we have less drug leaving the plasma going into the tissues. So we have a concentration of 3 milligrams per liter and a parent distribution value of only 4 liters.
So it’s important to understand that the volume is nothing more than comparing the dose given to the concentration that results and the primary cause of difference between one drug and another and a given patient is the extent to which the drug leaves the blood, giving the appearance of a much greater volume. Now, periodically, during this lesson what I’m going to do is present a question for you to answer.
That’s what I’d like you to do is when you see a slide like this: Pause the video. Try to answer the question and then when you restart the video, I will explain how we think through the answer to this question and you can see whether or not you’re correct. So pause the video and answer this question you should have answered that A is false. For volume of distribution, answer A says tends to be higher for a drug that is highly bound to albumin. Well, we just explained on the previous slide if that if a drug is highly bound to albumin that will remain in the plasma, doing a higher concentration which gives a lower volume of distribution.
B is also false. Volume of distribution tends to be lower for a drug that is highly tissue bound. That’s a false statement. It tends to be higher for a drug that is highly tissue bound because the drug leaves the blood and goes into the tissues giving a lower concentration and a larger apparent volume. C is a true statement. Volume of distribution tends to be higher for a drug that produces a low concentration in the serum because the drug leaves the the serum for the tissues and gives an apparent higher volume So the answer to this question is C. Now, let’s take a closer look at clearance using our fish tank model.
Now, most clearance is produced by the liver or kidneys we know that . The rate of elimination depends on the concentration that’s the hallmark of a first-order drug. Now that the difference, here is the clearance the volume cleared by the net as we pull the net through the tank does not change as clearance changed . Now, why is that? Well, if you if you go back to our model of the net being pulled through the tank as we pull the net through the tank, it goes through a certain volume of water and the volume that the net passes through per unit time is our clearance. That’s the definition of clearance, the volume cleared per unit time.
It doesn’t change when the concentration of fish in the tank changes. It’s dependent on the size of the net. So our schematic shows that if we have a clearance of four liters per hour the net is going to pass through those four liters. So even as the concentration of fish in the tank declines, yes, our rate of elimination will be changing. We’ll be pulling fewer and fewer fish out of the tank but we’ll always be clearing the amount of fish that are contained in those four liters. So let’s pause for another brain exercise, according to the fish tank model, clearance is… the rate at which fish are removed. That’s false.
Clearance is not the rate at which fish are removed; that’s the elimination rate. Its clearance the volume of the tank. No that’s also false. The volume of the tank is the volume of distribution this clearance the fraction of a tank’s volume that has fish removed over time. No that’s something we haven’t talked about yet. that’s coming up on the next slide we’re going to talk about the elimination rate constant. D says that clearance is the volume of water that has fish removed from it over time. That’s our definition of clearance. The volume that the net passes through, all the fish in that volume will be removed over time, but that number of fish may vary.
So the number of fish removed over time is the elimination rate. So that’s also false. Our answer is D. Now the elimination rate constant. Elimination rate constant is a very interesting pharmacokinetic phenomenon. It’s the slope of the natural log of concentration versus time curve Now one of the early slides, and in this lesson demonstrated that when we apply concentration of drug versus time we get a curved line which you can see it’s kind of a faint gray line on this graph but when we plot the natural log of concentration that curved line straightens out into a straight line which I’ve shown in red.
So it’s the natural log of concentration one and the natural log of concentration two over time there was a straight line, the application is rather obvious. If we’ve got a straight line and we’ve plotted natural log of our various concentrations that means based on the slope of that line we’re able to determine the concentration at any time in the future or what the concentration was at any time in the past during the elimination phase which means no drug is being given, only elimination is taking place. We can also determine the time that’s required to reach a given concentration. If we’re shooting for a particular concentration and we know what the concentration is now. we know the slope of the line.
we can determine how long it will take for the serum concentration to fall. So the target concentration, the equation that we would use is elimination rate constant. This is just the slope equation Delta Y over Delta X, natural log of concentration one minus natural log of concentration two divided by the time interval between concentration one and concentration two. Now there’s an easier way to represent this equation. In terms of executing it with a calculator and that would be natural log of concentration one over concentration two. Take the natural log of that value and divide that by time to get the elimination rate constant. It’s simply an easier process in terms of pushing buttons on your calculator.

Prof. Daniel Brown clarifies the definition of various parameters.

He also illustrates the meaning of Volume of Distribution (V), so we can understand how it relates to a body. Besides serum concentration (C) and a dose, we still need to consider other significant factors.

We can learn the concept of clearance (CL) and elimination rate constant (k) from this video as well. If you have any questions about the ‘brain exercise’ in the video, please feel free to leave them below.


Prof. Daniel L. Brown

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Clinical Pharmacokinetics: Dosing and Monitoring

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