The relationship between Volume, Clearance, and k

Taking a closer look at the elimination rate constant we can identify a different perspective for defining what the elimination rate constant is It’s the fraction of drug volume cleared per unit time. It’s typically represented by the equation clarence equal K times V elimination rate constant times the volume of distribution. Another way to represent that equation just algebraically rearranging it as K equals clearance over volume which actually is a better representation of what’s occurring physiologically between the clearance and the volume. Let me illustrate this.

If the K or the elimination rate constant is point 20 per hour which we can also designate as 0.20 hours to the -1, what this tells us is that 20% of the volume is cleared of drug every hour. It’s illustrated in the schematic below where 4 liters per hour as indicated in yellow are cleared by the net that we pull through the tank and since we have a tank of 20 litres 4 litres represents point 2 so every hour, if we pull the net through the tank at that rate, removing the fish from those those yellow liters, we have 20% of the volume of the tank being cleared Now if we represent this by actual serum concentrations.

If we start with 10 and we have a K of 0.2 hours to the minus 1 and then have 1 hour we will have a concentration of 8.2 then of 2 hours six point seven and the end of three hours 5 point 5. Now what’s interesting is that if we plot the natural log of those serum concentrations, we get two point three, two point one, one point nine and one point seven. So the elimination rate constant can be considered from two different angles. If we’re considering the plot of natural log versus time, it represents the decline in the natural log of the concentration per hour.

Note that every hour, the natural log concentration drops by point two which is the value of K. That’s the slope of the line. But it also represents the 20% of the volume is eliminated or excuse me cleared not eliminated but cleared every hour. So let’s illustrate how this relates, the volume the clearance, and the elimination rate constant. Keeping in mind our equation K is the proportion of volume that is cleared per unit time. Why does an increase in volume cause K to decrease? We can illustrate that very clearly.

There’s our schematic showing a clearance of 4 liters per hour a volume of 20 liters and we have a concentration of 8 milligrams per liter such that K is 0.2 because 20% of 20 is 4 4% excuse me 20% of that 20 liter tank is cleared every hour. Now if we double the volume without changing the clearance our clearance is still 4 liters per hour. Now our volume is 40 liters. Our concentration has dropped to 4 milligrams per liter and the K has dropped to 0.1 hours to the minus 1 because now it’s 4 in relation to 40 liters rather than 20 liters. So when volume changes, K will change even though clearance does not change.

We can also pose the question why does an increase in clearance also cause an increase in K? Well, they’re directly related. If we double the clearance to 8 liters per hour and we don’t change the volume of distribution. We still have a 20 liter tank but now every hour we’re clearing 8 liters instead of only 4. So K has now doubled to 40% or 0.4 hours to the minus 1. So let’s pause for another brain check here. According to the fishtank model k is __. Pause the video and answer that question. Well, if we look at our answers, according to the fishtank model K is the slope of concentration versus time curve.

No, K is the slope of the natural log of concentration versus time curve. The concentration versus time curve isn’t even a straight line. B says the fraction of volume cleared to the total volume of the tank that is our definition of K using the fishtank model. C says it’s inversely proportional to clearance. No, K is directly proportional to clearance as we Illustrated on the previous slide. So our answer is B. Let’s try an exercise to see if you can answer this question. Patient is taking a renal-excreted drug and develops renal failure. Which of the following would definitely change in this patient?

Consider our equation K representing the fact that elimination rate constant is the fraction of volume that is cleared per unit time. Our answer is both clearance and K when a patient’s renal function or for adequately metabolized drug liver function changes. Clarence will change and when clearance changes because the organ that eliminates the drug changes. K will also change. Volume is not going to change and since K represents the relationship between clearance and volume. K must change if the clearance changes. Take home points from that exercise. Changes in renal function directly affect clearance of really excreted drugs that’s an important point to keep in mind. Secondly, when clearance changes, the K will change in direct proportion.

A change in clearance will not cause a change in V.

K is simply the fraction of two other parameters: Clarence and volume. It cannot change anything. However, if clearance doesn’t change, a change in volume will cause an inversely proportional change. Okay it’s important to remember here in the relationship between clearance volume and emission rate constant. That the elimination rate constant is simply a representation of relationship between

two other important variables: clearance and volume. The clearance is inherently dependent on what’s taking place in the body. How the drug is being eliminated by the body? Volume is also dependent on things taking place in the body, but changes involume and clearance may be independent of each other. But the elimination rate constant is nothing more than an indication of the relationship between clearance and volume. so you can’t look at this equation from a purely mathematical perspective and assume that if K changes that clearance or volume would have to change in response. No it’s the other way around. A change in clearance or volume will cause a change in K.