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The concept of an accumulation factor

The concept of an accumulation factor
This the concept of percent loss is accumulation factor is a very significant point. So I want to take a closer look at how it relates to multiple dosing regimen and how the the one minus e to the minus KT factor comes into play once again as it did in continuous infusions Remember e to the minus KT is the percent remaining and 1 minus e to the minus KT is the percent lost. Now when we’re talking about a continuous infusion 1 minus e to the minus KT defines the accumulation from from time 0 up to a steady-state concentration.
So prior to that study said concentration 1 minus e to the minus KT tells us what percentage of the way we are to steady-state. In the case of an intermittent dosing regimen, a multiple dosing regimen, it’s a little different. Now the accumulation is not from a point 0 up to a steady-state concentration. The accumulation that results in multiple dosing is due to the fact that every time we give a dose, some drug is left over at the time we give the next dose.
So what happens is there’s a gradual accumulation and the more drug that’s left over after each dose, the more accumulation occurs as the patient increases from the serum concentration that resulted after the very first dose which is the concentration at time 0. How much that accumulates to the ultimate steady-state concentration and the accumulation factor that determines that difference is 1 minus e to the minus K tau where now T is the dosing interval or rather than the duration of the infusion.
So when we’re determining the C Max and C min from a multiple dosing regimen, we have the equation C max at steady state is equal to dose over volume which is C at time 0 over 1 minus e to the minus K tau. And we can see that the maximum concentration using the example shown of a 200 milligram dose, a 20 liter volume and a KA of 0.125 hours to the minus 1 200 milligram dose into 20 liter valium will give us a concentration at time 0 of 10 milligrams per liter and you can see that on the graph.
The dosing interval is 6 hours, so e to the minus KT where the dosing interval is 6 hours and K is 0.125 hours to the minus 1 is 0.47. One minus e to the minus K tau again where tau is 6 hours is 0.53. Obviously e to the minus K tau and 1 minus e to the minus K tau have to add up to 1. Now what’s interesting is that if we consider the C max that results from this example of 18.9 milligrams per liter. If you multiply 18 point nine times the value of e to the minus K tau which is 0.47, it arrives the Cmin of 8.9 milligrams per liter.
It’s all based on the accumulation factor of one minus e to the minus K tau. So if you take the dose divided by the volume and then divide that factor which is essentially C times zero of ten milligrams per liter and divide that by the value of one minus e to the minus K tau which is 0.53. That will give you the Cmax of 18.9 milligrams per liter. When we’re determining the Cmin concentration at steady state, we simply take the C max at steady state that entire equation and multiplied by e to the minus K tau. Because now each of the minus K tau is the percent remaining.
So if we take the CSS times the percent remaining at the end of the dosing interval, that represents Cmin at steady state. Let’s illustrate how this accumulation process occurs. We have 200 milligrams every six hours where they see at time zero of ten milligrams per liter That’s our starting point e to the minus KT is 0.47; one minus e to the minus KT where T is the Tao of six hours is 0.53. So we have a concentration of 10 milligrams per liter at the end of six hours that has fallen to four point seven milligrams per liter but we give a dose of 200 milligrams and every time we give a dose of 200 milligrams.
We’re going to increase the serum concentration by 10 milligrams per liter because we have a 20 liter tank that we’re putting 200 milligrams into. Therefore, the c-max after the second dose it’s going to be fourteen point seven milligrams per liter. We then have a certain concentration after 12 hours of six point nine milligrams per liter. Again 0.47 times fourteen point seven and the serum concentration maximum after the third dose is going to be ten again, based on putting 200 milligrams into a twenty liter tank.
After eighteen hours, that sixteen point nine will fall to forty seven percent, which is seven point nine times e to the minus K T where T is the Tao of six and C max again will increase by ten milligrams per liter up to seventeen point nine. Lastly the C min after twenty four hours will fall to eight point four and the c-max will increase after the fifth the fourth dose to eighteen point four milligrams per liter. Okay let’s take a look at the answer to this question. If a patient receives a multiple dosing regimen, the dose over tau is equal to clearance times steady state concentration. That is a true statement.
Cmin is equal to C max times e to the minus K tau. That’s also a true statement. and C max divided by C at time zero is equal to one over one minus e to the minus K tau. That essentially defines what happens during accumulation in a multiple dosing regimen. So it’s also a true statement. So the answer to this question is E. A, B and C are all correct.
Now let’s take a closer look at this first order process of dosing because the beautiful thing about first order drugs is that they manifest a linearity in that if we want to double the serum concentration or the average steady state certain concentration all we have to do is double the dosing regimen. And we want to cut it in half. We cut the dosing regimen in half. There’s a direct proportionality between the dosing regimen and the serum concentrations that result. In this example we’ve taken 200 milligram q 8 our regimen and doubled the dose to 400 milligrams q 8. So, in a sense we’ve doubled the dosing rate.
Now the patient’s clearance has not changed the volume has not changed, we can see that the concentration at time 0 would double because we’ve doubled the dose, the elimination rate constant has not changed. But the area under the curve would double. We also see that the C max will double the C min will double and the C average steady state would double and this is illustrated in the graphs to the left there the red curve representing the larger dose the blue curve the smaller dose.
So what we really have here as a direct proportionality and that the dose over tau of a new dose divided by the dose over tau of the old dose is the same as the average steady state concentration that resulted from the new dose divided by the average say state concentration that resulted from the old dose. If we have a 25 milligram per hour dosing rate as we do with 200 milligrams every eight hours, that yields a serum concentration and average serum concentration at steady state of 12.5 milligrams per liter. If we want to double that to 25 milligrams per liter we would simply double the dosing rate to 50 milligrams per hour.
And we would then obtain the average steady-state concentration of 25 milligrams per liter. So first-order drugs make dosing changes very simple when we’re dealing with direct proportionality changes. The same thing would apply with continuous infusion as as we’ve shown here with the dosing rate of dose divided by tau. The rate of infusion change will be proportionate to the steady-state concentration change that we’re trying to achieve. Now one thing I want to emphasize here is that our dosing regimen does not affect the time to steady state. As we change the dosing regimen, the actual concentrations will change. But the time that it takes to achieve steady-state will not change.
In the first example here of 200 milligrams every six hours as a multiple dosing regimen 95% steady-state is a level of 19.1. Now in the case of the they curve on the right is 200 milligrams q 8 hours, it doesn’t show on the slide, but the maximum concentration is only sixteen point six. But the time to ninety-five percent of the steady state which is 15 point eight is still the same same thing applies for the rate of infusion below, whether it’s 25 milligrams per hour or 50 milligrams per hour. At 95% of steady state the serum concentration is ten point three whereas a 95 percent of steady state, the center concentration at 50 milligrams per hours twenty point six.
But the time it takes to reach that 95 percent of steady state concentration is still the same. It’s still four point three half-lives. So what I would like you to remember is that if you’re trying to achieve 95% of steady state, that’s your standard for a reaching steady state. It’s going to take four point three half-lives regardless of the dosing regimen. It doesn’t matter what the size of the dose is it doesn’t matter what the dosing interval is. It’s based on the half-life because the half-life relates directly to the elimination rate constant.
Okay and the half-life of four point at the half-life times four point three is ninety five percent if you’re satisfied shooting for ninety percent as achieving steady-state you would only require three point three half-lives. Now let’s consider bioavailability we mentioned in the very first lesson. That we’re going to assume that bioavailability most of the time is a hundred percent so we don’t have to worry about it for most of our examples. But there are times when you have to factor in bioavailability.
When you give me the world dose and the patient doesn’t get all of the dose, so we have to assume that the functional dose a patient receives refers to the amount of drug that actually contributes to the serum concentration. Remember the the blood serum the plasma concentration is our total frame of reference. So this would be the same as if we gave a dose intravenously. All the dose gets into the blood. Okay, but when we give an oral or intramuscular injection or an oral dose that may not be the case. Now the functional dose of patient receives will be less than the oral dose if the bioavailability is less than one hundred percent.
So we can represent this by the equation dose IV equals dose PO times F, the bioavailability factor. What’s most important here is that you remember that if the patient has to take a dose orally. It’s going to be larger than what the IV dose would be unless bioavailability is 100 percent. Now if the bioavailability is 100 percent and then the IV dose would be the same as the PO dose because the F value would be equal to one. But if the bioavailability is anything less than a hundred percent or F is anything less than 1, then the PO dose is always going to be more than the IV dose.
So rather than remember equation, just remember that relationship To keep things simple, if the F is less than 1, the oral dose is going to have to be larger than the IV dose and it will be proportionately larger based on the value of bioavailability or the value of F. Now let’s summarize the dosing equations that we’ve covered. For continuous infusion CSS is equal to the rate of infusion divided by the clearance.
If we want to find a concentration some time prior to steady state for a continuous infusion that would be equal to the rate of infusion times the accumulation factor for continuous infusions which is 1 minus e to the minus K T where T is the time of infusion divided by the clearance. For a multiple dosing regimen and which we’re zeroing in on a see average steady-state concentration. That’s equal to dose over tau or the dosing rate in milligrams per hour divided by clearance. For the c-max multiple dosing regimen that is equal to the dose divided by the volume over 1 minus e to the minus K tau That’s our accumulation factor for multiple dosing regimens.
For single dose C at time 0 would simply be dose over volume and the C min at steady state for multiple dosing regimen is simply the C max at steady state times e to the minus K tau. Let’s try another exercise. A patient is on a continuous infusion with an infusion rate of 40 milligrams per hour and a CSS of 12 milligrams per liter. Which oral regimen would achieve a C average steady state of 15 milligrams per liter? If the oral product comes as 200 and 250 milligrams capsules and it has an F of0.80. So it’s only 80% bioavailable. The answer to this question… is 500 milligrams every eight hours.
Now let’s walk through this to make sure you understand how we apply these principles. To a problem like this first of all based on our first-order linearity principle. We know that 40 milligrams per hour. If that achieves a certain concentration of 12 in order to achieve a certain concentration of 15. We would need to give the patient 50 milligrams per hour That’s intravenous That’s not factoring in bioavailability. For 50 milligrams per hour of an intravenous medication If we’re gonna give it orally. We’re gonna have to scale that up by a factor of 80%. Because our our bioavailability is 0.8.
So instead of 15 milligrams per hour which is what we would give IV, we’re gonna have to give 62.5 milligrams per hour as the oral dose. So that means we’ve got two options now we can figure out should we give that 62.5 milligrams per hour based on a q6 regimen or a q8 hour regimen? Knowing that the strengths that we have available to us are 200 and 250 milligrams capsules. If we take sixty two point five milligrams per hour times six hours, that’s 375 milligrams and we would give every six hours. If we take sixty two point five milligrams per hour times eight hours, that comes out to 500 milligrams.
So that actually gives us our best dosing option because we simply have the patient take two of the 250 milligrams capsules and we’re right on the money at 500 milligrams whereas the 375 milligram dose that would be required on the q6 regimen or if we doubled it on a q12 regimen doesn’t match the 200 and the 250 milligrams capsules that we have available to us. So, it’s a straight linear proportion but we have to scale up the dose based on the bioavailability.

We can regard % lost as an “Accumulation Factor”.

With dose, volume (V), elimination rate constant (k), C times zero (C0), and tau, we can calculate e to the minus KT. Next, we can easily know that one minus e to the minus KT, which is the Accumulation Factor. Then we can obtain Cmax,ss and Cmin,ss .

Besides, Prof. Brown gives a clear illustration of drug accumulation and examples of the first-order linearity.

What is bioavailability? And how do we apply it? Please leave your answer below.


Prof. Daniel L. Brown

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Clinical Pharmacokinetics: Dosing and Monitoring

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