Understanding Compartmental Models in Pharmacokinetics

So this morning, what we're going to be doing is to look at how we describe the compartmental models using mass balance equations. So what we're going to do is we're going to look at different types of models and, you know, in PQ, and you know, in PQ. And we're going to look at, key components of compartmental motor, and we're going to look at mass balance issues, you know, specifically. So we're going to be looking at equations a lot, you know, in this vector. And I would say that, to describe them. Okay. So let's make a stats. So in general, I'm sure you've seen this a lot with you. We talked about AGME when we talk about pharmacokinetics. So we talked about we talked about the drug administered goes in, you know, into the system, and you have it being distributed. Execution. But when the drug is also being distributed, it goes to the sites of it goes to the site of action because that's where it has to produce its effects. If the And then once it gets to the site of action, then it it produces whatever we want it to produce, especially in terms of efficacy. But we know that we can also have off target effects. Effects, those are what you really want to call design effects. The drug will go to muscles, you know, go to other places where it can, you know, produce effect that we do not want. And those are all go and design effects. So we can have efficacy on the we're not necessarily talking about just the efficacy. We're also talking about, toxicity. Now in compartmental modeling, there are 2 types of models that we use. So we use what is known as the typical compact metal models, and we also use what is known as the piezoelectric monocokinetic models. So in this model, our focus is going to be mostly on the compact metal models. I will leave Dan. I'm sure you've started already with, PBPK 1 to sort out these electrochemical based, orders. And, of course, you've seen it in fact, you know, in lectures in PBPK 1 already on, you know, tissues and organs that are represented as compartments in PBPK models. And that's actually how it is structured. For the way we describe or we use equations, mass balance equations to connect these tissues and organs are exactly the way we do with compartmental models. So a lot of the things that we're going to look at today is also relevant for this analytical based monoclonal model. And that is why with Dan, I think in another 1 or 2 weeks, then you'll be looking at compartmental models, the equations behind physiologically based on our magnetic models. We've done. And when we talk about compartmental modules, I'm sure Leon also introduced, you know, this to you a little bit. We're talking about just using 1 or 2 compartments to represent the body. So we're just everything in the body, we put them together into 1 or 2 compact means. Okay? And sometimes we can say 3, you know, or more. You know? But these are usually very, very limited number of compact means. And our assumption is that drugs distribute and eliminate from this compartments. So we have drug distribution and elimination from this compartment. Now the number of compartments that we use for each drug is not fixed. It is determined by the data. We look at the data, then we decide how many compartments do we need to describe this data. And that is why it is often described as being data driven because it is the data that will tell you how many compartments that you need. You need. Online, these are the things that you need. Online, these are the things that you need. Online, these are the things that you need. I just allow the data to tell us, you know, how many of those are needed. And these compartments do not necessarily this is very important. It will not necessarily correspond to specific organ or reflect physiological processes. You cannot say the central compartment is just blood or central compartment is just muscle or the peripheral compartment is just adipose. We don't we do we do not say, yes, it may reflect what is going on in plasma. It may reflect what is going on in adipose, but you cannot say specifically that this is just blood or plasma or it's just adipose. Okay? And because this compartment do not necessarily correspond to specific organs or tissues of the body, that means that when we get parameters from our fitting of the border to this data, then we have to be very careful. So the parameters that we get from this then have to be interpreted with caution because you cannot say that the volume of the central compartment is definitely the volume of the log or definitely the volume of, you know, I don't know, you know, for adding for a peripheral compartment, you cannot say it's definitely the volume of adipose. It gives us the extent of distribution, but you cannot say categorically that this is what it actually means. Now there are advantages and disadvantages to using this very simple models. The advantages, I'm sure you can easily tell, is that they are very easy to implement. Okay? In maybe 10, 15 minutes, you can come up with 1 or 2 compartment model that describe a data. But if you are going to do that for a physiologic based model, then you need more than 10 minutes. You know, because you need to gather a lot of parameters. So advantages are, one of them is a very easy to implement. Mostly, they do not require prior information. You don't need to have prior information to have the data, then you let the data tell the department and panelists. Okay. They do not require theoretical parameters, for instance. They can be adjusted to focus on specific processes when orders are not necessary. Okay. If I want to focus on, say, the liver, for instance, I can have a compartment for liver, and then, say, compartment for liver, compartment for blood, and then the rest of the body. If I need it, then I can include that as well. So I can also incorporate specific compartment. You know, I can incorporate compartment for specific organ of tissues in this setting as well. And they are useful for relating parameter values to. Okay? That's one of the advantages of them. If I see differences in plasma concentration between individuals, and I know that this is due to the differences in their genotype of c 2 c 19, I can easily bring that information into the model to explain why I have the differences in the plasma concentration. And it can be used to simulate what other scenarios easy. If I want to look at what happens if I give the drug at twice the dose that I've studied, I can easily simulate that scenario and look at, what happens. But the disadvantages are, it is data driven. Okay? It is data that tells you what's going on. If your experiment has not been conducted very well, then you may end up with a wrong model or the model that is may not necessarily adequately describe your drug. For instance, if your drug is such that there is a very, very long terminal phase to your plasma concentration. There's a long terminal phase, you know, to the, you know, to your plasma concentration. A very typical example is methotrexate. You can look it up, you know, maybe after the lecture, you know, compartmental models for methotrexate. You will see that some people will say that for methotrexate, it is 4 compartments. Some people will say it's the 3 compartments. If you consider to collect your data for a bit longer, not you may see 3 or 4, then if you collect it for a very long time, then you may see actually true number of compartments that describes your plasma concentration, profile. If I go to, you know, children, then the is likely going to be different. The plasma concentration is likely going to be different. So I cannot extrapolate that model to children. So I have to be very careful when it comes to, you know, the way I used in order because of that. Okay. So now let's think about and let's start to follow, you know, move towards how we can put these models together. Now the first thing I want us to imagine is that we have, you know, maybe a big, you know, system like we have here. It it's just a big tank. If you get where I mean, it's just a big tank. If you say that tank is just our body. Okay? That tank is our body. And what we want to do, you know and for that tank, we don't we don't know the volume. Let's pretend that we do not know the volume of that tank. We do not know the volume of that tank. Okay? But we want to conduct an experiment that will help us to determine the volume of that tank. So our aim is actually to determine the volume of that tank, which we, you know, believe is just a one compartment system. So we are talking about 1 compartment now. You know, those are the terminologies that we're using. That system, you know, is just a single compartment. So that is just, you know, the partition that describes what we are looking at. In this case, there's no partition. It's just a single kind. So compartment. And then we can talk about steady states. And the steady state means that we what we have in terms of concentration is uniform throughout the system. There's no net change in concentration. You know, this the system is at steady state. Okay? And, also, we can also talk about border strokes. And what that means is that if we add, say, you know, you know, a dye, you know, dye, you know, pouring agents into that tank, you know, Then when we add it, if we assume that we add it all at once, then that is a glucose dose. As you know, you know, when you add, the, the. So the method that we're going to use that we're going to use a dose of a known amount of a chemical, which is going to be a dye. And we're going to conduct an experiment to measure the concentration of the drug in the stamp. And our calculation then is going to be that if we assume that when we add the dye to this compartment, everything distributes equally. There's no next, you know, change in concentration. The concentration measured in any part of this tank is the same. Okay? The concentration measured in any part of this tank is the same. So that means that if we know the amount of the dye that we have added, and if we know the concentration that is now measured in any part of this tank, okay, if we conduct an experiment, I will take a sample from this tank and we determine the concentration of the drug in any part of this tank. And we can then use the concentration that is now measured and the dose, which we know of the chemical that we have added to this tank to work out what the volume of this tank is going to be. So the volume of this tank is just going to be the dose that we've added divided by the concentration that we have measured in any part of the tank. Okay? Now, again, you know, going back to it, there are certain facts or an assumption that we have made, you know, with this system. One of them is that the system is well stirred. That is equal distribution of the chemical, the dye, you know, which means there is instantaneous distribution of the chemical within the system. That's our assumption. One of the assumptions. And the chemical is not able to leave the system. Everything is contained within that system. And that is why it doesn't matter when we take that, you know, sample, you know, because everything's contained and contained it anywhere in in any part of the tank. But there's also some facts that we can establish. And one of them is that only the concentration of the sample is useful. Okay? Only the concentration of the sample is useful. Amount of the chemical in the sample is not. If you determine the amount of the drug in the sample that we have taken, it's not useful to us because we cannot relate that to the dose to work out the concentration. I mean, the volume. So it's only the concentration of the chemical in the sample that we have taken that is important to us because that has some element of volume that we're interested in it. And then once we relate it to the dose, then we can get to the volume of the system that we're interested in. And, also, we know the dose, which is the amount of the chemical that we added initially. And the concentration is also determined, you know, from our sampling. And the volume that we have now, you know, we can say that it's actually operational. It does not reflect the actual volume of the system. It's just based on what we can determine from this experiment. Okay? For us, for instance, if we have, you know, some of these chemicals talk to different parts of maybe the, you know, the the the container of the tank, then that may affect what we measure in terms of the concentration. So the volume of that we have measured in this step is actually operational. And the calculation only works if the compound is rapidly and uniformly distributed. Okay. If those 2 as you know, if those 2 are not, you know, are not there, you know, it is not rapidly distributed and not uniformly distributed, then the calculation will not work for us. Okay? These are some of the things I'm sure that you learned when you are looking at one compartment model, for instance, with. Okay. Now let's not talk about when we assume that we can also have our chemical leave system. Okay. It's not just that. We add the chemical to the tank, and it stays there. Now we add chemical to the tank, and the chemical can also leave. And that is what we add when we talk about dosing of the drug into human body, for instance. Okay? So in this case, our tank is human body. And the chemical that is added is the dose of our drug. And the drug leaving the system is just elimination of our drug, either by metabolism or by exclusion. Okay? So now in this system now, as you can see, we are assuming we are still assuming a one compartment system. We are assuming that we're going to give a dose of the drug instantaneously as a bolus dose. We are assuming that we are interested in the following of distribution of the drug in that time. We are also assuming that the drug is able to leave the system. Okay? What that means is that we can also do the same experiment, you know, and take concentration of our drug, you know, but now it means that when we take the concentration of the drug at different times, the the concentration will be different because the drug is able to leave the system. Okay? So we have to now conduct an experiment that will take the drug concentration at different time points. And also, it's important for us to know that for this system now, it is only at time 0 when there is no ability of the drugs, of system to get rid of the drug that we can work out the form of distribution of that time. Okay? It's only at time 0 that we have a system that is similar to what we had in the previous slide. Because when the system is attacked as at time 0, then there is no elimination. So if we can determine the concentration of the drug at that time 0, and we can assume that this the drug is able to distribute equally within the system at that time 0, then things that you did with Leon when you have to extrapolate that y to time 0 when it crosses the y axis. Basically, what you're trying to do is to work out the concentration of the drug at times 0 because that is the only time that the concentration of the drug can help us to get the form of of the system that we are interested in. Okay. And we have 2 parameters describing the system. We have clearance, which is a parameter of elimination, and we have form of distribution as well. Okay? Clerance is the volume clear of drug per unit time. I'm sure you know that. And, you know, now we can also look at the plasma concentration profile, you know, in this system. You know, if we say this system is like, you know, what we have when a drug a a dose of a drug is administered intravenous intravenously as a bonus drug. You see the plasma concentration profile on the right hand side. Okay. That's what you see, where you measure plasma concentration at different time point, and you plot that against time. Now as I've said, the volume of distribution is going to be now equals to the dose of the drug divided by the concentration at time 0 because that is the only time we can determine the concentration of the drug. And if we look at the AUC of our probe, the area under that plasma concentration curve, then we can integrate our plasma concentration profile with respect to time, and that will then give us our AUC, which will be equals to the dose divided by kR if we solve that integral. And then we can also relate our clearance to volume of distribution by using elimination rate constants. Okay? So these are some of the things that you've already covered with Leon, so we are not going to spend a lot of time on that. Okay. So now what we're going to be looking at next are what we call the rate equations. Now what we call rate equations in PP are basically equations that allows us to to describe drug movement in the body. So when we want to describe how drug is moving in the body, we can use what is known as rate equations. Okay? So rate equations are used to describe drug movement in and out between our compartments. Okay. So and that is what we're going to be looking at a lot today. Now in terms of how the drug is getting into the body. Okay? In terms of how the drug gets into the body into in the first instance, there are a number of input systems that we can use. And one of them is the, which we just looked at. The means that the drug is added into the system all at once. And there is also what is known as the 0 order. Okay? The 0 order means that the drug is added into the system at a constant rate. Okay? And the rate of the the at which the drug is added is added into the system is just constant over time. Okay? And that is what you have with an infusion. You know? So when you have a 0 order input into the system, then you have constant rate of the drug going into the body. You can also use what is known as first order input into the body. First order input into the system just means that the rate at which the drug is going into the body is proportional to the amount that you have in the system where you're adding the drug from. Okay? So there's a bit of proportionality there when you're talking about first order inputs. We we're we're going to see example of that as well. And for outputs, you can have either 0 order inputs, you you know. And also, you can also have 1st order and outputs. You know, for outputs, you can have 0 order. You can have 1st order as well. We're going to look at some examples of this, but our focus is going to be mostly on 1st order inputs and output and maybe bonus and 0 order. In terms of outputs, we're not going to look a lot at 0 order. We're going to look more at 1st order. But what does 0 order process means? 0 a 0 order process means that the rate is constant and does not depend on drug concentration or amount. The rate at which the drug is going in or out of the system is constant, and it does not depend on the concentration or the amount that you have in the system. But when you talk about first order, the rate is proportional to truck concentration or the amount that you have in the system. And when you talk about first order, usually, we use this to describe what is known as we usually use it to describe oral drug absorption, for instance. Okay? And if you look at the equation that we have there for rates for 0 order, we're saying the rate is just equals to k multiplied by concentration raised to the power of 0. Okay. Because concentration raised to power of 0 is just 1, then it means that the rate is just equals to k. So the rate of drop inputs or output from the system is just described by that k, which is just the rate of drug in or out of the system. But for first order, you will see that the rate is just equals to k, which is the rate constant describing the first order process multiplied by the concentration or the amount. In this case, it's concentration raised to the power of 1, and that is just equals to k multiplied by the concentration. Okay. So the rate is proportional to the amount or the concentration that you have in the system. In this case, it's concentration. The rate is proportional to the concentration that you have in the system. And rate constant is actually the proportionality constant that describes that weight. Okay. So in addition to this, we can also have a second order process. A second order process just mean that the rate is also proportional to concentration, but it's proportional to the to concentration raised to the power of 2. Okay. So it's more or less, you know, similar to 1st order, but now the proportional, you know, the rate is proportional to the square of the concentration that you have in the system. Okay. Now another rate that we can another rate, you know, of input or output that we can have into the system is what is known as the saturable process. Okay. And this is probably what you've looked at maybe in your PBPK, you know, model with with with that. Okay. What this means is that and usually, actually, we use this to describe inputs and outputs, which are usually dependent on carrier mediated absorption or distribution or saturable elimination, which can be, you know, for metabolism or exclusion. They are usually described using the equation. And in that case, the rate is equals to as you can see here, if the rate is equals to 3 max multiplied by concentration divided by concentration plus our kilometers parameter. So the system is actually saturable. So at very high concentration, you can see that we are approaching. Okay. So what it means is that as concentration increases, rate also increases. But as concentration increases further, that rate, approaches a plateau, and that plateau is equivalent more or less to our. And that is the saturable, process of, input or outputs into a system. But for 1st order, as you can see, this is just a broken line here. This is the 1st order process. As concentration increases, the rate of 1st order increases indefinitely because it's just the rates equals to proportional constant, which is our rate constant multiplied by the concentration. So if concentration increases, that rate will continue to Okay. So now let's go back to this model that we were looking at. Okay? So what we are going to do now is we're going to look at rate equation for the system that we have here. Okay. So this is the model that we that we have. This is the system that we looked at previously. So what you have there is a 1 compartment model. You have a 1 compartment model that is that you input into the system that system is not a bonus dose, and the outputs from the system is described by the first order elimination. And in this case, we are assuming the body is a work of partner system. Okay. So what it means is that the model that we have here is therefore a work compartment is by a first order process, then we can use our parameter that is describing that first order process, which is the elimination rate constant So in this case, what we know is that the rate of change of drug in this system, which is the a b t. Okay. The rate of which is through the initial condition. And we have also described how the drug is changing in the system by using the entity. And that also allows us to incorporate the first order process of any relationship of a drug from the system. Okay? And therefore, we have a mass balance system. Okay? So that is your first, probably, I don't know, first differential equation. Hopefully, that is straightforward and simple enough to understand because we are going to start to make this complicated as we go along. Okay. Now one right or not, which means the units that I have on the left hand side has to be the same with the unit of what I have represented on the right hand side. Okay. In this case, by the a to t, a is in amount, probably milligrams, and my t is in time, probably in hours. So on right on the left hand side, I have milligram per hour. Then on the right hand side, I need to have milligrams per hour too. Okay. Any questions so far? Okay. Everything looking straightforward so far? Okay. Now what we can also look at is a situation where we are not just we are not we are not writing the mass balance equation. We are not writing the mass balance equation in terms of the amount of You know? And that's what we're going to do here. So it's the same process. Okay. So our o d now is not going to be an amount, but it's going to be in concentration. And what it means is that because we want to write the o d in terms of concentration, this is our problem. Okay. So we're saying that the rate of change of concentration is this in the system, which is the c d t. It's equals to minus clearance multiplied by concentration. Minus clearance because clearance, you know, is going away from the system. I know multiply by our per liter. Right? So what do we have on this side? So on this side, we have milligrams per hour. Right? So we need the volume. Where do we need the volume? You need the volume here. And therefore, this means the of this is liters. And if we have the liters here, then this can cancel that. And if we have the liters here, then this can cancel that. And then we got milligrams per hour and milligrams per hour on this side. And that is why we act to have black volume on the left hand on the left hand side there. Because we need that volume to give us mass balance so that we can have equation is now waiting in terms of concentration. Okay. Our ordinary differential equation is in terms of concentration. So we cannot use just the amount as the initial condition for that differential equation. So condition for that differential equation. So we need to start solving that differential equation from a point that is based on concentration. And that is why the concentration of the drug at time 0 is going to be equals to the dose divided by, which is going to be the initial condition for that rate equation. And the output that we're interested in is just the concentration of that compartment. Okay? Good. So it's very, very important for you to note to notice that when you are writing your differential equation in terms of concentration, it's important for you to note that it's better you write it with. Okay? Concentration goes with. And Okay. So we're going to see, you know, series of examples of this, and we're going to write OD a lot, for the next one hour or so. Again, if you look at this, you know, this is what I've just explained. You know, now once the mass balance once we've taken the volume to the other side, so our density is milligrams per liter per hour, and our Okay. So with this work, mass balance also confirmed in terms of the units on the left hand side and on the right hand side of that differential equation. Okay. So we can actually link all this to what you did with you. You know? You can link all this back to everything that you did with you. T equals to minus clearance over bottom multiplied by clearance, which is the same thing as minus k multiplied by c because clearance over bottom So you get concentration of time 0. And if you look at those, you can work out the volume of distribution. So you can see how this is now coming back and connected. Based on what you did with young and also work with. Because the assumption that you also use to deal with this work of as well. Now what I want to emphasize for this one compartment, IV, clearance, elevation with constant, and volume. Okay? But we we only need 2 of them Okay. So now let's go to a 2 compartment system. Okay? And this is where we are going to begin to make this really, really interesting. Or clearance of the drug from that compartment. Okay. So we have 2 parameters describing our elimination from that of family, which would be, you know it doesn't mean that they are doing you know, those the first compartment, our central compartment, also into the very, very compartment. And drug is also coming back from the very, very compartment into the central compartment. Drug is coming back from 42 also into far pathway into the central compartment, drug is coming back from body 2 also into body 1. And drug is going from body 1 into body 2. So the grid that I described, drug transfer from body 1 to body 2 is k 1 2. And that k one to b k, drop going from 1 to 2. Does that make sense? Okay. K 12. This is transfer from compartment 1 to compartment 2. That is why it is k 12. Okay. Now in terms of drug transfer from Okay? So if they are also describing drug transfer within this compartment in terms of amount, then it's better you use this k 12 or k 21, you know, or k 12 and k 21, you know, because they're both k terms of drug transfer, then you can use what is known as k 12 is actually equals to q divided by v 1. Okay. K 12 is equals to q divided by v 1. And k 21 is also equals to q divided by v 2. Okay? So all these parameters can describe drop movements in and out of, Okay. In total that we are talking about. But really, we only need 4 of them. Because once we have 4 of them, if we want the rest, we can work them out To describe this system, either in terms of amount or either in terms of constitution, we can use those 4 to work out the rest. So what we're going to say now is that if we want to describe this system in terms When terms of micro, what we need are kvonek12andk21. And you will notice that from the macro, we can derive the macro. 4 of those parameters. You know? So if we have p 1, p 2, and q, we can work out k, v 1, k 12, and p 21. From p r s and v 1, we can work out k. K. From and v 1, we can work out a 12. From q and v 2, we can work out k 21 as well. Okay? So macro and micro parameters can be used to describe this system just using 4 parameters. Now in terms of writing ordinary differential equation for this system, Okay. Let's let's make some attempt. Now let's write it, first of all, in terms of amount. Okay. So we're going to write it for this compartment in terms of amount, and we're going to write it in terms of 2 multiplied by a 1. Okay? And now don't forget, we are focusing now on the first compartment, which is body 1. And the rate at which it's coming in is k 12 multiplied by a 1 because that's where it's coming from. And the second arrow is going out, Okay? Now let's now look at how we can write it in terms of concentration. Now in terms of concentration, So you can convert from micro to macro and then begin to write your own. Okay? So now let's start with the first compartment. It's it's actually the is c 1. Plus, again, q now for the arrow that is coming into that compartment. Okay. Plus q because the parameter is coming into that compartment. It's standard parameter. So it's like micro is like the base. You know, macro is like what you derive from the macro. Okay. Micro is like basic parameter. Macro is what you derive from the micro. Okay? So now, hopefully, this is not making sense because you will soon start to write your own ODs. Okay. Now what I want us to look at now is sometimes sometimes we can decide to mix this up. Sometimes we can decide that actually, we don't want to, you know, have, you know, the Okay? So I I hope this is clear because this is where we mix this up, and, hopefully, this explains in this kind of Okay? So, hopefully, this is this is clear. And, also, I try to look at situation where instead of writing that first compartment in terms of constitution and the second compartment in amount, I now switch it. Okay? Any problem? Okay. So now let's look at another situation. Now for this system, we'll be talking about PRI, volume, and KE. And for Michael, we'll be talking about volume KE and k. Okay? Now let's look at how we write ODE for this system. Okay. So write ODE for this system is very straightforward. Okay. Again, it's all about granting ODE for each of those compartments. So we can start with our different compartment. We have to write the ODE for So the initial condition for e will be d. So that is why n d zero is equals to d as the initial condition. But the r is going out of it, which is minus. Okay? So, hopefully, this is something you know, because as I said, you you you have to begin to write your own ODE at some very, very, very soon. Okay. So now let's look at what happens when we talk about infusion. Now the problem that we have with infusion is that our our situation is highly complicated because It's not for us this time. No. It's not for us to that. The drug is going into the body, into the system at a constant rate, and then we stop. And then the body, the system continues. So there are 2 phases. Which is the rate of drug infusion into the system minus over b multiplied by a. And it is over b because we need to use k there. The and then a will be in milligrams. So you have mass balance in that equation. Okay? Now because we are interested in the concentration of the drug, You know? So we have to get the solution of that into concentration by taking a and dividing it by volume of distribution. Okay. So that that's what happens during the infusion. But when we switch off the infusion when we switch off the infusion, then that is when time is greater than t in. That is time greater than the duration of the infusion. Then our ABT is equals to p minus is elimination of what we already have in the system. Okay? But we also have to be mindful of the fact that we are not starting from 0. There is already something in the system. So the initial condition that will have in the system when we switch off the infusion to get a t a. Okay? So we need to solve Okay. So when we talk about so I think this is actually the same thing as that slide. Oh, sorry. Yeah. Sorry. This is sometimes, what we add is that sometimes, what we add so so far, actually sorry. Yeah. What is different here is that sometimes we as what we did previously is when we are using a 0 order input Okay? Sometimes, we can have this our system described by what is also known as zero order absorption. This is something that we can also use to describe absorption when a drug is given orally. So this system that we use to describe our the equation that we use to describe our system, when the drug is given intravenously by infusion is different. You know, that's what what we saw before. It's just when the drug is given by intravenous infusion. Okay? And for that, we know that we know the dose. We know the duration of the infusion. Now sometimes we can use the same set of equations to describe what you got is given for right. In which case, it's because our first order absorption system is not working. So So in addition to the clearance and volume that we are interested in, we are also interested in the duration of the absorption, which is t apps. Okay? But the way the system is set up is entirely the same. So we have 2 systems. You know, 1, And then the only is essentially the same. You know? And then the r zero is now those divided by t apps instead of divided by t e. So we use this sometimes just to describe our absorption instead of first order or, you know, that we you know, that we that we talked about. So you some of you have probably thought of this, maybe your project or something like that, but I don't think you'll have any exercise in in in the rest of the workshop. Okay. So I think this is probably coming to the end now. Now this is a 2 compartment system, you know, which is, I think, probably the most complicated we look at. I think it's not as complicated as the intuition, but So now, arrow coming in from the central compartment is actually k a multiplied by a g. A g is the amount in compartment where the drug is coming from. And because that is an amount, Okay. So what you will notice is actually this is the system that we have used, you know, all with all the model that we have looked at today, and this is what is known as mapillary compartment system. Okay. What it means usually is that you have a system where you have a central compartment and you have series of compartment attached to this compartment. So you have drug distributed into this other compartment and you have elimination from that central compartment. Okay. So this system is what is known as mammillary compartment system. And it's widely used in pharmacokinetics as you have seen in a lot of the models that we've looked at today. Okay. Just for us to summarize, so we've been able to see compartmental models, 1, 2, 3 compartments represented, you know, used to represent what we have in our system, a number of mass balance equations today and how they are used to express absorption or describe absorption, distribution, and elimination based on compartmental orders. Okay. And red equations, as I've said, are very, very important to us. We use that with ordinary differential equations to describe our system. Okay? And they can be implemented in softwares. You know? And we're going to be looking at how we implement them in different softwares. Next week or so, we'll start to implement them in, in MATLAB, you know, and then you will see how we actually describe this.

Okay. So let's make a stats. So in general, I'm sure you've seen this a lot with you. We talked about AGME when we talk about pharmacokinetics. So we talked about we talked about the drug administered goes in, you know, into the system, and you have it being distributed.

Execution. But when the drug is also being distributed, it goes to the sites of it goes to the site of action because that's where it has to produce its effects. If the And then once it gets to the site of action, then it it produces whatever we want it to produce, especially in terms of efficacy. But we know that we can also have off target effects. Effects, those are what you really want to call design effects.

The drug will go to muscles, you know, go to other places where it can, you know, produce effect that we do not want. And those are all go and design effects. So we can have efficacy on the we're not necessarily talking about just the efficacy. We're also talking about, toxicity. Now in compartmental modeling, there are 2 types of models that we use.

So we use what is known as the typical compact metal models, and we also use what is known as the piezoelectric monocokinetic models. So in this model, our focus is going to be mostly on the compact metal models. I will leave Dan. I'm sure you've started already with, PBPK 1 to sort out these electrochemical based, orders. And, of course, you've seen it in fact, you know, in lectures in PBPK 1 already on, you know, tissues and organs that are represented as compartments in PBPK models.

And that's actually how it is structured. For the way we describe or we use equations, mass balance equations to connect these tissues and organs are exactly the way we do with compartmental models. So a lot of the things that we're going to look at today is also relevant for this analytical based monoclonal model. And that is why with Dan, I think in another 1 or 2 weeks, then you'll be looking at compartmental models, the equations behind physiologically based on our magnetic models. We've done.

And when we talk about compartmental modules, I'm sure Leon also introduced, you know, this to you a little bit. We're talking about just using 1 or 2 compartments to represent the body. So we're just everything in the body, we put them together into 1 or 2 compact means. Okay? And sometimes we can say 3, you know, or more.

You know? But these are usually very, very limited number of compact means. And our assumption is that drugs distribute and eliminate from this compartments. So we have drug distribution and elimination from this compartment. Now the number of compartments that we use for each drug is not fixed.

It is determined by the data. We look at the data, then we decide how many compartments do we need to describe this data. And that is why it is often described as being data driven because it is the data that will tell you how many compartments that you need. You need. Online, these are the things that you need.

Online, these are the things that you need. Online, these are the things that you need. I just allow the data to tell us, you know, how many of those are needed. And these compartments do not necessarily this is very important. It will not necessarily correspond to specific organ or reflect physiological processes.

You cannot say the central compartment is just blood or central compartment is just muscle or the peripheral compartment is just adipose. We don't we do we do not say, yes, it may reflect what is going on in plasma. It may reflect what is going on in adipose, but you cannot say specifically that this is just blood or plasma or it's just adipose. Okay? And because this compartment do not necessarily correspond to specific organs or tissues of the body, that means that when we get parameters from our fitting of the border to this data, then we have to be very careful.

So the parameters that we get from this then have to be interpreted with caution because you cannot say that the volume of the central compartment is definitely the volume of the log or definitely the volume of, you know, I don't know, you know, for adding for a peripheral compartment, you cannot say it's definitely the volume of adipose. It gives us the extent of distribution, but you cannot say categorically that this is what it actually means. Now there are advantages and disadvantages to using this very simple models. The advantages, I'm sure you can easily tell, is that they are very easy to implement. Okay?

In maybe 10, 15 minutes, you can come up with 1 or 2 compartment model that describe a data. But if you are going to do that for a physiologic based model, then you need more than 10 minutes. You know, because you need to gather a lot of parameters. So advantages are, one of them is a very easy to implement. Mostly, they do not require prior information.

You don't need to have prior information to have the data, then you let the data tell the department and panelists. Okay. They do not require theoretical parameters, for instance. They can be adjusted to focus on specific processes when orders are not necessary. Okay.

If I want to focus on, say, the liver, for instance, I can have a compartment for liver, and then, say, compartment for liver, compartment for blood, and then the rest of the body. If I need it, then I can include that as well. So I can also incorporate specific compartment. You know, I can incorporate compartment for specific organ of tissues in this setting as well. And they are useful for relating parameter values to.

Okay? That's one of the advantages of them. If I see differences in plasma concentration between individuals, and I know that this is due to the differences in their genotype of c 2 c 19, I can easily bring that information into the model to explain why I have the differences in the plasma concentration. And it can be used to simulate what other scenarios easy. If I want to look at what happens if I give the drug at twice the dose that I've studied, I can easily simulate that scenario and look at, what happens.

But the disadvantages are, it is data driven. Okay? It is data that tells you what's going on. If your experiment has not been conducted very well, then you may end up with a wrong model or the model that is may not necessarily adequately describe your drug. For instance, if your drug is such that there is a very, very long terminal phase to your plasma concentration.

There's a long terminal phase, you know, to the, you know, to your plasma concentration. A very typical example is methotrexate. You can look it up, you know, maybe after the lecture, you know, compartmental models for methotrexate. You will see that some people will say that for methotrexate, it is 4 compartments. Some people will say it's the 3 compartments.

If you consider to collect your data for a bit longer, not you may see 3 or 4, then if you collect it for a very long time, then you may see actually true number of compartments that describes your plasma concentration, profile. If I go to, you know, children, then the is likely going to be different. The plasma concentration is likely going to be different. So I cannot extrapolate that model to children. So I have to be very careful when it comes to, you know, the way I used in order because of that.

Okay. So now let's think about and let's start to follow, you know, move towards how we can put these models together. Now the first thing I want us to imagine is that we have, you know, maybe a big, you know, system like we have here. It it's just a big tank. If you get where I mean, it's just a big tank.

If you say that tank is just our body. Okay? That tank is our body. And what we want to do, you know and for that tank, we don't we don't know the volume. Let's pretend that we do not know the volume of that tank.

We do not know the volume of that tank. Okay? But we want to conduct an experiment that will help us to determine the volume of that tank. So our aim is actually to determine the volume of that tank, which we, you know, believe is just a one compartment system. So we are talking about 1 compartment now.

You know, those are the terminologies that we're using. That system, you know, is just a single compartment. So that is just, you know, the partition that describes what we are looking at. In this case, there's no partition. It's just a single kind.

So compartment. And then we can talk about steady states. And the steady state means that we what we have in terms of concentration is uniform throughout the system. There's no net change in concentration. You know, this the system is at steady state.

Okay? And, also, we can also talk about border strokes. And what that means is that if we add, say, you know, you know, a dye, you know, dye, you know, pouring agents into that tank, you know, Then when we add it, if we assume that we add it all at once, then that is a glucose dose. As you know, you know, when you add, the, the. So the method that we're going to use that we're going to use a dose of a known amount of a chemical, which is going to be a dye.

And we're going to conduct an experiment to measure the concentration of the drug in the stamp. And our calculation then is going to be that if we assume that when we add the dye to this compartment, everything distributes equally. There's no next, you know, change in concentration. The concentration measured in any part of this tank is the same. Okay?

The concentration measured in any part of this tank is the same. So that means that if we know the amount of the dye that we have added, and if we know the concentration that is now measured in any part of this tank, okay, if we conduct an experiment, I will take a sample from this tank and we determine the concentration of the drug in any part of this tank. And we can then use the concentration that is now measured and the dose, which we know of the chemical that we have added to this tank to work out what the volume of this tank is going to be. So the volume of this tank is just going to be the dose that we've added divided by the concentration that we have measured in any part of the tank. Okay?

Now, again, you know, going back to it, there are certain facts or an assumption that we have made, you know, with this system. One of them is that the system is well stirred. That is equal distribution of the chemical, the dye, you know, which means there is instantaneous distribution of the chemical within the system. That's our assumption. One of the assumptions.

And the chemical is not able to leave the system. Everything is contained within that system. And that is why it doesn't matter when we take that, you know, sample, you know, because everything's contained and contained it anywhere in in any part of the tank. But there's also some facts that we can establish. And one of them is that only the concentration of the sample is useful.

Okay? Only the concentration of the sample is useful. Amount of the chemical in the sample is not. If you determine the amount of the drug in the sample that we have taken, it's not useful to us because we cannot relate that to the dose to work out the concentration. I mean, the volume.

So it's only the concentration of the chemical in the sample that we have taken that is important to us because that has some element of volume that we're interested in it. And then once we relate it to the dose, then we can get to the volume of the system that we're interested in. And, also, we know the dose, which is the amount of the chemical that we added initially. And the concentration is also determined, you know, from our sampling. And the volume that we have now, you know, we can say that it's actually operational.

It does not reflect the actual volume of the system. It's just based on what we can determine from this experiment. Okay? For us, for instance, if we have, you know, some of these chemicals talk to different parts of maybe the, you know, the the the container of the tank, then that may affect what we measure in terms of the concentration. So the volume of that we have measured in this step is actually operational.

And the calculation only works if the compound is rapidly and uniformly distributed. Okay. If those 2 as you know, if those 2 are not, you know, are not there, you know, it is not rapidly distributed and not uniformly distributed, then the calculation will not work for us. Okay? These are some of the things I'm sure that you learned when you are looking at one compartment model, for instance, with.

Okay. Now let's not talk about when we assume that we can also have our chemical leave system. Okay. It's not just that. We add the chemical to the tank, and it stays there.

Now we add chemical to the tank, and the chemical can also leave. And that is what we add when we talk about dosing of the drug into human body, for instance. Okay? So in this case, our tank is human body. And the chemical that is added is the dose of our drug.

And the drug leaving the system is just elimination of our drug, either by metabolism or by exclusion. Okay? So now in this system now, as you can see, we are assuming we are still assuming a one compartment system. We are assuming that we're going to give a dose of the drug instantaneously as a bolus dose. We are assuming that we are interested in the following of distribution of the drug in that time.

We are also assuming that the drug is able to leave the system. Okay? What that means is that we can also do the same experiment, you know, and take concentration of our drug, you know, but now it means that when we take the concentration of the drug at different times, the the concentration will be different because the drug is able to leave the system. Okay? So we have to now conduct an experiment that will take the drug concentration at different time points.

And also, it's important for us to know that for this system now, it is only at time 0 when there is no ability of the drugs, of system to get rid of the drug that we can work out the form of distribution of that time. Okay? It's only at time 0 that we have a system that is similar to what we had in the previous slide. Because when the system is attacked as at time 0, then there is no elimination. So if we can determine the concentration of the drug at that time 0, and we can assume that this the drug is able to distribute equally within the system at that time 0, then things that you did with Leon when you have to extrapolate that y to time 0 when it crosses the y axis.

Basically, what you're trying to do is to work out the concentration of the drug at times 0 because that is the only time that the concentration of the drug can help us to get the form of of the system that we are interested in. Okay. And we have 2 parameters describing the system. We have clearance, which is a parameter of elimination, and we have form of distribution as well. Okay?

Clerance is the volume clear of drug per unit time. I'm sure you know that. And, you know, now we can also look at the plasma concentration profile, you know, in this system. You know, if we say this system is like, you know, what we have when a drug a a dose of a drug is administered intravenous intravenously as a bonus drug. You see the plasma concentration profile on the right hand side.

Okay. That's what you see, where you measure plasma concentration at different time point, and you plot that against time. Now as I've said, the volume of distribution is going to be now equals to the dose of the drug divided by the concentration at time 0 because that is the only time we can determine the concentration of the drug. And if we look at the AUC of our probe, the area under that plasma concentration curve, then we can integrate our plasma concentration profile with respect to time, and that will then give us our AUC, which will be equals to the dose divided by kR if we solve that integral. And then we can also relate our clearance to volume of distribution by using elimination rate constants.

Okay? So these are some of the things that you've already covered with Leon, so we are not going to spend a lot of time on that. Okay. So now what we're going to be looking at next are what we call the rate equations. Now what we call rate equations in PP are basically equations that allows us to to describe drug movement in the body.

So when we want to describe how drug is moving in the body, we can use what is known as rate equations. Okay? So rate equations are used to describe drug movement in and out between our compartments. Okay. So and that is what we're going to be looking at a lot today.

Now in terms of how the drug is getting into the body. Okay? In terms of how the drug gets into the body into in the first instance, there are a number of input systems that we can use. And one of them is the, which we just looked at. The means that the drug is added into the system all at once.

And there is also what is known as the 0 order. Okay? The 0 order means that the drug is added into the system at a constant rate. Okay? And the rate of the the at which the drug is added is added into the system is just constant over time.

Okay? And that is what you have with an infusion. You know? So when you have a 0 order input into the system, then you have constant rate of the drug going into the body. You can also use what is known as first order input into the body.

First order input into the system just means that the rate at which the drug is going into the body is proportional to the amount that you have in the system where you're adding the drug from. Okay? So there's a bit of proportionality there when you're talking about first order inputs. We we're we're going to see example of that as well. And for outputs, you can have either 0 order inputs, you you know.

And also, you can also have 1st order and outputs. You know, for outputs, you can have 0 order. You can have 1st order as well. We're going to look at some examples of this, but our focus is going to be mostly on 1st order inputs and output and maybe bonus and 0 order. In terms of outputs, we're not going to look a lot at 0 order.

We're going to look more at 1st order. But what does 0 order process means? 0 a 0 order process means that the rate is constant and does not depend on drug concentration or amount. The rate at which the drug is going in or out of the system is constant, and it does not depend on the concentration or the amount that you have in the system. But when you talk about first order, the rate is proportional to truck concentration or the amount that you have in the system.

And when you talk about first order, usually, we use this to describe what is known as we usually use it to describe oral drug absorption, for instance. Okay? And if you look at the equation that we have there for rates for 0 order, we're saying the rate is just equals to k multiplied by concentration raised to the power of 0. Okay. Because concentration raised to power of 0 is just 1, then it means that the rate is just equals to k.

So the rate of drop inputs or output from the system is just described by that k, which is just the rate of drug in or out of the system. But for first order, you will see that the rate is just equals to k, which is the rate constant describing the first order process multiplied by the concentration or the amount. In this case, it's concentration raised to the power of 1, and that is just equals to k multiplied by the concentration. Okay. So the rate is proportional to the amount or the concentration that you have in the system.

In this case, it's concentration. The rate is proportional to the concentration that you have in the system. And rate constant is actually the proportionality constant that describes that weight. Okay. So in addition to this, we can also have a second order process.

A second order process just mean that the rate is also proportional to concentration, but it's proportional to the to concentration raised to the power of 2. Okay. So it's more or less, you know, similar to 1st order, but now the proportional, you know, the rate is proportional to the square of the concentration that you have in the system. Okay. Now another rate that we can another rate, you know, of input or output that we can have into the system is what is known as the saturable process.

Okay. And this is probably what you've looked at maybe in your PBPK, you know, model with with with that. Okay. What this means is that and usually, actually, we use this to describe inputs and outputs, which are usually dependent on carrier mediated absorption or distribution or saturable elimination, which can be, you know, for metabolism or exclusion. They are usually described using the equation.

And in that case, the rate is equals to as you can see here, if the rate is equals to 3 max multiplied by concentration divided by concentration plus our kilometers parameter. So the system is actually saturable. So at very high concentration, you can see that we are approaching. Okay. So what it means is that as concentration increases, rate also increases.

But as concentration increases further, that rate, approaches a plateau, and that plateau is equivalent more or less to our. And that is the saturable, process of, input or outputs into a system. But for 1st order, as you can see, this is just a broken line here. This is the 1st order process. As concentration increases, the rate of 1st order increases indefinitely because it's just the rates equals to proportional constant, which is our rate constant multiplied by the concentration.

So if concentration increases, that rate will continue to Okay. So now let's go back to this model that we were looking at. Okay? So what we are going to do now is we're going to look at rate equation for the system that we have here. Okay.

So this is the model that we that we have. This is the system that we looked at previously. So what you have there is a 1 compartment model. You have a 1 compartment model that is that you input into the system that system is not a bonus dose, and the outputs from the system is described by the first order elimination. And in this case, we are assuming the body is a work of partner system.

Okay. So what it means is that the model that we have here is therefore a work compartment is by a first order process, then we can use our parameter that is describing that first order process, which is the elimination rate constant So in this case, what we know is that the rate of change of drug in this system, which is the a b t. Okay. The rate of which is through the initial condition. And we have also described how the drug is changing in the system by using the entity.

And that also allows us to incorporate the first order process of any relationship of a drug from the system. Okay? And therefore, we have a mass balance system. Okay? So that is your first, probably, I don't know, first differential equation.

Hopefully, that is straightforward and simple enough to understand because we are going to start to make this complicated as we go along. Okay. Now one right or not, which means the units that I have on the left hand side has to be the same with the unit of what I have represented on the right hand side. Okay. In this case, by the a to t, a is in amount, probably milligrams, and my t is in time, probably in hours.

So on right on the left hand side, I have milligram per hour. Then on the right hand side, I need to have milligrams per hour too. Okay. Any questions so far? Okay.

Everything looking straightforward so far? Okay. Now what we can also look at is a situation where we are not just we are not we are not writing the mass balance equation. We are not writing the mass balance equation in terms of the amount of You know? And that's what we're going to do here.

So it's the same process. Okay. So our o d now is not going to be an amount, but it's going to be in concentration. And what it means is that because we want to write the o d in terms of concentration, this is our problem. Okay.

So we're saying that the rate of change of concentration is this in the system, which is the c d t. It's equals to minus clearance multiplied by concentration. Minus clearance because clearance, you know, is going away from the system. I know multiply by our per liter. Right?

So what do we have on this side? So on this side, we have milligrams per hour. Right? So we need the volume. Where do we need the volume?

You need the volume here. And therefore, this means the of this is liters. And if we have the liters here, then this can cancel that. And if we have the liters here, then this can cancel that. And then we got milligrams per hour and milligrams per hour on this side.

And that is why we act to have black volume on the left hand on the left hand side there. Because we need that volume to give us mass balance so that we can have equation is now waiting in terms of concentration. Okay. Our ordinary differential equation is in terms of concentration. So we cannot use just the amount as the initial condition for that differential equation.

So condition for that differential equation. So we need to start solving that differential equation from a point that is based on concentration. And that is why the concentration of the drug at time 0 is going to be equals to the dose divided by, which is going to be the initial condition for that rate equation. And the output that we're interested in is just the concentration of that compartment. Okay?

Good. So it's very, very important for you to note to notice that when you are writing your differential equation in terms of concentration, it's important for you to note that it's better you write it with. Okay? Concentration goes with. And Okay.

So we're going to see, you know, series of examples of this, and we're going to write OD a lot, for the next one hour or so. Again, if you look at this, you know, this is what I've just explained. You know, now once the mass balance once we've taken the volume to the other side, so our density is milligrams per liter per hour, and our Okay. So with this work, mass balance also confirmed in terms of the units on the left hand side and on the right hand side of that differential equation. Okay.

So we can actually link all this to what you did with you. You know? You can link all this back to everything that you did with you. T equals to minus clearance over bottom multiplied by clearance, which is the same thing as minus k multiplied by c because clearance over bottom So you get concentration of time 0. And if you look at those, you can work out the volume of distribution.

So you can see how this is now coming back and connected. Based on what you did with young and also work with. Because the assumption that you also use to deal with this work of as well. Now what I want to emphasize for this one compartment, IV, clearance, elevation with constant, and volume. Okay?

But we we only need 2 of them Okay. So now let's go to a 2 compartment system. Okay? And this is where we are going to begin to make this really, really interesting. Or clearance of the drug from that compartment.

Okay. So we have 2 parameters describing our elimination from that of family, which would be, you know it doesn't mean that they are doing you know, those the first compartment, our central compartment, also into the very, very compartment. And drug is also coming back from the very, very compartment into the central compartment. Drug is coming back from 42 also into far pathway into the central compartment, drug is coming back from body 2 also into body 1. And drug is going from body 1 into body 2.

So the grid that I described, drug transfer from body 1 to body 2 is k 1 2. And that k one to b k, drop going from 1 to 2. Does that make sense? Okay. K 12.

This is transfer from compartment 1 to compartment 2. That is why it is k 12. Okay. Now in terms of drug transfer from Okay? So if they are also describing drug transfer within this compartment in terms of amount, then it's better you use this k 12 or k 21, you know, or k 12 and k 21, you know, because they're both k terms of drug transfer, then you can use what is known as k 12 is actually equals to q divided by v 1.

Okay. K 12 is equals to q divided by v 1. And k 21 is also equals to q divided by v 2. Okay? So all these parameters can describe drop movements in and out of, Okay.

In total that we are talking about. But really, we only need 4 of them. Because once we have 4 of them, if we want the rest, we can work them out To describe this system, either in terms of amount or either in terms of constitution, we can use those 4 to work out the rest. So what we're going to say now is that if we want to describe this system in terms When terms of micro, what we need are kvonek12andk21. And you will notice that from the macro, we can derive the macro.

4 of those parameters. You know? So if we have p 1, p 2, and q, we can work out k, v 1, k 12, and p 21. From p r s and v 1, we can work out k. K.

From and v 1, we can work out a 12. From q and v 2, we can work out k 21 as well. Okay? So macro and micro parameters can be used to describe this system just using 4 parameters. Now in terms of writing ordinary differential equation for this system, Okay.

Let's let's make some attempt. Now let's write it, first of all, in terms of amount. Okay. So we're going to write it for this compartment in terms of amount, and we're going to write it in terms of 2 multiplied by a 1. Okay?

And now don't forget, we are focusing now on the first compartment, which is body 1. And the rate at which it's coming in is k 12 multiplied by a 1 because that's where it's coming from. And the second arrow is going out, Okay? Now let's now look at how we can write it in terms of concentration. Now in terms of concentration, So you can convert from micro to macro and then begin to write your own.

Okay? So now let's start with the first compartment. It's it's actually the is c 1. Plus, again, q now for the arrow that is coming into that compartment. Okay.

Plus q because the parameter is coming into that compartment. It's standard parameter. So it's like micro is like the base. You know, macro is like what you derive from the macro. Okay.

Micro is like basic parameter. Macro is what you derive from the micro. Okay? So now, hopefully, this is not making sense because you will soon start to write your own ODs. Okay.

Now what I want us to look at now is sometimes sometimes we can decide to mix this up. Sometimes we can decide that actually, we don't want to, you know, have, you know, the Okay? So I I hope this is clear because this is where we mix this up, and, hopefully, this explains in this kind of Okay? So, hopefully, this is this is clear. And, also, I try to look at situation where instead of writing that first compartment in terms of constitution and the second compartment in amount, I now switch it.

Okay? Any problem? Okay. So now let's look at another situation. Now for this system, we'll be talking about PRI, volume, and KE.

And for Michael, we'll be talking about volume KE and k. Okay? Now let's look at how we write ODE for this system. Okay. So write ODE for this system is very straightforward.

Okay. Again, it's all about granting ODE for each of those compartments. So we can start with our different compartment. We have to write the ODE for So the initial condition for e will be d. So that is why n d zero is equals to d as the initial condition.

But the r is going out of it, which is minus. Okay? So, hopefully, this is something you know, because as I said, you you you have to begin to write your own ODE at some very, very, very soon. Okay. So now let's look at what happens when we talk about infusion.

Now the problem that we have with infusion is that our our situation is highly complicated because It's not for us this time. No. It's not for us to that. The drug is going into the body, into the system at a constant rate, and then we stop. And then the body, the system continues.

So there are 2 phases. Which is the rate of drug infusion into the system minus over b multiplied by a. And it is over b because we need to use k there. The and then a will be in milligrams. So you have mass balance in that equation.

Okay? Now because we are interested in the concentration of the drug, You know? So we have to get the solution of that into concentration by taking a and dividing it by volume of distribution. Okay. So that that's what happens during the infusion.

But when we switch off the infusion when we switch off the infusion, then that is when time is greater than t in. That is time greater than the duration of the infusion. Then our ABT is equals to p minus is elimination of what we already have in the system. Okay? But we also have to be mindful of the fact that we are not starting from 0.

There is already something in the system. So the initial condition that will have in the system when we switch off the infusion to get a t a. Okay? So we need to solve Okay. So when we talk about so I think this is actually the same thing as that slide.

Oh, sorry. Yeah. Sorry. This is sometimes, what we add is that sometimes, what we add so so far, actually sorry. Yeah.

What is different here is that sometimes we as what we did previously is when we are using a 0 order input Okay? Sometimes, we can have this our system described by what is also known as zero order absorption. This is something that we can also use to describe absorption when a drug is given orally. So this system that we use to describe our the equation that we use to describe our system, when the drug is given intravenously by infusion is different. You know, that's what what we saw before.

It's just when the drug is given by intravenous infusion. Okay? And for that, we know that we know the dose. We know the duration of the infusion. Now sometimes we can use the same set of equations to describe what you got is given for right.

In which case, it's because our first order absorption system is not working. So So in addition to the clearance and volume that we are interested in, we are also interested in the duration of the absorption, which is t apps. Okay? But the way the system is set up is entirely the same. So we have 2 systems.

You know, 1, And then the only is essentially the same. You know? And then the r zero is now those divided by t apps instead of divided by t e. So we use this sometimes just to describe our absorption instead of first order or, you know, that we you know, that we that we talked about. So you some of you have probably thought of this, maybe your project or something like that, but I don't think you'll have any exercise in in in the rest of the workshop.

Okay. So I think this is probably coming to the end now. Now this is a 2 compartment system, you know, which is, I think, probably the most complicated we look at. I think it's not as complicated as the intuition, but So now, arrow coming in from the central compartment is actually k a multiplied by a g. A g is the amount in compartment where the drug is coming from.

And because that is an amount, Okay. So what you will notice is actually this is the system that we have used, you know, all with all the model that we have looked at today, and this is what is known as mapillary compartment system. Okay. What it means usually is that you have a system where you have a central compartment and you have series of compartment attached to this compartment. So you have drug distributed into this other compartment and you have elimination from that central compartment.

Okay. So this system is what is known as mammillary compartment system. And it's widely used in pharmacokinetics as you have seen in a lot of the models that we've looked at today. Okay. Just for us to summarize, so we've been able to see compartmental models, 1, 2, 3 compartments represented, you know, used to represent what we have in our system, a number of mass balance equations today and how they are used to express absorption or describe absorption, distribution, and elimination based on compartmental orders.

Okay. And red equations, as I've said, are very, very important to us. We use that with ordinary differential equations to describe our system. Okay? And they can be implemented in softwares.

You know? And we're going to be looking at how we implement them in different softwares. Next week or so, we'll start to implement them in, in MATLAB, you know, and then you will see how we actually describe this.

Transcript for:Understanding Compartmental Models in Pharmacokinetics

Transcript for:
Understanding Compartmental Models in Pharmacokinetics