After being away for almost one month due to graduation of my degree and my tight schedule in my home institute, now I have been able to sit behind my laptop again to continue what I have discussed in previous post about finite dose of toxic chemical in this case, barium, Ba. In the last post, finite dose for barium part 1, I have given the example how to measure a simple problem. Fatal dose in known, blood volume is known as well, so we can measure the Ba concentration in the blood associated with the fatal effect.

What I am going to explain in this post is more complex than the previous one. I will give the case of Ba, if entering the body at a time and if Ba is trapped inside the salt, dissolve in the blood stream for a certain period, and how we can measure the concentration of Ba for acute effect and for a certain period of time.

We have learned continous dose, where the the chemical was fed continously, everyday, every 4 hours etc to animal object. In finite dose, the dose is applied to the object test with a limited time. For example, accidentally we swallow barium, say 0.5 gram of Barium, if after that there is not further input, then the input of barium is only one time, not continous. On anthother case, for example we swallow a chunk material with a salt and trapped barium inside salt, and then the barium will dissolve as a function of certain period of time and absorbed by the blood. This “period of time” is considered as a one time input, because after that period of time, it is assumed that no more chunk material is swallowed. The elimination process between continous dose and finite dose is relatively the same.

So in this post I will propose two example of finite dose of barium, Ba. In the previous post, it has been known that

dBa/dt = k x [Ba concentration] or dBa/dt = k x [C], where k = -0.007/minute = -0.4/hour

First case, suppose somebody swallow 0.4 gram of Ba, barium, and we want to determine the time the body needs to reduce Ba to 0.04 g/liter (0.2 gram Ba in blood to cause subacute effect) and the concentration after one hour.

It would always be a good idea to create a simple flowchart, or model, or picture to aid us in imagining the problem as shown the picture below.

Notice the F_in and F_out, they refer to flow in and flow out. Both of them are functions of the concentration. The concentration of barium at any given time C(t) is determined by the initial concentration, the concentration that enters the body since that time and the amount that has been removed by the body since that time. It can be formulated as follows

C(t) = flow in - flow out

Two important elements here are flow in and flow out, you have to engrave this concept in your brain. Flow in may refer to initial concentration of barium on blood stream, whether absorbed instantaneously or within a period of time, it is symbolized as C(0) in this case for instant absorption. The elimination process is described by flow out. Flow out is formulated as follows:

Flow out = -kC(t), this will make the complete formula is

dC(t)/dt = C(t) = C(0) - kC(t)

C(t₁) = C(t₀) - kC(t₀)(t₁-t₀)

We already know that the initial concentration (C(t₀) is 0.4 gram/5 liter of blood in our body = 0.08 gram/liter, k, constant rate is 0.007/minute. We will calculate for one hour (60 min) increments.

C(t_{60 min}) = 0.08 gram/liter - 0.007/min x 0.08 gram/liter x 60 minutes
C(t_{60 min}) = 0.0464 gram/liter

C(t_{120 min}) = 0.0464 gram/liter - 0.007/min x 0.0464 gram/liter x 60 minutes
C(t_{120 min}) = 0.0269 gram/liter

You may continue this calculation until the Ba is completely removed.

Now I will calculate the concentration in two hours (120 mins) increments

C(t_{120 min}) = 0.08 gram/liter - 0.007/min x 0.08 gram/liter x 120 minutes
C(t_{120 min}) = 0.0128 gram/liter

C(t_{120 min}) = 0.0128 gram/liter - 0.007/min x 0.0128 gram/liter x 120 minutes
C(t_{120 min}) = 0.0020 gram/liter

Another example is for 30 minutes increments

C(t_{120 min}) = 0.08 gram/liter - 0.007/min x 0.08 gram/liter x 30 minutes
C(t_{120 min}) = 0.0128 gram/liter

C(t_{120 min}) = 0.0128 gram/liter - 0.007/min x 0.0128 gram/liter x 30 minutes
C(t_{120 min}) = 0.0020 gram/liter

You can try by yourself using another different time increments, 5 and 1 minutes. I have made the calculation on excel which you can download if you are interested and want to know the detail and complete calculation, click calculation of finite dose of barium. Below is the calculation over the hour 1 until 10 hours with 1 hour increments. As you can see there, the precision or the accuracy is less for higher time increments, 1 minute increments gives more accurate information about the barium concentraion contained in the blood. And you also know, the concentration of barium reaches 0.04 gram/liter of which subacute level occurs after 5-6 hours on 2 and 1 hours increments, 6 hours on 30 minutes increments, 7 hours on 5 and 1 minute increments. The concentration after one hour is at least 0.0535 gram/liter barium. This example assumes that the initial dose is 0.4 gram, which is 0.08 gram/liter in blood. How about we swallow 0.8 gram (0.16 gram/liter)? Try to solve this problem with the exact way that has been described above.

We have solved the problem by using simple method. However, this method although simple, require long time though. So calculus differentiation may offer shortest way to calculate the barium concentration.

The concentration is eliminated at the rate -k(t), it is an exponential solution. By learning the principle of calculus we may obtain that:

We can now solve the problem easily.

C(T) = C₀e^-kT

0.04 gram/liter = 0.08 gram/liter x e^-0.007.T

0.5 = e^-0.007.T, we know that ln e equals 1, then

ln 0.5 = ln e^-0.007.T

ln 0.5 = -0.007T x ln e

-0.693 = -0.007 T, T = 99 minutes, almost 100 minutes!

After one hour, the concentration of barium will be

C(T) = 0.08 gram/liter x e^{-0.007/min x 60} = 0.053 gram//liter

Second case, it has been briefly mentioned above, suppose the 1.2 gram barium is trapped on an unharmed salt, and as a function of time, this barium will dissolve and get absorbed by the blood within 10 hours. We will try to calculate the concentration at hour 10 and determine whether it is acute level or not.

We know that there is flow in and flow out. Flow in (F_in) in this case is a function of time (10 hours) when the barium is getting into the blood. So the formula now is

C(T) = F_in - F_out = F_in - kC(T)

by calculus method, the formula above will change to

by substitution from calculus method this result is obtained:

The flow in is expressed in gram/liter/min, for 1.2 gram Ba, it will be 1.2 gram Ba/5 liter blood/600 min equals 0.0004 gram/liter/min. We know k and t, so we can calculate as follows:

C(10 hours) = (0.0004 gr/l/min/0.007/min) - (0.0004 gr/l/min/0.007/min x e^{-0.007/min.600 min} = 0.056 gram/liter. With 5 liters blood, the concentration is 2.6 gram, obviously this is an acute concentration.

This entry was posted on Friday, August 3rd, 2007 at 3:10 pm and is filed under Risk Assessment I. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

Related Posts

PBPK Finite dose of barium part 1

After being busy with my theses now I am able to back to here again. Previous posts have explained the single time dose of a toxic chemical entering the ingestion tract. So this time, I would like to ...

Physiogically based pharmacokinetic (PBPK)

Physiologically based pharmacokinetic (PBPK) models are stock and flow models used to think about how chemicals move and concentrate within the body. While many dose-response experiments look only at ...

Bio Transfer Factor

Bio Transfer Factor (BTF) is a constant to describe the relationship between chemical substance in a tissue and the rate of ingestion to body of particular toxic agent (chemical substance). Often,...

Online seminar on PK/PD/PBPK Modeling using acslXtreme

I have received request to display this advertisement, I hope you find this useful: AEgis Technologies would like to invite you to attend an online seminar on PK/PD/PBPK Modeling using acslXtreme®...

Physiologically based pharmacokinetics example part 1

Stock and flow model is a practical method to predict and calculate chemical or substance concentration in a building with a simple model. However, more sophisticate approach has to be done to measure...