I think that volume replacement is good practice. Interesting that it takes 72 hours to come back to baseline. That suggests that the model has a long right tail, longer by far than the usual exponential models would likely predict. Thus, it may be a fractal process with a monotonically decreasing gamma density function, as wait times often are.

Thank you very much for your answer. It was not possible to take more controls after 12, 36, and 48 hours as the study was performed with donors who went home after donor plasmaphereses. I suppose that there is no linear decrease of Hgb- and Hct-levels but a rapid decrease in the first 24 hours and a slow decrease after that due to volume shits and fluid replacement.

Hello Carl, would you be so kind as to explain what a "fractal process with a monotonically decreasing gamm density function" is. The half-life of saline infusion is about 20 min: so the influence of an infusion of saline infusion should be away after about 5 half lives or after 2 hours. Can you explain why it takes 72 hours until the hemoglobin levels return to baseline? Does ist matter that donors donated 600-700 ml plasma?

Sincerely

Josef

OK. In "Wanasundara SN, Wesolowski MJ, Puetter RC, Burniston MT, Xirouchakis E, Giamalis IG, Babyn PS, Wesolowski CA (2015) The early plasma concentration of 51Cr-EDTA in patients with cirrhosis and ascites: a comparison of three models. Nuclear Medicine Communications 36: 392–397," we see that the early kinetics of a bolus (after peak) is a logarithm of time more so than an exponential. That is, the primary method the body has to mitigate the effects of disturbances proceeds as the logarithm of time for linear concentration, and not, as we have been told for 100+ years as the logarithm of concentration for linear time. That, in effect, means something like the following, that if the first half-life takes 20 minutes, the second one takes 40 minutes, the third 80 minutes, the fourth 160 minutes etc. Now eventually, this new "law" breaks down and the fourth half-life might be 200 minutes instead of only 160. As you can see, we are adding times geometrically not linearly. So, with a more fractal motivated model, like a gamma variate, we can duplicate the early timing, at least for the first 24- h or so. Thereafter, half-life gets extremely slow.

Now, why is this happening? In short, the volume in which the drug is contained gets larger in time, and it can be shown (see Wesolowski CA, Wesolowski MJ, Babyn PS, Wanasundara SN (2016) Time varying apparent volume of distribution and drug half-lives following intravenous bolus injections. PLoS One 11: e0158798.) that a drug will eventually be completely eliminated from the body BEFORE the terminal volume is achieved, at least for a single dose experiment. In other words, the slowest half-life is not "elimination" in the body, it is late redistribution, and redistribution eventually gets to be extremely lethargic.

As far as I understood, it does not mean a simple 1st order kinetics with a  one-compartment model, but a more complicated kinetics in a multi-compartement system with delayed half-life or elimination at the end. Correct?

Hi Josef,

What I am saying is that people are built more like trees than to steam locomotives. It is hard to stop thinking in terms of "compartments," but there is an alternative that makes doing so worthwhile. If we had two (or more) pressure vessels with leaks between them with one vented to atmosphere we could reasonably (neglecting turbulence) argue for compartmental construction. For most drugs, there is either an impenetrable diffusion barrier (hydrophilic drugs) at cell membranes, or not (lipophilic drugs). Even in the lipophilic case, one would have to show that drug transport across a "diffusion" barrier is rate limiting before one can reasonably suggest the existence of an isolated compartment. Even that latter would not make it reasonable to assume instant mixing. In effect, independent instantly mixed compartments are one dubious interpretation of sums of exponential terms, there are others, for example, a variable volume of distribution in time model. And, there are better mathematical models. Let me argue this from the viewpoint of statistics. Sums of exponential terms, as density functions, assume mixture distributions composed only of scaled exponential distributions. The likelihood of that type of model being the statistical model of choice for an arbitrary curve is very small. Even if we ignore better single models, like gamma distributions and insist that there are multiple independent processes that occur simultaneously, (think to two people cutting pencils to a specific length, if they are cutting to different lengths, the histogram of cut pencil lengths will be distributed as a mixture distribution) the likelihood that both distributions are exponential would be small. It would have to occur in a special case, like our pressure vessels, where the assumption of instant mixing of gas pressure within each vessel would not be too absurd. More likely would be a sum of a normal distribution and an inverse Gaussian, or about 1000 other common mixtures. Wise, working with power functions and gamma variates in 1985 racked his brain to find an actual two-compartmental model in the body, and came up with a trivial one, ferric ion kinetics (Wise ME (1985) Negative power functions of time in pharmacokinetics and their implications. Journal of Pharmacokinetics and Pharmacodynamics 13: 309-346.). That kind of introspection and testing of PK theory is too infrequent. When we tested for the partial probability of fit to a GFR marker for a biexponential (E2) some of its parameters failed being useful to the ordinary least squares regression result (Wesolowski CA, Puetter RC, Ling L, Babyn PS. Tikhonov adaptively regularized gamma variate fitting to assess plasma clearance of inert renal markers. J Pharmacokinet Pharmacodyn. 2010;37:435-74. doi:10.1007/s10928-010-9167-z).

We use washout models that assume maximum concentration initially that decreases with time. The other way of adding distributions is convolution, and when one process AND's with another the result is not a mixture distribution, it is a convolution of the two processes to make a more complete model. In effect, when we use a washout model to fit a concentration curve, we should be using inverse methods not goodness of fit to find the best washout that explains the convolution of the vascular signal with the washout response of the body. That concept led us to using Tikhonov regularization adaptively to find the best relative error of plasma clearance from gamma variates (Tk-GV).

First order kinetics does not imply exponentials, see the PLoS ONE article I cited above. For first order kinetics, gamma variates work better than sums of exponential distributions, which latter in statistical language would be "mixture distribution of exponential distribution type" First order only means that elimination is proportional to concentration. What it does not mean is that mixing is instantaneous, and the only way to make a reasonable model that agrees with simple thought experiments is to define a volume of distribution of a drug to be "drug-o-centric." That is, if we require that the volume of distribution of a drug is very small when it is still in a pill or in a syringe, and allow for the drug volume to expand with time, then we do not get exponential functions as a result, we wind up with other functions like C(t)=A-B ln(t) or numerous others. If we further require that the entire dose is delivered instantaneously, then the concentration is initially infinite, as it is contained in zero volume. There is no other way (for washout models) to have zero volume of drug distribution initially, but notice that instant mixing in zero volume is no mixing at all, at least initially. That solves the instant mixing problem, and uses an expanding drug volume in time to do it.

With the Tk-GV variable volume in time model, one can derive the time it takes to achieve 95% of terminal Vd, which is also 95% of terminal t1/2, and it is indeed just over 5 half-lives. For a biexponential variable volume in time model that time is only a fraction of a half-life to achieve 95% of terminal t1/2, which as you know is ridiculous. So, yes, there may be something else going on to make the effects last 72- h. However, there are two other effects worth mentioning. The 5 half-life rule applies to incremental changes, not bulk increases in body fluid. In the variable volume paper Table 1, attached, note that increases in Varea, and decreases in GFR have a profound effect on increasing the 95% times. And, one does not need a "first" half-life to be very long in time to produce very long lasting effects. Could this also be from the donors donating 600-700 ml plasma? That seems plausible.

Your explanation of a "fractal process with a monotonically decreasing gamma density function" does not explain me why plasma donors with normal renal function can not recover previous levels of hemoglobin in a shorter time. So, I agree with you, that "there may be something else going on to make the effects (of plasma donation) last about 72 hours." 

Well, plasma regeneration can take a long time.

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1202359/