@ Bidemi

Frankly, understanding your question is a challenge. Though, let me try.

1. The presented formula is used for calculating the mean value of the time to the first failure when the object is under influence of two competing risks simultaneously, under the assumptions:

the failure caused by  each of them appears suddenly independently at random, with the mean values  Tb  and  Tp, respectively, when active separately;

2. The same formula deals for the half life times, since they are proportional to the mean values;

3. When any therapy is applied, then such formulas are highly inappropriate, since then the half life time (in principle, at least) shold become longer!  Therefore for getting some reasonable formula we need to imagin the way the therapy is supposed to act. For instance, if the injections are supplied systematically, say every week, and if the aim is to slow down the evolution of a deadly-ending illness, then the simplest way is to assume, that there is a factor a (greater than 1) such that  TE = a Tb, where  Tb is the half life time in the absence of the medication.  BUT there are still working other cases of dead. Then, as modification I would suggest the following formula:

TE =  a Tb Tp / ( a Tb + Tp)  and then  a  becomes an unknown parameter to be estimated from observations.

Regards, Joachim

Thank you Joachim.

The effective half life of a radio-pharmaceutical is usually calculated from the values of biological half life (TB) and physical half life Tp. While physical half life is obtained from the source certificate, the biological half life has to be calculated from metabolic activity of the radionuclide within the patient. I am now asking whether there is a mathematical model or formula which can be used to calculate biological half life without analysis of sample taken from the patient. Once again, thank you sir for your explanation.

 The biological half-life (T½b) of a radiopharmaceutical is the time taken for the concentration of the pharmaceutical to be reduced 50% of its maximum concentration in a given tissue, organ or whole body, not considering radioactive decay.

 It should be experimentally determined.

Thanks for the last two answers. They helped me to better undersatand the subject of the question, but even now I am not sure that the meaning I have got is correct. Thus, I have realised that the question concerns the life of a radiopharmaceutical, not the patient:) Then it is obvious that the physical half life time is determined by the radioactive element of the pharmaceutical.  (some problems with describing it becomes more complex if the medicament contains more that 1 radioactive element; but I think this does not happen). Then, I've understood, that the biological half-lifetime is probably related to biological processes leading to removing radioactive element  from the aimed tissue (am I right?).

Now, if this is correct, then the question should be posed more precisely whenever a real value the calculations  is to be attained. According to the best of my knowledge, the first step of such investigations is to determine the exact goal of the calculations. Let be formulate one of the most useful though simple aims:  To determine the total  radiation left in the tissue within a given  time period T (say, a week)  from the instant of injection. The the formula cannot be as simple as the one presented in the question, or in my answer. The appropriate algorithm should involve an integral calculus as follows:

R[0,T] =  a*  integral0T b(t) *  2-(T-t)/p dt, where  p  is the physical half life time of the element, and  b(t)  describes the content of the element in the whole tissue at instant  t  after the injection,  and  a   denotes the coefficient describing the rate of increasing the radiation per one unit of the amount of the radioactive element.

In this model the biological half life of the element becomes a parameter of the function  b(t).  I agree with Mushtag, that this function has to be determined experimentally. This function depends of almost all physiological processes and likely it increases in the first hours, riches some plateau with maximal value od duration of anothe few hours, and finally it decreases to zero. I think that  b(t) it is not similar to decreasing exponential function which could justify usage of the biological half life time.  However, if it were, then the intensity of impact described by the product  b(t) * 2-t/p  would be again exponential and the Bidemi's formula would work well.  But even in this simple case we need experimetally established the biological  half life time  b  of presence of the element in the tissue (since then we can write  b(t) = 2-t/b).  

In more general cases I suggest to study data of pharmaceutical factories (whenever attainable), since medicaments are built in such a way, that the function  b(t) varies very moderately. This is not my subject, so I cannot advice anything concrete.  But if some information is supplied,  e.g  if we know the mean value and dispertion of the time of existence of the element in the tissue, then pretty valuable conclusions can be derived. It sufficies to replace then b(t)  by a gamma density function with the given moments and apply this to the above exemplary integral formula (perfomable e.g. via mathematical programms) leading directly to the obtained radiation. 

Regards, Joachim

ERRATUM (of the integral formula). This is the corrected sentence:

The appropriate algorithm for calculating the total dose in the interval [0,t] should involve an integral calculus as follows:

R[0,T] =  a * integral0T b(t) *  2-t/b dt,

where 

---   p  is the physical half life time of the element,

---   b(t)  describes the content of the radioactive element in the whole tissue  at  t  units of time after the injection, 

--- a   denotes the coefficient describing the rate of increasing of the dose of the radiation per a unit of the amount of the radioactive element.

A least this is what I wanted to communicate.  Further improvements are possible,  e.g. some analysis of the function  R[0,T]  in dependence of the shape of the function  b(t).

Regards, Joachim

Let me stress again, that for non-exponential change of the substance in the tissue, the half life time cannot be defined simply by the time of decrease to half of the initial amount, since usually the initial and the final  values are zero (only after some time it reaches maximum is starts to decrease). A more concrete suppositions are presented in the enclosed file. Any coments/questions are welcome.

Best regards, Joachim

Thank you so much Sirs for your explanations. Can we design a project on this or another on the basis of collaboration. What actually led to this question is the notion that the effective half life should determine the time of discharge of patient from the clinic and I found out that the effective half life is not easy to calculate in the clinic.