The Half-Lives: Physical, Biological, and Effective
Introduction: There are three half-lives that are important when considering the use of
radioactive drugs for both diagnostic and therapeutic purposes. While both the physical
and biological half-lives are important since they relate directly to the disappearance of
radioactivity from the body by two separate pathways (radioactive decay, biological
clearance), there is no half-life as important in humans as the effective half-life. As we
will see shortly, this half-life takes into account not only elimination from the body but
also radioactive decay. If there is ever a question about residual activity in the body, the
calculation uses the effective half-life; in radiation dosimetry calculations, the only half-
life that is included in the equation is the effective half-life. Let’s take a look individually
at the three half-lives.
Half Lives: Physical
Physical half-life is defined as the period of time required to reduce the radioactivity level
of a source to exactly one half its original value due solely to radioactive decay. The
physical half-life is designated tphys or more commonly t
1/2
. By default, the term t
1/2
refers to the physical half-life and t
phys
is used when either or both of the other two half-
lives are included in the discussion. There are a few things to note about the t
phys
:
• The t
phys
can be measured directly by counting a sample at 2 different points in
time and then calculating what the half-life is.
• For example, if activity decreases from 100% to 25% in 24 hours, then the half-
life is 12 hours since a decrease from 100% to 50% to 25% implies that 2 half-
lives have elapsed.
• One can also determine graphically what the half-life is. In the diagram below,
both of the lines have a half-life of 5 days, even though their activity levels are
quite different.
• The range of half-lives is boundless. There are isotopes with half-lives of nsec,
μsec, msec, sec, hr, min, days, weeks, months, years, centuries, millennia, and
even as long as a billion years (half-life of K-40 = 1.28 x 10
9
years). Most of those
time units would not be very useful for diagnostic or therapeutic studies and, in
fact, all commercially available isotopes range from 75 sec (Rb-82) to Sr-89 (50.5
days) with all others in between those two values.
• The physical half-life is unaffected by anything that we humans can do to the
isotope. High or low pressure or high or low temperature has no effect on the
decay rate of a radioisotope. Perhaps taking the isotope to absolute 0 or within a
degree of that temperature would affect the decay rate.
Half Lives: Biological
Biological Half-life is defined as the period of time required to reduce the amount of a
drug in an organ or the body to exactly one half its original value due solely to biological
elimination. It is typically designated t
biol
or t
b
. There are a few things to note about the
t
biol
:
• For radioactive compounds, we have to calculate the tbiol because the mass of the
isotope is usually on the nanogram scale and, when distributed throughout the
body, and especially in the target organ, concentrations are in the picogram/ml
range, much too small to measure directly.
• For non-radioactive compounds, we can measure the tbiol directly. For example,
assuming that a person is not allergic to penicillin, we could give 1,000 mg of the
drug and then measure the amount present in the blood pool and in the urine since
we administered such a large amount of the drug..
• One can also determine graphically what the biological half-life is. In the diagram
below, the drug has a biological half-life of 7 hr. Unlike t
phys
which is boundless,
t
biol
for commercially available radiopharmaceuticals is typically in the range of
sec (ventilation study) to days (phosphate based bone agents).
• t
biol
is affected by many external factors. Perhaps the two most important are
hepatic and renal function. If kidneys are not working well, we would expect to
see a high background activity on our scans. Also important is level of hydration.
A poorly hydrated patient, even with normal renal function, will have a high
background activity since limited urine is being produced, making it difficult to
eliminate isotope that has not localized in the target organ.
• Each individual organ in the body has its own t
biol
and the whole body also has a
t
biol
representing the weighted average of the t
biol
of all internal organs and the
blood pool. It is therefore very important to have a frame of reference. For
example, do you need to know the t
biol
of the drug in the liver or in the whole
body?
• All drugs have a t
biol
, not just radioactive ones. Drug package inserts often refer to
the half-time of clearance of a drug from the blood pool or through the kidneys.
• Since the whole body has a t
biol
representing the weighted average of the tbiol of
all internal organs, it will almost never equal that of an internal organ.
• It is a fallacy that physical and biological half-lives can never equal each other.
Consider Tc-99m MAA, for which both t
phys
and t
biol
are equal to 6 hours.
• Consider two extreme cases: for Tc-99m Sulfur Colloid, the t
biol
is considered to
be infinitely long; the t
biol
of Xe-133 gas in the lungs is measured in sec.
Half Lives: Effective
Effective Half-Life is defined as the period of time required to reduce the radioactivity
level of an internal organ or of the whole body to exactly one half its original value due to
both elimination and decay. It is designated t
eff
or t
e
. There are a few things to note about
the t
eff
:
• The t
eff
can be measured directly. For example, one can hold a detection device 1
m from the patient’s chest and count the patient multiple times until the reading
decreases to half of the initial reading. The patient is permitted to use the rest room
between readings as needed, so both elimination and decay are taking place. The
half-life being measured in this case is the t
eff
.
• The range of t
eff
typically varies from sec to hr.
• t
eff
is affected by the same external factors that affect tbiol since t
eff
is dependent
upon t
biol
.
Mathematical Relationship
There are three special cases that help to clarify the concept of effective half-life:
If t
phys
>>> t
biol
then t
eff
~ t
biol
e.g., for a Xe-133 for pulmonary ventilation study, where t
phys
= 5.3 days and t
biol
= 15
sec, the study is over within a few minutes. It did not matter whether the Xe half-life was
5 hr, 5 days, or 5 weeks. They are all so long compared to the biological half-life that the
effective half-life equals the biological half-life.
Calculation:
so... t
eff
= 15 sec
If t
biol
>>> t
phys
then t
eff
~ t
phys
e.g., for a liver scan with Tc-SC, t
phys
= 6 hr, t
biol
= Infinitely long. Because Tc-SC never
clears the liver, then the only half-life that matters is the physical half-life,
Calculation:
and... t
eff
= 6 hr
There is a third special case.
If t
biol
= t
phys
then t
eff
= 1/2 t
biol
= 1/2 t
phys
Example: for Tc-MAA for pulmonary perfusion imaging, t
phys
= 6 hr, t
biol
= 6 hr, then t
eff
= 3 hr.
Calculation:
and... t
eff
= 3 hr
In most other cases, one has to mathematically solve for the desired half-life since they
are not special cases.
Problem 1.
I-131 sodium iodide has a t
biol
of 24 d. What is t
eff
?
1/ t
eff
= 1/ t
phys
+ 1/ t
biol
= 1/8 + 1/24 = 1/6 so.... t
eff
= 6 d
Problem 2.
A Tc-99m compound has a t
eff
= 1 hr. What is t
biol
?
1/ t
biol
= 1/ t
eff
- 1/ t
phys
= 1/1 - 1/6 = 5/6 so... t
biol
= 1.2
Problem 3.
A radiopharmaceutical has a biological half-life of 4.00 hr and an effective half-life of
3.075 hr. What isotope was used?
1/ t
phys
= 1/ t
eff
- 1/ t
biol
= 1/3.075 – 1/4.00 = 0.0752033 ...Therefore t
phys
= 13.3 hr and the
radioisotope is I-123
Radiation Dosimetry Equation
The equation below is commonly used for radiation dosimetry calculations.
You’ll observe that the only half-life in the equation is the effective half-life.
Dγ = 73.8 . E
γ
. f
γ
. C
o
. 1.443 . t
eff
. φ Rads
