Halftime and time constant

Principle

Half-life or more practically halftime (t_½) is the time required for the decaying quantity N₀ to fall to one half of its initial value, ½N₀. Knowing t_½, at any time the remaining quantity can be calculated:

. (1)

When t = t_1/2, then N(t_1/2) = N₀2^-1 = ½N₀, hence the name halftime. Thus, after 3 halftimes there will be 2^─3 = 1/8 of the original material left.

N(t) decays with a rate proportional to its own, instantaneous value. Formally, this can be expressed as the following differential equation, where N is the quantity and λ is a positive number called the decay constant.

. (2)

The solution to this equation is:

N(t) = N₀e ^─λt . (3)

Generally, in (medical) physics and engineering, λ, a reciprocal time, is substituted by 1/τ. The Greek letter tau, τ, is the time constant of the decay process. The decay appears to be exponential (Fig. 1).

Fig. 1 Exponential decay with halftime t_1/2 = τ/ln2. Multiples of τ with their corresponding quantities are indicated.

Equation (1) and (3) look very similar and are strongly related. τ appears to be:

τ = t_1/2ln2. (4)

Substituting λ in (3) by 1/τ and using (4) equation (1) is obtained.

In practice (medical, physical, control engineering systems), after 8 halftimes or 5 time constants N(t) is supposed to be zero (< 0.01N₀). So, after this time the process is assumed to have reach its asymptotic value (here zero).

Application

Applications are innumerous in science, technology and medicine. Trivial examples in medicine are the clearance of a substance in the blood, the radio active decay of a tracer molecule and the way in which a thermometer reaches for instance the body temperature of a patient. It is also basic in optics (Lambert Beer’s law), spectroscopic applications, radio-diagnostics, radiotherapy, nuclear medicine, neurobiology etc.

More info

In (medical) physics and engineering τ characterizes the frequency response of a first-order, linear time-invariant (LTI) system (see Linear systems: general). A LTI system, simply said, has properties that do not vary. First-order means that the system can be described by a first order differential equation (2), an equation comprising the first derivative and no higher ones.

Examples include the electrical RC filter (comprising the resistor R and the capacitor C) and RL circuit (comprising the resistor R and the coil L). (see for a description of the circuit elements R, C and L Electrical resistance, capacitance and inductance). It is also used to characterize the frequency response of various signal processing systems, such as the classical magnetic tapes, radio and TV transmitters and receivers and digital filters – which can be modeled or approximated by first-order LTI systems. Other examples include time constant used in control systems for integral and derivative action controllers.

The frequency response is given by the amplitude and phase characteristic. The first one gives the gain of the system as a function of the frequency of a sinusoidal input. The second one gives the phase shift of the output of the system relative to phase of the sinusoidal input. The time constant is related to the cut off frequency ω₀ of the first order LTI system:

τ = 1/ω₀,

where ω₀ = 2πf₀ and f₀ the cut off frequency, i.e. the frequency of the sinusoidal input wave which output amplitude is 2 ^–0.5 (= 0.71) times the input amplitude. .

The time constant also describes the output to a very elementary input signal, the impulse function. Its output has the same shape as the exponential decay of Fig. 1. After one τ the decay is 63.2% of its original value, and 36.8% (about 100/τ %) remains. Another time of 4 τ is needed to reach the asymptote (zero).

See for more info about linear systems Linear systems: general and Linear first order system.