Lima J.J.Pedroso, de (ed.). Nuclear Medicine Physics

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408 Nuclear Medicine Physics

(a) (b) (c)

Recorder

1tt

cpm

FIGURE 7.37

(a) Tube with a liquid ﬂow and two radiation detectors applied to small volumes dV

and dV

(b) Responses of the two detectors D

and D

as a function of time are obtained in the same

record. (c) Responses of the detectors if a bolus of a radioactive tracer is injected at the entrance

of the tube and assuming the ﬂow in the tube is laminar.

Supposing that a single, small radioactive particle is injected at the entrance

of the tube, and the particle follows its path without incident, the responses of

the two detectors as a function of time in the same plot are as in Figure 7.37b.

The small particle has passed through the tiny volumes seen by the detec-

tors at times t

and t

, respectively. This experiment with just one particle is

probably the only situation in tracer dynamics in which the output curve is

not only predictable but also equal to the input curve.

If we inject not a particle but a bolus of radioactive tracer miscible in the liq-

uid stream and assume the ﬂow regime in tube is laminar, the output response

is different from the input one, as shown in Figure 7.37c. It was assumed in this

example that the input is a very fast injection which we call impulse function

(or impulse injection), or delta function.

The input and output curves are different but the output is predictable,

∗

that is, if the input is an impulse function and the ﬂow is laminar we can

predict what the output curve will be like.

In most of the situations of interest to biology and medicine, the systems are

much more complex than single tubes. The injected particles can not only have

a range of different paths but they can also be drawn by a range of different

velocities of ﬂow within the system. So these will have to be considered to be

chaoticsystems in whichthe processesarerandom in nature. We areinterested

in the overall statistical properties of the population of particles, and we are

dealing with density functions and statistical distributions. Contrary to the

preceding cases, one cannot deterministically predict the output curve.

Mean transit time is the average time necessary for tracer molecules to move

between two points, which may or may not include a compartment.

Mean residence time is the average time tracer molecules stay in a space

where they have been deposited or dragged to.

Both these times depend on the ﬂow that drags the tracer and on the volume

of the dispersing system.

∗

The response function for a catheter of length L and minimum transit T is h(t) =





Systems in Nuclear Medicine 409

Recorder

cpm

(a)

(b)

cpm

FIGURE 7.38

(a) Residue detection. (b) Responses of detectors D

and D

, in a real system. The output curve

is a widened curve showing the transit times of the particles distributed between times t

and t

Transit times are generally obtained by measuring at input and output, by

detection or direct sampling (Figure 7.38b). Since the tracer molecules can

have very different paths in a dilution system, their output concentration

varies with time to give a transit time curve (or a histogram in the case of

sampling). In other words, a time or activity or time or concentration curve

is obtained, with a proﬁle that depends on the input function proﬁle and on

the characteristics of the dilution system and that can be used to calculate the

mean transit time and volume.

Residence times are obtained by residue detection, with which the total

tracer activity that remains in the system is measured, generally by external

detection (Figure 7.38a). This type of detection is convenient and often the

only possibility available, but it has the disadvantage in that an accurate

deﬁnition and exclusive detection of the regions of interest are sometimes

hard to achieve, frequently leading to the need to make further corrections.

Suppose that for a spike of radioactive tracer in a real system curve 2 is

obtained at the output (Figure 7.38b).

The particles injected at time t

in an impulse injection came out as a

widened curve showing the transit times of the particles distributed between

times t

and t

. There is a statistical distribution of transit times.

We can see from the shape of the curve that the transit times which occurred

most often were those located about half way between t

and t

. The output

curve is an indication of how probable (frequent) a given value of transit time

has been.

If we call dV

the element of volume seen by the output detector 2, F the

ﬂow through the system C

(t) the tracer concentration at the output then the

amount of tracer present in dV

between t and t +dt is

(t) = dV

(t) (7.181)

410 Nuclear Medicine Physics

(t)

ttt + dt

FIGURE 7.39

The area of the shaded rectangle is f (t) dt = dq

and it represents the elementary amount of the

label that has crossed the system between instants t and t + dt.

Deﬁning a function f (t) = f(t) = (dq

/dt) which, for every value of t

represents the rate of change of the amount of tracer in volume dV

, is also

(t) = f (t) dt (7.182)

In Figure 7.39, the area of the shaded rectangle f (t) dt = dq

represents the

elementary amount of tracer that has passed the system between times t and

t +dt.

The area under f(t) gives the total amount Q of injected label

∞



f (t) dt =



= Q (7.183)

Instead of f (t), it is more convenient to use another function (normalized)

given by

h(t) =

f (t)

(7.184)

The product h(t) dt is not the absolute amount of tracer that has crossed

the system between times t and t +dt but the fraction of the total amount of

injected tracer that has crossed the system between t and t +dt. Of course,

if we carry out an injection such that Q = 1, the area under the curve equals

unity, the curve is then normalized and f (t) = h(t). The function h(t) is called

the frequency function of transit times or unit responsefunction of the system.

The dimensions of h(t) are [T]

−1

After an instantaneous injection, the fraction of the total amount of label

that has left the system through 2 until time t, as a function of time, can be

obtained by

H(t) =



h(t) dt (7.185)

Systems in Nuclear Medicine 411

We call H(t) the cumulative frequency function (Figure 7.40b). The fraction

of the total amount of labeled tracer that has left the system by time t is the

area under h(t), between time zero and t, because the total area under h(t) is

the unit

H(∞) =

∞



h(t) dt = 1 (7.185a)

An obvious relationship is

(H(t)) = h(t) (7.186)

Transit times and residence times are closely related to each other. For an

impulse input, the activity or time curve obtained by external detection at

the output that represents the transit times distribution can be converted

into the corresponding retention function r(t), that is, the activity or time

curve distribution of residence times (Figure 7.40c). In every time instant, the

fraction of the total tracer injected that remains inside the system is 1 minus

the fraction already eliminated H(t), leading to

r(t) = 1 −



h(t) dt =

∞



h(t) dt −



h(t) dt =

∞



h(t) dt (7.187)

h(t)

(a)

(b) (c)

(t) r(t)

m t

FIGURE 7.40

(a) Frequencyfunction of transit times. (b) H(t) is the cumulative frequency function. It represents

the fraction of the total amount of tracer that leaves the system up to time t. (c) r(t) is the retention

function, or the fraction of the total amount of tracer within the system as a function of time, that

is 1–H(t).

412 Nuclear Medicine Physics

Then for every time instant t, the integration of curve h(t) between t and

inﬁnity is a value of r(t), the retention function. Also true is

(r(t)) =−h(t) and h(t) dt =−dr(t). (7.188)

The retention function is the residue curve for a unitary impulse input, and

it can be converted to the transit time distribution through differentiation and

multiplication by −1.

Let us return to the deﬁnition of mean transit time for the case of an impulse

injection of an activity Q in a dilution system and enter with Equation 7.187

for residue detection. Then

t =

∞



tQh(t) dt

∞



Qh(t) dt



r(0)

−t dr



r(0)

−dr

∞



r(t)dt

r(0)

(7.189)

The mean transit time of a tracer,or the corresponding mean residence time,

respectively through or in an organ, space, or compartment is the quotient

of the total area under the residue curve and its initial height (t = 0 abscissa)

(Figure 7.41).

There is another way of obtaining the cumulative frequency function of

transit times, apart from the integration of h(t).

If the tracer is introduced into the system at a constant rate, after a sharp

rise (step function i(t)), the output will be the cumulative frequency function

(Figure 7.42).

If the input is a δ function, by deﬁnition the output is h(t). Then, since

δ(t) =

[i(t)] and h(t) =

[H(t)], (7.190)

r(t)

r(0)

Equal

area

FIGURE 7.41

The mean transit time of a tracer equals its mean residence time, and it is the quotient of the total

area under the residue curve and its initial high (t = 0 abscissa).

Systems in Nuclear Medicine 413

Input

i(t) H(t)

Output

FIGURE 7.42

If the tracer is introduced into the system at a constant rate, after a sharp rise (step function i(t)),

the output will be the cumulative frequency function.

we conclude that, by differentiationof the functions in Figure 7.42, at the input

and output, the same functions are obtained as in the case of an instantaneous

injection (Figure 7.43).

For a dilution system in which there is an impulse function of a tracer that

is exponentially cleared, the detected residue function of the system is of the

type Q exp(−αt). The mean residence time is

t = 1/α.

If the residue curve is a double exponential with equation

Q(t) = Q

exp(−α

t) +Q

exp(−α

t),

the mean residence time will be

t =

/α

(7.191)

7.2.5.1 The Stewart–Hamilton Principle

Let us consider a dilution system perfused by a constant ﬂow F and that at

t = 0 an instantaneous injection of an amount Q of tracer is administered.

The output function is Qh(t), and the amount of label leaving the system

between times t and t + dt is dq = Qh(t)dt. However, dq can also be given by

Input

i(t) H(t)

Output

FIGURE 7.43

If input is a δ function, δ(t) =

[i(t)] then the output is h(t) =

[H(t)].

414 Nuclear Medicine Physics

the product of the elementary volume of liquid that leaves the system in the

same time interval, dV = F dt, multiplied by the tracer output concentration

(t):

dq = F dtC

(t) dt

Then

(t) dt = Qh(t) dt. (7.192)

Integrating this equation between 0 and ∞, and considering the initial

conditions and that



∞

h(t)dt = 1

F =



∞

(t)dt

(7.193)

which expresses the Stewart–Hamilton principle that used to be a familiar

equation for blood ﬂow determination.

7.2.5.2 Volume Calculation

Let us suppose that in a labyrinth system, at instant t = 0, a liquid ﬂow was

initiated and simultaneously there was a step function U(t) of a tracer, accom-

panying the liquid displacement. It is assumed that there is no transition

regime in the ﬂow, that is, a stationary situation prevails from the beginning.

With these constrains, a function H(t) will be detected at the output.

For each value of t d[H(t)] is the fraction of the injected tracer with transit

times between t and t +dt. The value of d[H(t)] is also the fraction of the total

number of possible paths in the labyrinth with lengths whose crossing times

have to be between t and t +dt. Then the ﬂow at the output that corresponds

to the paths with these particular lengths is F d[H(t)], and the volume of this

set of channels is

dV = Ftd[H(t)]. (7.194)

The total volume of the system corresponds to all groups of paths with

length corresponding to transit times between 0 and ∞being, using Equations

7.189 and 7.193,

V = F

∞



th(t)dt (7.195)

Considering the integral on the right-hand side of Equation 7.184 and the

deﬁnition of the mean value of a function is



∞

th(t)dt



∞

h(t)dt

t (7.196)

Systems in Nuclear Medicine 415

that is, the mean transit time. Then Equation 7.197

V = F

t. (7.197)

The volume of the system is equal to the product of the ﬂow and the mean

transit time of the tracer through the system. Equality has been proved for a

stationary situation.

7.2.6 Distribution Systems and the Convolution Integral

Assume a series of two distribution systems in steady state (Figure 7.44) and

that there is an impulse injection of a tracer, δ(t), at the entrance of the ﬁrst

member, at t = 0. Let the frequency functions of transit times for the members

of the series be h

(t) and h

(t), respectively.

Each liquid particle that leaves the series between time t and t +dt needs a

time between τ and δτ + dτ to cross the ﬁrst element of the series and a time

between (t −τ) and (t −τ) +d(t −τ) to cross the second one.

The fraction of the total number of particles that crosses the ﬁrst element

between times τ and τ +dτ is h

(τ)dτ.

If at time τ,aδ function is introduced at the input of the second element a

shifted frequency function of transit times would be obtained at the output

(t −τ).

All the element volume h

(τ)dτ leaving the ﬁrst element can be considered a

δ function ofsmall amplitudethat is introducedat time t in thesecond element,

that is, h

(τ)dτd(t −τ), meaning a δ function with amplitude h

(τ)dτ applied

at instant t = τ.

The output for this function, which is the output of the series, is

(τ)h

(t −τ) dτ (7.198)

If all plausible τ values are considered for all t values, that is, if we consider

all the transit time values through element 1 which, added to possible transit

times through 2, make t, and if we calculate all the products h

(t) dτ assumed

as δ functions for the second element, with outputs h

(τ)h

(t −τ)dτ, the sum

of these output functions is the frequency function of the transit times through

δ(t)

FIGURE 7.44

Series of two distribution systems.

416 Nuclear Medicine Physics

the series, then

h(t) =



(τ)h

(t −τ) dτ (7.199)

If, for the same conditions, instead of considering the particles that leave

the ﬁrst element at time τ, we consider those that need the same time τ to

cross the second element, then these particles need time (t −τ) to cross the

ﬁrst element:

h(t) =



(τ)h

(t −τ)dτ (7.199a)

The right-hand sides of both Equations 7.198 and 7.199 describe the same h(t).

The physical meaning of this is that if we rearrange the system in such a way

that the second element is now the ﬁrst one, and if we inject a unit impulse

at the entrance of the new system the output will have the same frequency

function as in the former situation. Equations 7.199 and 7.199a are called con-

volution integrals, and we have just proved that the sequence of the functions

under convolution is immaterial. The simpliﬁed notation for convolution of

two functions was introduced on the last member of Equation 7.199b.

h(t) =



(τ)h

(t −τ)dτδ = h

(τ) ∗h

(t) (7.199b)

We can then write,

h(t) = h

(τ) ∗h

(t) = h

(τ) ∗h

(t). (7.200)

One property which could be anticipated from the equations just described is

that the convolution of one frequency function on another is itself a frequency

function.

Three time characteristics of frequency functions are additive:

1. The shortest transit time through the series must be the sum of the

shortest transit times through every member of the series.

2. The longest transit time through the series must be the sum of the

longest transit times through every member of the series.

3. The mean transit time through the series must be the sum of the mean

transit times through every member of the series.

This last property is not obvious, but it can be proved using some

mathematics.

Systems in Nuclear Medicine 417

Another important property derives from characteristic number 3 that, if in

a series of two elements, we know the ﬂow and the mean transit time through

either member then we can ﬁnd the volumes of the members.

The ﬂow through the series can be written:

F =

(7.201)

where V

, V

, and V = V

are the volumes of the elements and the

total volume, and

and

t =

are the mean transit times through the

elements and the total mean transit time.

Since

t −

and F is known, then

= F(

t −

)

and similarly

= F(

t −

). (7.201a)

Among the distribution systems in series found in the body are the

pulmonary and systemic circulations.

When samples are collected by means of a catheter, a new distribution

system is introduced into the series, the catheter itself.

7.2.6.1 Determination of Flow

We do not need to know the shape of the input function to ﬁnd the ﬂow and

probably the transit time, if the injection starts and ends during the course of

the experiment. It is

(t) = i(t)

∗

h(t). (7.202)

We can write i(t) = qf(t), where q is the total quantity of tracer injected

during the experiment and f (t) is the frequency function of the transit times

of injection, then

F = q

f (t) ∗ h(t)

C0(t)

(7.203)

If f (t)

∗

h(t) is the convolution of two frequency functions, then it is a

frequency function and its integral from zero to inﬁnity is unity.

Integrating the above equation for all times

∞



C0(t) dt = q

∞



f (t) ∗ h(t)dt (7.204)