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

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348 Nuclear Medicine Physics

wherex

is the population at year k, as a function of the maximum population.

This is the famous logistic equation for the population model that we will

further discuss later on.

Is it possible for the population to be constant over time in some equilibrium

state? In these circumstances, we would have x

k+1

= x

,or,

k+1

= Ax

(1 −x

) = x

⇒ Ax

−Ax

= x

⇒ (A −1 −Ax

= 0

⇒

A −1 −Ax

= 0

⇒

⎧

⎨

⎩

A −1

= 0

⇒

⎧

⎨

⎩

= 1 −

= 0

(7.22)

If A > 1, we have two equilibrium states. If A < 1, only the origin can be an

equilibrium state, because x

cannot be negative. Note that A > 0.

7.1.2.3 The Lotka–Volterra Predator–Prey Model

In nature, there are many examples of predator–prey pairs: lion–gazelle,

bird–insect, cat–rat, pandas–eucalyptus, etc. [13]. To describe the biologi-

cal interaction between two species in the particular relation where one, the

predator, eats the other, the prey, we can use the Lotka–Volterra model. For

this purpose, let us make the following simplifying assumptions:

• The predator is totally independent of the prey, and this is its only

food.

• The prey has unlimited food available.

• The prey has only one predator, the one that is being considered, and

has no other environmental constraints.

In nature, the reality is more complicated; but with these assumptions, we

can develop an easily understood model.

If there are zero predators, the prey has no limit on its growth and it grows

exponentially. If x(t) is the actual population at time t, its growth rate will be

Equation 7.23

dx(t)

= ax(t). (7.23)

However, with the existence of predators, growth will be lower; the

predators introduce a negative factor into the equation.

Ify(t) is the predatorpopulation attime t, eachof its individualshas a certain

probabilityof meeting prey. This probability is higher the morepredators exist

and also the more prey exists. So we will have the following:

• The rate of predator–prey contacts is jointly proportional to the size

of both populations.

Systems in Nuclear Medicine 349

• A ﬁxed portion of the contacts that occur results in the death of the

prey.

So we will have Equation 7.24.

dx(t)

= ax(t) − bx(t)y(t). (7.24)

Concerning the predator population, y, if there is no food (no prey), it will

die at a rate proportional to its size, as in Equation 7.25,

dy(t)

=−cy(t), (7.25)

because there is no energy for new births.

Fortunately for the predators,thereis prey, giving them energyto reproduce

themselves. For each predator–prey contact, there is a part resulting in food

for the predator; and its population will evolve according to Equation 7.26,

dy(t)

=−cy(t) +px(t)y(t), (7.26)

where p is a constant expressing the likelihood that the predator will eat the

prey.

Joining Equations 7.24 and 7.26, we ﬁnally get the Lotka–Volterra model

(Equation 7.27).

dx(t)

= ax(t) − bx(t)y(t)

dy(t)

=−cy(t) +px(t)y(t),

(7.27)

This model consists of two ﬁrst-order nonlinear coupled differential equa-

tions that cannot be independently solved and for which it is not possible to

ﬁnd an analytical solution; only numerical solutions are possible.

It can be shown that there is a pair (x, y) for which dt = dy/dt = 0, that is,

the populations are in equilibrium and do not vary with time. For that, we

can write Equation 7.28

0 = ax(t) − bx(t)y(t)

0 =−cy(t) + px(t)y(t),

(7.28)

Let us solve this system of algebraic Equations 7.29

[a −by(t)]x(t) = 0 ∧x(t) = 0

[−c +px(t)]y

(t) = 0 ∧y(t) = 0

⇒

[a −by(t)]=0

[−c +px(t)]=0

⇒

⎧

⎪

⎨

⎪

⎩

y(t) =

x(t) =

. (7.29)

350 Nuclear Medicine Physics

This state is called steady state or stationary state. Its value depends on the

constants a, b, c, and p, the parameters of the model.

We will come back to this model later to analyze its phase trajectories.

There are more complete models of two populations. For example, in [8] we

can ﬁnd a model that includes competition (ﬁghting for the same resources)

among species, aggressive interaction (ﬁghting each other, although not eat-

ing each other), symbiotic cooperation (mutual help), predator–prey relation,

and weak–strong interaction (one species is better prepared to survive).

7.1.2.4 A General Model for the Interaction of Two Populations

Let us consider two biological populations interacting in a certain environ-

ment. They can be modeled by the two differential equations below [7]:

x = ax +bxy

y = cy +dxy,

(7.30)

with the four parameters a, b, c, and d (real numbers).

The linear terms ax and by describe the growth or decline of the populations

x(t) and y(t) separately considered from one another. For a > 0, x(t) grows

exponentially; for a < 0, x(t) decreases exponentially. For b > 0orb < 0, the

same happens with y.

The interaction between the two populations is modeled by the nonlinear

termsbxy and dxy. The interaction dependson thenumber ofcontacts between

the elements of x and the elements of y. The constants b and c express the

likelihood of an interaction if there is a meeting. Several forms of interaction

between the two populations can be described by this model, depending on

the signs of the four constants a, b,

c, and d, as seen in Table 7.2, assuming that

a predator has only one kind of prey.

TABLE 7.2

Signs of the Constants in the General Model of Biological

Interaction and Interaction That They Express

ABCd Interaction

− ++−Predator(x)-prey(y) models

+ −−+Prey(x)-predator(y)

+ +++Mutualism or symbiotic models

+ −+−Models of competion

Systems in Nuclear Medicine 351

If we now consider the logistic equation for each of the populations and, at

the same time, their interaction, we will have Equation 7.31:

x = a



1 −



−

y = a



1 −



−

xy,

(7.31)

where the constants a

, a

, b

, and b

have speciﬁed meaning.

7.1.2.5 Modeling Epidemic Phenomena

Letus nowconsider thecase ofan inﬂuenzaepidemic. Oneinfected individual

is introduced into the population. The infection spreads by contact between

an infected and a healthy person. The healthy person may or may not be

infected by this contact. After some time (about two weeks in the case of ﬂu),

the infected person recovers and remains immune to the particular virus;

so the person will not be infected again in our model. The following state

diagram (Figure 7.5) illustrates these interactions.

The differential equations governing the dynamics of these populations are

Equation 7.32

S =−βIS +γR

I = βIS −αI

R = αI − γR.

(7.32)

Susceptibles

healthy

Infected

Recovered

FIGURE 7.5

Interactions in the ﬂu epidemic model.

352 Nuclear Medicine Physics

If the immunity period is inﬁnite, then γ = 0, and this model results in the

Kermak–McKendrick model [7] given by Equation 7.33.

S =−βIS

I = βIS −αI.

(7.33)

After a while, the population divides into three groups: the susceptible, the

infected, and the recovered. The recovered can eventually become susceptible

if their immunity ends (they are only immune for a certain time).

7.1.2.6 Physiological Systems

Physiological systems include the organs of animals, in general, and human

ones, in particular. They are the most complex systems that nature has ever

created. They are largely composed of various kinds of systems (mechani-

cal, ﬂuidic, chemical, etc.), which, by including living organs, acquire new

properties through synergy.

Example 7.1 A Blood Vessel

Figure 7.6 represents a blood vessel.

Assume that the vessel is working in a steady state where all quantities are

constant. In these conditions, it has to be that Q

= Q

, as there is no blood

accumulation because the volume is constant. Let Q = Q

= Q

be that blood

ﬂow.

To ﬁnd a relation between Q, P

, P

, and V, two properties are considered:

its resistanceto the ﬂow and its compliance with the pressure (which increases

the volume of the vessel). To simplify the analysis, consider the following

assumptions:

1. A vessel without compliance, rigid, of constant and well-known vol-

ume. For that we will have, as in a ﬂuidic system, the ﬂow equal to

ext

= 0

V = Volume of the vessel

= Input ﬂow

= Pressure at the input

= Onput ﬂow

= Pressure at the output

ext

(Outside pressure, the reference) = 0

FIGURE 7.6

A blood vessel. (Adapted from F.C. Hoppensteadt, C. S. Peskin. Modeling and Simulation in

Medicine and the Life Sciences, 2nd ed., Springer-Verlag, 2004.)

Systems in Nuclear Medicine 353

the pressure difference at the two ends of the vessel divided by the

ﬂuidic resistance of the vessel, as in Equation 7.34.

Q =

−P

, (7.34)

where R is the vessel resistance; we have a resistive vessel here.

2. An elastic vessel without any ﬂuidic resistance. In these conditions,

the pressure at the input is equal to the pressure at the output, for

any value of Q. The relation between the pressure P and the volume

of the vessel can be approximated by Equation 7.35,

V = CP ⇒ C =

, (7.35)

where the constant C is the compliance of the vessel. We can also

write Equation 7.36,

V = V

+CP, (7.36)

where V

is the volume of the vessel for zero pressure (dead volume).

C is again the compliance of the vessel.

We can check the analogy of the following two situations:

Vessel: Due to the compliance C, the vessel enlarges with the pressure P until

it reaches a ﬁxed volume V, which depends on the compliance. Then we will

have V = CP.

Capacitor (electric): Due to thecapacity C,the capacitor chargeswhen applied

to a ﬁxed voltage V, until its charge reaches a ﬁxed value, which depends on

the capacity. Then we will have Q = CV.

These two theoretical situations are reasonably close to reality. A real ves-

sel has simultaneously resistance and compliance. The relation among the

variables is not linear.

In the human body, the circulation separates the vessels of resistance from

the vessels of compliance. The big arteries and veins are mainly compli-

ance vessels, because small pressure differences are sufﬁcient to produce the

blood ﬂow and their volumes change considerably. The small arterials (arteri-

oles) and the small veins (venules) are mainly resistive vessels: their volumes

change very little, and big pressure differences are generated there.

Consideringboth properties,wecan conceive anelectrical circuitequivalent

to a vessel, with the analogies of Table 7.3, as observed in Figure 7.7. From

the circuit laws, we obtain Equation 7.37.

=−

, C

−v

⇒

=−

, (7.37)

354 Nuclear Medicine Physics

TABLE 7.3

Analogies among the Physiological and

Electric Systems and Variables

Vessel Electrical Circuit

Pressure, PV, Voltage

Volume, VQ, Charge

Flow, QI, Current

Resistance, RR, Resistance

Compliance, CC, Capacity C

ext

= 0

i(t)

variable V

variable

V +

–

FIGURE 7.7

Analogy between a vessel and an electrical circuit.

V(t)

(t)

I(t)

FIGURE 7.8

The biomechanics of inhalation. (Adapted from E.N. Bruce. Biomedical Signal Processing and Signal

Modeling, John Wiley & Sons, Inc., 2001.)

Systems in Nuclear Medicine 355

Example 7.2 Biomechanical Model of Pulmonary Inhalation/Exhalation

Inhalation and exhalation can be described by a biomechanical system

illustrated in Figure 7.8.

In a relaxed state, without any air ﬂow, the pressure inside the lungs is equal

to the atmospheric pressure, P

. During inhalation, the respiratory muscles—

the diaphragm and the thoracic muscles—contract and lower pleural pressure

(t) around the lungs inside the chest.

Due to their elasticity (described by the compliance C, the inverse of plas-

ticity), the lungs expand as the transmural pressure decreases; and as a

consequence, the pressure inside the lungs P

(t) momentarily decreases. The

pressure in the ﬂuidic resistance R

of the respiratory tract falls, resulting in a

ﬂow of air to the inside of the lungs. As long as the muscles contract increas-

ingly strongly, this process is repeated and inhalation continues. At the end

of inhalation, the abdomen and chest muscles contract and force P

(t) to

exceed P

(atmospheric pressure), thus expelling the air from the lungs. In

resting respiration, the exhalation muscles do not need to contract, because

the relaxation of the inhalation muscles alone is enough to make P

(t) greater

than P

This biomechanical process can be mathematically described based on the

physical properties of the physiological structures involved. It is a mixed

ﬂuidic-mechanical system.

Theair ﬂow depends on the pressure difference though theﬂuidic resistance

and on the resistance itself. If I(t) represents the air ﬂow rate of inhalation

or exhalation, we will have Equation 7.38.

I(t) =

−P

(t)

, or, P

−P

(t) = I(t)R

. (7.38)

If V(t) is the volume of the lungs during the time interval Δt, the volume

variation will be Equation 7.39, assuming I(t) to be constant during Δt, then

Equation 7.39,

(t +Δt) − V

(t) = I(t) Δt ⇒

(t +Δt) − V

(t)

Δ t

= I(t) ⇒ I(t) =

(t),

(7.39)

Substituting Equation 7.39 in Equation 7.38, we get Equation 7.40.

−P

(t) =

(t)R

. (7.40)

The compliance of the lungs C

is deﬁned by the relation between the

volume of the lungs and the transmural pressure. Mathematically, it is

Equation 7.41.

(t)

−P

(t)

. (7.41)

356 Nuclear Medicine Physics

from which we get Equation 7.42.

−P

(t) =

(t)

⇒ P

= P

(t) +

(t)

. (7.42)

Now substituting P

in Equation 7.42, we obtain Equation 7.43.

−P

(t) =

V(t)R

⇒ P

−P

(t) +

(t)

V(t)R

⇒ P

−P

(t) =

V(t)R

(t)

(7.43)

Since we consider the atmospheric pressure constant, we deﬁne the

differential pressure (Equation 7.44),

P(t) = P

−P

(t), (7.44)

and

P(t) =

(t)R

(t)

(t) +

(t) =−

P(t)

(t) +aV

(t) = bP(t)

(7.45)

Finally, Equation 7.45 is a ﬁrst-order linear differential Equation relating

the lungs’ volume, V

, with the relative intramural pressure, P(t). From the

systemicpoint of view,it canbe representedby theblock diagraminFigure7.9.

Note, further, that Equation 7.46

V(t) = I(t) ⇒ V(t) = V(0) +



I(t) dt, (7.46)

where V

(0) is the resting volume. Since it is a constant characteristic of each

individual, it can be considered null; and V

(t), thus, represents the volume

of variation in the lungs relative to V

(0).

P(t)

Respiration system

(t)

FIGURE 7.9

A systemic representation of the respiratory system.

Systems in Nuclear Medicine 357

7.1.2.7 Compartmental Analysis

Compartmental analysis in physiology makes it possible to monitor the distri-

bution of biological substances inside the human body [11]. To deﬁne a correct

dosage of medication, we need to know the rate of drug absorption by the

different tissues and organs. Other natural substances (hormones, metabolic

substrates, lipoproteins, peptides, etc.) have very complex distribution pat-

terns. Radioactive tracers are used to discover these patterns.Atracer is mixed

with the intended substance and the mixture is introduced into the body.

Its concentration in the blood can be sampled at successive instants of time

throughradioactive analysis. This radioactivity can also be measuredthrough

radiation imaging. The physiologist can then compute the rate of absorption

and release of a substance by a tissue, or the rate of degradation or elimination

from the bloodstream (by the kidneys, for example) or from the tissues (by

biochemical degradation).

In compartmental analysis, the body is divided into a set of homoge-

neous (well mixed) interconnected compartments that exchange substances

between themselves and degrade those substances according to simple linear

kinetics. Let us consider, for example, the case of two compartments, the ﬁrst

representing the circulatory system and the other representing all the tissues

relevant to the analysis (not necessarily an organ or a single physiological

entity). Figure 7.10 illustrates the situation.

The variables in Figure 7.10 have the following meanings:

• u

, u

rate of injection of substance into compartments 1 and 2.

• K

rate of ﬂow of substance from compartment 1 to compartment 2.

• K

rate of ﬂow of substance from compartment 2 to compartment 1.

• K

rate of degradation of substance in compartment 1.

• K

rate of degradation of substance in compartment 2.

Mass M

Concentration x

Volume V

Mass M

Concentration x

Volume V

FIGURE 7.10

Two compartments exchanging substances between themselves and the environment.