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

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338 Nuclear Medicine Physics

Consider, for example, a two-population biological system, the predator–

prey pair, where the prey is a herbivore and the predator is a carnivore. If the

vegetation increases in the environment, the herbivore population increases

exponentially;as a consequence,the carnivorepopulation also increasesexpo-

nentially, which leads to a fall in the herbivore population, because there are

more predators. Therefore, the initial increase in the herbivore population

retroacted through its relation with the carnivore population.

The sugar regulation in blood can be represented by Figure 7.1, where

multiple feedback paths exist to allow the self-regulation of a healthy body.

Stimulus Message

Receptor

Message

Response

Controller

Feedback

Actuator

Cerebral cortex

(a)

(b)

Regulator

Diencephalon

Hypophysis

Vagus nerve

B-cells

Pancreas

A-cells

Glucagon

Glucocorticoids

Insulin

Corticotropic

Medulla

Adrenals

Cortex

Adrenalin

yroxin

yroid

yreotropic

Somatotropin

Corticosteroids

Adrenalin

Retroalimenta

Insulin

Regulation

Glycosensible

receptors in

pancreas

C.N.S.

(hypothetical)

Blood sugar

level

Liver storage

Muscle energy

consumption

Kidney overﬂow

valve

Disturbance

FIGURE 7.1

Feedback allows the control of blood sugar concentration. (Adapted from L. Bertalanffy. General

Systems Theory, George Brazillier, NY, 1969.)

Systems in Nuclear Medicine 339

In self-regulation, the feedback happens with a negative sign and allows

the comparison by the difference between two values: the desired value (the

reference) and the observed (measured) value. This difference comprises the

system error at that moment, and corrective actions are taken (by a controller)

to reduce the error to zero (or nearly). It is thus, said to be a negative feedback.

7.1.1.3 Environment, Closed Systems, Open Systems, and Homeostasis

A system is always surrounded by its environment and communicates with

it through its boundary. There is a dense relationship among the system

elements, as observed in Figure 7.2.

The intensity of communication between the system and its environment

varies from case to case. Achemical reaction in a closed vessel where reagents

are mixed is described, from the outside perspective, as a closed system. The

laws of thermodynamics apply to these systems.

However, if the communication between the system and its environment

is intense, then we have an open system, exchanging materials, energy, and

information with its environment througha boundary.Aliving cell is a typical

case of an open system.

Sometimes, it is difﬁcult to identity the boundary of an open system exactly,

as it is permanently and continuously communicating with its environment.

Living beings are open systems: they constantly receive energy from and

send energy to the environment. Their internal structure is permanently

undergoing change and so they are never in a stationary state (in the sense

that all the attributes of all its elements have a constant value over time);

instead they are in a state of chemical and thermodynamic equilibrium,

and their attributes display well-deﬁned patterns over time, maybe with

small transient changes. This equilibrium state is homeostasis and is illus-

trated in Figure 7.3. The concept was created by the French physiologist

Claude Bernard (1813, 1878) [3], when he famously wrote, “La ﬁxité du milieu

intérieur est la condition de la vie libre,” which today remains the concept

that underpins homeostasis. The concept was afterward established by the

Boundary

Input

Output

FIGURE 7.2

(a) A set of independent objects does not build a system. (b) A few scattered relations are not

enough to make a system. It needs a dense collection of relations as seen in (c). (Adapted from

R. Flood and E.R. Carson. Dealing with Complexity, An Introduction to the Theory and Applications

of Systems Science, Plenum Press, 1993.).

340 Nuclear Medicine Physics

Input

Output

Cell at instant t

Cell at instant t + Δt

FIGURE 7.3

Homeostasis. The inner components of the cell (m

represents molecules) have been replaced

between two successive instants of time, t and t +Δt, but the cell remains in a steady viable

state. (Adapted from R. Flood and E.R. Carson. Dealing with Complexity, An Introduction to the

Theory and Applications of Systems Science, Plenum Press, 1993.)

American physiologist Walter Cannon (1971–1945), in his book The Wisdom of

the Body (1932), where he describes the general characteristics of homeostasis

though his famous four propositions [4]:

1. Constancyin anopen systemsimilar tothat represented by our bodies

requires mechanisms which act to maintain this constancy. Cannon

based this proposition on insights into the ways by which steady

states such as glucose concentrations, body temperature, and acid–

base balance were regulated.

2. Steady-state conditions require that any tendency toward change

is automatically faced with factors that resist change. An increase

in blood sugar results in thirst as the body attempts to dilute the

concentration of sugar in the extracellular ﬂuid.

3. The regulating system that determines the homeostatic state consists

of a number of cooperating mechanisms simultaneously or succes-

sively acting. Blood sugar is regulated by insulin, glucagons, and

other hormones that control its release from the liver or its uptake by

the tissues.

4. Homeostasis does not occur by chance: it is the result of organized

self-government.

7.1.1.4 Entropy and Negentropy

Things show a tendency toward disorder and disorganization; entropy

measures this tendency. Entropy, as previewed by the second principle of

thermodynamics, conveys the irreversible law of degradation of materials

and energy: systems tend toward disorder.

This conclusion was stated by Clausius [5] with his famous principle of

maximum entropy (“the entropy of the universe tends to a maximum”), or

Systems in Nuclear Medicine 341

mathematically we have Equation 7.1, where S represents the entropy.

dS ≥ 0 (7.1)

However, there is an apparent contradiction in that homeostatic systems

maintain order and improveorganization.This was the problemaddressed by

Ilya Prigogine, who in 1945 proposed the theorem of the minimal production

of entropy. This applies to stationary nonequilibrium states and explains the

analogy relating the stability of thermodynamic equilibrium states (the clas-

sical notion) with the stability of biological systems similar to that expressed

by homeostasis. Prigogine won the Nobel Prize for chemistry in 1977 [6]. His

results are given by Equations 7.2 and 7.3.

dS = d

S +d

S, (7.2)

dS ≤ 0, (7.3)

S represents the entropy internally produced by an irreversible degradation

process (chemical reactions, diffusion, heat transfer, etc.) and d

S represents

the entropy imported by the living open system, in the form of matter and

energy. From Clausius’ law, we have Equation 7.4:

S > 0, (7.4)

which, replaced by Equation 7.2, gives Equation 7.5.

S ≤ 0 (7.5)

Due to Equation 7.5, we call entropy negative to d

S negentropy. Open

systems import negentropy. Figure 7.4 illustrates the difference between a

closed system and an open one.

Closed system

dS ≥ 0

Open system

dS = d

S+d

(a) (b)

FIGURE 7.4

(a) Closed system (Clausius equation) and (b) open system (Prigogine equation).

342 Nuclear Medicine Physics

As a consequence, the thermodynamic laws are only applicable to closed

systems. Entropy thus, appears to be a force against homeostasis. With the

aid of homeostasis, systems import materials, energy, and information from

the environment; and with them, they offset the tendency toward disorder

by creating and maintaining the order, resisting the second thermodynamic

principle, and even being capable of evolving toward states of greater order

and organization.

The structure of a system expresses the way its elements relate to each other

and ﬁxes the foundations of the processes evolving within it.

Ifwe observea system at successive instantsof time, we notice thatit evolves

in some way. The behavior of the system is, in fact, this perception. This

behavior is goal oriented if it can be understood and assessed with regard to

some particular purpose.

In open systems, a certain ﬁnal state may be reached from different initial

states, following different trajectories. The ﬁnal equilibrium state is said to

have an equiﬁnal value, and the system has the property of equiﬁnality.

7.1.1.5 Autopoiesis, Adaptation, and Cybernetics

One living cell produces its own components, which, in turn, produce the

actual cell. This is the autopoietic property of living systems: the ability to

self-reproduce. Living systems are autopoietic, because they are organized so

as to allow their own processes to generate the components needed for these

same processes to continue.

A system is adaptive if it is able to adapt. Adaptation emerges as a need

to respond to environmental changes in order to survive and keep the sys-

tem performing well. It can be said that the Darwinian theory of evolution

constitutes a theory of adaptation. The ability to adapt is fundamental when

the environment frequently and profoundly changes. Adaptation requires

the ability to regulate and control to cope with rapid changes and long-term

changes (short-term and long-term regulation). Any living being requires

short- and medium-term multiple regulation and control systems in order

to survive, sometimes called the law of requisite variety.

In our lives, we often look at things only partially, seeing just some parts

and forgetting to look at the holistic view. That is why the results of our

actions are sometimes contrary to our expectations, because we dismissed or

forgotthe phenomenological complexity. We need systemic thinking; we need

models and methodologies to deal with complexity. Without formal systemic

views, only parts, only simplistic explications and simplistic solutions are

attainable.

Cybernetics studies adaptation, regulation, and control. It has been deﬁned

as the science of control and communications in animals and machines (pro-

posed by Wiener in his 1948 book Cybernetics, or Control and Communication

in the Animal and Machine).

Systems in Nuclear Medicine 343

7.1.1.6 A Hierarchy of Systems

Ina hierarchyof systems and control,we maydeﬁne systems on several levels:

we will have a hierarchy of systems. A metasystem is a system above other

systems in a hierarchy. A subsystem is below another system. In humans, the

brain or consciousness is the highest level metasystem, and the atom is at the

lowest level. We may deﬁne a hierarchy: atom, molecule, cell, tissue, organ,

functional system, …, consciousness.

The hierarchical organization is a logical representation of the phenomena,

systems, and subsystems. Going down hierarchy, we increase the resolution

of the analysis and see more and more details. The choice of an appropri-

ate resolution level for the analysis of a given phenomenon is crucial to

ﬁnally deciding the questions we will work with in the study. The chosen

level becomes the system in focus.

A system scientist must be simultaneously holistic, looking for the system

as a whole, and a reductionist, that is, capable of understanding the system

in its details, in each of its multiple parts.

Going up in a systems hierarchy, we ﬁnd a property fundamental to all of

systems theory: the whole is greater than the sum of the parts. This is the

emergency property: the hypersystem has (sometimes unexpected) proper-

ties that do not exist in any of the subsystems, they just emerge when the

hypersystem is built. This is the synergy effect.

Cells are grouped in different organs (kidney, liver, heart, lung, etc.), each

with a speciﬁc irreplaceable function and with properties different from its

cells. The organs together build up a body with emerging properties: they are

organized through communication and (cybernetic) control into a hierarchy

of body parts that allow the capacities of sight, hearing, smell, locomotion, and

emotion, that is, of the human being. Neither is a human being an aggregate

of body parts nor is a society an aggregate of human groups. Systems join

together to make a hypersystem whose properties differ from those of the

subsystems.

Emergence is a characteristic of phenomena that we cannot explain any

other way. The emergence of a society is its culture.

Synergy is a concept used to describe the emergence of unexpected and

interesting properties, for example, the theory of organizations (the synergy

of group work, for instance).

According to Bertalanffy [1], we can deﬁne a system hierarchy as shown in

Table 7.1.

During the last century, several metaphors appeared for systems, as the

system theory was tentatively applied to human groups and societies.

First came the machine metaphor, looking to a closed system with

a set of well-deﬁned objectives and a rigid control structure to reach

them (technological servomechanisms are a good example). This metaphor

inspired the development of industrial systems and their manage-

ment.

344 Nuclear Medicine Physics

TABLE 7.1

A Hierarchy of Systems According to Bertalanffy [1]

Level Description and Examples

Symbolic systems Language, logic mathematics, sciences, arts, morals, etc.

Sociocultural systems Populations of organisms (humans included); symbol-determined

communities (cultures) in man only.

Humans Symbolism; past and future, self and world; self-awareness as a

consequence of communication by language.

Animals Increasing importance of trafﬁc of information (evolution of receptors,

nervous systems, etc.); learning; beginnings of consciousness.

Lower organisms “Plant-like” organisms:increasing differentiation of systems (“division

of labour”); distinction of reproduction from individual functions.

Open systems Fire, cells, and organisms in general.

Control mechanisms Thermostat, servomechanism, and homeostatic mechanisms in living

organisms.

After Prigogine, this vision was superseded by the organic metaphor,

describing complex networks of elements and relations with a transformation

affected by feedback.

However, the organic metaphor cannot be used to describe emotions. The

brain metaphor naturally appeared.

The culture metaphor means that a network of values, beliefs, and norms

can be included.

Finally, the political metaphor allows the interaction of networks of interests

that individuals aim at reaching.

In our study of biological and physiological processes, we will concern

ourselves with the development of computable mathematical models, that is,

models which we can program in a computer and use to simulate, analyze,

and predict the behavior of systems in the ﬁrst two levels of the Bertalanffy

hierarchy.

Our analysis will be based on the physical, chemical, thermodynamic, prop-

ertiesof phenomena.Using theappropriatelaws,we obtaina setof differential

equations that enable us to computationally describe and simulate the respec-

tive dynamic behavior. The differential Equations also serve to obtain very

practical mathematical representation: the state equations both linear and

nonlinear.

Some dynamic properties, such as stability and type of transient time

response, are easily obtained from these representation tools.

Some nonlinear systems have special importance in the biological and phys-

iological context: chaotic systems. As we will see, a system is chaotic if we

cannot predict its behavior from observable initial conditions. We will look at

some examples and a short introduction to its theory.

Systems in Nuclear Medicine 345

7.1.2 Mathematical Modeling and Analysis of Biological Systems:

Population Models

Mathematical models of biological species ﬁnd a wide spectrum of applica-

tions, ranging from environmental studies to disease propagation and viral

epidemics, tumour growth, etc. These models, if realistic, have a wide practi-

cal application for the understanding of population dynamics and to forecast

population evolution.

7.1.2.1 The Growth of a Single Species’ Population

Let N(t) be a population of a single species at time t, in a given environment.

At the next time instant, t +Δt, the population will be N(t +Δt), given

by the mass conservation equation: the population in the next period will be

the population of the current period plus those born until that point, minus

the dead ones, plus those entering from abroad, minus those leaving, as in

Equation 7.6.

N(t +Δt) = N(t) + the born during Δt −the dead during Δt

+the imigrated during Δt −the emigrated during Δt.

(7.6)

Manipulating Equation 7.6, and assuming that there are no migrations, one

obtains Equation 7.7:

N(t +Δt) −N(t) = ΔN(t) = born during Δt −dead during Δt. (7.7)

Assuming now a birth rate a and a mortality rate b per time interval, the

born and dead during Δt are aN(t

)Δt and bN(t)Δt, respectively. Substituting

in Equation 7.1, we get Equation 7.8.

N(t +Δt) −N(t) = ΔN(t) = aN(t)Δt −bN(t)Δt. (7.8)

Dividing both sides of Equation 7.8 by Δt, we get Equation 7.9.

N(t +Δt) −N(t)

Δt

ΔN(t)

Δt

= aN(t) − bN(t). (7.9)

Next, applying to Equation 7.9 the limit when Δt tends toward zero

lim

Δt→0

ΔN(t)

Δt

= aN(t) − bN(t). (7.10)

we obtain the deﬁnition of the derivative of N with regard to t Equation 7.11.

dN(t)

= aN(t) − bN(t). (7.11)

346 Nuclear Medicine Physics

7.1.2.2 The Logistic Equation

Let us assume that there are no deaths according to Equation 7.11. Then the

population will grow without limits, according to Equation 7.12

dN(t)

= aN(t). (7.12)

The solution of Equation 7.12 is an exponentially growing function. In real-

ity, a population in a given environment with ﬁnite resources cannot exceed a

certain limit, when all the available resources are used to sustain the popula-

tion. To take this into account, a limitative term is introduced in Equation 7.12,

resulting in Equation 7.13,

dN(t)

= aN(t) − bN

(t) = aN(t)



1 −

N(t)



= aN



1 −



, (7.13)

known as the logistic equation or the Verhulst equation [7]. The constant a is

the intrinsic growth rate, and k = b/a is the maximum population value that

the environment can support, the so-called carrying capacity. If N = k, the

population remains constant.

We can also develop a discrete model, giving the population values only at

well-deﬁned times.

Let us suppose a biological species whose individuals live a whole number

of time intervals (e.g., years) and are born and die at the end of the year.

Initially, there are N

individuals. At the end of that year and the start of the

next year, there are N

individuals; after two years, there are N

individuals;

and at the end of k year, there are N

individuals.

In principle, we can say that at the end of the ﬁrst year, the population N

is proportional to N

, through a proportionality constant A that depends on

the environmental conditions (amount of food and water, climate, etc.), as in

Equation 7.14

= AN

. (7.14)

If A > 1, the population increases and can be considered the reproduction

rate; if A < 1, it diminishes until total extinction, and it is the mortality rate.

If A > 1 is constant over the years, the population will grow without limit,

leading to a population explosion.

However, if the population increases without limit, on the one hand, there

will not be enough food and, on the other hand, any predators living in the

same environment will ﬁnd moreindividuals of the species in question, which

will, thus, limit the population growth. This limiting effect can be introduced

into the equation though a subtractive term, which is very low for small

population values and very strong for high population values, for example,

= AN

−BN

. (7.15)

Systems in Nuclear Medicine 347

If B is so much less that A, A, B  A, the subtractive term will only be

relevant for high values of N

, corresponding to the desired effect. In the

succeeding years, we will have, assuming that A and B are constant over the

years,

= AN

−BN

= AN

−BN

k+1

= AN

−BN

(7.16)

The time will come when the population cannot grow any more, because

it has reached its maximum values N

max

. Let m be that year and N

= N

max

The next year, the population will be Equation 7.17

m+1

= AN

max

−BN

max

. (7.17)

Forcing N

m+1

to be positive, we will have Equation 7.18.

max

−BN

max

≥ 0 ⇔ N

max

(A −BN

max

) ≥ 0 ⇒ N

max

≤

. (7.18)

The relation A/B must be less than the maximum reachable population

ﬁgure. Expressing the population as a fraction of its maximum values, as in

Equation 7.19

max

⇒ N

= x

max

for all values of k, (7.19)

and subtracting Equation 7.19 from Equation 7.17, we obtain Equation 7.20a.

k+1

= AN

−BN

⇒ x

k+1

max

= Ax

max

−Bx

max

. (7.20a)

Solving for Equation 7.20a for x

k+1

, we get Equation 7.20.

k+1

= Ax

−Bx

max

⇒ x

k+1

= Ax

−Bx

⇔ x

k+1

= Ax

(1 −x

(7.20b)

We have ﬁnally achieved the model for population growth, in the form of

a nonlinear ﬁrst-order difference Equation 7.21.

k+1

= Ax

(1 −x

) = f (x

) x ∈[0, 1], (7.21)