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

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378 Nuclear Medicine Physics

x´= Ax + By + u

A = 2 B = 2 C = 2

D = –3 u = 0

y´= Cx + Dy + u

–2

–4

–6

–6 –4 –2 0

246

FIGURE 7.18

Saddle point (eigenvalues: 1; -2).

x =



−2 −3



x +





u = 0.

(7.130)

7.1.2.11.4 Nonlinear Systems with Several Singularities and Local Stability

Example 7.6 The Lotka–Volterra Model

In Example 7.5, we compute the singular points of the Lotka–Volterra

model, to give Equations 7.110 and 7.111. Let us consider the nonnull singu-

larity. Linearizing the nonlinear differential equations around it, the Jacobian

Equation 7.131 is obtained.

A =

∂f

∂x



a −bx

−bx

−c +px









0 −b



. (7.131)

Systems in Nuclear Medicine 379

Calculating the eigenvalues of A, its characteristic polynomial is given by

Equation 7.131,

λI − A



λ +b

−p



= λ

+pb

= λ

+ac, (7.132)

whose roots are (Equation 7.133):

λI − A

= 0 ⇔ λ

+ac = 0 ⇒ λ = 0 ±j

√

ac. (7.133)

For given initial populations of each species, how will they evolve? The sin-

gularity is a center, because the eigenvalues are imaginary conjugates. Using

pplane7, the phase curves of Figure 7.19 are obtained, for a = 0.4; b = 0.01;

c = 0.3; and p = 0.005.

The two singular points are clearly visible in Figure 7.19: one saddle point

(the origin, eigenvalues 0.4 and −0.3) and a center (eigenvalues +0.3464i and

−0.3464i). The dark surface represents a trajectory that is gradually opening

in the basin of attraction of the origin.

7.1.2.11.5 Uncertainty in Initial Conditions

Let us now see what happens at the boundary of the regions of conver-

gence for a certain singularity. Take the example of the nonlinear system

(Equation 7.134).

x = (x −1)

−u

y = (y −2)

−u.

(7.134)

(t)

q(t) q(t)

(t)

FIGURE 7.19

Phase curves for Lotka–Volterra model.

380 Nuclear Medicine Physics

For the input u = 1, this system has 4 singularities, given by Equation 7.135,

















; (7.135)

to which the following Jacobians correspond Equation 7.136.

A =





repeller(source)

A =



0 −2



seal repeller

A =



−20



seal repeller

A =



−20

0 −2



attractor (sink)

. (7.136)

Drawing the phase plane with pplane7, the interesting Figure 7.20 is ob-

tained, exhibiting the different types of singularities.

The line y = 3 is a decision boundary for the singularity [0.3]. Above 3, the

trajectories diverge to inﬁnity; whereas under 3, they converge to the stable

node [0.1]. We can see this boundary in more detail, as in Figures 7.22 and

7.23.

For example, in Figure 7.21, the line y = 3, between x =−2 and x = 0, is a

boundary.

Reducing the scale of the vertical axis, we obtain Figure 7.22.

x´

= (x–1)

–u

u = 1

y´

= (y–2)

–u

–1

–2 –1 1

2340

FIGURE 7.20

Phase plane trajectories for system (Equation 7.134). It has two saddle points, one stable node

and one unstable node (Equation 7.136).

Systems in Nuclear Medicine 381

3.1

x´

= (x–1)

–u

u=1

y´

= (y–2)

–u

3.08

3.06

3.04

3.02

2.98

2.96

2.94

2.92

2.9

–2 –1.8 –1.6 –1.4 –1.2 –1

–0.8 –0.6 –0.4 –0.2 0

FIGURE 7.21

Phase trajectories for the system (Equation 7.134) around a boundary.

Reducing the vertical scale still further, we can again see a well-deﬁned

boundary. In Figure 7.23, the vertical interval shown is [2.9999 3.0001].

Here we may raise the following question: if the initial conditions are mea-

sured by an instrument and if the measurement error is greater than 0.0001,

is it possible to predict the behavior of this nonlinear system?

No, it is not. This illustrates a characteristic of the chaotic behavior of non-

linear systems: the impossibility of predicting their behavior.An inﬁnitesimal

difference in the initial conditions may make the difference. This illustrates

the famous butterﬂy effect ﬁrst used by E. N. Lorenz in 1992 [15]: “Predictabil-

ity: Does the Flap of a Butterﬂy’s Wings in Brazil set off a Tornado in Texas?”

which afterward has been repeated in multiple contexts and geographic ver-

sions. This does not mean that the system is stochastic. The system is, in fact,

deterministic, but the uncertainty in the initial conditions prevents us from

predicting its evolution.

The future of a chaotic system is indeterminate, although the system is

deterministic.

The trajectories that deﬁne a boundary are called separatrices, and they

divide the state space into regions of different dynamic modes.

The set of points giving rise to trajectories leading to a certain singular point

is called the basin of attraction of that point, by analogy with the concept of

a hydrographic basin.

382 Nuclear Medicine Physics

3.01

x´

= (x–1)

–u

u=1

y´

= (y–2)

–u

3.008

3.006

3.004

3.002

2.998

2.996

2.994

2.992

2.99

–2 –1.8 –1.6 –1.4 –1.2 –1

–0.8 –0.6 –0.4 –0.2 0

FIGURE 7.22

Moredetailon the boundary.Notethe different scale of the verticalaxes with regardto Figure 7.21.

7.1.2.11.6 Bifurcations and Chaos

Let us look again at the logistic equation of the population growth of a species,

repeated in Equation 7.137, where A is the fertility rate of the population.

k+1

= Ax

(1 −x

) = A(x

−x

), x ∈[0, 1], (7.137)

This model has its two singularities deﬁned by Equation 7.138

= 1 −

= 0.

(7.138)

The convergence toward one or the other depends on the initial state.

When A < 1, the single possible singularity is the origin, x

= 0 (the species

vanishes).

We can implement this equation in Simulink [17], as in Figure 7.24, using

the memory block as being equivalent to a pure time delay.

Simulating with several values of A, we obtain the results shown in

Figure 7.25.

The system exhibits a strange behavior: its period depends on A, the fertility

rate. As A increases, the population is periodic with a period depending on

A. For A in the interval:

Systems in Nuclear Medicine 383

3.001

2.999

–2 –1.8 –1.6 –1.4 –1.2 –1

–0.8 –0.6 –0.4 –0.2 0

x´

= (x–1)

–u

u = 1

y´

= (y–2)

–u

FIGURE 7.23

An inﬁnitesimal difference in the initial conditions produces very different system behaviors: in

one case, it goes to inﬁnity; whereas in another case, it goes to a stable node.

[3; 3.34495] oscillates with period of 2 generations;

[3.4495; 3.54408] oscillates with period of 4 generations; and

[3.54408; 3.56440] oscillates with period of 8 generations.

–

3.5

Memory

Squared

Scope

Popula

Population

FIGURE 7.24

The logistic discrete equation implemented in Simulink.

384 Nuclear Medicine Physics

A = 2.8

Population

A = 3

A = 3.3 A = 3.5

A = 3.55 A = 3.6

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0246810

Generation

A = 2.8; reaches a ﬁnal steady state

A = 3, tends towards a ﬁnal steady state

A = 3.3; oscillates, period 1 generation

A = 3.55; oscillates, period 8 generation A = 3.6; chaos (there is no period)

A = 3.5; oscillates, period 2 generations

Population

0.9

0.8

0.7

0.6

0.5

0.4

0.3

Generation

12 14 16 18 20

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

0246810

Generation

12 14 16 18 20

Population

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0246810

Generation

12 14 16 18 20

Population

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0246810

Generation

12 14 16 18 20

Population

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0246810

Generation

12 14 16 18 20

Population

0.9

0.8

0.7

0.6

0.5

0.4

0.3

FIGURE 7.25

Evolution of the populations for different values of A.

Let us now perform the following graphical construction:

For each value of A, we calculate the values of the population x that

belong to the respective cycle. These values can be obtained by simulating

in Simulink, for example, and then reading the values of the population

vector that compose one period afterward in the workspace. For example,

Systems in Nuclear Medicine 385

0.9

0.8

0.7

0.6

0.5x

0 0.5 1 1.5 2.52 3.534

0.4

0.3

0.2

0.1

FIGURE 7.26

Bifurcation map of the logistic function of population growth.

for A = 3.3, we will have x = 0.8236 or x = 0.4794; for A = 3.5, it will be

x ∈{0.3828; 0.5009; 0.8269; 0.8750}; and for A = 0.55, x ∈{0.3548; 0.81265;

0.54049; 0.88168; 0.37034; 0.82781; 0.50601; 0.88737}. For values of A less than 3,

we will have only the steady state value; for A less than 1, the steady state is the

origin. Graphically, we obtain the famous bifurcation diagram of the logistic

function, as seen in Figure 7.26.

If we amplify the ﬁnal part of the ﬁgure, between 3 and 4, interestingly

we obtain a ﬁgure in which the same patterns in Figure 7.26 are repeated

but on a different scale. If we amplify in this new ﬁgure again, we would

obtain another ﬁgure with the same patterns on another scale. Therefore, by

increasing the detail of the observation, we always get the same pattern as in

Figure 7.27. This is a characteristic of fractals.

We ﬁnd that as A increases,the oscillation period is duplicated until periodic

behavior is no longer seen. Behavior in which there is no repetition of a pattern

is chaotic. A point on the A axis in Figure 7.27 where there is a duplication of

the period is called a bifurcation point.

7.1.2.11.7 Feigenbaum Numbers

Let A

be the value of A at which the period 2n originates the period 2n +1.

The number Equation 7.139

−A

n−1

n+1

−A

, (7.139)

is called the Feigenbaum delta. We can compute it for different values of n

and obtain Table 7.4.

The point after which there is no more periodic behavior and chaotic

behavior appears is called the accumulation point.

386 Nuclear Medicine Physics

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

3.4 3.6 3.8 4

FIGURE 7.27

The fractal characteristics of the bifurcation map in Figure 7.26. (http://www.vanderbilt

.edu/AnS/psychology/cogsci/chaos/workshop/BD.html).

Feigenbaum studied the evolution of his delta for several systems at the

Los Alamos National Laboratory, USA; and in 1975, he found the following

result for all of them [15]:

lim

n→∞

= 4.66920161 ... (7.140)

Accordingto Hilborn [15], this number is destined to belong to the Pantheon

numbers of physics (as does π, the golden number (

√

5 −1)/2, etc.), and it

has been considered a universal constant.

TABLE 7.4

The Sequence of Feigenbaum Numbers

2 3.44949

3 3.54409 4.751479915

4 3.564407 4.656199242

5 3.568.750 4.668428309

6 3.569692 4.664523044

7 3.569891 4.688442211

8 3.569934 …

9 3.569943

10 3.5699451

11 3.56994556

Inﬁnity 3.56994567 4.669201…

Systems in Nuclear Medicine 387

7.2 Tracer Kinetics Modeling

7.2.1 Introduction

NM methodologies have progressively achieved a central role in the research

of physiological and pathological processes, in the discovery or assessment

of drugs, and in the diagnosis and treatment of a variety of medical condi-

tions. Recent developments in human physiology have been ﬁrmly based on

the molecular approach of metabolic processes and on the use of radiotracer

kinetic methods to study the associated functions, inspired by paradigms of

biological and pharmaceutical sciences.

Radiotracers are distributed and located in the body as a result of physical–

chemical interactions with living tissues in the course of speciﬁc physiological

processes,and they can be used to convey information. However, the complex

chain of events of various kinds that involves the dynamics of radiotracers is

a sequence of processes governed by laws, which are generally well known.

These processes may consist of mechanisms such as ionic exchange, diffusion,

metabolic incorporation, binding to receptors, immunological reactions, etc.

The PET and SPECT are imaging techniques that use minimal amounts

of radiolabeled molecules as tracers to study molecular pathways and

molecular interactions, in vivo, without interfering with the processes being

studied.

The detection of changes in the tracer distribution and concentration in bio-

logical tissues is the directaim of the NM experimental method. By processing

series of maps of tracer concentrations in a particular region of interest, with

reasonable spatiotemporal resolution (millimeters and seconds), the charac-

teristic biochemical and physiological dynamic parameters can be ascertained

with good approximation.

Important characteristics of a good tracer are afﬁnity, selectivity, ade-

quate distribution in physiological compartments, and low participation in

nonspeciﬁc bonds.

Quantitative approaches to the function of human organs and tissues based

on modeling or data analysis are the goal in most of the studies on molecu-

lar functional imaging. The role of modeling is to apply appropriate analysis

paths to data in order to evaluate the physiological and kinetic behavior of

tracers.

Although the models used in NM involve different analysis of the sys-

tems, they generally lead to differential or integral equations or to differential

equation systems. The solutions of these equations yield expressions that are

supposed to recreate the experimental data and the behavior of the system in

relation to the tracer used.

Broadly speaking, every methodology in NM is a systems theory exer-

cise, that is, an input function is used to study a system through an induced

response.