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

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368 Nuclear Medicine Physics

This result is full of information. It tells us that the state trajectory for a

given initial condition is a weighted sum of exponentials of the eigenvalues

of the A matrix (the terms e

λt

). The weightings depend on the eigenvalues. It

can be said in this regard that the eigenstructure of the A matrix is the genome

of the autonomous system.

What happens when one of the eigenvalues of the A matrix is positive? Its

exponential tends to inﬁnity with t, and the same happens for the term with

which it contributes to the state evolution in Equation 7.89.As a consequence,

if the initial condition is nonzero, even if very small, the state tends toward

the inﬁnite, even if all the other terms in Equation 7.89 tend toward zero. The

system is unstable!

Here we come across the problem of the stability of the system with regard

to the initial conditions. The eigenvalues of A are also called the modes of the

system. The solution (Equation 7.89) of the state equation is called the modal

solution, because it is expressed by the modes.

7.1.2.10.2 Solution of the Complete State Equation

The complete state Equation 7.90 has the input part

x = Ax +Bu, (7.90)

and it is not so easy to ﬁnd its solution.

Consider the property of derivation of time-varying matrices (Equation

7.91).

d[M(t)N(t)]

= M(t)N(t) +M(t)

N(t), (7.91)

in which the derivative of a matrix is the matrix whose elements are the

derivatives of the elements of the initial matrix, that is,

M =

)



−At

x(t)



= e

−At

x(t) − Ae

−At

x(t) = e

−At

[

x(t) − Ax(t)]. (7.92)

However, since,

x(t) = Ax(t) +Bu(t). (7.93)

then substituting Equation 7.93 into Equation 7.92, we have Equation 7.94.



−At

x(t)



= e

−At

Bu(t). (7.94)

Integrating both sides of Equation 7.94, we have Equation 7.95.

−At

x(t) =



−Aτ

Bu(τ)dτ +K. (7.95)

Systems in Nuclear Medicine 369

where K is a constant (matrix) of integration. Making t = 0 in Equation 7.95,

we get Equation 7.96.

−A0

x(0) =



−Aτ

Bu(τ)dτ +K ⇔ x(0) = K. (7.96)

Multiplying both sides of Equation 7.96 by e

, grouping the terms, we get

Equation 7.97.

−At

x(t) = e



−Aτ

Bu(τ) dτ +e

x(0). (7.97)

Multiplying the two matrix exponentials, adding the exponents At–At = 0,

knowing that the exponential of the null matrix is the identity matrix, we

obtain Equation 7.98.

Ix(t) = e

x(0) +



A(t−τ)

Bu(τ) dτ. (7.98)

As we can see in Equation 7.98, x(t) is the sum of two contributions

x(t) = x

(t) +x

(t) (7.99)

one coming from the initial conditions (zi—zero input) and the other coming

from the external input (zs—zero state).

7.1.2.10.3 Solution of the Two-Compartment State Equation

We have seen that the Equations of two compartments are Equations 7.100

and 7.101.

=−k

(7.100)

=+k

−k

(7.101)

If we assume that a substance of mass M

is instantaneously introduced into

the blood ﬂow (bolus injection) at t = 0 and that it mixes quickly, we have

= 0ew

= 0. The homogeneous state equation will be Equation 7.102,







−k





. (7.102)

370 Nuclear Medicine Physics

whose solution is Equation 7.103.

x(t) = c

−λ

x(t) =



(t)



e c





(7.103)

and λ

are the eigenvalues of the A matrix, and the coefﬁcients a

and b

depend on the right and left eigenvectors. The eigenvalues are Equation 7.104,

1,2

=−



−k

)

+4k

. (7.104)

7.1.2.11 Nonlinear Systems, Singular Points, and Linearization

Many natural systems, particularly biological and physiological ones, are

describedbynonlinear differential equations. Thisposes thequestionwhether

the nonlinear systems can be approximated to linear ones, if we want to apply

the linear systems theory. We call this mathematical operation linearization,

and it is done at the points wherethe derivatives vanish, called singular points

or singularities.

7.1.2.11.1 Singular Points or Singularities

In many practical situations, the ideal operating conditions are characterized

by the constancy of inputs, states, and outputs, that is, a steady state or an

equilibrium situation. Suppose that the system is described by the nonlinear

model in Equation 7.105.

x = f (x, u). (7.105)

Linearization is mathematically represented around equilibrium points or

singular points, also called steady states if the system is stable. We can say that

the vector x

(n×1) is a singular point (or an equilibrium point) corresponding

to the constant input u(t) = u

, if and only if Equation 7.106 is true.

f (x

, u

) = 0. (7.106)

The corresponding equilibrium output is the vector y

(r×1) such that

= g(x

, u

). (7.107)

Systems in Nuclear Medicine 371

Example 7.5 The Lotka–Volterra model

To compute the singularities of the Lotka–Volterra model in Equation 7.14,

we eliminate the derivatives as in Equation 7.108.

0 = ax

(t) −bx

(t)x

(t)

0 =−cx

(t) +px

(t)x

(t).

(7.108)

Solving for x

and x

we obtain 7.109

[a −bx

(t)]x

(t) = 0 ∧x

(t) = 0

[−c +px

(t)]x

(t) = 0 ∧x

(t) = 0

⇒

[a −bx

(t)]=0

[−c +px

(t)]=0

. (7.109)

The two solutions of Equation 7.109 are Equations 7.110 and 7.111.

⇒ solution 1 :

⎧

⎪

⎨

⎪

⎩

(t) =

; (7.110)

⇒ solution 2 :

(t) = 0

. (7.111)

We can conclude that the Lotka–Volterra model has two singularities. We

will see the consequences of this fact later.

7.1.2.11.2 Linearization Methods: Approximation to the Taylor Series

The linearization (of a model) of nonlinear equations in the neighborhood of

a singular point deﬁned by the triplet x

, u

, y

consists of approximating the

vectors of functions f , for example, to linear functions, in the neighborhood of

that singular point. Analytical linearization is based on the approximation of

the functions f

and g

to a Taylor series (in the neighborhood of the singular

point), where Δx

and Δu

are the deviations with regardto the singular point,

as in Equation 7.112.

(

+Δx

, x

+Δx

, ...+ x

+Δx

, u

+Δu

, u

+Δu

, ..., u

+Δu

)

= f

(

, x

, ...+ x

, u

, ..., u

)



k=1

∂f

∂x

Δx



k=1

∂f

∂u

Δu

+termos de ordem superior. (7.112)

372 Nuclear Medicine Physics

For the output equations, in the same way we have Equation 7.113.

(

+Δx

, x

+Δx

, ...+ x

+Δx

, u

+Δu

, u

+Δu

, ..., u

+Δu

)

= g

(

, x

, ...+ x

, u

, ..., u

)



k=1

∂g

∂x

Δx



k=1

∂g

∂u

Δu

+termos de ordem superior. (7.113)

Neglecting the terms of a higher order (meaning that the linearization

will be valid only in a small neighborhood) and considering Equations 7.114

through 7.117, the linearization is almost ready.

A =

⎡

⎢

⎣

∂f

∂x

···

∂f

∂x

. ···

∂f

∂x

···

∂f

∂x

⎤

⎥

⎦

= F

; (7.114)

B =

⎡

⎢

⎣

∂f

∂u

···

∂f

∂u

. ···

∂f

∂u

···

∂f

∂u

⎤

⎥

⎦

= F

; (7.115)

C =

⎡

⎢

⎣

∂g

∂x

···

∂g

∂x

. ···

∂g

∂x

···

∂g

∂x

⎤

⎥

⎦

= G

; (7.116)

D =

⎡

⎢

⎣

∂g

∂u

···

∂g

∂u

. ···

∂g

∂u

···

∂g

∂u

⎤

⎥

⎦

= G

. (7.117)

The derivatives in Equations 7.114 through 7.117 are computed in the

singular point, substituting the values of x and u with x

and u

, respectively.

The matrices of the ﬁrst derivatives of the functions in order to each of their

arguments are called Jacobians.

Substitutingin the truncated Taylor series (in theﬁrst-orderterm), we obtain

a linear model. The singularity itself is a point of the system and so respects

the state Equation 7.118.

= f (x

, u

). (7.118)

Systems in Nuclear Medicine 373

If we consider a point in the neighborhood of the singularity x

+Δx,we

will have Equation 7.119, because the derivative of the sum is equal to the

sum of the derivatives.



•

+Δx



+Δ

x. (7.119)

However, now approximating the Taylor series (only ﬁrst-order terms), we

have Equation 7.120.

(

+Δ

x) = f (x

+Δx, u

+Δu). (7.120)

Taking into account Equations 7.114 through 7.117, we obtain Equa-

tion 7.121.

(

+Δ

)

= f (x

, u

) +A Δx +B Δu. (7.121)

or the state and output equation in standard form (Equation 7.122).

x = A Δx +B Δu

Δy = CΔx +D Δu.

(7.122)

The validity of Equation 7.122 can only be assumed when the terms of sec-

ond and higher orders of the Taylor series are negligible, that is, when the

deviations from the singular points are very small. To simplify the notation,

we can represent the linearized system eliminating the Δ

in Equation 7.122

but not forgetting that the input, state, and output variables represent devia-

tionsfromthe singular point and notabsolute values. Thematrices A, B,C, and

D are the Jacobians of f and g (in order to x and to u) given by Equations 7.114

through 7.117.

7.1.2.11.3 Types of Singularities

Alinear system, except for the special case where det(A) = 0, has only one sin-

gularity. If the input is null, the singularity will be the origin of the state space.

Letus consider this case, without loss of generality (in the linear case, the input

displaces the singularity in the state space but does not change its nature).

The stability of the systems is determined, as we saw, by the eigenvalues of

the state matrix A.

There are several types of singularities, some stable and some unstable [15].

Let us look at an example of each type.

1. Stable node (sink, attractor): when the eigenvalues of A are negative

and distinct, the state trajectory approaches the singularity with-

out oscillations, as seen in Figure 7.12, for a second order system.

The state equations in the example are Equation 7.123. These tra-

jectories, called phase curves, were computed using pplane7 [16] in

374 Nuclear Medicine Physics

Matlab

. The arrows indicate the direction of the evolution of the

trajectory.

x =



−20

0 −3



x +





u = 0.

(7.123)

2. Stable focus (focus attractor): when the eigenvalues of A are com-

plex conjugates with negative real parts, the trajectory tends toward

the focus with oscillations. Figure 7.13 shows the trajectories for the

system of Equation 7.124.

x =



−3 −1



x +





u = 0.

(7.124)

3. Unstable source (repelling source, unstable node): both the eigenval-

ues are real positive. The trajectory is repelled without oscillations.

x´ =

K = 3 m = 1

d = 1

v´ = –(

kx+dv)/m

0.5

–0.5

–1

–1 –0.5 0

0.5 1

FIGURE 7.13

Stable focus (eigenvalues: −0.5000 +1.6583 i; −0.5000 −1.6583 i).

Systems in Nuclear Medicine 375

The system is unstable, given by Equation 7.125 and Figure 7.14.

x =





x +





u = 0.

(7.125)

4. Unstable focus (repelling focus): when the eigenvalues of A are

complex conjugates with a positive real part. The trajectory is

repelled with oscillations, as shown in Figure 7.15, for the system

(Equation 7.126).

x =



−31



x +





u = 0.

(7.126)

5. Center: when the eigenvalues are both imaginary. The trajectories are

closed curves, periodic, around the center, as seen in Figure 7.16, for

the system (Equation 7.127).

x =



−30



x +





u = 0.

(7.127)

x´ = 2x + u

u = 0

y´ = 3y + u

0.5

–0.5

–1

–1 –0.5 0

0.5 1

FIGURE 7.14

Unstable node (eigenvalues: 2;3).

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x´ = y

u = 0

y´ = –3x + y + u

0.5

–0.5

–1

–1 –0.5 0

0.5 1

FIGURE 7.15

Unstable focus (eigenvalues: 0.5000 +1.6583 i; 0.5000 −1.6583 i).

x´ = y

u = 0

y´ = –3x + u

0.5

–0.5

–1

–1 –0.5 0

0.5 1

FIGURE 7.16

Center (eigenvalues: +1.73 i; −1.73 i).

Systems in Nuclear Medicine 377

A linear oscillator has the differential equation 7.128

+ay = 0, (7.128)

where a is a positive constant.

Choosing x

= y and x

= dy/dt as state variables, Equation 7.129

is the state equation for Equation 7.128.

= x

=−ax

(7.129)

The phase plane of Equation 7.129 is drawn in Figure 7.17. It is

similar to Figure 7.16, which is an oscillator in which a = 3, as can be

seen in Equation 7.129.

6. Saddle point: when one eigenvalue is positive real and the other is

negative real, as in the system, which is seen in Equation 7.130 and

Figure 7.18.

x´

= ya = 1

1.5

0.5

–0.5

–1.5

–2

–2 –1.5 –0.5 0

0.5–1 1 21.5

–1

y´

= –ax

FIGURE 7.17

Phase plane of the oscillator (Equation 7.128).