A MODEL OF THE TUMOUR SPHEROID

RESPONSE TO RADIATION

Identiﬁability Analysis

F. Papa

Dipartimento di Informatica e Sistemistica “A. Ruberti”, Sapienza Universit`a di Roma, Via Ariosto 25, Rome, Italy

Keywords:

Tumour spheroids models, Radiotherapy, Global identiﬁability.

Abstract:

A spatially uniform model of tumour growth after a single instantaneous radiative treatment is presented in

this paper. The ordinary differential equation model presented may be obtained from an equivalent partial

derivative equation model, by integration with respect to the radial distance. The main purpose of the paper

is to study its identiﬁability properties. In fact, a preliminary condition, that is necessary to verify before

performing the parameter identiﬁcation, is the global identiﬁability of a model. A detailed study of the identi-

ﬁability properties of the model is done pointing out that it is globally identiﬁable, provided that the responses

to two different radiation doses are available.

1 INTRODUCTION

The mathematical literature on solid tumour growth is

very wide. Looking through it, this evolution line can

be recognized: the earliest models were focused on

avascular tumour growth; then models of angiogene-

sis were developed; more recently, models of vascular

tumour growth are starting to emerge (Byrne, 2003).

With reference to mathematical models of avascu-

lar tumour growth we can underline the presence of

two different kinds of models: the spatially uniform

models and the spatially structured models.

The ﬁrst class of models concerns with models in

which details of the spatial structure of the tumour are

neglected and the attention is focused, for instance, on

the tumour overall volume or on the total number of

cells present within the tumour itself.

On the other hand, the second class concerns with

models in which the spatial coordinates are taken into

account, in order to investigate the role of rate limit-

ing, diffusible growth factors on the tumour develop-

ment.

In this paper a spatially uniform model of tumour

growth, after a single instantaneous radiative treat-

ment is presented, with the main purpose of studying

its identiﬁability properties. This model comes from

the integration with respect to the spatial coordinate

of the partial derivative equations of a spatially struc-

tured model (Bertuzzi et al., 2009), when it is possible

to neglect the distribution of oxygen concentration in-

side the tumour. In fact, the oxygen concentration is

generally very important in such models because it

inﬂuences the radiosensitivity of cells (Wouters and

Brown, 1997) and it determines the cell death when

its level is too low. Nevertheless, when the tumoral

spheroid, during all its growth, remains smaller than

a critical dimension at which an internal necrotic re-

gion starts to develop (‘small spheroids’), then it can

be assumed that:

1. the oxygen concentration is higher than the mini-

mum value necessary to the cell life

2. the initial distribution of oxygen inside the

spheroid is sufﬁciently uniform to be assumed

constant

In view of 1. the cell death for insufﬁcient oxy-

genation can be neglected and the radiation is the only

cause of death. Moreover for 2. it can be assumed that

the radiosensitivitycoefﬁcients are constant for all the

tumoral cells inside the spheroid. With these two as-

sumptions, the ODE model presented in this paper is

completely equivalent to the original PDE model pro-

posed by Bertuzzi et al. (2009), and it may be ob-

tained from the latter by integration with respect to

the radial distance, as mentioned above.

419

Papa F. (2010).

A MODEL OF THE TUMOUR SPHEROID RESPONSE TO RADIATION - Identiﬁability Analysis.

In Proceedings of the Third International Conference on Bio-inspired Systems and Signal Processing, pages 419-423

DOI: 10.5220/0002721304190423

 SciTePress

2 AN ODE MATHEMATICAL

MODEL OF THE TUMOUR

SPHEROID RESPONSE TO

RADIATION

Although quiescent cells have been evidenced in

tumour spheroids (Freyer and Sutherland, 1986),

(Sutherland, 1988), for simplicity we will assume that

all viable cells proliferate with the same rate and this

assumption is reasonable because the model is for-

mulated under the assumption of ‘small spheroids’,

where the oxygen level is sufﬁciently high and uni-

form. So in a spheroid we will distinguish: viable

cells, lethally damaged cells and dead cells.

Under the hypothesis of ‘small spheroids’, let us

consider the following ODE model (Papa, 2009), ob-

tained by integrating the PDE equations of the model

proposed by Bertuzzi et al. (2009):











V(t) = χV(t),

(t) = (χ

− µ

(t) ,

(t) = (χ

− µ

(t) + µ

(t) ,

(t) = (χ

− µ

(t) + µ

(t) ,

(t) = µ

(t) − µ

(t) ,

(t) = µ

(t) − µ

(t) ,

(t) = µ

(t) − µ

(t) ,

(1)

where V(t) is the volume of viable cells, V

(t),

(t) and V

(t) are the volumes of three subcom-

partments of lethally damaged cells andV

(t), V

(t)

and V

(t) are the volumes of three subcompartments

of dead cells (Bertuzzi et al., 2009), (Papa, 2009);

with χ and χ

we denote the constant proliferation

rates, respectively, of viable cells and of the three sub-

compartments of lethally damaged cells (that we sup-

pose to progress across the cell cycle and to divide

until they die), with µ

and µ

, respectively, the death

rate of lethally damaged cells and the degradation rate

of dead cells. All these dynamic parameters are pos-

itive and, since lethally damaged cells eventually die,

it is necessary to assume that µ

> χ

. The output of

the model is the total volume of the spheroid, obtained

by summing the state variables:

y(t) = V(t) +

∑

i=1

(t) +

∑

i=1

(t) . (2)

Without loss of generality, cells are assumed to oc-

cupy all the volume of the spheroid.

Considering only impulsive irradiations, both the

direct action and the effect of binary misrepair will be

considered instantaneous and described by a non lin-

ear relation named linear-quadratic (LQ) model (Bris-

tow and Hill, 1987). Denoting by δ the surviving frac-

tion of cells after a single impulsive irradiation, the

LQ dose-response relation has the form:

δ = e

[−αd−βd

]

, (3)

where d is the dose, α and β the radiosensitivity pa-

rameters related, respectively, to the direct action of

radiation and to the binary misrepair of DSBs. Then

the initial conditions for the basic model, according to

(3), are:











V(0

) = e

[−αd−βd

]

V(0

−

) = (1 − e

[−αd−βd

]

)V(0

−

) = 0, i = 2, 3,

) = 0, j = 1, 2, 3,

(4)

where V(0

−

) is the spheroid volume before irradia-

tion.

Equations (1), with their initial conditions (4), de-

ﬁne a linear time-invariant dynamical system and (2)

is the corresponding linear output equation.

3 PARAMETRIC

IDENTIFIABILITY OF THE

MODEL

There are different methods for studying the identi-

ﬁability of dynamical systems. For the model pre-

sented above it has been used the similarity transfor-

mation method (Travis and Haddock, 1981), that can

be only used for linear dynamical systems. In gen-

eral, some parameters of a linear stationary dynam-

ical system are not known. Therefore the similarity

transformation method allows to determine the iden-

tiﬁability properties of system parameters when they

correspond to the elements of the model matrices or

when there is a univocal relationship between them.

It is easy to understand, looking at the structure of

the matrices given below, that a univocal relationship

exists between the parameters (χ, χ

, µ

) and the

elements of the system matrices whereas it does not

happen for the radiological parameters (α, β). Con-

sidering the parameter δ, given by (3) and depending

on the radiological parameters (α, β), even if it was

identiﬁable, the parameters α and β would not be uni-

vocally determined from its value. It will be shown

that α and β can be univocally identiﬁed by exploit-

ing model responses to at least two different radiation

doses.

Let us study the identiﬁability of the parameter

vector

θ =



χ χ



, (5)

ranging in the admissible set Θ ⊂ R

, where

Θ ={θ ∈ R

| χ, χ

, µ

> 0, µ

> χ

and 0 < δ < 1} .

(6)

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420

Taking model equations (1) - (4) into account, let us

denote by

x =



V V



(7)

the state vector and by A(θ), c

(θ) and b(θ) respec-

tively the model dynamical matrix, the state-output

matrix and the fraction of the initial state vector inde-

pendent of the spheroid initial volume. In particular,

for the elements of A(θ), c

(θ), and b(θ) we have

that:











(θ) = χ,

(θ) = a

(θ) = χ

− µ

(θ) = a

(θ) = −µ

(θ) = a

(θ) = µ

(θ) = a

(θ) = µ

all other elements of A(θ) are equal to zero,

(θ) = 1, i = 1, . . . , 7,

(θ) = δ, b

(θ) = 1− δ,

(θ) = 0, i = 3, . . . , 7.

(8)

Then the model (1) - (4) can be written in a compact

form:

(

˙x(t;θ) = A(θ)x(t;θ), x(0

;θ) = b(θ)V(0

−

y(t;θ) = c

(θ)x(t;θ).

(9)

It is useful to observe, at this point, that the output

y(t;θ) obtained by the model (9), in which no input

acts, is the same output obtainable by the following

model:











¯x(t;θ) = A(θ) ¯x(t;θ) + b(θ)u(t),

¯x(0

−

) = 0,

¯y(t;θ) = c

(θ)¯x(t;θ)

(10)

with u(t) = u

(t)V(0

−

), where u

(t) is a Dirac unit

pulse function. In fact:

y(t;θ) = ¯y(t;θ) = c

(θ)e

A(θ)t

b(θ)V(0

−

). (11)

Therefore, it is easy to understand from relation (11)

that the identiﬁability problem of θ for the model

(9) is the same one for the model (10). In partic-

ular we can talk about controllability of the couple

(A(θ), b(θ)), since the role of the matrix b(θ) in the

model (9) is equivalent to the one in the model (10).

The similarity transformation method is based on

the following theorem (Papa, 2009):

Theorem 1 . Let the triples (A(θ), b(θ), c

(θ)) and

(A(φ), b(φ), c

(φ)) be observable and controllable.

Then

(θ)e

A(θ)t

b(θ) = c

(φ)e

A(φ)t

b(φ), t ∈ [0, T] (12)

if and only if a nonsingular matrix P exists such that







PA(θ)P

−1

= A(φ),

(θ)P

−1

= c

(φ),

Pb(θ) = b(φ).

(13)

Proof . It is immediate to see that (13) implies (12) by

taking into account the power expansion of the expo-

nential. The inverseimplication, that requires the con-

trollability and observability properties, was proved

by Kalman (Kalman, 1963), (Kalman et al., 1969).



From Theorem 1 it is easy to understand that given

an indistinguishable couple (θ, φ) ∈ Θ for the system

(9), if it exists, the corresponding system matrices,

(A(θ), b(θ), c

(θ)) and (A(φ), b(φ), c

(φ)), have the

same structure and are linked by the algebraic rela-

tions (13). It is easy also to see that if (13) have a

unique solution (θ, I) then indistinguishable couples

do not exist. So the following obvious lemma follows:

Lemma 1 . Let the triples (A(θ), b(θ), c

(θ)) and

(A(φ), b(φ), c

(φ)) be observable and controllable.

Then the system (9) is globally identiﬁable in Θ if and

only if the equations (13), for all ﬁxed vector θ ∈ Θ,

have the unique solution (φ, P) = (θ, I).

The observability and controllability properties of

the triple (A(θ), b(θ), c

(θ)) of (9) have been proved

by Papa (2009). Now we can prove the following re-

sult.

Theorem 2 . The model (1) - (4) is globally identiﬁ-

able with respect to the unknown parameter vector θ

given by (5) and ranging in the set Θ deﬁned by (6).

In fact ∀θ ∈ Θ it does not exist in Θ another parameter

vector φ that gives the same output.

Proof . Given θ ∈ Θ, let us consider a (1× 5) vector

φ = [χ

∗

] ∈ Θ and the (7 × 7) matrix P.

From the ﬁrst equation of (13) it is easy to obtain the

following equation system:









(14)

where r

and c

are, respectively, a (1× 7) row vector

and (7× 1) column vector:

A MODEL OF THE TUMOUR SPHEROID RESPONSE TO RADIATION - Identifiability Analysis

421







(χ

− µ

) + p

(χ

− µ

) + p

(χ

− µ

) + p

−p

+ p

−p

+ p

−p













∗

(χ

∗

− µ

∗

)

∗

+ p

(χ

∗

− µ

∗

)

∗

+ p

(χ

∗

− µ

∗

)

∗

− p

∗

− p

∗

− p

∗







, i = 1, ..., 7.

Furthermore, using the second and the third equation

of (13) it is easy to obtain, respectively, the following

relations:

+ p

+ ... + p

= 1, with i = 1, ..., 7,

(15)







δ+ p

(1− δ) = δ

∗

δ+ p

(1− δ) = (1− δ

∗

δ+ p

(1− δ) = 0, with i = 3, ..., 7.

(16)

By solving equations (14) - (16), it can be shown

that (13) admits only the trivial solution (θ, I). For de-

tails see the proof of Theorem 4 given by Papa (2009).

Then, from Lemma 1, we can say that the model (1)

- (4) is globally identiﬁable with respect to the ﬁve

considered parameters. 

Theorem 2 does not establish the global identiﬁa-

bility of the model (1) - (4) with respect to the radio-

logical parameters. In order to study the identiﬁability

of the radiological parameters α and β it is necessary

to consider two different doses d

and d

and the cor-

responding parameters δ

and δ

(

= e

[−αd

−βd

]

= e

[−αd

−βd

]

(17)

It is easy to verify that the relation between (δ

, δ

)

and (α, β) is one to one if d

6= d

. Therefore, let

us consider two different initial states x

(1)

;θ) and

(2)

;θ) related to two different doses and let us

observe parallely the two corresponding system re-

sponses. Let us deﬁne a new parameter vector

θ =



χ χ



(18)

ranging in the admissible set Θ ⊂ R

, where

Θ ={θ ∈ R

| χ, χ

, µ

> 0, µ

> χ

0 < δ

< 1 and 0 < δ

< 1}.

(19)

Let us deﬁne by x

∈ R

the total state vector



(1)

(2)



, (20)

that is the union of two state vectors of the type (7),

(1)

and x

(2)

, related to the two different initial states,

and the block matrices

(θ) =



A(θ) 0

0 A(θ)



(θ) =



(θ) 0

0 c

(θ)



(θ) =



(1)

(θ) 0

0 b

(2)

(θ)



(21)

where A(θ) and c

(θ) are the same matrices deﬁned

in (8), b

(1)

(θ) and b

(2)

(θ) are such that

(1)

(θ) =



(1− δ

) 0 . . . 0



(2)

(θ) =



(1− δ

) 0 . . . 0



(22)

Obviously, A

(θ), C

(θ) and B

(θ) are, respectively,

(14 × 14), (2 × 14) and (14 × 2) matrices and the

model (1) - (4) can be written as:











˙x

(t;θ) = A

(θ)x

(t;θ),

;θ) = B

(θ)



V(0

−

)

V(0

−

)



(t;θ) = C

(θ)x

(t;θ),

(23)

where y

(t;θ) ∈ R

is the union of the two outputs

related to the two different initial states.

Using the observability and the controllability

properties of (9), it can be easily shown that the sys-

tem (23) is observable and controllable too (Papa,

2009).

Now we can prove the following result.

Theorem 3 . The model (1) - (4) is globally identiﬁ-

able with respect to the unknown parameter vector θ

given by (18) and ranging in the set Θ deﬁned by (19),

exploiting the outputs of the model obtained from two

different radiation doses.

Proof . Given θ ∈ Θ, let us consider a (1× 6) vector

φ = [χ

∗

] ∈ Θ and the (14× 14) matrix

P. Dividing the P matrix into four (7× 7) blocks

P =





(24)

it is easy to show that from the matrix equations (13),

developing the block products, the following subsys-

tems are obtained:







A(θ) = A(φ)P

(θ) = c

(φ)P

(1)

(θ) = b

(1)

(φ)

with i = 1, 2,







A(θ) = A(φ)P

0 = c

(φ)P

(2)

(θ) = 0

with i, j = 1, 2, i 6= j.

(25)

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422

The ﬁrst two subsystems of (25) are similar to the one

studied in the proof of theorem 1. So both the subsys-

tems have only the trivial solution (θ, I). Therefore it

results that

∗

= χ, χ

∗

= χ

, µ

∗

= µ

, µ

∗

= µ

∗

= δ

, δ

∗

= δ

and P

= P

= I.

(26)

It is simple to verify that, with the results (26), the

latter two subsystems of (25) give the solutions P

= 0.

Therefore we have that system (13) admits only

the trivial solution (θ, I). Thus, from Lemma 1, we

can say that the model (1) - (4) is globally identiﬁable

by exploiting the model response y

(t;θ) to at least

two different doses d

and d

. 

4 CONCLUDING REMARKS

In this paper we have considered a spatially uniform

dynamical model of tumour growth after a single in-

stantaneous radiative treatment. In this model the

details of the spatial structure of the tumour are ne-

glected and the attention is focused on the temporal

evolution of tumour overall volume after the radia-

tive treatment. The model can be used for different

applications. For instance, to asses the efﬁciency of

the radiotherapeutic treatment, but for this applica-

tion it is necessary to identify the unknown param-

eters. A preliminary condition, that is necessary to

verify before performing the parameter estimation, is

the global identiﬁability of the model.

In this paper a detailed study of the identiﬁabil-

ity properties of the model is done, pointing out that

it is globally identiﬁable, provided that the responses

to two different radiation doses are considered. This

important property assures a correct formulation of

the parametric identiﬁcation problem. The paramet-

ric identiﬁcation of the model and the corresponding

validation, with respect to both the ﬁtting and the pre-

diction capability of the experimental data, are treated

by Bertuzzi et al. (2009).

REFERENCES

Bertuzzi, A., Bruni, C., Fasano, A., Gandolﬁ, A., Papa, F.,

and Sinisgalli, C. (2009). Response to radiation of tu-

mour spheroids: Modelling and parameter estimation.

Bulletin of Mathematical Biology. Accepted.

Bristow, R. and Hill, R. (1987). Molecular and cellular ba-

sis of radiotherapy. In Tannock, I. and Hill, R., edi-

tors, The Basic Science of Oncology, pages 295–319.

McGraw-Hill.

Byrne, H. (2003). Modelling avascular tumour growth. In

Preziosi, L., editor, Cancer Modelling and Simulation,

pages 75–120. Chapman and Hall/CRC.

Freyer, J. and Sutherland, R. (1986). Regulation of growth

saturation and development of necrosis in emt6/ro

multicellular spheroids by the glucose and oxigen sup-

ply. Cancer Research, 46:3504–3512.

Kalman, R. (1963). Mathematical description of linear dy-

namical systems. SIAM Journal of Control and Opti-

mization, 2:152–192.

Kalman, R., Falb, P., and Harbib, M. (1969). Topics in

Mathematical System Theory. McGraw-Hill, New

York.

Papa, F. (2009). Models of the tumour spheroid response

to radiation: identiﬁability analysis. Technical Re-

port 9, Department of Computer and System Sciences.

http://padis2.uniroma1.it:81/ojs/index.php/DIS Tech-

nicalReports/index.

Sutherland, R. (1988). Cell and environment interactions

in tumor microregions: The multicell spheroid model.

Science, 240:177–184.

Travis, C. and Haddock, G. (1981). On structural identiﬁ-

cation. Mathematical Biosciences, 56:157–173.

Wouters, B. and Brown, J. (1997). Cells at intermediate

oxygen levels can be more important than the “hy-

poxic fraction” in determining tumor response to frac-

tionated radiotherapy. Radiation Research, 147:541–

550.

A MODEL OF THE TUMOUR SPHEROID RESPONSE TO RADIATION - Identifiability Analysis

423