Feedback Spreading Control Applied to Immunotherapy

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Feedback Spreading Control Applied to Immunotherapy

KHALID KASSARA ^a

a University Hassan II , Morocco Published online: 22 Sep 2006.

To cite this article: KHALID KASSARA (2005) Feedback Spreading Control Applied to Immunotherapy, Mathematical Population Studies: An International Journal of Mathematical Demography, 12:4, 211-221, DOI: 10.1080/08898480500301819 To link to this article: http://dx.doi.org/10.1080/08898480500301819

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Feedback Spreading Control Applied to Immunotherapy

Khalid Kassara

University Hassan II, Morocco

How should immunotherapy be controlled so as to eliminate cancer cells from a tissue? The populations of immune cells, tumor cells, chemokine and complexes are governed by four semilinear partial differential equations controlled by both the dosage of effector cells and the therapy zone. The corresponding control pro- blems are formulated in the framework of feedback spreading control (FSC) seek- ing to expand the zones without tumor cells to the entire tissue. Algorithms for computing FSC laws are used to demonstrate how the dosage of effector cells and the therapy zones are determined in order to provide feedback therapy proto- cols which keep the patient healthy.

Keywords: Immunotherapy; mathematical modelling; cell population dynamics;

spreading control

1. INTRODUCTION

In the last three decades, cancer research has involved mathe- maticians in the modeling of effective therapies for cancer patients (Matzavinos et al., 2004; Friedman, 2000; Kirshner and Panetta, 1998; Webb, 2002). Such models enable a deeper understanding of the mechanisms associated with tumor initiation and progression, and open the way to elaborate control of cancer (Swan, 1990; Murray, 1990; Kirshner and Panetta, 1998; Fister and Panetta, 2000; Kimmel and Swierniak, 2003; Burden et al., 2004).

Immunotherapy consists of the use of natural and synthetic sub- stances helping the immune system to compete with tumor. One of these therapies (Matzavinos et al., 2004) consists of attacking the avascular tumor by the so-called tumor-infiltrating cytotoxic lympho- cytes (TICLs) leading to the semilinear PDE model which is under

Address correspondence to Khalid Kassara, Department of Mathematics and Computer Science, University Hassan II Ain Chok, P. O. Box 5366, Casablanca, Morocco.

E-mail: k_kassara@yahoo.com

Mathematical Population Studies, 12:211–221, 2005 Copyright # Taylor & Francis Inc.

ISSN: 0889-8480 print=1543-5253 online DOI: 10.1080/08898480500301819

211

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consideration here. Its finite dimensional version, an ODE model, is investigated in Arciero et al. (2004).

The concept of spreading control is used (El Jai and Kassara, 1994) to model and control expansion phenomena arising in spatially distrib- uted processes. It consists of a distributed parameter system (i.e., a control system governed by PDEs), a map to be spread which describes the zones to be expanded and selection procedures (Kassara, 2000) for deriving feedback spreading control laws.

Kassara (2002) introduced the speed of a spread and studies feedback spreading control under speed constraints. Aubin and Kassara (2004) investigated the algorithmic aspect and showed how to build control laws generating spreads slower or quicker than a desired given speed.

Here, I apply spreading control techniques to derive immunother- apy protocols (the rate of supply of effector cells and the therapy zone) leading to clear cancer from the tissue.

In the next section I present the model of immunotherapy and state the control problem. Section 3 is devoted to an overview on feedback spreading control. In Section 4, I derive an algorithm for computing therapy protocols.

2. STATEMENT OF THE PROBLEM

Matzavinos et al. (2004) describe the evolution of a tumor in an avas- cular tissue X R ⁿ with n ¼ 1; 2 or 3 subject to TICLs attack by four state variables: density ‘ of TICLs, density T of tumor cells, density a of chemokine and density f of TICL-Tumor cell complexes. These den- sities vary according to the semilinear partial differential equations:

@‘

@t ¼ d 1 r ² ‘ k ₁ ‘ k ₂ ‘T k ₃ r ð‘raÞ þ k ₄ f

k 5 þ T þ k ₆ f þ uv _H

@T

@t ¼ d 2 r ² T þ k 7 ð1 k 8 TÞT k 2 ‘T þ k 9 f

@a

@t ¼ d ₃ r ² a k ₁₀ a þ k ₁₁ f

@f

@t ¼ d 4 f þ k ₂ ‘T

ð1Þ

where r denotes the gradient operator and v _H stands for the charac- teristic function which is given by

v _H ðxÞ ¼ 1 if x 2 H 0 if x 62 H

ð2Þ where H denotes a measurable subset of X.

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The d i and k i are positive parameters. The terms d 1 and d 2 are random motility coefficients for the corresponding populations and d 3 measures diffusion of chemokine. k ₃ is the chemotaxis coefficient, k ₇ and k ₈ express the logistic growth of the tumor and k ₂ ; k ₆ ; k ₉ and d 4 define the terms of local kinetics. Now k ₄ and k ₅ compose the pro- liferation term and k ₁ is a decay coefficient. For precise information on the previous coefficients and their values I refer to (Matzavinos et al., 2004) and the references therein.

System (1) is augmented with boundary conditions of Neumann type:

@‘

@n ðx; tÞ ¼ @T

@n ðx; tÞ ¼ @a

@n ðx; tÞ ¼ 0 8x 2 @X and t 0 ð3Þ

where @=@n stands for the outward normal derivative on the boundary.

According to Matzavinos et al. (2000), at each time t, the term uðtÞv _HðtÞ ðxÞ represents a source or supply of effector cells infused into the tissue over the zone HðtÞ, with a dosage uðtÞ. The zone HðtÞ refers to the therapy zone at time t, see Figure 1. Thus the control in system (1)–(3) is represented by

v : t 2 ½0; t _f L ðuðtÞ; HðtÞÞ 2 R M X

where u 2 L ² ð0; t f Þ and M X stands for the set of measurable non empty subsets of X. The interval t _f > 0 and the map v respectively stand for the time horizon and the protocol of the therapy. We denote by V the set of such controls v.

FIGURE 1 Spread of the zones without cancer cells x

at times t

, t

and t

with an arbitrary location of the initial zone of therapy Hð0Þ.

Feedback Spreading Control 213

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Now assume that there is a positive small number e such that:

Tðx; tÞ e () no tumor cells near x at time t

ð4Þ Subsequently the zones without tumor cells are modelled as:

x t ¼ fx 2 X j Tðx; tÞ eg for each t ð5Þ and thereby the problem of eliminating tumor cells before time t _f is set as:

Find a protocol v ¼ ðu; HÞ 2 V so that:

x t x s if 0 t s t _f and x t

¼ X:

ð6Þ In order to keep the toxicity to the normal tissue acceptable, both the rate of effector cells uðÞ and the area of the therapy zone H during the therapy session should be reduced as little as possible. This leads to a trade-off (Bellomo et al., 2003) between the amount of external source of effector cells and the objective to eliminate cancer cells, lead- ing to the optimal control problem:

Find a protocol v h ¼ ðu h ; H h Þ 2 P e such that:

ðu; HÞ 2 P e ¼) uðtÞ u _h ðtÞ and HðtÞ H h ðtÞ for each t such that 0 t t _f

ð7Þ

where P e denotes the set of solutions v ¼ ðu; HÞ of problem (6).

3. FEEDBACK SPREADING CONTROL

I present the concept of feedback spreading control. Let X R ⁿ be an open and bounded domain with sufficiently smooth boundary @X.

Assume A is an unbounded densely defined linear operator which generates a C 0 analytic semigroup ðSðtÞÞ _t0 on Z ¼ L ² ðXÞ (Balakrishnan, 1981), and consider the semilinear control system:

@z

@t þ Az ¼ uðz; vÞ in X ð0; 1Þ ; ð8Þ

with initial data

zðx; 0Þ ¼ z ₀ ðxÞ in X ð9Þ

where z ₀ 2 domðAÞ and u denotes a nonlinear operator mapping D V to Z, with V another Hilbert space and D a closed subset of Z. For t _f > 0 and a measurable function v: ½0; t _f ! V, we denote by zð; vÞ a solution, when it exists, of System (8)–(9) on the interval ½0; t _f .

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Moreover let x be the map to be spread, defined as x: D Z ! 2 ^X :

A measurable function v: ½0; t _f ! V is called a spreading control with respect to map x if

zðt; vÞ 2 D for all t 2 ½0; t _f ð10Þ and

ðxðzðt; vÞÞÞ _0t<t

is non-decreasing. ð11Þ To define the speed (Kassara, 2002) of the spread ðxðzðt; vÞÞÞ _t generated by a spreading control v we set the expression for each t 2 ½0; t _f ,

speedðt; vÞ ¼ : lim inf

h#O

lðxðzðt þ h; vÞÞ n xðzðt; vÞÞÞ

h 0; ð12Þ

where l stands for a measure on X. Next, I define feedback spreading control.

Definition 3.1. The mapping 1: D ! V is said to be a feedback spreading control (for short FSC) law with respect to x if, for all z ₀ in D; v ¼ 1ðzÞ defines a spreading control for system (8)–(9) i.e., satisfy- ing conditions (10) and (11).

Then for each couple ðy; zÞ 2 Z D, consider the tangential con- dition

8 d > 0; 9 0 < h < d and kpk d such that SðhÞz þ hðy þ pÞ 2 D and

xðSðhÞz þ hðy þ pÞÞ xðzÞ:

ð13Þ

Then define the set-valued maps T x and F x for each z 2 D as as:

T xðzÞ ¼ :

fy 2 Z j Eq: ð13Þ holds with ðy; zÞg and

F x ðzÞ ¼ :

fv 2 V j uðz; vÞ 2 T x ðzÞg: ð14Þ Also we need to let

R x ¼ :

fðy; zÞ 2 D ² j xðyÞ xðzÞg: ð15Þ I present the basic result which characterizes FSC laws as selec- tions of the feedback map F x ðÞ:

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Theorem 3.1. Let 1 : D ! V be a measurable function and assume that (i) The semigroup SðÞ is compact and R x is closed.

(ii) uð; 1ðÞÞ sends convergent sequences of D into weakly convergent ones in Z.

Then 1 is a FSC law with respect to x iff 1 is a selection of the map F x given by (14), ie. 1ðzÞ 2 F x ðzÞ for each z 2 D.

To compute the speed of the spread generated by a FSC law, I use the formula of (Kassara, 2002),

speedðt; vÞ ¼ hðuðzðtÞ; vðtÞÞ; zðtÞÞ ð16Þ where z ¼ zð; vÞ and where h denotes the speed functional defined as,

hðy; zÞ ¼ :

lim inf

h#0;kpk!0

lðxðSðhÞz þ hðy þ pÞÞ n xðzÞÞ

h ð17Þ

for each z 2 D and y 2 T x ðzÞ. In addition, when D and s x are convex, then, we have

hðy; zÞ ¼ ds x ðzÞðy A zzÞ _ ð18Þ for each y 2 T x ðzÞ and z 2 D \ domðAÞ, where ds _x denotes the direc- tional derivative of s _x . Due to (16) we can see the role played by the func- tional h of (17) in evaluating the speed of the spread generated by a spreading control. For sake of simplicity we then define the functional

qðz; vÞ ¼ :

hðuðz; vÞ ; zÞ for each z 2 D ; v 2 F x ðzÞ ð19Þ To design an FSC law, we propose the following algorithm sub- sequent to a convenient use of expressions (13) and (14). Let n: D 7! R ^þ a measurable function standing for the speed to exceed, see substep 2.a below, then

Algorithm 3.1.

Given h > 0; N 2 N such that h ¼ t _f =N.

1. Initialize r ¼ z 0 and r ¼ xðz 0 Þ.

2. Iterate for k ¼ 1 to N.

2.a Find v such that:

xðSðhÞr þ huðr; vÞÞ r and qðr; vÞ nðrÞ: ð20Þ 2.b Let v _k ¼ v and z _k ¼ SðhÞr þ huðr; vÞ.

2.c Put r ¼ z _k , r ¼ xðz k Þ and go to 2.a.

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3. At each time t k ¼ kh:

The approximated state is z k , the spreading control is v _k , the generated spread is ðxðz k ÞÞ _k .

Convergence of Algorithm (3.1) means that:

ðiÞ v ^h ! v _s in L ² ð0; t _f Þ when h ! 0 ðiiÞ v _s is a spreading control:

ðiiiÞ z ^h ! zð; v _s Þ when h ! 0

where v ^h and z ^h are given as,

ðv ^h ðtÞ; z ^h ðtÞÞ ¼ :

ðv 1 ; z ₁ Þ on ½0; h½ .. .

ðv N ; z _N Þ on ½ðN 1Þh; t _f

Although Algorithm (3.1) was tested successfully with respect to the heat equation (Aubin and Kassara, 2004), its convergence, according to the above sense, is not proved mathematically. Furthermore, it should be convenient to stress that a crucial sequence in this algor- ithm is to be done at the level of (20), where it is required to get a con- venient selection determining an approximate value of the spreading control.

4. ALGORITHM (3.1) APPLIED TO IMMUNOTHERAPY

First, I begin by restating the problem in the framework of Section 3.

For that purpose, let X ¼ :

ð0; ‘ Þ and put

z ¼ ðz 1 ; z ₂ ; z ₃ ; z ₄ Þ ⁰ ¼ ð‘; T; a; fÞ ⁰ and v ¼ ðu; HÞ; ð21Þ then system (1) can be rewritten as in (8):

zz _ þ Az ¼ uðz; vÞ where for each z ¼ ðz 1 ; z ₂ ; z ₃ ; z ₄ Þ ⁰ and v ¼ ðu; HÞ:

Az ¼ :

d 1 r ² z ₁ ; d 2 r ² z ₂ ; d 3 r ² z ₃ ; d ₄ z ₄ ⁰

ð22Þ with domain dom ðAÞ ¼ W ₀ ¹ ðXÞ ⁴ , where

W ¹ ₀ ðXÞ ¼ :

f 2 L ² ðXÞ j f ⁰ 2 L ² ðXÞ and @f

@n ¼ 0 on @X

: ð23Þ

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and

uðz; vÞ ¼ :

gðzÞ þ ðuv _H 0 0 0Þ ⁰ : ð24Þ While the function gðzÞ coincides with:

k 1 z 1 k 2 z 1 z 2 k 3 r ðz 1 rz 3 Þ þ _k ^k

_þz ^z

þ k 6 z 4

k ₇ ð1 k ₈ z ₂ Þz 2 k ₂ z ₁ z ₂ þ k ₉ z ₄ k 10 z ₃ þ k ₁₁ z ₄

k ₂ z ₁ z ₂ 0 B B

@

1 C C

A ð25Þ

Here, the state space is Z ¼ L ² ðXÞ ⁴ and the control space is V ¼ R M X .

The map x to be spread is given by xðzÞ ¼ :

x 2 Xjz 2 ðxÞ e

f g ð26Þ

for each z belonging to the set D ¼ :

z ¼ ðz 1 ; z ₂ ; z ₃ ; z ₄ Þ ⁰ 2 Zjz i 0 for each i

:

We know by Balakrishnan (1981) that for c > 0 the operator cr ² with domain W ₀ ¹ ðXÞ generates a compact analytic semigroup ðS c ðtÞÞ _t>0 on L ² ðXÞ. Then the operator A generates the semigroup

SðtÞz ¼ ð S _d

ðtÞz 1 ; S _d

ðtÞz 2 ; S _d

ðtÞz 3 ; expðd 4 tÞz 4 Þ ⁰ ð27Þ for each z ¼ ðz 1 ; z 2 ; z 3 ; z 4 Þ ⁰ 2 Z and t > 0. The therapy protocol follows Algorithm 3.1. I show now how to implement the key sequence to execute at substep II.a.

Let h be a small positive number and consider algorithm 3.1 in which, for sake of convenience, the approximated state z _k is replaced by z ^k . Also let v _k ¼ ðu k ; H k Þ and suppose that z ^k and v _k are determined

for k ¼ 1; . . . ; L 1. Let, for each k,

z ^k ¼ :

ð‘ k ; T _k ; a k ; f _k Þ ⁰ ð28Þ then we have,

T _L ¼ S _d

ðhÞT L1 þ hðk 7 ð1 k ₈ T _L1 ÞT L1

k 2 ‘ _L1 T L1 þ k 9 f _L1 Þ ; ð29Þ and

f _L ¼ expðd 4 hÞf L1 þ k ₂ h‘ _L1 T _L1 : ð30Þ

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Define for each v ¼ ðu; HÞ 2 R M,

‘ ^v _h ¼ :

S _d

ðhÞ‘ L1 þ h

k 1 ‘ L1 k 2 ‘ L1 T L1 k 3 r ð‘ L1 ra L1 Þ þ k 4 f L1

k 5 þ T L1

þ k 6 f _L1 þ uv _H ð31Þ that we rewrite in the form

‘ ^v _h ¼ a h ðz ^L1 Þ þ huv _H where

a _h ðzÞ ¼ :

S d

ðhÞ ‘ þ h

k 1 ‘ k 2 ‘T k 3 r ð ‘ raÞ þ k 4 f

k ₅ þ T þ k 6 f ð32Þ for each z ¼ ð ‘; T; a; fÞ 2 Z. Then, according to (20), the protocol v L and the approximated associated state z ^L is updated by seeking ðu; HÞ ¼ v L

such that

S d

ðhÞT L þ hðk 7 ð1 k 8 T L ÞT L k 2 ‘ ^v _h T L þ k 9 f _L Þ e on the zone xðz ^L Þ. This is equivalent to

k ₇ ð1 k ₈ T _L ÞT L k ₂ ‘ ^v _L T _L þ k ₉ f _L e S _d

ðhÞT L

h on xðz ^L Þ. This yields

‘ ^v _h T L b h ðT L ; f _L Þ e

hk ₂ on xðz ^L Þ where

b _h ðT; fÞ ¼ :

S _d

ðhÞT þ hðk 7 ð1 k ₈ TÞT þ k ₉ fÞ ð33Þ for each ðT; fÞ 2 L ² ðXÞ ² . Then by considering (31) we get,

uT _L v _H b _h ðT L ; f _L Þ hk ₂ T _L a _h ðz ^L1 Þ h ² k 2

on xðz ^L Þ ð34Þ This implies that u may exist whenever H xðz ^L Þ. However, in concordance with (7) we take the therapy zone as

H _L ¼ xðz ^L Þ ð35Þ

Feedback Spreading Control 219

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It follows that the dosage of effector cells at time L, u L must satisfy the inequality,

u L T L b _h ðT L ; f _L Þ hk ₂ T _L a _h ðz ^L1 Þ h ² k 2

on H L ð36Þ

In sum,

Theorem 4.1. Assume that

z ^k ¼ ð‘ k ; T _k ; a k ; f _k Þ ⁰ and v _k ¼ ðu k ; H k Þ ð37Þ are determined for k ¼ 1; . . . ; L 1. Then

z ^L ¼ ð‘ L ; T _L ; a L ; f _L Þ ⁰ and v _L ¼ ðu L ; H L Þ ð38Þ are provided as:

Put:

c L1 ¼ :

k 7 ð1 k 8 T L1 ÞT L1 k 2 ‘ _L1 T L1 þ k 9 f _L1 ; T L ¼ S _d

ðhÞT L1 þ hc L1 ;

a L ¼ S d

ðhÞa L1 þ hðk 11 f _L1 k 10 a L1 Þ;

f _L ¼ expðd 4 hÞf _L1 þ hk 2 ‘ _L1 T L1 : Then let:

xðz ^L Þ ¼ fx 2 XjT L ðxÞ eg and H L ¼ xðz ^L Þ:

Find u such that;

T _L u ^b

^ðT

^;f

^Þhk _h

k

^T

^a

^ðz

^Þ on H _L ; And put:

u L ¼ u and ‘ _L ¼ a _h ðz ^L1 Þ þ hu L v _H

:

ð39Þ

where a _h and b _h are respectively defined by (32) and (33).

5. CONCLUSION

Mathematical modeling and control contribute to solve the problem of clearing cancer from a tissue. The model used corresponds to a special case of immunotherapy and consists of four semilinear PDEs con- trolled by the dosage of effector cells and the zones where they act which stand for the therapy zones. The zones without tumor cells are to be expanded to the whole tissue using the concept of feedback spreading control. The latter, by the implementable algorithm that it generates, leads to the desired therapy protocol laws. At each time

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and according to (35), these consist of taking the zone without tumor cells as the therapy zone and infusing on the effector cells with a rate determined by (36) by a suitable selection procedure.

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