Modeling of Sarcoma Cells Circulation in Mice

Copyright © IFAC 10th Trie nnial World Co ngress, Munich, FRG , 1987

MODELING OF SARCOMA CELLS CIRCULA TION IN MICE A. Swierniak*, Z. Duda*, E. Skierska** and

J.

S. Skierski**

*Department of Automatic Cuntrol, Silesian Technical Univenity, Gliwice, Pstrowskiego 16, Poland **Department of Biophysics, Medical Centre of Postgraduate Education, Wanaw, Marymoncka 99, Poland

Abstract.A model of Janik warsawian SarcO/lJ3. cells circulation in the mouse Is presented. A quantitative estimation of thR model is based on data obtained in the in vivo experiments. The model is composed of the compartments representing the most important organs de3cribed by their simplified transmittances • Keywords. Biomedical modelling; parameter esti/lJ3.tion ; identification l1near systems ; compartmental models •

INTrD DOCTION

The problem of dissemination of tumor cells is the key question in oncology. The fornation and localisation of secondary tll'nor foci depends on the interaction of the tumor cells with the individU3.1 organs of the organism in the direct rranner. It is ','/ell known that the living neoplastic cells are released from the primary turnor to the circulation. Due to the interaction of the cells flow, they undergo acc umulation and either find conditions to division and g l''Owth ( metastasis) or undergo destruction. The fact tmt in the majority of turnors nearly all circulating cells undergo disintegration is also widely known. Only few cells have a cmnce of settling down and forming a secondary turnor ( Dinge/lJ3.ns,1973 Roos and others, 1977 , 1979) • The blood stream spreads the cells uniformly OVer the whole organiSM but USU3.1ly metastases is loca lised only in strictly organs e.g. lungs, liver, spleen, lymph ,nodes etc.

Neoplastic cells circulation in organism and their elimination from the blood and other organs are the key problems in understanding of cancer kinetics. In this paper we present a simple model of these processes whose parameters are found basing on in vivo experiments performed in the Hedical Centre of PostgradU':l.te Education. (Skierska , 198 5). The model of Janil{ Warsawian Sarcoma cells circulation in mice is discussed but the objective is to present a methodological aspects of modelling and simulation of the phenomena connected with tumor cells kinetics taking JWS cells as an example. The experiments consist in intravenous injection of the JWS cells traced by radioactive Cr tracer. After 5 min ,90 min and 24 h a part of aninals is decapitated and the radioactivity of nain organs is measured. These data are used to estinate the parameters of time responses of contents of the organs which are in turn originals of the simplified transmittance models of the whole close-loop system. The method of estination is based on the simulation of time responses with parameters cl1lnged by tr1.a l-and-error procedure. Since the data have broad range of changes the estinated transients should belong to the intervals found in experiments •

In the experiments the cells initi~ting thl? process are injected to the lu.'1gs and are etimin.ated or desintegrateci by organs while circula ting tn the blood. 'l'he qwntitative model of this process is proposed ba s ing on the so called co:npartmental models i dea (Brown, 1900, Godfrey ,1983, Sandberg , 1978). 'l'hp specific organs are treated a s dynamic tanks (compartments) connected by noninertial links • Since we assurn2 that thl? compartments are

73

71

A. S"icrlliak ,./ (/1.

linear a nd causal systems the dynamic my be described by transmittanc;, oodels \':hich seem to be very si~le and convenient both for .sil'T[llific3. :ion 3. n-; bi.olo.:;ical interpre-

Contents of the compartment "/hich is measured in experiments is the c ifference of integr
tation •

R(t) =

~ \x\u)-

Y(U))dU

(2)

o

After I.apla ce tl"8.ns:orrn ',:e obta in TRANsmTTANCr.: MODE L.": 0" COHPAKT~.o:NTS

~(X (s)- yes)) = ; (X\S)(1 - K(s)))

R(s)= A compartment is a component element of

the whole system. It is built by lumping in one element some properties of distributed system. In the case of the compartment:tl nodel of the organism all the compartments model important (from the considered point ef view) or6&. ns, Our model ClSS U::1 2 1 i nearity a nd causality of th" compartments • The second c.ss'Ju"'":ption is eb'/ i ~ us \,;hi le the fir s t one seems to be reasonablf' ir. the case of short cbservati o n horizon s \"hen G.ccumulation. desintegration and transport are the most important phenomena • The problem of "penetrE\tion" inside 3. CO::1pa.rtme nt ::.tnd the s ame identification of the structure o f the compa rtment is ,~ry difficult. The more every compartment should be trea ted as two-input-two-output e~ . e me nt since the dyna mic of cells flow i s different t h3. n the dynarr.ics of ri isintcgrated cell fragments flow and they both d iffer fro m the dyna mic ef cisintegra tion ( Ki mmel. 31-: i e rski. ';983) • In the exp e riments however we are able to measure only the tot:tl radioactivity of the organ and thus in the model only a global response of the compa rtment for a g lobal input is con s idered. The simplest model aS3L:ming only accumul a tion and rJisintegration of cells 'w-ith parallel ideal mixing ~s a form of the first order i:1ertial ele ment. Thus it may be descY'ibed b y the following transfer function : r~

(s )

y(s ) =-X (s)

k

(1 )

s T + 1

where Y (s) and X (s) are Laplace transforms of the output and the input of the compartment, k is a gain and T is a time constant •

Although the first inertial model of the compartment is oversimplification,since it :!leans tha t outflow curves are approximated by the single exponential signals, it may be used in the analy s is of the whole compa rtmental model. A very g ood ap p roxima_ tion of the compartment dynar.!ics is g iven by an inerti a l model \·11 th time delay (S",iemiak. 1986) as follo ws

•

Since T « T . o ~ omitted.

the time delay !!'Ily be

The organs considered in our model are as follows : lungs - 1. blood - 2, spleen - 3, liver - 4, kidneys - 5 and other organs - 6. The arr:ount of the orga ns a nd their role imply following relation between time constans

The literature da ta ( Quintana. Hqezk a , 1979) shows th9.t cardiac output distributions fulfil follo lling ineqUllities

These rel·3. tions emble to a pproximate the transmitt:t'1.ce of the whole organism in the closed loop by the transfer function : G (s)

=

1 1-K (sj o

where K (s) is a simplified transmittance o of a n open system given by the following formula :

\I()(\ c ling ()f Sarcoma Cells Circulati()1l ill \Ii ct'

K

(s)

o

c~r2.cter i­

The kidneys co:np2 rtment may b ·? zed a s follo'.o/s :

k

1+ sT o

Th us G (s)

1+sT

1+sT

=

0

1

1

(4 )

l' 1+5 _1_ a (1-k)

1+sT -k o

1 s

1 - k

1-k ..:.....:2 1-k

1+sT

~1+ST

k k a

1 2 5

1 +s

o

"""'1+'S'1'1

( 9)

The other organs lI)3.y be desoribed in Lapla _ Since the input signal may be considered as w(t) S(t) its transform is W(s) .. 1

ce transform

doll)3.in by the model

and thus 1

t 10 \ Hence we obtain for the lung compartment :

R1 (5)=

1-K \s) 1 1-K (s)

1

S

~

1-k

1 s

1+sT 11

1

0

(5)

1+sT

1-k

For s implifica ti o n we a s s i g n where

=

T

_ 1

a

T

1-k

1-k

_T _1_

T

1

11

1-lc

A

s

1-K (s) 2 1-K (s)

K1(s)~~

__ 2

2

1-k 1-k_ k _ 2

A

3

1+sT _ _2_1

1-k

1-k

1-k

k a 1 2 3

A

__ 4 k k (a,+a kJ. 1-k 1 2 · 1 3)

1:1

(6 )

1+sT

0

1-k

A

where

5

-----2 1-k

,

k k a 1 2 5

A

1-1<1'; = __ J

6

kt2\

1-k

~

T21

Thus th e timp responces of t he c 0ntents 01

1-k 2

For the spleen compartment \ve obtain : 1(3 \s)

1-k

A

~

1

For the blo od compe' rt :nent we obtc'1 in R2 (s) = 1

1

= 1s 1

s

t1-K3ls )) a 3 K2\s)K1 (s)

3

k1k 2 a

1-k

1+sT

1-k

1 1-K (s) :::::

the compa rtments the origins of Laplace transforms presented in previous chap ter are g ive n as fo llo ws

0

3

(7) T R2 {t ) - A2 ( 1 +

-T

_ t

eT)

2;

Since the time constant of the liver compartment i s l a rge enough

"If>

obtair. a fol-

lowing transform of the liver contents

t

R (t)= A \1-e- T

3

3

)

1-k

R/, (S)= ~

1 __4 ( a s

1-1<

4

+a k )k k 3 3 1 2

T -T ~e T-T 4

( 1+sT4)(1+sT) (8)

where

t

t R5 \ t) .. \

\ 1-A (e -

T_

- '1' e

1

»)

76

A. Swierniak 1'1 !I/.

t

- T

)

Simulation of the time responses of the compartments contents by trial-and-error algorithm enable estimation of the paraceters which ensure that the values of the functions in the given time moments lie in the intervals obta1ned during experiments. Our est1mitions are as follows: T .. 150 min T

:I

10 m1n

system. The use 0 f transfer fune tions 0 f speciftc compartments of the complex system enables the simple method of the time responses assessment for the input signal (the cells injection) \';i thout need of sir.JUltaneous computer simUlation of great number of runs • Moreover it allows to esti~te possible simplifications by dominant dynamics retllntion only and Otnittance of the elements with Smill influence on the transient states. It stays in good agre_ ement with the role of particular organs in elimination kinetics.

They ensure t!-at all the responses of the eompart~ents cuts the sets of avu i'_ablp. data. Moreover the sizes of the parameters are in good agreement ·.,1th intuitive as-

The transmittance oodel is therefore much simpler and ras b8tter biolo g ical interpretation th3.n the model in the form of differential equations. Such a model was applied by(Kimnels and Skierski ,1983) to describe the kinetics of L-1210 leu!{emic cells circulation in the mouse. They assumed in this paper complex comp3rtl'lent' model in the form of twodimensional system in which the dynamics is divided eJl.-pl1cite into two parts: a trHnsition one and. a disintegration one. It seems however that data from the in vivo experin:ent does not allow to such "penetration" of the compartment behaviour • The meaSuI'Cment of the total compartments content enables only assessment of the global dynamics of the cells and the cell fragments and the introduction of additional degrees of freedom in the model implies the rranipulation of the model parameter.}lol"2over -..ve are not able to find which part of accumulated cells in organs results from destruction and which from

sumptions given in the previous section.

diss'?mination •

1

A1 .. 0.35 a .. 0.85 A2 .. 0.0155 T21 '" 270 min

T2

0.1 min

"3 .. 0.028 A4 .. 0.15 T41

= 90

min

T4 .. 30 Inin "5

1.2

A

0.57

A6 . 0.4

T61 = GC r:.1Ii

CONCUJ3ION The compartmental mOdel of the sarcoma ce;lls circulation in the mouse organism has been presented. Parameters of the model have been estimated basing on the to1;3.1 l"'3.dioactivity measurements .Only ::;hort t1me horizons of obserw.tion have been taken into accotmt whose duration fro:n the injection mOllent is no longer th3.n one day. Only few cells proliferate and it enables the use of the linear time invariant models. It implies in turn the possibility of the transmittance description of the

Among other worns with preceeded our study we ID9ntion the model proposed by(Senator, Lurowski Cl nd Dorosze\l5-l{i , 1970) who investigatet the in vivo kinetics of injected lymphoid cells. Their model 'NaS simple and elegant but took into account only three organs.

ACKND '.'il2DG~lENT

This worn

\\EI.s sLlpported by

Polish Aca_

demy of Sciences, grant no .I'ffi II-1.5.4.

'.[o
RSFERENCSS Brown, R. ? 1930. Compar1:ment:l l syste :n analysis : sta te 0f th ~ a rt. lEE TrBns. Biomed. Engng , B~!E- 2 7, 1. o inger:!3. ns, K . f' . 1973 • 3eha vio ur of intrBvenously injected malignant lymphoma cells • A morphologic study. ~. Cancer In s t. ,-21L 1983 • Godfrey, K.R. 1983 • COmpartmental l10dels and their Application, Academic Press, New York. Kimmel, H.,B. Kim;.')el, J.S. Sk ierski 1983. JV"a the r
//

R.oos , 2 ., K.F . ::J inge mans, I.V. van de ?a vert, 11.vdn den Berg-\
2§, 399 Sandberg, I;Ii. 1978 . On the mathematical foundations of compartmental analysis in biology, medicine and ecology, ~ TrBns. Circ.Syst. ,0'.5-25, 2:73-2:78 Senator,K., S .Zurowski,J .Doroszewski 1970. Mathematical nodel of the kinetics of transplanted lymphoid cells, Int.J. Apple Rad. Isotopes., ?J.., 253-259 , Swierniak,A., J .S.Skicrski 1986. ;·!odel1ng of letUemic cells elininat i,)n f rom isola t ed ;]louse organs • In r:: isenfe Id .; .and I'I .fatten ed. :·10delling of Biomedica l Systens,No rth- Holla nc. , pp . lf7_ 5:'.