A Discrete-Time Model for Acute Leukemic Process, Its Parameter Estimation and Clinical Applications

Copniglll Budape~l ,

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El\DOCR Il\E-METABO LI C SYSTEMS - I

I FAC 9th Triennial World Congre!'l'> Hungan , 19H4

A DISCRETE· TIME MODEL FOR ACUTE LEUKEMIC PROCESS, ITS P ARAMETER ESTIMATION AND CLINICAL APPLICATIONS S. Kitamura*, A. Takekawa**, H . Mori*** and N. Yamaguchi t 'Departmmt oJ I nstrumentation EngineeTi"", Faculty oJ Enginuring, Kobe Vniversit)" Kobe, japan "Graduate School oJ Science and Technology, Kobe Vniversit,V, and Departmf1lt oJ Mathematics, Kobe Technical College, Kobe, japan ***Department oJ PhysiololO', School oJ Medicine, Kobe Vniversit)" Kobe, japan tCmtral Clinical LaboratOlY, School oJ Medicine, Kobe Vnivenit)" Kobe, japan

Abstract. A discrete-time model for describing the dynamic relation between the administered drugs and the number of leukemic and normal white blood cells in the peripheral blood is presented . The unknown parameters in the model are estimated from the clinical laboratory test data by using the least square method . The model agrees well with the clinical data and enables us to predict the future conditions of the patient and to evaluate the clinical parameters like the apparent cell increasing rate in the living body . Keyword. Biomedical, Leukemia , Medical information processing, Modelling, Parameter estimation

MODEL FOR DYNAMIC PROCESS OF LEUKEMIC AND NORMAL WHITE BLOOD CELLS INTRODUCTION Discrete-time model. The variables measured of the peripheral blood from leukemic patients were the number of platelets(PLTS) , red blood cells (RBC), hemoglobins(Hb) , reticulocytes(Ret) , normal white blood cells (normal WBC), and leukemic cells. The drugs administered to the patients were P1ednisolone, Cytocine arabinoside( Ara- C), 6- Mercaptopurine(riboside )(6-MP(R )) , Behenoyl Ara - C(BHAC), Aclacinomycin, and Daunomicyn, three or four of which were conbined for chemotherapy . Figure 1 shows, as an example, the course of change in the condition of an acute myelocytic leukemia patient . Normally, clinical state can be divided into three phases, i.e . , the remission after the first treatment (phase I and phase I' respectively denote complete remission and partial remission), the remission after relaps (phases II and II', likewise), and the final state (phase Ill). In the following analysis, administered drugs are considered as input variables to the living body and two output variables are adopted: the first one is the numbe~ of leukemic cells which represents abnormal hematogenesis and the second one is that of normal white blood cells which represents normal hematogenesis. For a more detailed classification of blood cells in relation to the various types of leukemia, see Table 1 As in references (Mori (1982), Takekawa (1983)) , taking into account the existing experimental results on the cell-killing action of chemo- therapeutic agents (Shimoyama, 1978) and under appropriate assumptions, we can construct the following differential equation describing the relation between the input (D . 1 and D .) of one dose at t=to and the output 2 (N(t) ,Jt :;;;. t ) ' (The notation used is given in o Table 2.)

Acute leukemia is a disease in which immature blood cells proliferate in the marrow and peripheral blood. The therapeutic strategy for acute leukemia is normally to (1) kill totally the leukemic cells in the marrow, blood , and organs by combined chemotherapy (total cell kill) and bring the subject to a remission state, and (2) let normal white blood cells recover, suppressing the reproliferation of leukemic cells. Although this strategy has remarkably prolonged the surviving period of patients , the skill still depends much on doctors' experiences . It seems that no system theory- oriented work on such problems has yet been reported. We have already reported an approach for describing the behavior of platelets and white blood cells of subjects under treatment by using an auto-regression model It succeeded in simulating the behavior, with time, of platelets and also in predicting their trend for the next 10 days or so. However, the result for leukemic cells was not satisfactory. In this paper, a discrete - time model is proposed for describing the dynamic relation between the administered drug and the number of leukemic and normal white blood cells. The unknown parameters in the model are estimated from time - series laboratory data by the least square method. The results show that the model agrees well wi th the clinical data and that it can predict the cl inical state for 5 to 14 days. Clinical parameters such as the doubling rate constant of cells, the drug sensitivity, and the duration of drug effects can also be evaluated.

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S. Kitamura et aL .

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I-ABLE

1.

1

Course change in condition of acute myelocytic leukemia patient

Cl asification of Normal White Bl ood and Leukemic Cel l s Leukemic

Normal white blood

Type acute myelocytic leukemia (AML) acute promyelocytic leukemia (APL) smoldering acute leukemia (SAL) acute lymphocytic leukemia (ALL) acute monocytic leukemia (AMoL) leukemic reticulosarcomatosis (LRS)

TABLE

10 10 10 10 10 June J~IY Aug. Sep. Oct. Nov. +--- ]I - - t +--][----t

2

gr anulocyte monocyte

I

granulocyte granulocyte monocyte

k- th sampling time (day)

N

number of leukemic or normal white blood cells dosage of drugs (mg) doubling rate of cells (l/day) drug sensitivity (l/(day. mg) drug satu r ation cons t ant(l/mg) drug extinction rate (l/day)

a Y. B:

a

e

lymphoblast prolymphocyte monoblast promonocyte ,monocyte sarcoma cell

Notation

tk

D

myeloblast promyelocyte

subscript 1 cell cycle non-specific type dr ug 2 cell cycle specific type drug i index number for drugs of type 1 index number for drugs of type 2 m total number of ccs drugs n total number of ccn drugs q index number for sampling point K frequency of sampling

Di sc r e t e -time Mode l fo r Acute Le u kemi c P r ocess

n

H (t-t O»

= a - i :l Y Dli exp (- 9 i m

-

1.: 8 / j=l

a j (1-exp(- ajD2jexp( -B 2j (t-t ») O (1)

The second term on the right-hand side of Eq . (1) represents the effect of the drugs of cell cycle non-specific type (e . g. Daunomycin, Aclacinomycin) and the third term represents the effect of the drugs of cell cycle specific type (e . g . Ara- C, BH AC, 6- MP(R), Predonisolone) . Next , Eq . (l) is approximated in a discrete form with a time step of 1 (day). For the case of intermittent dosing at K temporal points, we o b tain the following difference equation .

- 9 (t -t H k q

»-

m

k

1.: 8 / a (l-exp(- a.1.: j=l j j J q=O (2)

Note t h a t the concentration of each drug in t h e ma r row and blood is replaced by its dosage . When two or more drugs are used in combination, their effects should be the represented by the dominant one . For example , for the case of Fig . l , we set n=l and m=O for phases I ' and 11' , since Aclacinomycin dominates , and n=O and m=l for phase Ill , since 6- MPR dominates .

Paramete r estimation. I t is assumed that all the parameters in Eq . (2) ( a , Cl , 8, y , and 9) are constant in each phase : I (1 ' ) , 11(11 ' ) , or Ill . Note that this is a drastic assumption for tractable parameter such parameters would estimation , because actually vary with the the rapeutic course of time . In this case , the parameter identifiability can be checked under ideal circumstances . For the estimation of unknown parameters, the method of minimization of "o ne - step" predicti on error is used with the squared criterion

N'( tk

) is the "one- step" predicted value by substituting the laboratory data in Eq . (2) and M is the number of data used for the parameter estimation. For practical computation, the program package SALS (Nakagawa (1982)) was used . For the applicati on of this method , the data should be taken at a constant sampl ing rate . This is , however, very rare in clinical situation. Therefore , the data were pre processed by interpolation so as to have a constant period ( normally 1 day) before the parameter estimation. In Table 3 , the estimated values for the where

given

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subject in Fig . l are summarized for each clinical phase. Also , Figs. 2 to 4 show the measured data and the simulated results by Eq . (2) for each clinical phase ( M=26 for I ', 25 for Il', and 34 for Ill) . As shown in these figures , the model of Eq . (2) fits the measured data very well . By applying the method here , we can get almost the same results as for ten other subjects which are categorized in Table 1 .

CLINICAL APPLICATIONS Prediction of future behavior . The model in Section 2 is used for the prediction of future behavior of leukemic and normal white blood cells . The results are shown in the region "Prediction" of Figs.2 to 4 . The prediction here means to compute successively the "one- stepahead" state (N(t )) from Eq.(2) by using the kl known input quantities (D . and D . ) and the 1 2 previous state (N(t )). It ~~ be saIa~rom these k figures that the model succeeded in predicting the cell behavior of the next 5 to 7 days . This technique can be useo to predict the reappearance time of leukemic cells in the peripheral blood. When leukemic cells are killed and decreased by drug administration, it becomes impossible to get a correct count of them from the image under the microscope, a normal laboratory test . Results f or a smoldering leukemic patient(SAL) are shown in Fig . 5. The unknown parameters in Eq.(2) were estimated by using the data from the start to the 32nd day, and the states after the 33rd day were predicted by the model . Note that we lacked the number of leukemic cells from the 32nd day until the 45th day; however , the theoretical curve of Eq . (2) (with " dosing " ) predicted the reappearance of leukemic cells in the blood two weeks before . ( "No dosing" stands for the case where no dosing is assumed . Note also that the hematogenesis for normal cells seems to have been broken on around the 40th day . ) Evaluation of drug effect. From the parameters in Table 3, we can evaluate the drugs ' effect on a living body. Define the apparent cell increasing rate for the cell cycle non- specific drug by

and for the cell cycle specific drug by

R

ccs

(t ) - Ft

"21

( t-t ») 0

(5 )

Figure 6 shows R (t) (t=O to 5 days) for phases I ' and II ' wi th t~gnparameters in Table 3 and D11 =20(mg) of Aclacinomycin , and R cs(t) for phase III wIth D =80(mg) of 6-MPR . T~e apparent cell 21 IncreasIng rate decreases by dosing from point p( corresponding to the value o f a) to point S , and gradually recovers. We may say that (i) The drug sensitivity, which is represented by the distance between P and S in Fig. 6, for leukemic cel ls becomes small with the transition of clinical phase , while that for normal cells becomes large . This suggests that actual therapy becomes difficult as the phase proceeds . (ii) The reco very is faster for leukemic cells than normal cells. This suggests also the difficulty to kill leukemic cells without destroying normal cells and implies the partial remission of the subject .

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REFERENCES CONCLUSION

Mori,H ., S . Kitamura,T . Nagaoka,A . Takekawa , and N. Yamaguchi (1982) . Simulation and prediction of clinical states of 1eukemia by the auto-regression model, Mathematics and Computers in Simulation XXIV , North- Holland Pub. pp 564- 568 Takekawa , A. , S . Kitamura,H .Mori and N. Yamaguchi (1983) Modelling and evaluation of the clinical states of acute leukemia by a mathematical model , Preprint MEDINFO 83, Amsterdam. Shimoyama, M. (1978), The cell -ki lling action of chemo - therapeutic agent(in Japanese), Medicine Mook, No.5 Kanehara Pub. Nakagawa, T. and Y. Koyanagi (1982) Analysis of experimental data by the least square method (in Japanese), University of Tokyo press.

The discrete-time model proposed in this paper succeeded in simulating the clinical course of change in the condition of leukemic patients . It provides a means to evaluate and predict the clinical parameters which will be useful for practical therapy. There is possibility to improve the model by taking cell cycle kinetics into account. Acknowledgment The authors would like to thank Dr . Katsuyasu Saigo of the School of Medicine, Kobe University for his helpful discussions .

3

TABLE

Type

Estimated

Phase

Cell

r'

LC NC LC NC LC NC

I I"

AML

III

Parameters for the Case in Fig . 1

a

0.13 0.24 0.36 0.36 0 . 21 0.31

LC : Leukemic cell

y (xlO

1. 90 0.52 1. 64 0.84

NC

-2

) 9,

3/ a

a (xlO

0.71 0.50

0.45 2.12

2

)

1. 65 0.23 0 . 50 0.26 1.10 0.15

Normal cell

LC (

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50

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Discrete- time Model fo r Acute Leuk emic Pr ocess

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