Многозначная идентификация модели роста раковой опухоли и методика многокритериального

анализа эффективности воздействия лекарства

Лотов А.В. (Вычислительный центр им. А.А.Дородницына

РАН, ВМК МГУ им. М.В.Ломоносова)

Фатеев К.Г. (ВМК МГУ им. М.В.Ломоносова)

ПЛАН ВЫСТУПЛЕНИЯ

1. Введение2. Идентификация параметров модели на основе визуализации в случае неустойчивого решения задачи идентификации 3. Аппроксимация множества критериальных точек, достижимых при всех допустимых параметрах; поддержка многокритериального выбора варианта 4. Многозначная идентификация модели роста раковой опухоли 5. Методика многокритериального анализа эффективности воздействия лекарства в случае неоднозначных параметров

б)

а)

в)

Характерные формы графика функции ошибок

2. Идентификации параметров на основе визуализации в случае неустойчивого решения задачи

идентификации

Let the dynamics of the system under study to be described by

Where is the state vector, is the control vector, all at the time-moment k, are the vector of unknown

parameters, is the time-step. The initial state is assumed to be given.

nk Rx )(

rk Ru )(

0)0( xx

1,...,0),,,( )()()()1( Nkuxfxx kkkk

pR

Computing the error function .

Let a control function be given.

Let be the set of observations, where are observable values at the time moment .

Let be the trajectory of the system for the given control and a vector  .

Let be a given relation between trajectories and the observable values.

The error function is a function of differences between and .

)(

1,...,0,ˆ )( Nku k

}),,{( )( KkvkV k )(kv

k1,...,0),(ˆ )( Nkx k

)()(kv

,)),(ˆ()(ˆ )()( Kkxv kk

)(ˆ )( kv

Computing the graph of the error function and its visualization

1. The value of the error function is computed for a large number (M) of random vectors .

2. The set of M points is approximated by a relatively small number of p+1-dimensional boxes.

3. The system of boxes is visualized by its two-dimensional slices.

),...,( 1 M

)(

MiR pii ,...,1,))(,( 1

Approximating of the graph of the error function by boxes)(

Identifying a region in parameter space

• An expert points out such a region in the parameter space (identification set) , that the solution of the parameter identification has the form .

• In such an approach, the model parameters can be identified by using a synthesis of observations and non-formal experience of the expert.

• The further study examines the case when the region contains more than one point.

3. Аппроксимация множеств критериальных точек , достижимых при всех

допустимых параметрах; поддержка многокритериального

выбора варианта решения

The dynamics of the system under study is described by

Here . We assume that and does not change in time.

For given a control function and a given vector  , the equation allows constructing the trajectory of the system .

The trajectory tube for the entire set and for a given control can be approximated by a population of trajectories generated for M random vectors .

0)0( xx

1,...,0),,,( )()()()1( Nkuxfxx kkkk

M

lu

The multi-criteria finite choice problem Let us consider the problem of selecting one of L

of feasible control functions , where .

Suppose that the decision problem is described by m criteria, denoted by z and associated with the trajectories by a given mapping .

Then, the set of criterion uncertainty for a feasible control is approximated by the set of criterion points for .

By approximating this set by a system of boxes, its visualization is provided. Then, the most preferable control is selected by comparing .

),,( xuFz

Luu ,...,1

)1,...,0,( )( Nkuu kll

lZlu

)),(,( ll xuFz M

lZ

4. Многозначная идентификация модели роста раковой опухоли

Simeoni M., Magni P., Cammia C. Predictive Pharmacokinetic-Pharmacodynamic

Modelling of Tumor Growth Kinetics in Xenograft Models after Administration of Anticancer Agents //

Cancer Research, 2004.

To identify parameters of the model, experiments with nude (young) mouse were performed: the tumor is implanted and hailed by using several anticancer agents.

The scheme of the pharmacokinetic model

The pharmacokinetic model

The pharmacokinetic model for the time-moments between the injections is

where is the concentration of the anticancer agent in the central part of the body (lever, lungs, heart, etc.), and is the concentration of the anticancer agent in the peripheral part of the body (marrow, brain, etc.).

)()(

)()()(

2211122

221110121

tCktCkdt

dC

tCktCkkdt

dC

1C

2C

The pharmacokinetic model-2

There is a discontinuity of at the moments of injection

where DOSE is the quantity of injected agents and V is the volume of the central part of the body. The variable is continuous.

VDOSEtCtC kk )()( 11

1C

2C

The scheme of the pharmacodynamic model

The pharmacodynamic model

The identification problem

One has to identify the parameters

in the case without injections, i.e.

The result of a standard identification procedure is given by the red line.

010 ,, w

Standard identification

Computing the approximation

In general, the error function was computed for about 500 000 combinations of the parameters. The set of these points in parameter space was approximated by 3761 boxes.

010 ,, w

Dependence of error function on the parameters and 0 1

Feasible values of all three parameters for 0.15<psi<0.5

Feasible values of all three parameters for 0.15<psi<0.23

The identification set for and 10

Visual identification of 0w

Visual identification of 0

Visual identification of 1

The values of the parameters

The obtained values of the parameters are:

1) By using standard method we obtain

2) By using the visualization-based method we obtain

085.0,334.0,146.0 010 w

]11.0,34.0.0[],44.0,23.0[],18.0,13.0[ 010 w

5. Методика многокритериального анализа эффективности воздействия

лекарства в случае неоднозначных параметров

Strategies being studied when point-wise parameter estimates are used

The following strategies have been selected from the

list of strategies in the process of multi-objective screening of 140 strategies of drug application by

using the Pareto frontier visualization

Instability of strategies

Criteria

Here

y1 is W(10),

y2 is W(20),

y3 is the total dose of drug,

y4 and y5 are c1 and c2.

Methods for selecting from a large number of strategies with uncertain outcomes

• Lotov A.V. Visualization-based Selection-aimed Data Mining with Fuzzy Data. International Journal of Information Technology & Decision Making. Vol. 5, No 4 (December 2006). P. 611-621.

• Lotov A.V., Kholmov A.V. Reasonable goals method in the multi-criteria choice problem with uncertain information, Doklady Mathematics, 2009, vol. 80, no. 3, 918-920.

• Lotov A.V., Kholmov A.V. Reasonable goals method in the multi-criteria choice problem with stochastic information, Artificial Intelligence and Decision Making, 2010, № 3, с. 79-88 (in Russian, to be translated in Scientific and Technical Information Processing).

Summary of the talkWe propose a graphic method for constructing the sets

of uncertainty for model parameters. This knowledge is used in the framework of our methods for approximating the trajectory tubes by their slices (the reachable sets or the sets of uncertainty). The slices inform on the possible deviations from the non-perturbed trajectory.

Approximating the set in criterion space accessible for all possible parameters can be carried out as well. Thus, the technique also offers supporting the decision making, including the multi-objective decision problems with models, which parameters are known not precisely.

Our Web site

• http://www.ccas.ru/mmes/mmeda/

Дополнение. Покрытие многомерных невыпуклых множеств

параллелотопами

Remark: Covering a multi-dimensional set

Let be a non-convex set. Let be a finite set. Then .

Let be the Tchebychev distance among points , i.e. . Then, -neigh-hood of the point is the set

.

i.e. a box. If , then provides a (full) covering of the set A.

nRA AT AxTxTAh :),(max),(

),( yxniyxyx ii ,...,1,max),( yx,

x

iin yxRyxU :)(

),( TAh TxxUTU )(),(

Approximating a multi-dimensional set

If , then the set covers the set A only partially. The set T is called the covering base. Let H be a sample of M points of A. Let .

Then, is the completeness function of the covering provided by the base T. The Deep Hole of the set H for the covering base T is the set

),( TAh ),( TU

MmT )()(

),(),(:),( THhTxHxTHDH

)),(:()( TUxHxcardm

Application of the Deep Holes method for approximating a multi-dimensional set

Let describe the j-th iteration of the DH method. On the previous iterations, the covering base must

be constructed. 1. Let generate a sample H of M points of A.2. Compute and display the function ;3. If the expert is satisfied by the completeness for the

covering base and some value of , then stop else let ,

where ; 4. Start new iteration.

1jT

jjj xTT 1

),( 1 jj THDHx

)(jT

1jT

Detailed description of the method is provided in

Каменев Г.К. Визуальная идентификация параметров моделей в условиях неоднозначности решения, Математическое моделирование, 2010, т.22, № 9, с. 116-128.