Solving systems prognosticating disease course

MATHEMATICAL

Solving

BIOSCIENCES

Systems

243

Prognosticating

S. YA. ZASLAVSKY AND Ukrainian Academy of Science Kiev, USSR

K, A. IVANOV-MUROMSKY

Communicated by V. Glushkov

ABSTRACT A class of learning systems called solving systems (S-systems), which can be considered models of the thinking elements of a man in prognosticating his activity, is investigated. Ratios are deduced to estimate S-system efficiency with assigned laws for changing the fixed parameters. The results of using S-systems for prognosticating the operation course and the postoperative period of subjects of surgery are described with Fallot’s tetrad as the basis of the scheme employed.

INTRODUCTION

If complex problems are dealt with when the precise solution cannot be obtained owing to lack of data, or if a method is unknown, or if the method is laborious and shortage of time makes it practically unacceptable, the expert is compelled to estimate approximately the values sought. If a very small quantity of data is available, the expert often proceeds by intuition without knowing with certainty by which way he arrives at the result. To solve such problems the methods of heuristic programming [l , 21 are utilized. Ordinarily, heuristic programs do not guarantee an optimal solution, and they sometimes do not guarantee that a solution will be attained at all. In most practical problems, however, they yield good results [3]. One such problem is an important one in practical medicine; its solution has become possible only recently owing to development of computers and medical data retrieval systems-the problem of prognosticating the course of the pathological process. This process is understood to be a state of self-controlling system-organism unstable conditions

Copyright @ 1970 by American Elsevier Publishing Company, Inc.

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S. YA. ZASLAVSKY AND K. A. IVANOV-MUROMSKY

developed as the result of unusual or excessive external and internal effects

VI. Various authors have suggested techniques for solving problems of prognosticating the result of surgery in removal of the gallbladder 651, lung respection [6], and myocardiac infarction [7,8]. However, all these methods, being of a particular nature, lend themselves to discrete cases only. All the works cited are based on an attempt to simulate the particular disease, employing some understanding of its mechanism. In this case the prognosis is obtained as a result of studying the informationexchange process among separate organs, characterized by values of some parameters. For example, in [7] the analytical description of conduction of the heart musc!e pulse suggested by Wiener and Rosenblueth [9] is used. The approach described has some grave shortcomings. The main one is that mechanisms of many diseases are not yet clear, and in most cases quantitative characteristics of some organs are not determinable. Moreover, since the disease course is dependent on norm, we may speak about types of models corresponding to certain types of constitution. This paper discusses a general approach to solving prognosticating problems (irrespective of given class of diseases). A simplified procedure of reasoning by the doctor is laid down as the basis for the approach. The class of disease in question is considered to be a set of states of the ill organism and connections among them. As distinct from the methods based on examination of models of a disease, the states of the subject are described by means of a set of their external indications. Since the equilibrium of the organism is disturbed and restored in passing through a number of states, we can speak about a certain order of altering normal working conditions of individual (fixed for each disease) organs being influenced by pathological irritants. For instance, the weakening of blood flow and decrease in oxygen supply to some sections of the heart muscle results in metabolism disturbance in a cell, which in turn conditions a change in excitable properties of the cell: threshold, duration of refraction periods, shapes of action voltage, etc. [lo]. Prognostication of the disease course is understood to be an indication of a certain sequence of changes in the organism. For the doctor, the initial factor is a ‘equence of essential states observed in the subject. The problem of the doctor reduces to continuation of the initial sequence of states and application of his knowledge and experience. Sometimes the initial sequence can contain states which have not been detected in a given subject, but from the doctor’s standpoint they are essential for vital activity of the ill organism. Then the determination of a series of sequential states, which incorporates all initial states, is of interest. Mathematical Biosciences 8 (1970), 243-263

DISEASE COURSE

It is postulated

245

that the doctor solves such problems in two stages:

i . Preliminary selection of data from memory. 2. Compilation of prognosis on the basis of selected data and their various characteristics, part of which can be subjective in nature. Such an approach is more general than described above because, being not directly connected with the disease mechanisms and based only on the reasoning of the doctor (partly intuitive), it practically renders itself applicable to any disease. In considering various classes of diseases, and accordingly the reasonings of different doctors (experts), the postulated stages of solution are left unchanged. Only the knowledge and experience of the doctor change, i.e., the content of his memory. The task of this investigation consists in formalizing the technique used by the doctor in prognosticating development of diseases. This task acquires a particular interest in connection with creation of medical information retrieval systems. Abstracting from the nature of mechanisms bringing about a solution of the prognosticating problem in the doctor’s brain, we arrive at construction of formal solving systems. Here the preceding approach to solution of the prognosticating problem needs some specification connected with the absence of any guarantee concerning the problem solution. The following errors are possible in establishing the prognosis: 1. The doctor’s idea about an initial sequence of states may be wrong: 2. Preliminary selection of data may be wrong. 3. The data selected may result in a wrong prognosis. We exclude errors of the first kind from our considerations, considering the initial sequence to be right if there exists in the memory even one row of sequential states containing the initial sequence. However, in this case the wrong initial sequences may include some which are right for the more experienced doctor. In other words, the memory has insufficient data to construct the prognosis sought. Therefore, in the course of accumulation of new cases of developing diseases of the class in question, memory enlargement is provided for. Errors of the second an3 third kinds necessitate construction of learning solving systems. Learning rules are introduced; they come from the prognosis correctness criterion, which requires that the prognosis, a series of sequential statrs of the subject, contain an initial sequence of states. Of course, according to this criterion, the memory may contain several correct prognoses for one such sequence, and they may have not a single one which could be affirmed by later clinical observations. TO eliminate this shortcoming, even to some degree, in practical use of the Mathematical Biosciences

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solving svstems, the doctor’s interference is admitted as a teacher. learniig Lies in this case do not change much.

The

1, PRl NCIPAL DEFINITIONS

solving system (S-system) is a device answering questions coming from outside. Answers change in time. The interaction of the S-system with an outside medium can be considered according to the scheme sugges$ed by Glushkov [ 111. Two modes of operation are stated by this scheme: the examination mode determined by the sequence QUESTIONANSWER, and the learning mode determined by the sequence QUESTIONpi

ANSWER-ESTIMATION.

vital part of any S-system is its memory. We shall discuss an S-system the memory of which is built on the basis of an information model of the man’s associative memory described in [12]. The set ‘8 of information elements stored in the memory with a binary ratio specified in it is associated with the oriented connected graph G Liuced by associative connections among various elements with one-to-one correspondence. There corresponds to each information element tii (a, E 2l), the node i of the graph G, and any two nodes i,j (i,j E G) are linked with an arc then, and only then, if the corresponding elements u‘, zI (ai, a$E 21)are in a relation determined in 91. It is implied that the set 81 and, consequently, the graph G are finite. Moreover, assume G to be a graph with no loops. The last assumption, adopted for simplicity, is nevertheless quite natural for many practical problems where a graph node can be interpreted as a state of the process in question. For instance, considering a problem of simultaneous operation of several computers [13], where the process state corresponds to a number of operating computers, we should take into account only jumps between several states, since the jump from a certain state into the same state would correspond to simultaneous failure of computers and repairs. This is also true for the problem of prognosticating the course of the pathological process where a node of the graph G is interpreted as a state of an organism [14]. The jump from some node to itself would contradict the understanding of the nature of processes occurring in a living organism, according to which states of the organism dialer from one another at different moments in time [15]. The operation of the S-system consists of: EXCITATION of some set of graph nodes by means of an external irritant (question), transfer of EXCITATION pairwise to its incident nodes, and choosing from the set of’ EXCITEDnodes a sequence satisfying certain conditions (construction of an answer). The

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Associated with each node x of the graph G is a pair of real nunnegative numbers : Ssand vZ, the weight and threshold, respectively. Questions are represented by k and dimension vectors v = (vi, . e . , vJ, 1 where vc = , and k = the number of graph G nodes. (0 An answer given by the S-system is also a k-dimension vector w = ( Wl,--9 w,) the components of which take on either of two values, 0 or 1. Questions defined this way can be interpreted as follows. Since the dimension of the vector v equals the quantity of nodes of the graph G, there may be one-to-one association of each node i of the graph with the components vi of the vector v (with fixed numeration of the nodes). If an event in which vi = 1 is interpreted as excitation of the node i, then, as distinct from [121, the presence of several simultaneously excited nodes is admitted here, the excitation being maintained until an answer is constructed. Each arc ij of the graph G is associated with the real nonnegative number pij (0 < pij < 1) which is the price of the association a,-aj in the set a.* After corresponding normalization, the quantity pij can be interpreter as the probability of a jump from the node i to the node j. Fixing a set of the graph G nodes having not one arc, we form a zero level. The set of the graph G nodes linked to the nodes belonging to the zero level form the 1st level. The 2nd, 3rd, . . . , rth levels are formed analogously. If the node of the ith level (i = 1, . . . , r) is linked to the nodes belonging to the i 1st level randomly and independently, the jump distribution law being specified in advance, we obtain a class of graphs to be considered a set having a probability field conditioning the probability of choice of each partic - representative of the class. This set of graphs containing the probability criterion is called the random graph. Any S-system incorporates two operators: F and Q. Operator F serves as a preliminary choice in G of information referring to the question put and associates the latter with the graph T’. (TV c G). With the aid of Q in graph TV,the answer w of the S-system is built. A system consisting of F, Q, and G is labelled B. When F, Q, and G are constant, the system B, interacting with the external medium, * The price of the association a-b depends on a number of preceding joint excitations LJand b, on the length of time elapsed from the instant of the last excitation of u and 6, and on the excitation frequency of elements connected with a and b by association (context) (see [ 12, 161). Mathematical Biosciences

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always gives the same answer to the fixed question. To obtain the Ssystem we must point out its parameters which change in learning. DeJinitiorz

1.1. Th.e solving system (S-system) is said to be a system B complemented by a set of finite parameters involved in G with an explanation of how these parameters change during the learning (reinforcement rule). For each node x of graph G a numerical function vl is determined depending on the weight of nodes which are the beginning of arcs, of which the end is x and the question is v. The operator F can be realized by a sequence of functions E = k(x)>,
& e,(x)

= +

0, = 1,

otherwise,

where YI = {y/x -* y,y f~ G Y; > q,I.* Assume the functions e,(x) to be determined for all HZ\< i. Then, forxE Yi, %,I@) = Yi+19 where Y2+1 = {y/x n+ y,y~G 1~; > qj). The value of the operator F-graph TV, the nodes of which pertain to the set ii 'j

l
and the arcs connecting these nodes are the same as in the case of G. If T, does not contain even a single path to which all nodes x belong such that v, = 1, the S-system is said not to be giving an answer to question v. Operator Q is realized by the arithmetic function. It was pointed out in the Definition 1.1 that a change in answers is connected with a change in some fixed parameters. For a random graph G such parameters are chosen to be s, and rZ (X = 1, . . . s k). The case in which the s, are changed in learning is discussed in [17], which treats the quantitative rra:as changeable parameters. Let A be an S-system having memory G as a random graph. We denote a random learning sequence (1s.) by L = (v,, . . . , v+~). A question put to system A is labeled v, and a corresponding answer, w. * x -+ y means that the nodes x, y are the beginning and the end of the arc. Mathematical Biosciences 8 (1970), 243-263

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Definition 1.2. If v, = 0, y is said to belong to answer w (symbols: v, = 1vJ. If 0, = 1, v, c y, then, and only then, when UI = 1. If 0, = M’,for all y (y = 1, . . . , k), the question is said to correspond to the answer (symbols: v c w). DeJhitiolz 1.3. Suppose system A is first given by 1.~.L in the learning mode. Then, in the examination mode a certain question v is put and the answer w is determined. With v C-Zw, the result of the v experiment is positive; otherwise its result is negative. The necessary condition that the result of the v experiment be positive follows directly from the property of the G. It is necessary that nodes of graph G corresponding to components of the vector equaling unity belong to different levels.

2.

OPERATION

MECHANISM

OF S-SYSTEMS

The S-systems taught by means of random 1.~., i.e., a class of I.s., is stated with a probability field determined over it for choosing any sequence. The 1.~. L is determined by its length, i.e., by a number of questions put in it, and by stating the probability distribution of questions appearing in L at thejth place. It is supposed that each question appears independently from all others with probability ltVand this probability is not dependent on j. The law for changing the quantities 7~~(i = 1, . . . , k) in learning is stated as follows. Let x and y be the least and the greatest component number of the vector v, respectively, such that v, = v, = 1. Then we use Z&(X,y) for a set of the graph G nodes belonging to even a singIe path which connects the nodes corresponding to x and y and contains all nodes j such that 2‘.= 1. * The notation Z’,(X,y) -+ z is understood to be a set of the graph G nodes which are ends of arcs the beginnings of which belong to ZL(x, y). Assume that the 1.~.L’ is obtained from L by addition of the vector v: L’ = Lv. Then vi(L) is a threshold of the node i [i E G) after the 1.~.L has been given. Then ri(L) is enlarged by unity if i $ &(x, y); the ni(L) is not changed otherwise. That is, 7Ti(L’)=

ni(L)9

if

+4

otherwise.

+ 1,

i E lyU(x,y),

(1)

A more genera1 reinforcement rule, according to which the graph node * If only one component vzof the vector v equaling unity exists, then we may take in graph G a node of the last level, i.e., that which has the largest number, for y. Mathematical

Biosciences

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threshold cannot only increase but also decrease, is stated by the relation

Ti(L’) =

Ti(L) + l9

if

I&(x, y) A* i,

TJ_ld) - l)

if

i E Ib(X, y), 7Ti(L) # 0,

TiiL),

unless otherwise specified.

(

(2)

Find the function I,U:for each u (11E G) by the relation

WY‘ =

ZI

%dJz -

zEz(g(x#?/~

zezuiz p4z,s,. I

Here c0zzl =

1, (0,

with probability pij, with probability

1 - pii,

where pzU = P(z -* u). Suppose the graph G nodes of each level (beginning from the 2nd level) are ends of arcs the beginnings of which are chosen ‘Jn the preceding level randomly and independently. Assume that from er Th node x of the it b level a jump is possible to not more than two nod&s y and _ of the (i + 1)th level. A ssociate the pair of nodes (x, z) with a random quantity k& depending on the manner of connecting the not s, the I.s. L (which by supposition is determined by the probability distribution h, and length N), as well as on a sign of difference ly: - n,. According to [ 171, h’,,(L) = %4(L)%(L),

where

(3)

a:(L) is a random quantity determined by the relation a:(L) =

1

with probability g;(L),

0

with probability

1 - g’,(L),

g:(L) is the probability of z E 7’.‘.after learning the S-system with the help of the 1.~. L; s,(L) is the value of the node weight z (z E G) after learning the Ssystem with the help of the 1.~.L.

Let AB = b,, - b,,. The operator Q is stated by the equality Q(x) = AB. This means that, for AB > 0, the node y (1~~= 1) belongs to the S-system answer and, for AB < 0, the node z (~1~= 1) belongs to the answer. The case when a jump is possible from node x to r nodes il, . . . , ir is reduced to the previous one by the sequential dichoto*ny method, i.e., to a sequential comparison of bzil with bgiz, . . . , bst, with bsin, bziz, with bxi39 and bzi,,. 9 bin-l l

l

l

Mathema ticai Biosciences

DISEASE COURSE

251

Let the question be put to the fixed S-system A after learning by means of the 1.~.L the length of which equals iV. Then trees will be selected from each node i for which vi = 1 through use of operator F in graph G. Now the node i, which is incident to i or to the node belonging to the tree, belongs to the tree if yi > vi. Thus a number of trees selected in G equals a number of the vector v components equaling unity. A totality of all selected trees forms the graph T.. Constructing the answer w is reduced to finding in T, the path connecting the nodes x and y corresponding to the least and greatest number of the v components equaling unity. An answer of the S-system to the question v is constructed through the operator Q in the following way. Let the node m (m E T.) belong to the answer; and let a jump to the nodes zl, . . . , z, (zi E TV; i = 1, . . . ,n) be possible from m. Then to the answer belongs the node zi for which s = max1<3 rri (~1 is not dependent on vi) holds true. 3.

APPLICATIONS

OF S-SYSTEMS FOR PROGNOSTICATING

DISEASE COURSE

The course of diseases depends on many factors of which the main ones are [HI: 1. Sex, age, the disease, state of brain, nourishment. 2. Virulence of the microorganism causing the disease. 3, Correctness of diagnostics and treatment; this covers all external effects on the organism during the course of the disease. The accuracy of the prognosis is conditioned by the level on which the disease is described. The fullest description entails registration of changes 4Iathematical Biosciences

in the molecules and cells covered by the pathologica process takin in the paganism, The infu~~tion of the next tevet is thophysio~ogica~, i.e., changes occurring in individual organs are studie stratio~ of the dynamics of the j~a~ly~a less accurate des~riptiu~ of its external mani ment of the disease implies representing consequences of disturbances in discrete organs and cells, i.e., the c~i~i~a~ crjption of the disease which is written in its rn~~~ the pathophysio~og~~a~ d~s~ripti~~ of a ~nfo~ation co can be obtained from special literature and partly from the disea consider a fixed S-system A. To use it in prog~usti~t~ng a some @lassit is necessary first to I the memos of the system with descriptions of possible courses of the sease. Here the choice of a description level the disease be~~rnes essential. the 41 level owing to tack of s ease in molecules and cells caused by this di diseases would

practical solving of the disease prog to utilize the e~i~ie~~disease descrip i.e., that which is put down in histories of diseases, which is the only source of information. The description is made in the futm of the scheme of all aunt variants of the course of iseases. In this case each varia series of sequential states charac rized by a set of certain indic Since each succeeding state accrues from the prec ng one, then, for ind~~at~~~scha~a~teri~~ng both states, i4r a quantitative estimation which is a statistical characteristic of a jump from one state to the other. ~~~seq~ent~y the rn~~o A ~~~tai~s a description of a certain class of diseases, The graph node is associated with a state of the subject; two nodes are by means of an arc i jump is feasible from the state corr=spending to oue node to the sta orres~ond~ng to the otfper node. Sametimes the literature and histories of disease have i t in~o~atj~~ for describing an essential state of the di~ased Then either a set of s~ptoms ~urrespo~ding to the given state or other indicatior,s externally characterizing the given state are used, For instance, W SC for r nting a state called “trae ~otom~” is prepared) tr the tar of o~~ratio~ and not state of the subject, Yet there is no misunderstanding here, w mean the subject s. rte in which he was subjected to tracheotomy, •~~~~~~~U~~C~~ ~~~~~~en~~~ 8 (1 ml),

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Such a scheme, called Fallot s tetrad, was drawn up* at the Chest Surgery Clinic of F. CZ. Yanovsky Research Tuberculosis a-<1 Chest Surgery Institute in Kiev on the basis of data borrowed fr< speci 1 literature on chest surgery [ 181 axl results of analyses of 100 h orie: I disease of subjects operated on ‘because of one form of conge. 31 heart disease. Obviously, the following sequence of work io convenient. A “skeleton” scheme based on information borrowed frc tn the literature is drawn up first. Then the scheme is made more exact and is suppiemented by Data taken from the history of disease. The drawing up of the scheme takes into account the pathological shifts whic’ra occurred in the organism during disease development, as well as the skill of the surgeon. The latter is most important both at thf: time of operation and during the postoperative period. In this case the prognosis incorporates aftereffects w! *ch may* develop in the subject organism during a given time period. To calculate statistical characteristics and f r: ;Isitions frcm state to state it is necessary to use the truest ixiformavitiu. The rlegree 4 I truth of the information ut;lized will finally have its efkc: JI tte accuracy of the established progno: s. Thereforr tht; histories G ’ diseases the diagnoses of which had bee!- verified (e , during the operation, autopsy :tc.j are selected from .he 5xhivcs .I ‘ni=clinic for tkir: case. Consider some s +aij : L te i (i = 1, . . , , k), contained in the memory of S-system ‘. S dp?oTe the number of disease histories containing i equals IVi* Let P,-~be th2 aumber of cases when a transition from state i to state j takes place. Obb ’ ausly , 2

ri, =

A$.

1.1-j

Then the quantity

af ;1;ansition from i to j. And

may be used as the statistical charxteristic Es

j. i-* f

ii =

1;

then, if N ---*00, according to the rule uf larp wmbers, qi, *pija The quantities qij can be calculated b )th su
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in any case the information stored in histories of disease must be ordered. This is attained by making a standard history of disease; such a history is now being developed at the Institute of Cybernetics AS Ukr. SSR. When we took on solving the disease problem, we had no standard forms for describing the disease development. As z result we had to process manually histories of disease borrowed from the chest surgery The work done clinic. Quantities qii were calculated simultaneously. enabled US to make some recommendations for ordering the information, and they were utilized in constructing and selecting the way of coding the standard history of disease. After the graph (memory of S-system A) has been constructed and statistical characteristic of transitions qii have been calculated, the next stage in preparing initial data for prognosticating is the determination of weights si and thresholds ri for each node of graph G. The thresholds vi are first assumed to be equal to zero for all i (i E G). Weight si (i E G) of graph G nodes has been determined as follows. Competent experts of the clinic were given a list of all states corresponding to nodes of graph G and were asked to estimate each state from the standp0in.t of its effect on the further development L. a disease. The data of each expert were evaluated by a decimal scale: Each expert has assigned an integer from 0 to 10 for a state. Thus in the program realization we suppose that the upper limit of values Si, C,, equals 10. Weight Si of node i equals an average magnitude of values assigned to a corresponding state by all experts. Finally we want to point out that, apart from materials of the chest surgery clinic, we processed data of Kiev Research Institute on Neurosurgery on swelling diseases in any sections of brain. (For the preliminary results of progcoses and a brief description of the scheme refer to [22].) The program realizing S-systems by means of the M-20 computer consists of four units and 1~ so organized that changes introduced into individual units do not require changes in the whole program. The main characteristics of the program are as follows. Graph G can contain not more than 300 nodes if not more than 5 arcs issue from each node. The number of levels in graph G must not exceed 10. Each of these requirements may be applied as reinforcement laws. With such restrictions, all initial and intermediate data are contained in the immediate access memory of the computer. The program consists of 800 instructions. The time of processing one of question is from 1.O to 2.5 sec. A series of experiments with random S-systems has been effected by means of the compiled program. The experiments have been aimed at verifying the correctness of some theoretical inferences and failsafety of S-systems for practical use in prognosticating diseases. Mathematical Biosciences 8 ( 1970)) 243-263

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The general scheme of the experiments is as follows. First, the Ssystem is given an examination sequence consisting of 20 questions, Then the system is taught by means of a random:- teaching sequence with length N = 5. After that the S-system is taught again by a random teaching sequence of length N, and so on. Such a system is a certain mcdiiication of the learning system suggested independently by Starinetz [23] and Overton [24]. The totality of the examination and teaching sequence will be referred to as the teaching cycle. The S-system is said to be given an i-question if i, its components, equals unity. If a l-question is put, then any answer of the S-system will be correct in accordance with the requirement that the question belong to the answer. The examination sequence contained five l-questions. After the initial message to the examination sequence S-system, it was found that only three questions of five had given answers. For instance, for the question in which unity equals a componeilt corresponding to the state-aortic insufficiency-the answer obtained was: aortic insuficiency, hemorrhage, brain hypoxyia, death. After 1 teaching cycle, by means of a learning sequence of length 5 and reinforcement law 2, the answers to all five l-questions were obtained. In using reinforcement law 1 the answers to all l-questions were obtained after 3 teaching cycles. The following results ‘Nere obtained for 2-questions. After the first message of the examination sequence of fifteen 2-questions, answers were obtained for twelve, of which four were correct. When using reinforcement law 1, twelve questions were answered after 11 cycles; nine answers were correct. In utilizing reinforcement law 2, fourteen answers were obtained after 10 cycles had been correct. Consider an example. For the 2-question in which components equaling unity correspond to states: 1. Considerable anastomosis between lung and bronchial arteries. 2. Tracheotomy-with a given value of weight Si and thresholds 7~ of graph G and a given probability distribution pgj graph TVobtained as a result of preliminary selection. After the first message of the 2-question, v, an answer was not obtained. The correct answer was obtained after 7 cycles: Considerable anastomosis between lung and bronchial arteries:

weakness, tracheotomy,

heart

hemorrhage from trachea, death. Mathematical Biosciences 8 (1970), 243-263

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The second series of experiments involved forty i-questions for i > 3. An increase in summary lenb,h IV of the teaching sequence was required here for obtaining correct answers. Consider several examples. For the 3=question, components equaling unity complied with states: 1. Lung aftereffects. 2. Lung squeezing by exudate. 3. Hemorrhage from trachea. An answer was obtained after only 4 teaching cycles. The answer reads: Lung afrereffecls, hemorrhage from lungs, death. The correct answer was obtained after 5 cycles: Lung aftereffects, cessation of drainage functioning, lung squeezing bY exudate, tracheotomy, hemorrhage from trachea, death. In the second question the components equaling unity corresponded with states: l., blood 2. 3.

Imperfect performance of a device for artificial (DiACB). Disturbance of capillary permeability. Hemorrhage into front mediastinum.

circulation

of

The correct answer reads: Imperfect performance of DACB, acidosis, reduction in albumen fractions of blood, disturbance of capillary permeability, hemorrhage

into front mediastinum. When utilizing reinforcement law 2, this answer was obtained after 18 cycles and, when using law 1, the correct answer was not obtained. Some questions put to the S-system under examination conditions were unanswered despite rather great length of the summary teaching sequence. The analysis of causes has shown that, as a rule, in such cases the S-system memory contains insufficient information to construct an answer. In other words, the S-system memory has no path containing a question. This is clue mainly to the small number of histories of disease handled. Summing up the discussion, we see that: On the whole, under examination conditions, the S-system received sixty varied questions which were not involved in any teaching sequence. After teaching by means of the teaching sequence, the summary length (1970)) 243-263

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of which on the average equals 50, the S-system gave answers to fifty seven questions, fifty-two or 86 ‘A, were correct. Most important is the verification of the prognosis by clinical observations. Of fifty-two correct answers, fifty were verified by observations. The remaining two answers had the fatal issue (this took place in reality) and, since the setting of some intermediate states had been associated with additional examinations, the question of verifying these answers has been left open. APPENDIX ANALYSIS

OF ONE CLASS

OF S-SYSTEM

Dejnition A. 1. The teaching efficiency G(v, L) of a random Ssystem A by means of teaching sequence L according to question v is called the probability that the result of v experiment on system A will be positive. To find an answer it is necessary to ti a class of teaching sequences, a reinforcement law, and a random S-system. Efficiency E(v, L) is to be determined for S-system A with reinforcement law 1. It should be noted that, since in this case Theorem 1 of [171 may be used, the problem is reduced to determining the distribution of random value b& Without loss of generality we can assume si = 1 for all i (i = 1, . . . , k). Then, according to (3), we have bii(L) =

1,

if node i belongs to T, and is incident to a,

otherwise. If g:(N) is a probability that i E T, after teaching by means of teaching sequence L of length N, then 0,

MKiW) =

Paiduv,

DbXi = PLldw - Pa&3

Probability g:(N) can be written in the form g;(N) = P{q(N)

P{q(N)

< y’i> =

= YG, y,‘i > k)

k=O

P(y; > k}P(q(N) k=r,

= k}.

(4)

Here vi(N) is the value of node threshold i after system A is provided with a teaching sequence L of length N. Mathematical

Biosciences

S ( 1970), 243-253

S. YA.

258

ZASLAVSKY

AND

K. A. IVANOV-MUROMSKY

The distribution of random quantity ~1 is determined by the probabilities of jumps in graph G al?d by questions. Suppose a fixed question v is put to system A. Let node i (i E G) belong to the tth level, and let the (t - 1)th level make nodes z,, . . . , z,, z n-j-19

l

l

l

9 z n+m

with

21,

l

l

l

I) 2, E I;;(x,

y),

zn+l,

9.

l

3 z,p+m $ I&(X, y)*

Then + = + q*_i)* 9%= co,,i -I- ’ + m&&i - @&i Owing to the independence of jumps, all quantities COAX are distributed in the same way. Therefore pti = pie The generating function of random quantity # can be written l

l

iFi

=

[PiS

+

(1 - pi)ln

[J3iSw1 +

l

(1 - pi)]“,

(9

and the probability P{qf > k} is determined from the relation P{vi > k} = 2; Fi(s), (6) where Z; designates summation of coefficients attached to powers of s which exceed k. To calculate g:(N) we are to find values P{r,(N, = k). LEMMAAA. PiTi(

The relation k}=

N-_(k-2)

ml=1

m2=ml+l

2

. (1 -

holds true ProoJ

iv-UC-1)

2

l

**

P{i E l;l(x,

2

rl[

mk=mk-1$-l

16 j
y)})

l

n

l

(1 -

P{i E cg& Y)} P{ i E lzk(x, y)})

(7)

For k = 0, from (1) we have P{77i(N) = 0} =

P(i E 1x(x, y)}. l< j
(8)

For k = 1 we obtain expression (7) if in the product (8) each of the factors P(i E /2(x, y)} is consecutively replaced by 1 - P{i E Zz(x, y)} for all nz (I < nz \< N). Add the products transformed in the above way P{Ti(N) = 1> = 5

nr=l

P{ i E 12(x, y)}(l - P{ i E 12(x, y)}).

1-G j_(iv, if m

We use induction over k to establish the truth of relation (7) for each k > 1. Suppose that, for all k \< t, we have P{q(N)

= t} =

*v- ( t-1

1

P(i E I&, ml=1

mt=mt-I+1

y)}

I< jdx.

{i E l>‘(x, y)})

l

l

* (1 -- P{i E l;;lt(x, y)}).

(9)

DISEASE

259

COURSE

From (1) it follows that t + 1 factors of the form 1 - P{i E /2(x, y)) should be in the expression for P{vi(N) = t + l}. Therefore, in each of the summands from (9), we replace each of the factors P{i E 22(x, y)} so thatj > m, + ion 1 - P{i E 2,$(x, u>>. After replacement, each summand in (7) will contain a product of the form

n:

l< jGN. 3-m .....mr.mt+1

P{i

E

ZZ(x,y)}(l - P{i

121(x,y)})’

E

x (1 - P{ i E l?t(X, y)})(

1 -

l

l

P{ i E zpl(X,

y)}).

In this case, obv:otlsly, each succeeding summand in (9) will give new summands the quantity of which will be the same as that of the preceding summand decreased by unity since, in passing to the next summand, j is increased by unity with the top limit unchanged. On comnletion of the described procedure, we obtain ml changes from 1 to N-

t;

1~ changes from ml + 1 to N .

‘

.

.

.

.

.

.

.

l

*

m, changes from m,_l + 1 to

Assuming m,,. 1 =m,+

1);

(t .

N -

.

.

.

1.

1,wehave

N-t

P{q(iv)=t+l)=

p** ml=1

l

hi’

2

mt=mt-1+1

mtt1=?nt+l

(1 - P{ i E Zzl(x, y)})

P{ i E 1X(x, y)} l
0 (1 -- P{ i E Zzt+l(x, y)}), l

which proves the lemma. Probabilities of the form P{i E Z(x, y)} are represented in the form W E 0, y)> = P{Z(x, i) # #}P{Z(i, y) # +}. This, in turn, enables us to use the method described in [17] for their calculation. Substituting (7) and (6) in (4), we determine g:(N). Probability distribution pij is stated. Therefore

and the efficiency concerning question v on teaching S-system A by means of teaching sequence L, the length of which equals N, according to [17] Mathematical Biosciences

S. YA. Z,ASLAVSKY AND K. A. IVANOV-MUROMSKY

260 is

SO

determined from the relation

we have proved THEOREMA.. 1.

If, in teaching S-system

A with the aid of teaching

sequence L, the length of which equals N, reinforcement rule 1 is used, then the epciency of system A concerning question v is determined by (10).

Determine efficiency 6(v, L) for S-systems with reinforcement law 2. Suppose si = 1 for all i (1’E G) and that it does not change in the process of teaching. Then B’,(N) = WjiaI(N) and g:(N) = P{yI > ri(N)}

= 5 P(Ti(N)

= n>P($

> II>.

n=O

For calculation of probabilities P{vi(N) = H)-we can write a relation similar to (7), but it results in laborious expressions which made us resort to describing the process of changing rri(N) through use of a Markovian finite chain (since 0 < ni < N). Assume 7ne72+1 = P{vT~(N) = n + l/‘ri(N - 1) = II), 7n.n-1 7 n.n

=

P(ri(N)

= n -

l/Ti(N - 1) = n>,

=

P(ni(N)

= n/ni(N -

1) = II).

Then the transition matrix for node i has the form

Mi =

700

701

0

0

...

710

711

712

0

...

0

721

722

T22

. . .

(11)

-

. . . . . . . . . . . . . . . . . . . . . . . . . . ..*.......

From (2) we have an expression for

T,,,+1,

Tn,+-l,

7 n,n+l

=

P(gyx,

7 n.n-1

=

P(i E lFf’l(x, y)} = P{ ly+l(x, i) $ 4)

7

=

r2.n

y) “% i} = PniP{l~+l(X~ 0) # +},

1 .- Tn.n+l -

Mathematical Biosciences

T,* la

Tnen-l*

l

P{ lz”(i,

y)

#

$},

DISEASE COURSE

261

LEMMA A.2. If, in teaching a random S-system A with the he@ of teaching sequence L, the length of which equals N, reinforcement rule 2 is used, the probability of i E T, for a fixed question v and i E G is determined from the relation

where M(n) is characteristic polynomial of matrix (11) ; is the algebraic complement of element of determinant M(1) found on the intersection of the (t + I)st row and the (n + I)th coluw*~; 3‘1, ’ - 9 A, arp the characteristic numbers of matrix (I 1); mj is the multiplicity of lj; t = vi(O) is the initial value of the threshold.

Mt.,(A)

l

Proof

Let V&I)) = 0. Then we obtain P(ni(N) = n> if we perform

the operation of exponentiation of matrix Mi to power N. Unknown probability element ~~~~is placed on the intersection of the first row and (n + 1)st column ’ Then we have’ g&V)

=

$.ktP{$ n=O

> n}.

Using Perron’s formula [24], (N)

To,?l =

w r

dAmi_’ Wb II 9

1 dmv-l lNMo,Jn)(A - ,,Jmv I)! ( m,

we obtain relation (12) for vi(O) =

A=Av

t.

Summing up the above, we state THEOREMA.2. If, in teaching S-system A by means of teaching sequence L, the length of which equals N, reinforcemerit rule 2 is used, then the eficiency of system A concerning question v is determined by relation (lo), where g:(N) is found from (12).

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S. YA. ZASLAVSKY AND K. A. IVANOV-MUROMSKY

262

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Mathematical Biosciences 8 (197Oj, 243-263