Application of multivariate autoregressive modelling for analysis of immunologic networks in man

Malhl Comput. Mod&q. Printed in Great Britain

I I. pp. 192-796,

Vol.

APPLICATION

OF

IMMUNOLOGIC

Department Tokyo;

Akaikew,

Internal

the Institute

Orthopedics,

School

AUTOREGRESSIVE

Haruyasu

Medicine,

of Medicine,

Gunma

We tried to apply Akaike's for

coefficients

were

contribution.

suggested slower

that,

a

the analysis with

of the movements

including

to a normal

control

thereby

subject.

to simulate

conditions.

The

approach

Autoregressive

indicating

permeated

well

immunology,

In

(Jilek

(Jilek &

many

fact,

using

immunologic

Marchuk, models

1978;

Bruni, but

1984), such

difficult satisfactory

as

and

system:

Bell,

Doria,

Wada

does

(Jilek

Perelson,

&

of

in

study

pathologic method

et al. (AR)

as formulated various

Sudo et al., al.,

1987:

Present

models

& Koch,

the solving

sometimes

of

already

of

T8

also

under

biofeedback

body (Wada, Akaike,

1985).

have

a variety

coefficients

a standardized

assessing

of most

approaches

immunologic

solutions

the

analysis

yet

field

Prikrylova,

1985;

& Prikrylova.

Pimbley,

general

investigators

studies

not

to be one of

perspective

immunology

published

the

seems

and

progressive modern

into

it

of

for

in the body.

modelling;

has

(SLE),

when compared

of the system

Recently,

modelling

was

almost

the malfunction

of the system

may provide

it

are governed

autoregressive

by

relative

concept,

in such a patient

response

data

Akaike's

erythematodes

autoregressive mathematical

Autoregressive

chronologic

lupus

of

Japan.

autoregressive

man.

obtained

representation

an impulse

networks

in

with this latter

T-cells,

University*

and Department

of multivariate

systemic

INTRODUCTION

Although

.OF

Udagawa-

Maebashi,

used to compute

The obtained space

present

immunologic

Keywords.

ANALYSIS

Keio

Tokyo;

o f T8+ lymphocytes

suppressor

us to make a state

analyzing

and were

patient

Medicine,

networks

by those of T4+ lymphocytes,

lymphocytes,

enabled

method

from clinically

method

By

in

components

exclusively

and

obtained

of

University*,

immunologic

of a least squares

power

FOR

and Eiichi

Mathematicsw,

modelling

analyzing

Yamada-.

School

of Statistical

Abstract.

means

MODELLING

IN MAN

Hirotugu

of

0895.7177,:88 $3.00 + 0.00 Pergamon Press plc

MULTIVARIATE

NETWORKS

Takao Wada",

1988

&

study,

we

demonstrated to

by Akaike

is useful

& Kato, Wada. &

1968;

Akaike.

Ozaki.

in

in

1986: Wada,

Akaike.

Aoyagi,

to

In

apply

Ishiguro,

the

Kojima,

1986).

(Akaike,

of immunologic

an

spectral

regulations

Wada,

tried

of AR modelling

that

approach

feedback

method

for analysis

1979;

1986; Matsuo

system

et the

Akaike's

1967;

Akaike.

et al., 1985)

networks

in man.

complex

networks

is

not

to

lead

METHODS

1. Statistical

Prikrylova,

Akaike.

1985).

1985)

792

Procedures

1968; Akaike,

Ozaki,

(Akaike, Ishiguro.

1967; et al.,

Proc. 6th Int. Conf. on Mathematical

a. Akaike's

The

Relative

basic

model

Power Contribution

equation

is given

for a

k

(ARPC)

multivariate

AR

Xi(S)'

aij(m)xj(s-m)

+ ni(s).

(i=l,Z ,...I k).........(I),

that

(m) is a weighting

matrix

of A(f),

this equation

that

all

correlation

coefficients

rij

of

system, the

between

variable

qij

of

the

ni (s) and

determine (m)3

the coefficients,

and

.

Akaike's 1973),

Akaike

algorithm Information

thus making

utilized

b. Drawing

the

repeated

ni(s) and nj(s).

computed

under

equations

from the equation

the

assumption

correlation

between zero

ni

that

to

spectrum

pij (f) is expressed

its decomposed

(i+j).

fractions

by the following

with

unnecessary.

noise covariance

(I). there

and hence

Response

Curves

{aij (m)i resemble

functions,

can

which

as bij (s)'s or { bij(m)]

when a noise impulse

will below,

the output

is put into xj.

of

be in xi

However,

if

ia. .(m)] are to represent the true impulse '.I response functions in an open loop system, they should

have been determined

the coefficient output

aij (m).

xi when a noise

itself,

becomes bij

impulse

zero.

estimate

j

(m);

following

equations.

in such a way that

which determines

Akaike

from

the

is given to was

iaij

able

(m))

by

xi to the

However, is

(s) and nj (s).

equated

of

in principle,

coefficients

response

the sense that they determine

(Akaike,

calculations

{aij(m)f obtained,

sij between be

Two Kinds of Impulse

to iaij

value

(AIC)

at

Pii (f)

expressed

a Levinson-type least

Criteria

the use of Yule Walker's Using

order

aij (m)'s or

and the

xi

(f)

by

the order M for the above equation

simultaneously, recursive

In

method.

variable

by the following.

nj

aij (m) can be estimated

least squares

(ARPC).

rij(f)=-------

impulse

a

of

of contribution

xj to another

Autoregressive

coefficients

contribution

the degree

of f. and is given

of

(s) is zero.

using

(not a inverse

the

on

values

(f)v

a

power

expresses

and

certified

can be used for analysis

feedback

condition

The

scalar

of the

and that sij is a variance

relative

frequency

coefficient Akaike

noise for xi.

multivariate

a

an element

of

m=l j=l

aij

is a complex

representing

Akaike's

k

I: c

ni (s) is white

Note that (A(f)-l)ij matrix)

n .. J

by

M

where

793

Modeling

the

no

bij(l) = aij(l)

sij is power

as a summation

m-l

of

bij(m) = aij(m) +

qij (f) and is defined

Z

aii(k)bij(m-k)

k=l

(m=2,3 ,....M)

equation. M

k 'ii

k

bij(m) = Z aij(k)bij(m-k) k=l

(f)=Zqij(f)'L

j=l

(m=M+l,M+Z I..... )

j=l The

where

response

impulse

qijCf)’

of

the open

equivalent

to

be

plotting

the

response

to the impulse equivalent

the

standard

lbij(m)i

loop system

1.0 can

deviation

.

If

(*SD),

one

2SD

to

drawn needs to 2 x

an by the

times

{ bij(m)j

are plotted. M A(f)=

- (I

- z m=l

A(m)exp(root(-1)2nfm)),

The

response

simulated equations.

of the closed

with

the

use

loop system of

the

can

be

following

794

Proc. 6th Int. Conf. on Mathematical

---____---(IIa)

qJ x Z(s-l)+V

Z(s)=

Modriling

lymphocytes. cyto-flow

_----__---(~I~),

X(s)= H x Z(s)

Their

meter

(Hachioji. their

Tokyo,

Japan)

percents

lymphocytes.

where

levels were measured

in Special

in

It

Reference and were

the

.A(&1)

A(M) ’

I

0

........

0

0

0

I

........

0

0

T-cells

0

........

0

1

Fig.

compares

0

T8+

of

those of helper respectively.

.

the

.

.

frequency

0

0

I

0*

the relation

in the control

.

I

T-cells,

(cycle/week)

the SLE patient

........

of

RESULTS

frequency

0

as

number

is known that the levels

and supressor

a

expressed

total

T4+ and T8+ fairly well reflect

A(1) A(2) .......

with

Laboratory

maximal

the

fluctuation

of

subject

(left panel)

(right panel).

peak of

frequency,

of ARPC to

of the

of

ARPC

0.2

almost

was contributed

and in

In the control, was

found

cycle/week.

At

at

20% of the fluctuation

by that of serum

a

this of T8+

IgG level,

but

not by that of T4+, thus 80% of the fluctuation of

U(s) 0 0

v=

H=[

and

On using

system

01.

the

frequency

of 0 cycle/week,

T8+

from zero matrix,

0, simulating

the

at a standstill.

contribution

level

2.

The

One

immunoloqic

of the authors

from

Another patient

a

erythematodes was being

prednisolone

the of

study,

IgG

venous once

subpopulations

and

IO:30

very

cannot

signal

Some

examples

shown

in Fig.

a.m.

of

the systems

equated

to

to

initial

level of each variable

housewife). and was

mg/day

of

IgG on

clinical

was not changed 1

week.

The

lymphocyte

included

T4+

and

T8+

stay

in

the

control

impulses, at

a

indicates

an

open

shows impulse system (left

the

whereas loop

the

was T8+

in

dotted

line

of

and

or

T8+.

either

concerned,

panel),

in

the

the

to

particular

a

SLE

100~

closed indicates

in

that

at the top

into

which

As far as the open

given.

in were

was assumed

The figure

system.

4.4%

systems

note that the solid

response

response

was is

the

inpulse

of the and

the

the

impulse

standstill,

Also, one should

variable.

in

2.6%

are

only the

T8+

equal to the mean value of that

system,

Maebashi.

around

be

line

of Orthopedics,

of

any

curves

2SD of the fluctuation

assumed

the 45 weeks

at

namely

to the

lupus

University,

The

be reflected

The size of

to the level.

(i.e.,

Gunma

low

interests,

Before giving

IO

completely

we present

level

as an outpatient

the

T4+.

response

Again,

special

of

responses given

of impulse 2.

SLE).

with

of

by

to it.

used

(a 31 year old

the

that the increased

were

blood was taken at a

was of

and his data were

the dose of prednisolone

her

was

in the SLE patient

findings

taken

levels

systemic

at 100%

that the movement

set of data analyzed

at the Department

Throughout

was seen

movement

indicating

than

patient,

the

healthy

blood was

at around

with

treated

of Medicine.

Japan.

p.m.

(SLE)

administered

being

School

for

as a

immunoglobulin

for 42 weeks

as controls.

served

His venous

serum

subpopulations

once weekly

and

(T.W.)

subject.

checking

lymphocyte

She

utilized

study

control for

data

the

and almost

implying

T8+ as an effective

present

peak

low frequencies

by

frequencies

In

of T8+ was contributed

of T4+,

with

regulated note that

maximal

fluctuation

of

by any other

of itself.

however,

fluctuation

one should

unexplained

fluctuation

the

equations,

these

Z(s) starts

IO....

1

0:

T8+ remained

the

the

patient

an loop

control (right

Proc. 6th Int.

panel),

the response

impulse

since the signal to

coming

it eventually

showed

system,

with

level showed subjects.

up-and-down

In

the

of T8+,

movements

loop system

compared

with

indicating

tend to be

those of the closed

that the former

of

the impulse,

represents

secondary

or network

movements.

and that the

effects

obtained

of

involving

noteworthy

the

open

Especially,

loop

descending

system

movements

waves

latter

the

feedback

implying

that the direct

and T4+ is highly to

the closed

gradually suggesting

abnormal

the

no

50

between

the response of

of some

for the abnormal

In

As

compensatory

responses

the present that

of T4+ to

assumption

correlation

between

there

each white

able to decompose

is

noise,

the degree

xi to

is and

the

of

other

study,

the analysis

with

the slower components of

T8+

are

level

by those

Although

it is not entirely

1).

whether

the movements

express

those of suppressor

T4+

the

almost level clear

of T8+ and T4+ faithfully

respectively,

cells,

of

ARPC

of

T cells and

the observation

helper is at

SLE %

no

Akaike

the power spectrum,

Control

Thus

variables

determine

variable

00

that

a projected

other.

governed

DISCUSSION

the

principle, independent

(Fig.

T

and

bidirectional,

Control

Under

in

exclusively

T8+.

was

of one

fluctuations

waned

clinical

the 2

rather than

effect

x. not x. to xi. J' J

suggested

T8+

in

on the

between

can quantitatively

variable

weeks,

25

a

variable

and

and tries to detect

relationship

unidirectional

weeks,

in the patient.

period

the existence

mechanisms

practically

relation

loop system,

over

in

pii

much

or observations

ARPC,

fields.

of one variable

for

showed

both

variable

of T4+

for more than

cause

component

the response

and

how

by another

variables

direct

one

was the response

the SLE patient.

to

(Akaike.

qij(f)

the shape of 2 seemingly

compares

contribution The most

between

sequentially

experimental

when

loop system,

represents

possible

relationships

both

delayed

xj

which describes

xi is contributed

delineating

IgG

in

ARPC,

ascribable

variable

x. at various frequencies. From medical J standpoint, ARPC appears to serve a purpose of

the

respectively.

the movement

gives

variable

as in

It can be seen that the responses

open

effects

however,

and 17 weeks

and in the patient,

(f)

qii (f).

of another

Thus the ratio between

1967).

on

fluctuations

for 10 weeks

consonance

the

loop

noise

white

return

19s

into its fraction

(f).

bar,

from T8+ cannot

prolonged

aftereffects control

by a hatched

to cause any aftereffects

In the closed

itself. T8+

of T8+ was the same as the

itself as indicated

Modding

Con/: on Mathrmatical

pii

SLE

~~A__& TO’(self)

T8’(self)

0,

02

0.3

0.4

0.5

4--7ni-

IgG

T4*

0.1

02

0.3

0.4

05

Fig. 2

was

Fig.

1

The map of relative

explaining

the behavior

and in the patient.

power contribution

of T8+ in the

control

Impulse response

curves

given to the T8+ level.

indicate while

-b

the response

loop system.

weeks

when an impulse

The dotted

for the open

the solid lines indicate

the closed

20

lines

loop system,

the response

for

Proc. 6th Inl. Conf. on Mathematical

796

least compatible including

with

supressor

the notion T cells

part of their

autonomy.

The

response

that T8+ cells

have lost a

Bruni.

Modelling

C..

Systems

large

Notes

Doria,

G.

Theory

G. Koch, et al. (1979).

in Immunology.

in Biomathematics.

In

Lecture

Springer-Verlag.

32. impulse

certain

important

visualize system

of

and

of

closed

increase of

a

applied

but

certain

and

the

variable

to

loop.

such effects

in an open

(Fig.

suggests difference

between

loop

system

that

other

cytokines

the responses

factors,

the

of

T8+. the

loop system

compensate

for such

ACKNOWLEDGEMENTS

This

study

was

Cooperative

carried

Research

and was partly

supported

New Drug Development

out

under

Program

and Welfare,

Takahashi

Foundation,

from the Ministry

Japan, Tokyo,

ISM

for

by a grant-in-aid

Research

of Health

the

(87-ISM-CRP-20)

and a grant

from

Japan.

REFERENCES

H. (1967).In

Akaike.

/Series

Spectral

B. Harris,

Analysis

editor,

of Time

John Wiley,

New

York, 81-107. Inst. Statist.

Math.,

425-439.

20. Akaike,

Ann.

H. (1968).

Akaike,

H. (1973).

Information eds.).

F.

Proc.

2nd

Theory(Petrov. Akademiai

Int.

Symp.

on

B.N. and Csaki,

Kiado,

Budapest,

267-

281. Akaike,

H.. T. Ozaki,

M. Ishiguro

et a1.(1985).

Monographs:

A Publication

Computer

Science

of

Institute

tics

The

No.23,

Bell, G.I., A.S. (1978). Dekker.

of Statistical

TIMSACPerelson,

Theroretical

Part

2,

Mathema1-168.

G.H. Pimbley.

et al.

Immunoloqv.

Marcel

Notes

and Epidemi-

65. 8-14.

Optimization

Division,

N. and T. Wada

Nephroloqy, Wada,T.,

H.

Models

Software,

distributed

in Pub-

by Springer

Wada, T.,

H.

T., al.

T.

Proceedinqs

Symposium

of

of Pediatric

and E.

Kato (1986).

&

28, 263-268.

Akaike,

Jap. J. Nephrol. Wada.

(1986).

September,Tokyo

Akaike

J. Nephrol.

256.

an overresponse.

Immunoloqy

Mathematical

-the International

implies

including

possibly

or interleukins,

In Hoffman,

Verlag. Matsuo,

the open

(1985).

eds, Lecture

Springer-Verlag.,

Immunoloqy.

of

finding

T..

G.I. (1984).

lication

long

for a

this

of T4+ to

and the closed

that

evanescence

Although

the overresponse

loop

the elevation

loop system

2).

oloqy, Marchuk,

by

open

and Hraba.

in Biomathematics.

to

others

It is noteworthy

the

us

M. and D. Prikrylova

G.W.

the

indirect

to two types of system;

T4+ level without

period

of

also enable

effects

Jilek,

They not only

of the T8+ level caused

the

proposed

characteristics

interest, the direct

effects

also

information.

the dynamic

separate

being

curves

Aoyagi,

(1987).J.

S. Sudo et al.

(1986).

28, 1237-1243. F. Kojima,

Clin. Nutr.

H. Yamada

Biochem.

et

1:247-