Math1 Compur. Modelling, Vol. 14, pp. 610-613, 1990 089%7177/90 $3.00 + 0.00 Printed in Great Britain Pergamon Press plc

CLINICAL USEFULNESS OF MULTIVARIATE AUTOREGRESSIVE (AR) MODELING AS A TOOL FOR ANALYZING T-LYMPHOCYTE SUBSET FLUCTUATIONS

Takao Wada*, Udagawa** ,

Haruyasu Yamada**, Hiroshi Inoue**, Takenobu Iso**, Eiichi Shigeomi Kuroda*** 1

Department of Medicine, Keio University School of Medicine, Tokyo, Japan*; Department of Orthopedics, Japan**;

Gunma University School of Medicine, Haebashi, Department of Medicine, Okura National Hospital, Tokyo, Japan***

Abstract Using Akaike’s method of multivariate autoregressive modeling (AR), we analyzed the fluctuation of the CD4 and CD8 lymphocyte subsets in the peripheral blood and the plasma level of immunoglobulin G (IgGl in two control healthy subjects, one systemic lupus erythematosus (SLEl patient and two rheumatoid arthritis (RA) patients, and three hemodialysis (HDl patients. Akaike’s relative power contribution (ARPCl and impulse response curves were found to be useful to demostrate the features of malfunction of T lymphocytes in the IgC regulation in each pathologic state: the instability of the IgG regulating system in SLE, the delayed reponses of IgG regulation in RA and HD patients.

&y words _* Autoregressive modeling; T-lymphocyte; immunoglobulin

Introduction

Our previous papers (l-7) have shown the usefulness of Akaike’s method of autoregressive modeling for analyzing feedback-network systems in the body. The present study is an extension of the previous studies (4,5l to establish the clinical usefulness of the method for elucidating the pathophysiology of the patients suffering from rheumatic diseases and other immunologic disturbances.

Methods

We serially examined the time-related fluctuations of the subset composition of T- lymphocytes in 2 control subjects, an SLE patient, 5 patients of rheumatoid arthritis, and 3 patients of chronic renal failure undergoing hemodialysis. Venous blood was drawn from the patients once weekly for 30 to 55 weeks to examine the CD4, CD8, and the serum IgG levels. The time series data were fitted on a K-variate M-th order AR model as follows according to the Akaike’s method (8-11).

Xi(S) = 5 k m=l j=l

aiJ(m) xj (s-m) + n. (s). 1

Under the assumption that n.(s) is uncorrelated with nj(s) for i not equal to j, the power spectrum pii for the variable xi is expressed as a summation of qij(f). i.e. noise contributions from variables x.‘s. Akaike’s relative power contribution (ARP$l is expressed as the ratio qij(f)/pii(fl and can be used as an index of the degree of contribution of the fluctuation of xj,to that of xi in the feedback sys tern. The details of the computation process can be seen elsewhere (5).

Also, the AR coefficients aij(m) can be used to make the Akaike’s state space representation which works as a kind of differential equation, or its equivalent, proper to the system under study. Using such an equation, dynamic behaviors of the immunologic system can be described in time domain in the form of impulse response curves. The response of the closed loop system can conveniently be simulated by using the following equations.

zcs,=fi Zfs-ll+v

where X(s)=H Z(s)

A(l) A(2) . . . . . . . . . . ACM-1) A(H) . . . . . . . . . . 0 0 . . . . . . . . . . 0 0

0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i il

,and H=IIO . . . . . . . . . 0 1.

On using these equations, one should note that Z(s) starts from zero vector, 0, representing the system at standstill. Putting the i-th component of V(O) equal to 1.0 and keeping all the other components in V(s) (s = 0, 1, . . . . . 1 equal to 0, the output x.(s) (s = 0, l,....), which appears as the j- h i component of the matrix X(s), gives the impulse response of xj

610

Proc. 7th Int. Conf. on Mathematical and Computer Modelling 611

to xi.

Results

Fig. 1 shows the raw observational data in a control subject, T.W., of plasma IgG level and the fractional amount of CD4 and CD8 in peripheral blood. Only by looking at the data, one cannot find any definite relationship between these three clinical parameters. Accordingly, we tried to disclose such relationship by using the analysis with AR modeling.

Fig. 2 shows the ARPC maps for the two control subjects and three patients with rheumatic diseases. The x-axis in the figure represents frequency in terms of cycle/week whereas the y- axis the cumulative values of ARPC. The upper two pannels for the control subjects show that less than 10% (or less than one tenth) of the fluctuation of CD8 are contributed by the white noise for IgG and CD4. By contrast, the lower left panel for an SLE patient shows that, at least in the lower frequency range, more than 45% of the fluctuation of CD8 is contributed by the white noise for IgG and CD4. Since the one of CD4 occupied the major part of cumulative contribution, it seems that the slower fluctuation of CD8 is exclusively regulated by CD4, suggesting that a significant part of the autonomous function of CD8 is lost in this patient. The remaining lower two panels show the maps for the two RA patients. Al though there is much discrepancy between the two patients, it seems at least certain that much more part of the fluctuation of CD8 is contributed by the white noise for CD4 or IgG.

The results obtained with the impulse response curves will be shown in Figs. 3 to 7. Fig. 3 shows the impulse response curves for the response of IgG when an impulse was given to Igc. The size of impulse was 100 mg/dl. After the impulse, the plasma level once rose from the zero level to 100 mg/dl and, thereafter, it decayed in several weeks. Comparing the two control cases with the two RA patients, one wi 11 find that the decay curves for the latter two are slower, suggesting that the time constant is increased in the systems of the two patients. This probably indicates the lack of normal recovery of plasma IgG level when it becomes abnormally low or high in these patients. On the contrary, the case with SLE showed a prompt decay accompanied by a negative rebound response, suggesting that the stability of the system is significantly impaired in the patient.

Fig. 4 shows the response of CD4 when an impulse was given to IgG with a size equivalent to 2SD of white noise. It can be seen that in the two control cases CD4 responded negatively, but in a reverse manner in the two RA patients. In the case with SLE, however, the CD4 level showed a marked fluctuation, again suggesting the instability of the system.

Fig. 5 shows the response of IgG when an impulse was given to CD4 with a size equivalent to 2SD of white noise. The responses were positive in the two control cases, whereas the patients with SLE or RA failed to reveal such responses.

Fig. 1 TIE auIsE CF MNlRoL SEJECT 1

Fig. 2 Conlr0uUon of I@ and W to Fluctuation of CM

Fig. 3

EE6 _c, . . c.J . ..s. __R, _._1(2

Fig. 4 ItPuLsE BSPCNSE: Iai-co‘l

If'FULE=2sD

C ./’ -... D 1- .’ ‘. 4 .’ K? \.

! \. / ,/-?:, ..-._._._._.-.-._._~

R 0.5- !/RI ,,'

-.-.%____

% -1

1 ‘, ..‘Sl

-1.51, , , , , , , I I I , 0 12 3 4 5 6 7 8 9 18

IEEKS -cc1 . . ..Q ..s, __,Q, _._@

612 Proc. 7th Int. Conf. on Mathematical and Computer Modelling

Fig. 6 shows the response of CD4 when an impulse was given to CDB. A characteristic change was seen with regard to SLE, who showed a remarkably positive response of CD4, suggesting the overreaction of CDtl. Probably, this was the cause for the exclusive regulation of CD8 by CD4, which was seen in the ABPC maps (Fig. 2).

This abnormal response of CD4 in the SLE patient was compared to the responses in three HD patients. As can be seen in Fig. 7, the reversed responses occurred in the latter.

Discussion

In our previous studies we confirmed that the multivariate AR modeling is useful for the analysis of feedback-network systems (l-7). The principle of the feedback analysis with this approach is as follows. Let us take as an an example a simple feedback system, in which A regulates B and B regulates A (Fig. 8). In this system, the output signal from A is not only transmitted to B but also returns to A itself. The same is true as to the output signal from B. As a result, the two output signals become contaminated with the past histories of the both, making their fluctuations similar to each other and thus making it difficult to tell which regulates which.

The best way to tell “which regulates which” is to subtract the past histories from the fluctuations of A and B, leaving the white noise proper to each. By this way, it becomes possible for us to tell what part of the fluctuation of A is contributed by the white noise for B and vice versa. The ARPC shows how much of the power spectrum of A (or B) is contributed by the power, in terms of variance, of the white noise for B (or A). On the other hand, the impulse response curves are derived by putting a mathematical impulse into the white noise portion of the state equation after the system is initialized as the zero state.

The analytical results obtained with the AR modeling taught us a new way of defining the rheumatic diseases in terms of control theory. SLE seems to be explained as a defect in the stability of IgC regulation, whereas RA seems to be related to the delayed response to the abnormal fluctuation of IgC level. Comparing the CD8-CD4 response between the SLE patient and HD patients, we found a mirror image relationship between the two pathological states.

Since long an interesting relationship is known to exist between the above two pathologic states. When the patients suffering from SLE become uremic because of the SLE nephropathy, they have to be treated vi th HD therapy. Once such a therapy is initiated, the strength Of the SLE disease tend to subside in those patients. Many rheumatologists believe that this phenomenon can be explained by the immunologic disturbances caused either by uremia or the hemodialysis therapy itself. The mirror image relationship mentioned above suggest that the overreaction of CD4 lymphocytes to CD8 lymphocytes can be suppressed by such an immunologic state as possessed by the HD patients.

Fig. 5 IWULSE RESPCNSE: CM-la3 ImJL5E=23n

s! -3m- ‘._/’

d Rl

Fig. 6 lwuLER?Is?%xE:cD8-cM

IffULSS2SD 2,

x I -21, I < I , I

0 12 3 A 5 8 i i b 1;

I

4EECS __c, a . ..s. __R, _.-w

Fig. 7 lmJLsEREswNsE:cm-cDA

IWULSC23D 1,

Cl

5 _’ - ,7---

A0 \ c2 -: . . ---_

. . . .._____ :: ,____............_.__......... ........“. ‘.

. . -‘-:-.-.--____._.;:~

t <.-----’ -._,.._._.

S-I- P

f4

z-2- '. 'WI;. '.

x

-3 i123A56783;

EWS _ (., (-2 m, -- t,Q -.- m

Fig. 8

Feedback System Analysis with AR Modeling

Proc. 7th Int. Co& on Mathematical and Computer Modeiiing 613

In conclusion, the present approach clearly described the abnormal regulatory process of Igc production in various immunologic disturbances. Although the results have to be confirmed by some more additional observations, we believe that it was proven that the Akaike’s method is nicely applicable to the analysis of clinical data obtained not only in the field of immunology but also in others as well.

Acknowledgements

This study was supported partly by a grant-in- aid for New Drug Development Hesearch from the Hinistry of Health and Welfare, a grant-in-aid for scientific researches from the Ministry of Education (tt635711111, and a grant from Takahashi Foundation.

References

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