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Application of semilinear canonical correlation to the measurement of opioid drug effect

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Abstract

To examine the relationship between the electroencephalograph (EEG) and plasma opioid concentration, one would like to collapse the high-dimensional EEG signal into a univariate quantity. Such a simplification of the EEG is desirable because a univariate quantity can be modeled using standard nonlinear regression techniques, and because most of the information in the EEG is redundant or unrelated to drug concentration. In previous studies of the EEG response to opioids, the manner in which a univariate component was extracted from the EEG was ad hoc.In this paper, this extraction was performed optimally using a new statistical technique, semilinear canonical correlation. Data from 15 patients who received an intravenous infusion of the semisynthetic opioid alfentanil were analyzed. The components of the EEG that were nearly maximally correlated with plasma drug concentration were found, based on a standard pharmacokinetic-pharmacodynamic model. Two new EEG components were produced from the powers in the frequency spectrum of the EEG: a weighted sum of the logarithms of the powers, and a weighted sum of the powers expressed as percentages of the total power. These components both had a median R2 of 0.84, compared to median R2sranging from 0.37 to 0.83 for five commonly used ad hocEEG components. The new components also had less variability in R2 between subjects.

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Abbreviations

t:

Time

C E(t :

Effect-site concentration at timet

k e0 :

Rate constant for drug transfer from the plasma to the effect site

E :

EEG effect

E 0 :

Baseline effect, when no drug is present in the effect site

E max :

Maximum possible effect caused by the drug

IC 50 :

Effect-site concentration resulting in an effect equal to (E 0+E max/2

α :

Shape parameter

ε :

Additive error

q :

Number of dependent variables

Y 1,Y 2,...,Y q :

Dependent variables

γ 1,γ 2, ...,γ 1 :

Coefficients of the dependent variables

p :

Number of independent variables

X 1,X 2,...,X p :

Independent variables

f :

Nonlinear regression function

r :

Number of parameters off

Β 1,Β 2,...,Β r :

Parameters off

ε :

Additive random error

n :

Number of observations

y tk :

tth observation ofY k ,t=1, 2,...,n,k= 1,2,..., q

y t (γ):

tth observation of the univariate response obtained as a linear combination of the multivariate response,γ 1 y t1 +γ 2 y t2 +...+γ q y tq

x tj :

tth observation ofX j,t=1, 2,...,n,j=1,2,...,p

x t :

The vector (x t1,x t2, ...,x tp)

R 2 :

Proportion of variability in they t (γ) explained by thef(x t ;Β)

γ :

MaximumR 2 estimate ofγ

Β :

MaximumR 2 estimate ofΒ

γ (j) :

MaximumR 2 estimate ofγ, for thejth subject

γ :

Average of the (possibly renormalized)γ (j)

Y :

Dependent variable (i.e.,q=1)

p :

Number of independent variables, and the number of coefficients of the independent variables (i.e.,r=p)

X 1,X 2,...,X p :

Independent variables

Β=(Β 1,Β 2,...,Β p ):

Coefficients of the independent variables

ε :

Additive random error

n :

Number of observations

y t :

tth observation ofY,t= 1, 2,...,n

x tj :

tth observation ofX j,t= 1, 2,...,n,j=1, 2,...,p

b=(b 1,b 2,...,b p ):

Estimates (not necessarily optimal) of the coefficients

ŷ t(b):

Responses predicted using the estimatesb, b 1 x t1+b 2 x t2+...+b p x tp

SSE(b) :

Sum of squared errors from estimatingΒ byb,⌆ n t=1[y t −ŷ t (b]2

y :

Average of the observed responses,⌆ n t=1y t /n

SST :

Total sum of squares,⌆ n t=1 (y t−y)2

Β :

Estimate ofΒ that is least squares (minimumSSE) and maximumR 2

R 2 :

Coefficient of multiple determination, 1−SSE(Β)/SST

P :

Edge frequency percentage

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Author information

Authors and Affiliations

  1. Department of Statistics, Stanford University, Stanford, California

    Keith M. Gregg

  2. Department of Anesthesia, St. Elizabeth's Community Health Center, Lincoln, Nebraska

    John R. Varvel

  3. Palo Alto Veterans Administration Medical Center, Anesthesiology Service (112A), 3801 Miranda Avenue, 94304, Palo Alto, California

    Steven L. Shafer

Authors
  1. Keith M. Gregg
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  2. John R. Varvel
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  3. Steven L. Shafer
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Additional information

Supported in part by the Veterans Administration Merit Review Program, a Starter Grant from the American Society of Anesthesiologists, and Biomedical Research Support Grant RR05353 awarded by the Biomedical Research Support Grant Program, Division of Research Resources, National Institutes of Health, and the Anesthesia/Pharmacology Research Foundation.

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Gregg, K.M., Varvel, J.R. & Shafer, S.L. Application of semilinear canonical correlation to the measurement of opioid drug effect. Journal of Pharmacokinetics and Biopharmaceutics 20, 611–635 (1992). https://doi.org/10.1007/BF01064422

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