Surface Fitting with Custom Equations to Biopharmaceutical Data

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This example shows how to use Curve Fitting Toolbox™ to fit response surfaces to some anesthesia data to analyze drug interaction effects. Response surface models provide a good method for understanding the pharmacodynamic interaction behavior of drug combinations.

This data is based on the results in this paper: Kern SE, Xie G, White JL, Egan TD. Opioid-hypnotic synergy: A response surface analysis of propofol-remifentanil pharmacodynamic interaction in volunteers. Anesthesiology 2004; 100: 1373-81.

Anesthesia is typically at least a two-drug process, consisting of an opioid and a sedative hypnotic. This example uses Propofol and Reminfentanil as drug class prototypes. Their interaction is measured by four different measures of the analgesic and sedative response to the drug combination. Algometry, Tetany, Sedation, and Laryingoscopy comprise the four measures of surrogate drug effects at various concentration combinations of Propofol and Reminfentanil.

The following code, using Curve Fitting Toolbox methods, reproduces the interactive surface building with the Curve Fitting Tool described inSurface Fitting to Biopharmaceutical Data.

Load Data

Load the data from file.

data = importdata('OpioidHypnoticSynergy.txt'); Propofol = data.data(:,1); Remifentanil = data.data(:,2); Algometry = data.data(:,3); Tetany = data.data(:,4); Sedation = data.data(:,5); Laryingoscopy = data.data(:,6);

Create the Model Fit Type

You can use the适合typefunction to define the model from the paper, whereCAandCBare the drug concentrations, andIC50A,IC50B,alpha, andnare the coefficients to be estimated. Create the model fit type.

ft = fittype('Emax*( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n /(( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n + 1 )',...'independent', {'CA','CB'},'dependent','z','problem','Emax')

ft = General model: ft(IC50A,IC50B,alpha,n,Emax,CA,CB) = Emax*( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n /(( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n + 1 )

AssumeEmax = 1because the effect output is normalized.

Emax = 1;

Set Fit Options

Set fit options for robust fitting, bounds, and start points.

opts = fitoptions( ft ); opts.Lower = [0, 0, -5, -0]; opts.Robust ='LAR'; opts.StartPoint = [0.0089, 0.706, 1.0, 0.746];

Fit and Plot a Surface for Algometry

(f, gof) = ([Propofol, Remifentanil], Algometry, ft,...opts,'problem', Emax )

Success, but fitting stopped because change in residuals less than tolerance (TolFun).

General model: f(CA,CB) = Emax*( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n /(( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n + 1 ) Coefficients (with 95% confidence bounds): IC50A = 4.149 (4.123, 4.174) IC50B = 9.044 (8.971, 9.118) alpha = 8.502 (8.316, 8.687) n = 8.289 (8.131, 8.446) Problem parameters: Emax = 1

gof =struct with fields:sse: 0.0842 rsquare: 0.9991 dfe: 393 adjrsquare: 0.9991 rmse: 0.0146

plot( f, [Propofol, Remifentanil], Algometry );

Figure contains an axes object. The axes object contains 2 objects of type surface, line.

Fit a Surface to Tetany

Reuse the same适合typeto create a response surface for tetany.

(f, gof) = ([Propofol, Remifentanil], Tetany, ft, opts,'problem', Emax )

General model: f(CA,CB) = Emax*( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n /(( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n + 1 ) Coefficients (with 95% confidence bounds): IC50A = 4.544 (4.522, 4.567) IC50B = 21.22 (21.04, 21.4) alpha = 14.94 (14.67, 15.21) n = 6.132 (6.055, 6.209) Problem parameters: Emax = 1

gof =struct with fields:sse: 0.0537 rsquare: 0.9993 dfe: 393 adjrsquare: 0.9993 rmse: 0.0117

plot( f, [Propofol, Remifentanil], Tetany );

Figure contains an axes object. The axes object contains 2 objects of type surface, line.

适合表面镇静

(f, gof) = ([Propofol, Remifentanil], Sedation, ft, opts,'problem', Emax )

General model: f(CA,CB) = Emax*( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n /(( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n + 1 ) Coefficients (with 95% confidence bounds): IC50A = 1.843 (1.838, 1.847) IC50B = 13.7 (13.67, 13.74) alpha = 1.986 (1.957, 2.015) n = 44.27 (42.56, 45.98) Problem parameters: Emax = 1

gof =struct with fields:sse: 0.0574 rsquare: 0.9994 dfe: 393 adjrsquare: 0.9994 rmse: 0.0121

plot( f, [Propofol, Remifentanil], Sedation );

Figure contains an axes object. The axes object contains 2 objects of type surface, line.

Fit a Surface to Laryingoscopy

(f, gof) = ([Propofol, Remifentanil], Laryingoscopy, ft, opts,'problem', Emax )

General model: f(CA,CB) = Emax*( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n /(( CA/IC50A + CB/IC50B + alpha*( CA/IC50A ) * ( CB/IC50B ) )^n + 1 ) Coefficients (with 95% confidence bounds): IC50A = 5.192 (5.177, 5.207) IC50B = 37.77 (37.58, 37.97) alpha = 19.67 (19.48, 19.86) n = 37 (35.12, 38.87) Problem parameters: Emax = 1

gof =struct with fields:sse: 0.1555 rsquare: 0.9982 dfe: 393 adjrsquare: 0.9982 rmse: 0.0199

plot( f, [Propofol, Remifentanil], Laryingoscopy );

Figure contains an axes object. The axes object contains 2 objects of type surface, line.