IIR Third Annual Cell Based Assays Measuring Parallelism and Relative Potency In Well-Behaved and Ill-Behaved Cell-Based Bioassays Workshop Instructor: John R. Dunn, Ph.D. Chief Technical Officer Brendan Technologies 76-99-75 ext. 4 jdunn@brendan.com
Curve Fitting Linear Regression Requires: Weight individual points Least squares regression algorithm Nonlinear Curve Fitting Requires: Weight individual points Numeric processes to find the global minimum - the one set of coefficients that results in the lowest residual sum of squared error (RSSE)
Residual Sum of Squared Errors {(, ), } D= x y i = K N i i Data Points w = i variance i Weighting RSSE = N i= w i y i Y Fit Measure χ Prob = Dist, N - P Fit Probability χ ( RSSE ) 3
Nonlinear Regression Fitting R e s p o n s e 4 8 6 4 wsse Weights w Δ w Σ Δ w 3 Δ w 4 w 5 Δ w 6 Residuals (Errors).5..5 5 Dose Δ Δ Logistic Curve Being Fitted 4
3PL, 4PL and 5PL Logistic y = a* x b + c 3PL y y ( a d) = d + ( ) b + x c ( a d) = d + ( ) b + xc g 4PL 5PL The logistic model is a mathematical shape function only, its parameters do not correlate with any physical properties of the bioassay or immunoassay reaction. 5
5PL Regression Fitting wsse 4 g 4 6 5 5 5 c 6
7 5PL Coefs b vs g 5 4 3...3.4.5 g 3 4 5 4 3 b 3 4 SSE 5 4 3...3.4.5 g 3 4 5 4 3 b 3 4 SSE
Evaluating Curve Models Compute Assays With Pooled Weighting Model Fit using highest parameter curve model (e.g. 5PL) Fit lower parameter model, if desired, and compare Examine Assay Fit Metrics RSSE is a χ distributed number having (Number of Points Number of Parameters) degrees of unconstdom The χ probability (Fit Probability) is uniformly distributed from Fit Probability Average From All Assays A good curve fit model and good weighting model average in range of.3.7 (bioassays) and.4.6 (immunoassays) Higher fit probabilities caused by estimated variances too large Lower fit probabilities caused by inadequate curve models or estimated variances too small for some or all residuals 8
Asymmetric Curve (High Knee) 35 35 3 5PL (a>d) 3 5PL (d>a) 5 5 5 5 5 5 5 5 SSE:.59785 DF: 3 Res Var:.9995 Fit Prob:.89754 SSE: 4.655 DF: 3 Res Var:.48 Fit Prob:.47 35 3 4PL 4 3 3PL 5 5 5 5 SSE: 9.563 DF: 4 Res Var:.69 Fit Prob:.5988 5 5 SSE: 77.956 DF: 5 Res Var: 5.49 Fit Prob: <. 9
Asymmetric Curve (Low Knee) 35 3 5 5 5PL (a>d) 3 5 5 5PL (d>a) 5 5...5...5...5...5 SSE: 4.9946 DF: 3 Res Var:.66487 Fit Prob:.793 SSE:.7958 DF: 3 Res Var:.63753 Fit Prob:.85557 3 5 4PL 4 3PL 5 5 3...5...5 SSE: 49.796 DF: 4 Res Var:.499 Fit Prob: <....5...5 SSE: 899.534 DF: 5 Res Var: 79.97 Fit Prob: <.
Symmetric Curve 4 5PL (a>d) 4 5PL (d>a) 8 6 4 8 6 4.. SSE:.46 DF:3 Res Var:.87537 Fit Prob:.48944 SSE:.366 DF:3 Res Var:.775354 Fit Prob:.57546 4 75 4PL 5 3PL 8 6 4 5 75 5 5. SSE:.4384 DF:4 Res Var:.6596 Fit Prob:.65833. SSE: 556.3 DF:5 Res Var: 5.46 Fit Prob: <.
No Upper Plateau 5 5PL (a>d) 5 5PL (d>a) 5 5 5 5 5 5 5 5 SSE: 3.397 DF:3 Res Var:.366 Fit Prob:.3469 SSE: 3.379 DF:3 Res Var:.393 Fit Prob:.3465 5 4PL 5 3PL 5 5 5 5 5 5 SSE: 3.373 DF:4 Res Var:.8883 Fit Prob:.5698 5 5 SSE: 3.3339 DF:5 Res Var:.66678 Fit Prob:.6579
EEG Curve Metrics Method Controls: Reference Set Assay Coef a Coef b Coef c Coef d Coef g SSE Res Var Fit Prob EEG-43.6E+7.777 477 9.44 4.863797.837759.53 EEG-44.4E+7.853 3.6 5393.864.659783.339437.894 EEG-45.E+7.983. 37585 4.75 3.3984.64797.668 EEG-46.6E+7.854 65487 36786 4.5.458475.96495.98 EEG-47.3E+7.5 7.98 55777.393 7.36697 3.47394.4 EEG-48.4E+7.875 36.4 94.766.585367.36373.93 EEG-49.6E+7.8 59.7 7657 7.7 7.5354747.57949.84 EEG-5.6E+7.779 7.9 964 8.3.388836.477767.793 EEG-5.E+7.3 9.66 749.53 5.47568.8454.367 EEG-5.3E+7.67.8E+7 34744 3988.5.777587.355574.879 EEG-53.5E+7.67 5 67383 7.5.5749438.349888.94 EEG-54.3E+7.738 34. 7955 3.35 3.444434.6888487.63 EEG-67.7E+7.89 43.3 38994 4.554 4.733643.946749.449 EEG-68.3E+7.94 67.4 388 4.57 3.7953553.7597.579 EEG-69.E+7.3 4.85 68.778.599379.39876.9 EEG-7 58769.674 4.78 795737.59 3.385654.6773.679 EEG-7 4858643.73 6.348 893.455 5.37936.74587.37 EEG-7 584889.43 4876 88839 46.85.6435.3873.4 EEG-74.6E+7.68.37 9967.56 3.766358.7557.584 EEG-76.7E+7.875 395 396444.46 5.96567.9334.44 EEG-77.9E+7.837 933653 3866 658.3 8.65677.7355.4 EEG-78.E+7.843 54.6 469 7.63 5.984455.59689.38 EEG-79.E+7.38 3.6 44793.478 5.99.5838.4 EEG-8 848.365 5.34 5454.68 8.646579.79356.4 EEG-8.8E+7.774 33.7 673 5.659 3.8793498.77587.567 EEG-83 8878964.77 39.8 637883 3.7 3.4434.68863.693 EEG-87.7E+7.985 46.7 36399 5.44 4.7573.953464.447 EEG-86.6E+7.693 65599 45685 3369.4 4.5937.8586.546 EEG-85.5E+7.946 4.7 577.436 3.985578.65976.654 EEG-84.7E+7.865 64.66 54733 3.796 5.89398.578796.38 Assays 3 3 3 Average 4.75593.9586.5335667 Minimum.458475.96495.4 Maximum 7.36697 3.47394.98 3
4PL & 5PL Curves (VUT-9) VUT-9 (Current Assay) 4 Parameter Logistic Curve 4PL VUT-9 (Current Assay) 5 Parameter Logistic Curve 5PL 4
5PL Curve Metrics (VUT) Method Controls: Reference Set Assay Coef a Coef b Coef c Coef d Coef g SSE Res Var Fit Prob VUT- 435-3.943 48.7 76.3.7863.7863.39 VUT-4 345-5.894 49.4 96.5.66795.66795.796 VUT-5 349-8.39 5.34.4.584.584.47 VUT-6 3955-7.57 5.63 66.6 4.85986 4.85986.7 VUT-7 3484-7.738 49.73 7.46.599947.599947.6 VUT-8 445-4.45 47.5 5.35.6897.6897.435 VUT-9 484-5.747 48.6 6.99 3.479 3.479.65 VUT- 46-4.888 44.6 3.45.376677.376677.4 VUT- 43-5.466 5.7 56.6.8897.8897.368 VUT- 45-7.478 59.36 5.35.464443.464443.496 VUT-3 338-8.34 48.8 94.46.5395.5395. VUT-4 456-6.99 45.54 85.95.4679.4679.6 VUT-5 483-5.5 47 4.36.95E-.95E-.889 VUT-6 438-4.845 46. 7.38.548855.548855.3 VUT-8 456-4.56 47.93 8.64.74495.74495.78 VUT-9 479-9.55 5. 6.9.68847.68847.49 VUT- 46-6.9 48.96 9.9.653.653.4 VUT-3 46-8.3 5.97 5.39.85775.85775.355 VUT-5 4343-5.738 47. 4.3E-5.84.4868.4868.488 VUT-6 475-4.973 53. 66.8.5E-.5E-.874 VUT-9 4394-4.376 44.46 5.87.65497.65497.99 VUT-3 4366-4.97 46.84 3.8 4.99 4.99.4 VUT-3 4357-5.56 47.77 8.9.5.5.94 VUT-38 44-3.63 47.5 5.335.48665.48665.485 Assays 4 4 4 Average.386.386.35 Minimum.9.9.7 Maximum 4.859 4.859.889 5
4PL Curve Metrics (VUT) Method Controls: Reference Set Assay Coef a Coef b Coef c Coef d Coef g SSE Res Var Fit Prob VUT- 599 -.663 35.78 63 3.484 6.64. VUT-4 48 -.79 33.68 7 9.85853 4.996. VUT-5 454 -.76 34.69 88 37.453 8.76. VUT-6 485 -.79 34. 69 34.6 7.35. VUT-7 4 -.74 33.6 69 44.634.357. VUT-8 473 -.754 33.4 9 6.56495 8.8473. VUT-9 564 -.7 3.86 7 38.33 9.565. VUT- 486 -.84 9.75 8 7.7 3.86. VUT- 536 -.646 36.49 56.6883.344. VUT- 569 -.464 44.3 4 9.9438 9.54788. VUT-3 3876 -.839 3.8 74 5.8356 5.9678. VUT-4 4876 -.89 3. 94 4.4.. VUT-5 49 -.796 3.5 97 5.6444.63. VUT-6 579 -.734 3.84 85 7.4344 3.76. VUT-8 556 -.79 33. 73.97395.48698. VUT-9 566 -.648 33.8 58 48.73 4.87. VUT- 57 -.76 33.3 9 38.3739 9.637. VUT-3 536 -.686 35.53 6 43.758.5359. VUT-5 588 -.653 3.4 46 9.958 4.6479. VUT-6 5333 -.66 37.8 58 9.9 9.543. VUT-9 567 -.84 3.64 6 3.747.854. VUT-3 53 -.67 9.97 7.456.78. VUT-3 5 -.64 3.7 59 3.77983 5.3899. VUT-38 53 -.75 33.7 87.76 5.385.5 Assays 4 4 4 Average 9.93 4.965.3 Minimum.76 5.38. Maximum 5.84 5.97.5 6
VUT Residuals (4 Assays) VUT 4PL Weighted Residuals Fit Prob =.3 (4 Assays) VUT 5PL Weighted Residuals Fit Prob =.35 (4 Assays) 6 6. 4.4 6 Squared Residuals 5 4 3 Squared Residuals 5 4 3 5 5 5 3 35 4 45 5 5 5 3 35 4 45 7
5PL DEF-73 8
5PL Curve Metrics (DEF) Method Controls: Reference Set Assay Coef a Coef b Coef c Coef d Coef g SSE Res Var Fit Prob DEF-39 86.97 87.7 37 3.47 9.79647.63738.33 DEF-4 75.3 73.5 384 6.75 33.44339 5.573898. DEF-4 84.49 99.77 365 3.635.769.79389.96 DEF-4 689.475 74.78 4.84 3.7594.983.33 DEF-43 747.99 76 396 748.939.8766.479434.84 DEF-44 74.6 7.9 383 7.37 6.478 4.4968. DEF-45 963.9.7 47 3.393 4.668.43783.3 DEF-46 963.379 69.9 389.83.935.485584.8 DEF-47 943.64 59 44 4.898.84535.4745.88 DEF-5 456.9 7.9 43 79.755 3.5786.596344.734 DEF-5 44.7 38 3.434.45769.4949.96 DEF-5 45.76 89.5 387 3.455 3.793.5365.78 DEF-53 458.377 67.6 43.468 6.683648.394.35 DEF-54 45.35 53.6 37.9 3.3543.558739.763 DEF-55 44.97 7 49 7.37.4747 7.86E-.998 DEF-56 44.6 58.3 49 5.758.787.8644.83 DEF-57 465.93 36.5 39.69.8478.373.934 DEF-58 356.76 4.35 48.643 3.384545.5649.759 DEF-59 485.66 4.7 455.94 4.398466.73378.63 DEF-6 446.394 58 44 7.9.5795.6388.954 DEF-6 446.547 79.8 43 3.7 6.47934.78656.37 DEF-6 96.5.3 58 5.59 4.6935.4489.3 DEF-63 96.66 84. 54 3.437 6.6378.63.359 DEF-64 95.63 89. 58 3.546.7985.96548.978 DEF-65 558.43 66 46 8.4 4.8.3368.9 DEF-66 588.384. 475.58.36794.77989. DEF-67 578.56 76.44 448.99 8.488.35683.8 DEF-7 396.45 6.44 398.93 6.78.97.348 DEF-73 46.6 86. 45 7.3.874869.3478.93 Assays 9 9 9 Average 7.884.34.485 Minimum.47.79. Maximum 33.443 5.574.998 9
4PL Curve Metrics (DEF) Method Controls: Reference Set Assay Coef a Coef b Coef c Coef d Coef g SSE Res Var Fit Prob DEF-39 74.748 3.8 34 7.56895.5985.4 DEF-4 645.79 33.5 36 45.486 6.497466. DEF-4 747.7 3.33 3.4.86359.5 DEF-4 67.95 3.33 355.4657 3.65796.3 DEF-43 636.85 3.8 33 3.9884 3.46877. DEF-44 66.78 3.34 35 4.466 6.6689. DEF-45 9.68 37. 33.79 3.6843.4 DEF-46 9.694 35. 38 6.64886.9464.469 DEF-47 859.75 37.9 336.4588.636973. DEF-5 358.794 3.88 349.456 3.64438.3 DEF-5 37.645 3.36 3 7.7547.7.355 DEF-5 33.74 9.9 338.44965.4987.64 DEF-53 47.747 3.74 357.66539.666484. DEF-54 49.63 3.3 333 5.58473.79789.589 DEF-55 39.74 3.3 344.84475.697.6 DEF-56 33.8 33. 349 3.75886 3.3943. DEF-57 359.65 3.3 34 5.3.58757.35 DEF-58 33.64 9.7 388 5.7537.8875.569 DEF-59 445.45 5.9 433 7.85887.696.345 DEF-6 374.44 3.9 39 7.447.443496.7 DEF-6 394.73 3.74 39 5.797.8738.33 DEF-6 833. 3.96 46 3.855 4.5549. DEF-63 848. 34.6 459 6.695.37789. DEF-64 858.7 35.46 48..57457.39 DEF-65 48.3 3.8 44 3.4868 4.64974. DEF-66 5.9 3.4 44 3.789 4.46833. DEF-67 55.6 3.5 45 6.44735.3496. DEF-7 353.8 3.6 358.6395.5899.55 DEF-73 33.857 3.84 369 3.99736.99963.5 Count 9 9 9 Average 8..6.5 Minimum 5.585.798. Maximum 45.48 6.497.589
DEF Standard Response Residuals DEF 5PL Weighted Residuals Fit Prob =.485 (9 Assays) DEF 4PL Weighted Residuals Fit Prob =.5 (9 Assays) 6 6 Squared Residuals 5 4 3 Squared Residuals 5 4 3 5 5 5 3 5 5 5 3 Residuals - Residuals - - 5 5 5 3-5 5 5 3 Mean Response Mean Response
ABC Standard Curve
5PL Curve Metrics (ABC) Method Controls: Reference Set Assay Coef a Coef b Coef c Coef d Coef g SSE Res Var Fit Prob ABC-9 3.76.838 8. 6.93 6.964 5.43886.7943.66 ABC-3 74.9.597 55.4 6.79 3.549.535.5563.775 ABC-3 73.8 3.45 8.98 7.74 4.958.459.764.36 ABC-34 75.38.46 66.3.3 46. 5.7335.866575.57 ABC-35 67.56.548 57.74 548.5 7.89875 3.644937.6 ABC-36 95.84 3.3 34.58 4.65.7 4.993 7.45465. ABC-37 78.5 6.383 35.7 5.6.76 4.65946.3963.97 ABC-38 5..548 73.49 6.39 5.85.675.63357.53 ABC-39 9.79 4.464 37. 5.73.6.334866.667433.53 ABC-4 9.5. 49.6.4 93.734.66754.33377.76 ABC-4 57.78 4.3 4.85 9.99.48.4575.6638.537 ABC-4 8.4.78 97.3 6.3 7.8.4598.7959.99 ABC-43 98.6.97 75.67 7.6 6.648.65786.3893.7 ABC-44 75.66.856 85.75 5.97 5.99 6.57 3.856.44 ABC-45.89.638 8.9 8.7 5.687.59844.99.357 ABC-46 38.6.74 35.3 7.96 65.76 4.4358.79.3 ABC-47 9.4.53 4.6 8.8 3.358.4634.535.593 ABC-48 5.8.8 63.3 7.88 3.79.4744.7357.479 ABC-49.3.599 5.4 8.64 499.75.4956.4768.87 ABC-53 95.7.583 9. 8.9 965.4 9.33767 4.65883. Assays Average 3.75.876.38 Minimum.5.55. Maximum 4.99 7.455.775 3
4PL Curve Metrics (ABC) Method Controls: Reference Set Assay Coef a Coef b Coef c Coef d Coef g SSE Res Var Fit Prob ABC-9 84.9 3.693 38.73 5.3 6.39.637.97 ABC-3 59. 4.65 35.68 6.38.963.3767.8 ABC-3 65.9 5. 46.57 6.67 3.66.6687.36 ABC-34 74.35.67 9.38 7.5 6.48797.3666.93 ABC-35 65.87.99 3.86 8.97 7.96383.6546.47 ABC-36 94.5 3.58 34 4.56 4.949 4.97495. ABC-37 83.44 5.78 36.88 5.78 4.856633.68878.83 ABC-38 89.67 4.9 39.98 5.6.4.74735.58 ABC-39 9.9 4.749 36.36 5.66.344336.448.79 ABC-4 98.75.73 7.9 8.84.8743.6944.848 ABC-4 57.48 4.35 4.35 9.94.4579.4564.74 ABC-4 55.58 4.675 35.3 5.76 3.874.6693.84 ABC-43 7.6 4.86 35.78 6.75.4938.497743.684 ABC-44 64.99 5.3 45.3 5 7.87499.6467.49 ABC-45 68.69 3.34 3.5 7.86.4365.747875.53 ABC-46 86.37.863 7.38 6.67 4.73737.579.9 ABC-47 77.36 3.5 34.9 7.5.3555.4558.77 ABC-48 87.5 4.9 39.94 7.7.93434.64445.587 ABC-49 9.56 4.97 4.56 7.33.9934.997447.4 ABC-53 8.5 4.6 43.3 6.7.9344 4.383.5 Assays Average 4.48.493.379 Minimum.87.69. Maximum 4.9 4.97.848 4
ABC Standard Response Residuals ABC 5PL Weighted Residuals Fit Prob =.333 ( Assays) ABC 4PL Weighted Residuals Fit Prob =.379 ( Assays) 6 6 Squared Residuals 5 4 3 Squared Residuals 5 4 3 4 6 8 4 6 8 Residuals - Residuals - - 4 6 8-4 6 8 Mean Response Mean Response 5
Logistic Curve Fits 8 6 4 5 5 4PL 5PL a > d 5PL d > a Assay a b c d SSE Fit Prob a b c d g SSE Fit Prob a b c d g SSE Fit Prob FCD-98 87.939.38 5.559.78 58.38. 89.96.9 4.369.469.5 49.884..38-3.4 37.554 9.79.47 9.59.7 FCD-99 96.839.85 9.56.879 43.8..367.458 65.59.934 4.66 5.4..8-3.6 36.55 6.38.86 4.6.593 FCD- 95.988.55.86.37.8..38.378 78.476.4 4.85 3.364.38.9-3.49 4.344 5..95.5.868 FCD- 9.65.3 4.73.56 9.877. 96.649.378 58.389 3.87 9.857 8.395..949-3.6 49.7.47.45 8.68.9 FCD-3 9.63.59 5.8.485 95.878. 95.86.447 99.456.834 5.77 5.385..663-3.35 46.88 98.97.86 6.77.348 FCD-5 94.459.994 4.774.636 6.777. 97.54.44 67.3 3.43 3.4 35.743..84 -.843 44.5.7.348 5.579.47 FCD-6 9.69.4.79.44 5.54. 5.83.464 5.96.55 3.6 5.33..438 -.853 38.56 99.569.34 7.67.36 FCD-7 9.79.9..43 6.83. 94.46.564 43.68 3.59.589 9.75.38.88 -.738 34.664 96.976.49 3.75.7 FCD-8 9.997.99.85.53 4.97. 94.6.653 37.393.967.88 97.89..95 -.76 35.544 97.64.4.895.45 FCD-9 9.343.96.53.53 6.. 94.8.59 4.8 3.43.447 9.457.49 3.65 -.75 34.74 97.64.44 3.69.77 Average 6.99 <. 44.868.3 6.46.443 6
FCD Standard Response Residuals Squared Residuals 6 5 4 3 FCD 4PL Weighted Residuals Fit Prob =. ( Assays) 8. 9.5 7.6 7.6 5.8 4 6 8 Squared Residuals 6 5 4 3 FCD 5PL a>d Weighted Residuals Fit Prob =.9 ( Assays).7 6. 4 6 8 Squared Residuals 6 5 4 3 FCD 5PL d>a Weighted Residuals Fit Prob =.443 ( Assays) 4 6 8 4. 4.3.8 Residuals - Residuals - Residuals - - -4.8-3.8 -. 4 6 8 Mean Response - -. 4 6 8 Mean Response - 4 6 8 Mean Response 7