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Let S(t) and R(t) represent the number of cells that are sensitive or resistant to imatinib. During imatinib exposure, assume zero growth and a death rate δ for sensitive cells, µ << δ as the rate at which sensitive cells mutate to resistance, and no death and a growth rate β for resistant cells. The two cell populations then follow:
.0)0(;)0(
)()(
)(
0
RSS
tStRdt
dR
tSdt
dS
Nested in optimizations performed to estimate the parameters β, δ and µ, these equations were integrated numerically using lsoda of the R package odesolve. The parameters were estimated in exponentiated forms to constrain them and their confidence intervals (CIs) to positive values (see Table 1). CIs were estimated from Hessians (matrices of second derivates) of the SSEs (sum of squared errors) evaluated at the optimum (minimum SSE) using the function optim in R: Hessians were divided by 2, inverted, multiplied by SSE/(N - P) where N and P are the number of data points and parameters, and square roots of the main diagonal were then be multiplied by 1.96 to form the 95% Wald CI (in Table 1).
S Rµ
δ
β
Parameter Initial Value Final value optimized 95% Confidence Interval
β 0.80000 0.3600 TRUE (0.287, 0.453)δ2.5 1.00000 0.1780 TRUE (0.135, 0.235)δ5 1.00000 0.2570 TRUE (0.2, 0.33)δ10 1.00000 0.2300 TRUE (0.176, 0.301)μ 0.00005 0.0074 TRUE (0.00271, 0.02)S2.5(Day 2) 54.00000 54.0000 FALSE not fittedS5(Day 2) 57.87500 57.8750 FALSE not fittedS10(Day 2) 48.25000 48.2500 FALSE not fitted
Table 1. Parameter estimates of fit in Fig. 1. Units: rate constants=1/day, cells numbers=104.
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SSE = 223.5
CANCER RESEARCH 59, 4770–4775, October 1, 1999
Tumor Development under Angiogenic Signaling: A Dynamical Theory of Tumor Growth, Treatment Response, and Postvascular Dormancy
Philip Hahnfeldt, Dipak Panigrahy, Judah Folkman, and Lynn HlatkyV’=λV[log(K)-log(V)]
Model building datasets
Model validation data
V follows setpoint/carrying capacity K
With lambda2 fixed to zero and all else fitted to all of the data simultaneously, the fit was> g$params initial final opt CI95prctlambda1 1.92e-01 1.92e-01 TRUE (0.185, 0.202)lambda2 0.00e+00 0.00e+00 FALSE not fittedb 5.85e+00 5.78e+00 TRUE (5.47, 6.11)d 8.73e-03 8.73e-03 TRUE (0.00823, 0.00919)eT 1.30e+00 1.43e+00 TRUE (0.317, 6.42) ** 20 foldclrT 1.01e+01 1.02e+01 TRUE (3.22, 32.1) ** 10 foldeA 1.50e-01 1.55e-01 TRUE (0.0043, 5.64) **** very wide (>1000-fold) CI clrA 3.80e-01 4.34e-01 TRUE (0.00236, 79.8) **** very wide CI ****eE 6.60e-01 5.84e-01 TRUE (0.228, 1.5)clrE 1.70e+00 2.07e+00 TRUE (0.814, 5.26)V0 1.80e+02 1.90e+02 TRUE (178, 202)V0valid 3.00e+02 2.81e+02 TRUE (200, 395)K0 6.25e+02 7.93e+02 TRUE (742, 846)Initital SSE = 2,314,351; Final SSE=750,720.4 Note that the nice initial fit in Endo at 20 mg/kg is sacrificed for a better final fit in endo at 4mg/kg (next page).
Initial values take from here
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initial final opt CI95prctlambda1 1.92e-01 2.43e-01 TRUE (0.225, 0.262)lambda2 0.00e+00 0.00e+00 FALSE not fittedb 5.85e+00 5.56e+00 TRUE (4.85, 6.36)d 8.73e-03 9.70e-03 TRUE (0.00823, 0.0114)eT 1.30e+00 1.64e+00 TRUE (0.126, 21.1)clrT 1.01e+01 9.46e+00 TRUE (1.67, 53.5)eA 1.50e-01 9.30e-02 TRUE (0.0523, 0.165)clrA 3.80e-01 2.59e-01 TRUE (0.0963, 0.696)eE 6.60e-01 4.30e-01 TRUE (0.0993, 1.87)clrE 1.70e+00 1.25e+00 TRUE (0.264, 5.93)V0 1.80e+02 1.79e+02 TRUE (161, 198)V0valid 3.00e+02 3.11e+02 TRUE (290, 334)K0 6.25e+02 9.21e-03 TRUE (0.0076, 0.0112)
