Figure 1.
Effects of parameter variation on model output.
(A) Drastically different combinations of ionic conductances result in nearly identical action potential morphology. The bar graphs show log(G/Gcontrol) for each ionic conductance in the TNNP [15] model, where Gcontrol is the conductance in the published model (see Table 1 in Text S1 for full list) (B) Intracellular calcium (Ca2+) transients produced by the two parameter combinations are distinct, in terms of both amplitude and kinetics, suggesting that such information could be used to distinguish between the two parameter sets.
Table 1.
Physiological outputs in simulations with TNNP model.
Figure 2.
Schematic of input, output and regression matrix structures.
Randomly-varied model parameters are collected in an input matrix X with dimensions n, corresponding to the number of trials, by p, corresponding to the number of parameters. Simulation results define m outputs that are collected in the output matrix Y, with dimensions an n×m. Regression matrix B, with dimensions p×m, can be used to predict Y from X, the so-called “forward problem.” If m = n, and the outputs are linearly independent, then B can be inverted, and YB−1 should be a good approximation of X. This is our strategy for addressing the “reverse problem.”
Figure 3.
Predictions of the linear empirical model generated by reverse regression.
(A) Scatter plots are displayed for four input conductances: GNa (top left), GCa (top right), Gto (bottom left) and KNCX (bottom right). Each plot shows the value actually used in the simulations (abscissa) versus the estimate generated by the regression model (ordinate). The regression was performed on a simulated data set containing 300 samples. (B) R2 values for each conductance in the TNNP model in the reverse regression. The three cases shown correspond to regression performed with: all 32 outputs (blue); the sixteen “best” outputs (green), and the 16 rejected outputs (red).
Figure 4.
Parameter sensitivities for forward and reverse regression.
Values in the forward regression matrix B and reverse regression matrix B−1 are shown as “heat maps,” with white representing values near zeros, and blue and red indicating positive and negative values, respectively. (A) The forward regression matrix B, where each row represents the contributions of each of the conductances to a particular output. The bar graphs corresponding to two of these outputs (APD and diastolic [Ca2+]) are shown to the right. (B) The reverse regression matrix B−1, where each row represents the contributions of each of the outputs to a given conductance. The bar graphs corresponding to two of these conductances (GNa and GCa) are shown to the right.
Figure 5.
Application of reverse regression to constrain model parameters in heart failure.
Simulations were performed with the Hund & Rudy model of the dog ventricular myocyte [19], with changes made to 7 model parameters to replicate changes occurring in heart failure, as previously simulated by Shannon et al. [20]. (A) The differences between normal and pathological states is shown by contrasting the action potential waveforms and Ca2+ transients. The action potential is triangular in shape in heart failure while the Ca2+ transient is dramatically reduced in the failing cell. The directional changes in the 7 altered parameters are also indicated. (B) The true values of the changed parameters are shown alongside the values predicted by reverse regression. Each is represented as a multiple of the baseline parameter value, where no change is indicated by the dashed line. Note the break in the y-axis, reflecting the fact that the reverse regression procedure overestimates the change in the parameter Kleak. Similarly, the regression model overestimates the change in GKs, as the height of this bar, 0.86% of the control value, is difficult to visualize on this scale.
Figure 6.
Illustration of Bayesian probability approach.
(A) Distributions of GNa with different constraints. From left to right, histograms show GNa values in the complete data set; given that APD is in a particular range (from 295–298 ms, representing 10% of the samples); given that APD and Vrest (−84.96 to −85.02 mV) are in particular ranges; given that APD, Vrest, and Vpeak (37.05 to 37.81 mV) are in particular ranges. (B) Distributions of GCa, given the same constraints as in (A).