Research ArticleAPPLIED MATHEMATICS

Unlocking data sets by calibrating populations of models to data density: A study in atrial electrophysiology

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Science Advances  10 Jan 2018:
Vol. 4, no. 1, e1701676
DOI: 10.1126/sciadv.1701676
  • Fig. 1 Experimental recordings of AP biomarkers show significant differences between SR and cAF.

    Pairwise scatterplots of each unique pair of biomarkers in the SR data set (blue) and the cAF data set (red). Clear differences in how biomarker values are distributed between the two populations are seen, especially for the APD biomarkers. Overall, the data exhibit a large amount of variability, highlighting the importance of characterizing this variability in understanding cardiac electrophysiology.

  • Fig. 2 Calibration to biomarker distributions as opposed to their ranges reduces model bias.

    (A) Marginal distributions of the biomarkers in the SR data set (black) and POMs calibrated to biomarker distributions using the SMC algorithm (blue) or calibrated to biomarker ranges using LHS (red). The natural logarithm of APD20 values is used to better display their distribution. SMC for distributional calibration is seen to provide a significant improvement in agreement with the data. (B) Pairwise scatterplots of each unique pair of biomarkers in the SR data set (white) and the POMs constructed using SMC matched to distributions (blue) and LHS matched to ranges (red). The SMC-generated POM demonstrates good localization to the dense regions in the data but requires further calibration. An obvious correlation between APA and dV/dtmax is exhibited by the model, regardless of the sampling method used, but this correlation is not present in the data.

  • Fig. 3 Selection of an optimal subpopulation almost fully captures biomarker variability.

    (A) Marginal distributions of the biomarkers in the SR data set (black) and POMs selected as subpopulations of the SMC-generated POM that minimized ρ (red) or Embedded Image (blue). The simulated annealing algorithm succeeds at selecting a representative subpopulation, but the distributions of the APA and dV/dtmax are not quite captured. De-emphasizing dV/dtmax in the calibration process provides very good capture of variability in all other biomarkers. (B) Pairwise scatterplots of each unique pair of biomarkers in the SR data set (white) and the models from the SMC-generated POM that were accepted (light blue) or rejected (dark red) in the process of minimizing Embedded Image. Outside of dV/dtmax, especially its relationship with APA, the features of the data are very well represented by the final POM.

  • Fig. 4 Further variance in INa improves the realization of dV/dtmax values in the cAF data set.

    (A) Pairwise scatterplot of APA and dV/dtmax values in the cAF data set (white) and those accepted by distribution-calibrated POMs minimizing ρ (red). Allowing variance in the time constant significantly reduces the correlation between these two biomarkers in the POM, better realizing the spread of the data. (B) Marginal distribution of dV/dtmax values in the cAF data set (black) and the calibrated POMs varying only current conductances (red) or with additional variance in INa conductance and inactivation time (blue). This additional variance allows our calibrated POM to almost capture the marginal distribution of dV/dtmax values, where the original POM fails.

  • Fig. 5 Distributional calibration captures the morphological differences between SR and cAF atrial APs.

    Atrial APs produced by simulation of the populations of CRN models calibrated to biomarker data for patients exhibiting SR (blue) and cAF (red). In addition, the average of all traces for the SR (solid) and atrial fibrillation (dashed) populations is displayed. The increase in AP triangulation and reduced refractory period associated with cAF is demonstrated, especially by the averaged traces.

  • Fig. 6 Accepted parameter values for the SR and cAF populations predict well the changes in ionic behavior associated with the cAF pathology.

    Box plot of θ values comprising the POMs calibrated to the SR (blue) and cAF (red) data sets. Values are expressed in relation to the base parameter values for the CRN model. Current densities that show statistically significant differences (P < 0.001 from the Mann-Whitney U test) are indicated with “*.” The currents most well known as remodeled in cAF (Ito, IKur, IK1, and ICaL) all show significant differences in the correct directions.

  • Fig. 7 Calibration to distributions produces models that respond appropriately to antiarrhythmic treatment via IKr block.

    (A) Traces of the rapid component of the delayed rectifier K+ current for the models calibrated to the SR data set (blue, solid) and the cAF data set (red, dashed). The cAF population demonstrates an almost twofold increase in the activity of this current. (B) APs after treatment by 50% IKr block (gold) show significant AP prolongation compared to the same models without IKr block (red). This is also demonstrated by the averaged traces for both (black lines: treated, solid; untreated cAF, dashed). The restoration of atrial refractoriness in patients with cAF is demonstrated.

  • Fig. 8 Selection of parameter values using distribution-calibrated POMs produces updated models that correspond to provided data.

    (A) AP curves for SR (blue) and cAF (red) as predicted by the original and median CRN model. Experimental calibration via POMs produces a model that fits very well the general trend of the data (mean values of biomarkers from the data indicated by dashed lines). (B) cAF APs before (red) and after (gold) treatment via IKr blocker for the original (dashed) and modified (solid) CRN model. Our median CRN model predicts the antiarrhythmic effects of IKr block (via APD prolongation), whereas the original CRN model predicts little response.

  • Fig. 9 Variation of ±30% in current densities underestimates biomarker variance in the SR data set.

    Marginal distributions of the biomarkers in the SR data set (black) and distribution-calibrated POM using ±30% variance in ion channel conductances (blue). A reduced search space is still able to recover the general distributions of all biomarkers except for dV/dtmax, but the full extent of variation in the APD biomarkers and V20 is not present in the calibrated POM. Notably, the very low APD50 values recorded for some patients are completely unrepresented in the POM (arrow).

  • Table 1 Experimentally observed changes in current density associated with cAF are well predicted by POMs calibrated to distributions.

    Experimentally observed changes in current density associated with cAF are well predicted by POMs calibrated to distributions.. Changes in median current activities between the POMs calibrated to either the distributions or the ranges of the SR and cAF data sets, as compared with experimentally observed (Exp.) measurements of changes in current densities associated with this pathology. Experimental figures are taken from the specified references and rounded to the closest 10% to reflect the general uncertainty in their measurements and, in some cases, represent the combined result of multiple studies. The “↔” symbol indicates no significant change observed (P ≥ 0.01 from the Mann-Whitney U test for POMs). Distribution-calibrated POMs detect more of the differences in current densities that underlie the cAF pathology, correlating well with experimentally observed changes in the greatest majority of current densities.

    ParameterExp.POMs (distributions)POMs (ranges)
    gNa↕* (66)+11%
    gto~−70% (37)−85%−51%
    gKur~−50% (37)−40%−6%
    gKr (31)+10%
    gKs~+100% (41)
    gK1~+100% (37)+29%+33%
    gCaL~−70% (37, 67)−36%
    INaK(max)↔ (68)+10%
    INaCa(max)~+40% (40)+41%−18%
    Iup(max) (42, 43)−39%
    krel§ (42)

    *Peak INa current is reduced, but sustained INa increased.

    †Decreases in mRNA levels have been observed (37), but no direct experimental evidence for change in cAF has yet been provided.

    ‡Ca2+ uptake is reduced by decreased Serca2a levels but increased by enhanced phosphorylation of SERCA inhibitors.

    §Ca2+ release is increased but in a “leaky” fashion not necessarily best represented by changes to krel in the CRN model.

    Supplementary Materials

    • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/1/e1701676/DC1

      table S1. Summary statistics for the SR and cAF data sets are well recovered by the calibrated POMs.

      table S2. SMC with subsequent refinement produces POMs with very low divergence from the distributions in the data.

      fig. S1. Calibration to biomarker distributions, as opposed to their ranges, significantly reduces model bias for the cAF data set.

      fig. S2. Variability in the cAF data set is captured by a population of CRN models with varying current densities.

      fig. S3. Further variance in INa improves the realization of dV/dtmax values in the SR data set.

      fig. S4. Calibration to ranges fails to capture the morphological differences between SR and cAF atrial APs.

      fig. S5. Calibrating to data ranges does not identify all changes in ionic behavior associated with the cAF pathology.

      fig. S6. The distributions of parameter values selected for the SR and cAF POMs are distinct but regular.

      fig. S7. Variation of ±30% in current densities underestimates biomarker variance in the cAF data set.

    • Supplementary Materials

      This PDF file includes:

      • table S1. Summary statistics for the SR and cAF data sets are well recovered by the calibrated POMs.
      • table S2. SMC with subsequent refinement produces POMs with very low divergence from the distributions in the data.
      • fig. S1. Calibration to biomarker distributions, as opposed to their ranges, significantly reduces model bias for the cAF data set.
      • fig. S2. Variability in the cAF data set is captured by a population of CRN models with varying current densities.
      • fig. S3. Further variance in INa improves the realization of dV/dtmax values in the SR data set.
      • fig. S4. Calibration to ranges fails to capture the morphological differences between SR and cAF atrial APs.
      • fig. S5. Calibrating to data ranges does not identify all changes in ionic behavior associated with the cAF pathology.
      • fig. S6. The distributions of parameter values selected for the SR and cAF POMs are distinct but regular.
      • fig. S7. Variation of ±30% in current densities underestimates biomarker variance in the cAF data set.

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