Transit Time Measurement in Indicator Dilution Curves: Overcoming the Missing Ground Truth and Quantifying the Error

Figures

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  Figure 1. (Top) Sketch of a rigid pipe with an inner diameter of di containing a fluid flow with a constant mean flow velocity v. Two measurement points A and B with a distance of s are defined. (Bottom) Two indicator dilution curves (IDCs) obtained at the locations A and B by optical measurement. The transit time Δt is defined as the shift of the curves as indicated by the shift of the maxima.

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  Figure 2. Sketch of the rotationally symmetric geometry with a section containing a homogeneously distributed indicator (green area) and a blood analog (yellow area). After applying a laminar flow, concentration curves are obtained as the mean concentration at the pipe's cross-sectional area at the three marked locations (blue, red, and orange). The corresponding dilution curves are shown in Figure 3. Please note that the scale of the figure does not represent the scale used in the in silico model.

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  Figure 3. The three indicator dilution curves obtained from the fluid flow model with a sampling rate of 1,200 Hz. The relative concentration is normalized to the input concentration cICG. The blue, red, and orange curves show the dilution curves in a distance of 50, 100, and 150 cm to the injection side as shown in Figure 2, respectively.

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  Figure 4. (A) IDC1 in blue without any manipulations. In green, the same curve delayed by a transit time Δt = 160 ms. Both are sampled with 1,200 Hz. (B) Sub-sampling of the two curves from (A). Both curves are sub-sampled to 25 Hz. (C) Final data set containing two differently sub-sampled curves; one of them temporally shifted by the ground truth transit time Δt. White Gaussian noise is applied to both IDCs (SNR = 20 dB).

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  Figure 5. Visualization of the three different pre-processing (cutting) methods on the data. The cut level is set to 40% of the maximum. After applying cut method 3, the purple curve remains. Applying cut method 2 results in the curve consisting of purple and red. Application of cut method 1 results in the curve of all three colors purple, red, and blue.

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  Figure 6. (A) An example of a synthetically computed data set. It shows two indicator dilution curves with an delay of 1.5 s and a SNR of 37 dB. (B) An in vivo indicator dilution curve obtained from a cerebrovascular EC-IC bypass surgery by indocyanine green fluorescence angiography. The signal is obtained by averaging an area of 10 × 10 pixel which corresponds to one-third of the vessel's diameter.

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  Figure 7. Power spectral density of the in vivo and in silico data in the spectral rang of up to 5 Hz.

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  Figure 8. (A) Mean εframes using the raw data sets (fsampling = 25 Hz) with no mathematical fits. To obtain the transit time, the cross-correlation is computed after applying the cut method 3. (B) Mean εframes using the linearly interpolated data sets (fsampling = 25 · 100 Hz) with no mathematical fits. To obtain the transit time, the cross-correlation is computed after applying the cut method 2.

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  Figure 9. (A) Mean εframes using the gamma variate model on the data sets (fsampling = 25 Hz). Before fitting the model, cut method 3 is applied to the data set. To obtain the transit time, the cross-correlation is computed. (B) Mean εframes using the gamma variate model on the data sets (fsampling = 25 Hz). Before fitting the model, cut method 1 is applied to the data set. To obtain the transit time, the cross-correlation of the first derivative is computed. (C) Mean εframes using the LDRW model on the data sets (fsampling = 25 Hz). Before fitting the model, cut method 1 is applied to the data set. To obtain the transit time, the cross-correlation of the first derivative is computed.

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  Figure 10. (A) Mean εframes using the raw data sets (fsampling = 60 Hz) with no mathematical fits. To obtain the transit time, the cross-correlation is computed after applying the cut method 2. (B) Mean εframes using the linearly interpolated data sets (fsampling = 60 · 100 Hz) with no mathematical fits. To obtain the transit time, the cross-correlation is computed after applying the cut method 2.

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  Figure 11. (A) Mean εframes using the gamma variate model on the data sets (fsampling = 60 Hz). Before fitting the model, cut method 3 is applied to the data set. To obtain the transit time, the cross-correlation is computed. (B) Mean εframes using the LDRW model on the data sets (fsampling = 60 Hz). Before fitting the model, cut method 3 is applied to the data set. To obtain the transit time, the cross-correlation is computed. (C) Mean εframes using the LDRW model on the data sets (fsampling = 60 Hz). Before fitting the model, cut method 1 is applied to the data set. To obtain the transit time the cross-correlation of the first derivative is computed.

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  Figure 12. (A) The relative error of all data sets (fsampling = 25 fps) is shown in relation to the signal-to-noise ratio (SNR). The red line indicates the relative error of one frame. The blue dots show the single results obtained by the fitting the Local Density Random Walk (LDRW) model to the data cut at 40% by the cut method 1 and using its first derivative as input for the cross-correlation. The distribution of the dots within a column is irrelevant. All data sets had a relative error of less than one frame for SNR larger than 12 dB. (B) The relative error of all data sets (fsampling = 60 fps) is shown in relation to the SNR. The red line indicates the relative error of one frame. The blue dots show the single results obtained by the fitting the LDRW model to the data cut at 40% by the cut method 1 and using its first derivative as input for the cross-correlation. The distribution of the dots within a column is irrelevant. All data sets had a relative error of less than one frame for SNR larger than 18 dB.