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Edited by: Sanjay Ram Kharche, Western University, Canada

Reviewed by: Johannes H. Van Beek, Amsterdam University Medical Center, Netherlands; Stephen Michael Moore, IBM Research, Australia

This article was submitted to Computational Physiology and Medicine, a section of the journal Frontiers in Physiology

†These authors have contributed equally to this work and share first authorship

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The vascular function of a vessel can be qualitatively and intraoperatively checked by recording the blood dynamics inside the vessel via fluorescence angiography (FA). Although FA is the state of the art in proving the existence of blood flow during interventions such as bypass surgery, it still lacks a quantitative blood flow measurement that could decrease the recurrence rate and postsurgical mortality. Previous approaches show that the measured flow has a significant deviation compared to the gold standard reference (ultrasonic flow meter). In order to systematically address the possible sources of error, we investigated the error in transit time measurement of an indicator. Obtaining _{s} = 60

In the last two decades, optic-based medical systems were introduced to intraoperatively visualize vascular structures via fluorescence angiography (Raabe et al., _{i} the inner diameter of the vessel, and Δ_{i} and

The accuracy of the calculation of _{i}, and Δ

_{i} containing a fluid flow with a constant mean flow velocity

In this paper, we focus on evaluating the performance of methods ascertaining the value of Δ

Equation (1) is derived from the general definition of volume flow _{relative} is of parabolic shape and

Data sets containing two corresponding indicator dilution curves with a ground truth of the transit time Δ_{sampling} = 60

Therefore, we propose a method to compute highly adaptable data sets of two corresponding and differently sub-sampled IDCs with a ground truth of the transit time using an

Two hypotheses were set up as follows:

Using mathematical models to re-continualize the dilution curves, there is a decrease in the error in transit time measurements compared to using the raw data.

A sub-frame rate accuracy can be accomplished by combining a suitable configuration of a mathematical model, extracted features, and pre-processing of raw data.

The overarching goal is to correctly obtain the indicator bolus transit time. The accuracy is limited by the frame rate and a sub-frame rate accuracy is required. This can be solved by fitting an appropriate analytical function to the measured curves and obtain the time delay of two IDCs from the time delay of both fits. Methods to determine the most appropriate analytical function are described in section 2.2. The missing ground truth is a major problem to properly perform the proposed investigation. Therefore, IDCs are simulated, shifted by a defined time as the ground truth for the bolus transit time, and superposed with Gaussian noise. The methodology for the generation of the data set containing the ground truth is described in section 2.1. We are not seeking to perfectly model

All computation is done on a computer with an Intel i7-6500U processor.

The following requirements were set for the data sets of two corresponding IDCs:

The computed curves should mimic indicator dilution curves.

The curves' morphology should be highly configurable.

The ground truth transit time of the two curves in a data set must be known.

The samples of the curves should be shifted relative to each other (so using two curves with not identical samples ensuring the depiction of two curves taken at two different locations).

Different noise levels should be applicable.

The finite element-based software COMSOL Multiphysics (version 5.4) with the packages Computational Fluid Dynamics (laminar flow) and Chemical Reaction Engineering (transport of concentrated species and transport of diluted species) was used for this

Assuming laminar flow, an incompressible fluid and no external forces (such as gravity) the continuity equation yields

the term

The dye will spatially diffuse due to its concentration gradient. Hence, the direction of the diffusive flux can be normal to the fluid flow direction. This is expressed by the convection-diffusion equation (Equation 5 with _{Diff} being the diffusive flux and

The equations are fully coupled and solved by the PARDISO (PARallel DIrect SOlver) solver which is based on the LU decomposition (Lower-Upper). The relative tolerance in the calculation of each measuring point is set to 0.1%.

The modeling in this paper is performed using COMSOL Multiphysics as a tool for numerical simulation. Alternatively, other software can be used analogously (Ansys, OpenFOAM, or similar). Analytical approaches can be used as well for this investigation such as the Taylor–Aris approach. The numerical approach via COMSOL enables the implementation and calculation of more complex geometries and the ability to calculate spatially resolved concentrations. This is advantageous for future applications. In this paper, the used geometry and properties are simplified and not realistic representatives of the circulatory system and therefore merits of the methods used for computing are irrelevant.

The fluid was modeled as a homogeneous liquid. In contrast, blood is a cell suspension containing the liquid plasma and the solid cells. Since the indicator of interest (Indocyanine Green—ICG) binds primarily to the proteins in the blood plasma, it will be affected by a heterogeneous distribution of cells (Alander et al.,

Inner radius: _{i} = 2

Outer radius: _{o} = _{i}

Rigid vessel wall:

Applied volume flow:

Inflow modeled as a ramp function reaching the designated volume flow in 1 s.

Open outlet with:

Laminar fluid flow and convection (_{critical}

Incompressible fluid: ρ =

Non-Newtonian fluid behavior modeled by the Casson model (see

No wall slip: _{i}) = 0

No external pressure or hydrostatic pressure

No gravitational force.

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

The volume flow is chosen based on reports on physiological values of cerebral arteries of this size. Thereby, a linear regression of the weighted average (weighted on the sample size of each report) is performed and the value for the chosen diameter is rounded to the closest multiple of

For the simulation of the flow and the evaluation of the mean indicator concentration, a mesh with 175,176 elements was built. Each mesh element consisted of triangles with a radially decreasing side length ranging from 10^{−4} to 10^{−6} _{sampling} = 1, 200

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 _{ICG}. The blue, red, and orange curves show the dilution curves in a distance of 50, 100, and 150

The following steps were applied to fulfill the requirements:

Duplication of one IDC (see

Temporal shift of one IDC by a known value (ground truth) ranging from 1 to 4 frames in the case of 25

Different sub-sampling to the desired sampling rate (e.g., 25 or 60

Application of white Gaussian noise to both curves to represent Johnson–Nyquist noise (SNR ranges from 8 to 20

The motivation of duplicating one IDC is strongly linked to the expected temporal shift. The field of view in cerebrovascular surgery is ~3 × 3 ^{2}, the mean flow velocity is about ~

Different sources of noise will appear in measurements, e.g., thermal noise from the recording device and patient-related noise such as the heart rate. We chose to represent the Johnson–Nyquist noise in our data set by additive white Gaussian noise due to its similar characteristic. Additive white Gaussian noise is described by the probability density function shown in Equation (7) with μ as the mean value and σ as the standard deviation.

μ was set to zero and σ is calculated using the definition of the SNR (Equation 8).

With the correlation

of the noise power _{Noise}, the standard deviation σ and the approximated signal power

of the noise-free IDC samples

We did not add any other noise due to the lack of information on its characteristic since mostly the separation of desired signal and noise is not possible. In addition to Johnson–Nyquist noise, we could consider to include shot noise. The quantum fluctuation of the photons hitting the detector are dependent on the scattering, absorption, and fluorescence events in a tissue slab, which are stochastically determined processes. Hence, they can be interpreted as shot noise. Following the central limit theorem, the uncertainties of the interaction of a photon relies on the superposition of multiple randomized events and therefore shot noise can be assumed as normally distributed and modeled by additive white Gaussian noise (Wohland et al.,

The application of noise is done for each data set independently, so each time a new noise set is generated and added to the signal. The application of noise on the signal assumes that the error in transit time measurement is linked to amplitude noise and is therefore an appropriate representation of disturbing influences on the measurement.

The pipeline is shown in

In summary, different raw indicator dilution curves were simulated in COMSOL Multiphysics and imported to MathWorks MATLAB to compute different sub-samples, applying a temporal shift and noise. In total, 12,096 data sets were computed:

3 raw IDCs with different morphology;

2 sampling rates: _{sampling, 1} = 25 _{sampling, 2} = 60

12 different sub-sampling combinations;

7 different SNR levels (8–20 dB in 2

6 independent realizations for each noise level;

4 different temporal shifts (transit times).

As a result, a large amount of diverse data sets, each containing two corresponding IDCs [_{1}(_{2}(

MathWorks MATLAB R2019b was used for the following evaluation and analysis.

Typical analysis of transit time is based on feature-based methods such as peak to peak distances or using the cross-correlation of both signals as a tool to determine the transit time (Cimalla et al.,

We propose to fit a mathematical model to the data sets of both IDCs _{1}(_{2}(

Different features will be used to extract the transit time Δ

Before fitting mathematical models to the two curves of a data set, a pre-processing step is needed. Each IDC needs to be cut to ensure a robust fitting. Two examples are as follows: First, prior to the arrival of the indicator zeros are recorded. Including them into the fitting algorithm does not make any sense and can cause problems. Second, fitting the parabola model to the whole signal does not make sense either, therefore the peak is detected and only data points surrounding it are considered for fitting.

Addressing the problem of the first example, the data points prior to the appearance of the indicator should be removed since they contain no information and are basically zeros with noise applied to it. Equation (12) has shown to be empirically appropriate to find this cut point _{cut1} on the abscissa representing the appearance of the indicator despite of the impact of noise before the peak of the IDC (_{max} is the maximum concentration, _{up} only considers the data points left of the peak, and _{down} only considers the data points right of the peak). To determine the elements of the equation, a moving average with a window length of 20 samples is used to reduce the impact of noise on the signal. To ensure unambiguity, the search for the elements on the up slope is performed from left to right and on the down slope from right to left. This pre-processing is applied on all data sets independently to the further processing.

Addressing the problem of the second example, it is reported that cutting the data as a pre-processing step enhances the robustness and accuracy of the fit. Seven different levels of cutting are introduced: they range from _{max}) to _{max}) in steps of 0.1. Further, it is reported that an unequal cutting to the left compared to the right of the peak is favorable due to different information content in the data (e.g., exclusion of recirculation; Borges et al.,

No further cutting left of the peak and variable cutting levels right of the peak.

Cutting left of the peak at half the value of the variable cutting to the right of the peak.

Variable but identical cutting left and right of the peak.

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.

In summary, 21 different variations (7 cut levels times 3 cut methods) to cut the IDC as a pre-processing step are possible and will be evaluated in this analysis.

The mathematical models are described in the following sections as well as the initial estimation of their parameters _{init} = (_{1,init}, _{2,init}, …, _{n, init}). The initial estimation is needed to run a least squares optimization algorithm to determine the function's parameters

with _{lb} being the lower bound and _{ub} the upper bound of the respective parameters _{i}. To solve Equation (13), different algorithms like the “trust-region-reflective” or the “Levenberg–Marquardt” algorithms are suitable. The “trust-region-reflective” algorithm is used in our study (More and Sorensen,

The time shift of the functions can be determined after fitting the mathematical models to the pre-processed (cut) input data (two IDCs). The features used to determine the time shift are specific to the mathematical models and described in the corresponding following sections.

The parabola model is defined as shown in Equation (14). This model is fitted to the data points around the peak (according to the cutting method specified in section 2.2.1) of the IDC.

The parabola has a distinct morphology that facilitates setting up

No boundaries are needed for the optimization of the parameters

To determine the shift of the two fitted parabolas of one data set, the maximum of each parabola is used as feature.

The LDRW model function is defined as shown in Equation (18). This model can be fitted to the whole IDC.

According to the approach of Mischi et al. (_{1,init} represents the total amount of injected dye. Consequentially, it can be expressed as the integral of _{2,init} represents the slope of the ascending arm. _{3,init} represent the abscissa coordinate of the maximum. _{4,init} represents the zero time of the distribution and in our case it is slightly shifted away from the distributions maximum to be sure to not prune the distribution at the beginning because our function is defined only for positive values of _{4} and therefore optimization will be only performed for _{4}.

The boundaries for

To determine the shift of two fitted LDRW functions, the maximum as well as the maximum and minimum of the first and second derivative of each LDRW function are used as features. Further, the cross-correlation of the two LDRW functions is used to determine the time shift. It is applied to the two LDRW functions and their first derivative, as well as to the functions and their first and second derivative with an additional linear temporal interpolation with a factor of 100 (the functions are continuous but MATLAB is a vector and matrix based software and the calculation of the cross-correlation requires a discrete input and therefore the functions are sampled with a sampling rate 100 times higher than the input sampling rate for the fitting, e.g., _{sampling} = 25 _{sampling} = 2, 500

The gamma variate model is defined as shown in Equation (23). This model can be fitted to the whole IDC.

According to the approach of Madsen (_{1} and _{2} represent the coordinates of the maximum (obtained by differentiating the function and setting it equal to zero). _{3} represents the slope of the decreasing arm of the function. _{4} represents the zero time of the distribution and in our case it is slightly shifted away from the distribution's maximum to avoid syntax errors.

To calculate _{3,init}, an additional step is needed. Therefore, a moving average with a window size of 20 samples is applied and the endpoint [_{SW, n}, _{SW, n})] is used to calculate _{3,init}.

The boundaries for

To determine the shift of two gamma variate functions, the same features and methods are used as for the LDRW function.

The mono-exponential model is defined as shown in Equation (28). This model is fitted to the descending arm of the data set. Thereby, the descending arm is defined as the data points _{max}) after running a moving average on

According to Brands et al. (_{1,init} and _{3,init}, the smoothed data are used. _{1,init} and _{3,init} represent the coordinates of the maximum of the function, hence the start of the defined sector of the mono-exponential function. _{2,init} represents the slope of the function.

No boundaries are needed for the parameter optimization

Determining the shift of two Mono-Exponential functions is not trivial since their maxima and minima depend solely on the sections of _{max} and _{n} − _{max}). Finally, the cost function _{c} (see Equation 32) is minimized to obtain a time stamp _{eucl} for each IDC. The transit time can be calculated as the difference of both time stamps _{eucl} of one data set.

A control group is added to enable the assessment of the performance of the proposed methods. Using the peak of the raw data relies on a single sample point in each IDC and is very noise sensitive. The cross-correlation of the IDCs relies on a larger set of sample points is less noise sensitive and therefore used for this performance assessment. The cross-correlation is applied to the raw input data _{interpolate}(_{interpolate}(

In total, 22 combinations of mathematical models and features as well as 4 control groups are used to assess the transit time error on all 12,096

The performance parameter used to compare different methods ascertaining the transit time was defined as the absolute difference between the measured transit time Δ_{calc} of two IDCs of one data set and the ground truth transit time Δ_{frames} in frames to emphasize whether a sub-frame rate accuracy is accomplished or not. The calculation of the error in frames ε_{frame} is shown in Equation (34).

The mean value μ and standard deviation σ of ε_{frame} will be used for the evaluation. The μ and σ are calculated using 12,096 data sets

3 morphologically different IDCs;

7 different SNR levels;

6 cycles for each SNR level;

4 different transit times Δ

2 different sampling rates _{sampling};

12 different combinations of sub-sampling.

for each combination of methods and pre-processing (546 in total)

26 combinations of different mathematical models, features, and control group;

3 methods of cutting the data;

7 levels of cutting the data.

The absolute error at a certain SNR ε(

The following two sections show the results for the generation of the data sets containing two corresponding IDCs with a ground truth of the transit time Δ

An example of the results of the data generation is shown in

Power spectral density of the

The mean computation time from the input of two IDCs to the output of the transit time error was ~0.298 _{init} for both IDCs, the optimization of the parameters and finally the determination of the temporal delay _{calc} as well as the comparison to the ground truth transit time Δ_{frame}.

Visualizing all 8,036 combinations of methods ascertaining the shift of two IDCs is not appropriate. To facilitate the visualization of the results, a 7 × 7 matrix is used to show the mean ε_{frame}. The values for the mean ε_{frame} are determined for the values labeled at the axis, the space in between is interpolated. Each matrix is specific for a combination of a mathematical function, feature, cutting method, and sampling frequency _{sampling} and shows the mean ε_{frame} in dependency of the preformed cutting levels as a pre-processing step and the SNR. The color code in _{frame} and errors larger than 1 are presented in yellow. The performance of the cross-correlation on the raw data set with _{sampling} = 25 _{sampling} = 25 _{sampling} = 25 _{sampling} = 60

_{frames} using the raw data sets (_{sampling} = 25 _{frames} using the linearly interpolated data sets (_{sampling} = 25 · 100

_{frames} using the gamma variate model on the data sets (_{sampling} = 25 _{frames} using the gamma variate model on the data sets (_{sampling} = 25 _{frames} using the LDRW model on the data sets (_{sampling} = 25

_{frames} using the raw data sets (_{sampling} = 60 _{frames} using the linearly interpolated data sets (_{sampling} = 60 · 100

_{frames} using the gamma variate model on the data sets (_{sampling} = 60 _{frames} using the LDRW model on the data sets (_{sampling} = 60 _{frames} using the LDRW model on the data sets (_{sampling} = 60

In

Obtaining the ground truth transit time of two corresponding indicator dilution curves is a challenge. We have proposed and implemented a method to simulate adjustable IDCs with a known ground truth of the transit time in a statistical relevant quantity. Thereby, a high degree of freedom in the variation of the signal's morphology, sampling rate, sample positioning, transit time, and noise level is given. This allows a versatile use of the

The presented dilution curves were simulated by an _{ICG} (see

We have evaluated the performance of different methods ascertaining the transit time with varying complexity. Thereby, the transit time was calculated according to the systemic mean transit time theorem for single input and single output systems as defined by Perl et al. (_{frame} of <0.85 for all SNR (8–20 dB) and cut levels (see _{sampling} = 25

_{sampling} = 25 _{sampling} = 60

Mean absolute error ε and its standard deviation for two sampling frequencies _{sample} in dependency of the signal-to-noise ratio (SNR).

20 dB | 10.0 ± 7.1 ms | 7.2 ± 5.3 ms |

18 dB | 12.1 ± 7.9 ms | 8.3 ± 6.4 ms |

16 dB | 13.9 ± 9.5 ms | 10.7 ± 8.8 ms |

14 dB | 16.8 ± 12.1 ms | 11.4 ± 9.4 ms |

12 dB | 19.9 ± 15.1 ms | 15.3 ± 12.0 ms |

10 dB | 23.9 ± 18.5 ms | 18.5 ± 13.7 ms |

8 dB | 31.3 ± 22.9 ms | 23.5 ± 16.9 ms |

_{sample}_{sample}

Maintaining and restoring a sufficient blood flow is a crucial aspect in preventing post-surgical complications such as cognitive impairment (Lawton and Lang,

The proposed hypotheses are verified with some limitations regarding the methods used and present noise level in the signal. The investigated mathematical models and features represent a fraction of the possible methods. Therefore, more mathematical models and features can be implemented and tested on the generated data set. The investigated noise levels are chosen reasonably according to our experience but could be expanded to a larger range and finer gradation. A reduction of the noise has a large impact on the accuracy ascertaining the transit time, so different noise reduction techniques should be considered and evaluated. Nevertheless, care should be taken since the cross-correlation is depending on the data sets' morphology and obviously the morphology is affected by temporal filtering and thereby might affect clinical decision making (Lenis et al.,

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

The research related to human use complies with all the relevant national regulations, institutional policies and was performed in accordance with the tenets of the Helsinki Declaration, and has been approved by the authors' collaborators institutional review board or equivalent committee.

AN and WN conceptualized the research. AN, MR, and WN designed the research. AN and MR organized the software and database. MR performed the statistical analysis. AN wrote the first draft of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Priv. Doz. Dr. med. Uwe Spetzger of the Städtisches Klinikum Karlsruhe for the support by providing videos of neurovascular interventions. We thank Mr. Manfred Schroll, the IT responsible of the institute, for enabling a sufficient working environment during the Covid-19 outbreak. We thank Deborah Nairn and Mark Nothstein for proofreading this document. We acknowledge support by the KIT-Publication Fund of the Karlsruhe Institute of Technology.

The Supplementary Material for this article can be found online at: