Aortic dissection (AD), is a hazardous aortic disease with high mortality. The formation of AD is commonly initiated by the dilatation of the aorta or high blood pressures that tear the intima, allowing blood to flow into the aortic wall. The pulsatile pressure of the circulation then drives the blood. It separates the layers of the aortic wall, resulting in the formation of a true lumen and a false lumen  (figure 1.a). The false lumen represents the blood-filled space between the dissected layers of the aortic wall, while the true lumen is the usual passageway of blood.
The ADs are classified as Stanford type A (proximal) or B (distal) in the Stanford system (figure 1.b). If the ascending aorta is involved (Stanford type A), an acute condition with a high mortality rate within a few hours occurs in most cases due to the high blood pressure right after the aortic heart valve. On the other hand, Stanford type B cases (in the descending aorta) may become chronic, which means that the onset of the dissection dates back more than 14 days and patients can often be treated with medical therapy. In both cases, the symptoms of AD patients are sudden severe chest or upper back pain, which are not easily assignable to this disease.
Detecting an AD can be difficult because the symptoms are similar to those of a variety of health problems. Ultrasound Scanning (sonography), Magnetic Resonance Imaging (MRI) and Computerized Tomography (CT) are expensive techniques currently used for this purpose, with experts needed to read and interpret the images. Nevertheless, an easy to use and still reliable method for pre-identification of AD would be beneficial. Furthermore, tracking the development of the disease, such as false lumen expansion and false lumen thrombosis can be very helpful for the medical management of AD.
The presence of a false lumen alters the aortic haemodynamics and also changes the tissue distribution in the thorax. These changes can be identified and quantified by bioimpedance techniques such as impedance cardiography (ICG). In ICG, a current field longitudinally across a segment of the thorax is applied using a constant low magnitude and high-frequency alternating current. It is a non-invasive, safe, easy to use and low-cost method for measuring several cardiodynamic parameters (e.g. the Stroke Volume (SV) and the Cardiac Output (CO)) continuously . Besides, this method is portable, and the analysis could be automated.
By injecting a low-amplitude alternating current into the thorax and measuring the voltage drop ΔV on the thorax, impedance changes can be evaluated during a cardiac cycle. The negative of the first time-derivative of the impedance signal −|dZ/dt| is known as the impedance cardiogram. Since the conductivity of the blood-filled aorta is much higher than that of the surrounding tissue types, changes of the measured impedance are strongly related to changes in the aorta. Since both the blood volume and the blood flow will change in the case of an AD, an altered impedance cardiogram can be expected, which makes the ICG a good candidate for diagnosis and monitoring purposes [5,6,7].
A 3D numerical simulation model is used to compute the impedance changes on the thorax surface in case of the type B aortic dissection. A sensitivity analysis through the Global Sensitivity Analysis (GSA) technique is applied to investigate different electrode configurations in the simulation model with different input parameters to cover as many patient-specific cases as the dimension of the input space. The final aim would be finding the desired electrode configuration, which gives the highest difference between the impedance cardiograms of the healthy condition and the ones with the AD.
Sources of impedance changes during a cardiac cycle
The measured electrical impedance without respiratory or cardiac activity is known as static thoracic base impedance Z0. Upon ventricular ejection, a time-dependent pulsatile impedance change ΔZ(t) is obtained. When ΔZ (t) is superimposed on Z0, the time variable total transthoracic impedance Z(t) is registered.
By eliminating the oscillating cardiac-asynchronous respiratory component, Z(t) comprises a static DC component Z0 (22 Ω – 45 Ω) and a dynamic AC component ΔZ(t) (0.1 Ω – 0.2 Ω) which is synchronous to cardiac activity [8,9]. In many studies, sources of the thoracic impedance changes have been investigated, and a consensus is lacking in the origins of cardio-synchronous impedance changes due to different model assumptions. Hereof different approaches are listed in . Of course, simulation of transthoracic bioimpedance signals considering all possible time-dependent sources is impossible. Also, comparing experimental results obtained from dissected patients with earlier measurements in healthy states, is practically infeasible. Nevertheless, since the discrepancy between the healthy and the dissected state and not the evaluation of absolute measurement values is in the focus of this work, only the velocity induced blood conductivity variation and the volumetric changes of the aorta are considered as the sources of ΔZ (t) in a healthy case. It should be noted that the magnitude of Z0 not only varies among individuals and the frequency of the applied current but also with the electrode configuration used for signal acquisition.
Volumetric changes of the blood-filled aorta
The volumetric expansion of the blood-filled aorta changes corresponding to the cardiac pulse wave. For the sake of simplicity, a spatial average time-dependent cross-sectional radius of the aorta has been used in the simulation model for two sections separately, the aortic arch and the descending aorta, see figure 2. The data are based on measurements provided in  from a young, healthy male volunteer at rest.
Velocity induced blood conductivity variation
The electrical properties of resting blood mainly depend on the volume fraction of red blood cells (RBCs) called haematocrit, the temperature, and cell shape. However, the electrical properties of flowing blood are found to be influenced by the flow rate .
A spatial average time-dependent velocity of the blood flowing inside the aorta, taken from the experimental data provided in , has been taken into account for the aortic arch and the descending aorta (figure 3). During the systolic phase of a cardiac cycle, the heart contracts to pump blood into the aorta, and in the diastolic phase, the heart relaxes after contraction. This pulsatile blood flow causes the variation of blood conductivity inside the aorta. The reason is the orientation and deformation of the RBCs in case of flowing blood. At higher velocities, the shear stress increases, which consequently deforms the RBCs in the layer with the highest stress close to the vessel wall and also aligns them throughout the vessel. Both effects lead to a higher conductivity than the resting blood (figure 4) [13,14].
The Maxwell–Fricke equation for the conductivity of blood reads :
In [15,16], it is shown that the blood conductivity during pulsatile blood flow is not the same at any given velocity during acceleration and deceleration. This disparity is a consequence of the RBCs inability to achieve complete randomization at end-systole, which leads to less but still considerable conductivity changes during the cardiac cycle. However, for simplicity, conductivity changes which are shown in figure 5 have been assumed in the simulation model.
Geometry, physics and formulation
A 3D numerical simulation model is used to investigate the changes in the electric potential and the impedance changes on the thorax surface. The model has been set up in COMSOL Multiphysics  for the underlying time-harmonic current flow problem. Since the duration of the cardiac cycle is much higher than the period of the injecting current, simulations can be performed in the frequency domain. The electric potential drop is evaluated between the measuring electrodes by solving the Laplace equation for the electric potential V:
The model consists of a simplified geometry, as shown in figure 6. Three pairs of source (injection) electrodes are placed on the surface of the thorax (each pair in one vertical line) and inject an alternating current with a magnitude of 5 mA and a frequency of 100 kHz asynchronously. For each injection, the electric potential drop is evaluated between five measurement electrode pairs (each pair in one vertical line) which leads to the thoracic impedance
The boundary conditions are:
- V = constrant, on the top source electrode;
- ∫s[σ + jωɛ] ∇V · n dS = I0, on the top source electrode;
- V = 0, on the bottom source electrode;
- n. [σ + jωɛ] ∇V = 0, on the thorax surface
Modelling physiological changes in the presence of the false lumen
It has been shown in the literature [21, 22] that the blood flow is highly disturbed locally inside the aorta and changes to turbulent flow with strong recirculation. As depicted in figure 7, flow disturbances occur around the dissection, which inhibits the deformation and orientation of the RBCs. Thus, the flow shear rate and, consequently, the electrical properties of blood are altered. At the highest blood flow velocity and consequently the highest deformation and orientation rate of the RBCs, a remarkable difference in the electrical conductivity between the healthy (non-disturbed flow) and the aortic dissection conditions can be expected . Since no experimental or simulation data exist regarding conductivity changes of blood in this kind of disturbed flow, it is assumed that with a radially growing false lumen also the blood flow disturbances increase and the conductivity changes decrease. To quantify this assumption, a damage factor DF has been introduced, which is the ratio of the volume of the dissection to the maximum volume of the false lumen:
The damage factor DF is applied to the conductivity changes of blood in the descending aorta during the cardiac cycle to model the decrease in the conductivity changes of blood due to dissection (see the section on Implementation of global sensitivity analysis …, on the next page).
The aim of a Global Sensitivity Analysis (GSA) is to quantify the connection between the variance of the model output given the variability of its input. GSA is distinguished from a local sensitivity analysis since it investigates the whole input space of each random variable. This study aims to use a GSA technique to identify which electrode configuration has significant changes in the impedance cardiogram −|dZ/dt| given by the uncertainty on the developed status of AD. The discrepancies among obtained impedance cardiograms of the healthy and dissected models determine different states of the patient and are the criteria to simulate AD identification.
One of the most known techniques in GSA is the variance-based method. Here, the output variance is portioned in the sum of the contributions of each random variable. Consider a mathematical model output y as a function of an input random vector x of dimension n, the Sobol’ decomposition reads as 
A random variable xi is considered to be influential (non-influential) to the model output if the conditional variance V(E[y|xi]) is large (small) enough compared to the variance of the quantity of interest. Therefore, the first-order sensitivity index (or main Sobol’ index) can be derived as:
In this application, only the first-order and the total-order indices are considered. Any interaction between the input random variable can be derived by subtracting the first index to the total. Consequently, the difference will result in the amount of interaction present in the model.
The Sobol’ indices are computed from a Polynomial Chaos Expansion (PCE) of the model , which also represents a valid mathematical metamodel. PCE consists of the sum of orthogonal, multivariate polynomials ψα of increasing order up to some maximal polynomial order p. The polynomials are multiplied by expansion coefficients yα, which can be estimated with different methods . A PCE is written as:
Implementation of global sensitivity analysis (GSA) to the simulation model
To assess the sensitivity analysis, the input and output spaces together with the numerical models, have to be established. Since the aim of the study is to catch the difference between different health conditions, two numerical models are set. The first one refers to the healthy condition and the second one to the dissected condition. As described in section 2–2, the latter differs from the first one in the presence of the false lumen and another blood flow profile. Therefore, different input spaces are produced for each model. Besides, introducing variability in the input space of the models will guarantee the realization of as many patient-specific cases as the dimension of the input sample. Thus, a deeper understanding of the impedance cardiography for a human thorax can be revealed. From the models’ evaluations, the PCEs for the healthy and dissected conditions are constructed, and analyzing the differences between them will guide toward the choice of the best electrode configuration.
The input space of the healthy case is composed of only two random variables, namely the maximum radius of the true lumen RTL and the blood conductivity coefficient θH. RTL is considered uniformly distributed between 1.35 and 1.95 cm, according to the study . As it was shown in figure 2, the average radius of the aorta changes in time due to pressure changes over a cardiac cycle. Since the aorta in the simulation model is considered as a blood-filled lumen, different values of the RTL emulates different blood volumes dilating the aorta, in other words, different stroke volume values. It has to be mentioned that changes in the stroke volume will also vary the peak velocity of the blood passing the aorta. Since for higher velocities, almost all the RBCs are entirely aligned and deformed, the differences among the blood conductivity changes for different stroke volumes are not significant. Therefore, here, for simplicity, it is neglected. Based on the distribution's moments of RTL, the stroke volume changes approximately between 62 and 140 ml in the simulation model.
The blood conductivity changes as a function of the reduced average velocity 〈v/R〉 for five different haematocrit levels are shown in figure 5. For the parameterization of the haematocrit-dependent blood conductivity changes corresponding to other haematocrit values, the coefficient θH has been introduced. It emulates a scaling factor for Δσ at a haematocrit level of 35% up to 55% at maximum. Given h as the index that represents different haematocrit levels, the parameter θH,h is set as:
The dissected condition includes both the random variables of the healthy condition plus the radius of the false lumen RFL and the radial position of the false lumen to the true lumen αFL (representing different possible positions of the false lumen). The two new parameters have been considered uniformly distributed since knowledge regarding the dimension and position of the false lumen is not available prior to the measurement, and the uniform distribution better represents the lack of knowledge regarding a model variable. The description of the input space for both case studies is given in Table 1. Furthermore, since in the case of aortic dissection the damage factor DF also affects the conductivity changes, the blood conductivity results in:
Input space description for the healthy and dissected study cases.
The two models produce a measurement of the impedance cardiograms for each source electrode pairs Ninj and for each measuring electrode pairs Nmeas at each time step Nt, as described earlier. The time interval is limited to the first half of the cardiac cycle. After setting the input and output space characteristics, the next step is to set up the options for the PCE. It is essential to notice that a minimum number of point evaluations Ns is needed to have an accurate expansion. Ns is defined to satisfy
Two PCE functions
Informed consent has been obtained from all individuals included in this study.
The conducted research is not related to either human or animal use.
Results and discussion
Three source electrode pairs (A, B, and C) and five measurement electrode pairs (m1 to m5) are considered in the simulation models of the healthy and dissected conditions (figure 9).
Each simulation contains an injection from one of the source electrode pairs and measuring from all the five measurement electrode pairs. For each simulation, the impedance cardiogram − |dZ/dt| is computed and through all the simulation results (for different input variables), two metamodels,
Combinations of injections have been applied, and the results of calculating
In figure 12, the results of the sensitivity analysis on
From the sensitivity analysis in the dissected case in figure 12.b, RTL is less sensitive than in the healthy case. The next most sensitive parameter is RFL, which emulates the blood conductivity changes due to the existence of the false lumen. The sensitivity analysis of the difference between the healthy and the dissected cases
In figure 13,
To summarize, in the first stages of the AD in which the existence of the false lumen does not make apparent changes to the rheology of the blood flow, the presence of the disease by impedance cardiography might not be noticeable. However, as soon as the dissection creates remarkable pathological changes in the cardiovascular system, the changes in the measured impedance cardiogram due to the development of the disease such as false lumen expansion and false lumen thrombosis, might be trackable.
This study aims to investigate different electrodes configurations concerning the discrepancy between the healthy case and type B aortic dissection case. For this purpose, a numerical simulation model using a simplified geometry of the thorax has been set up. Since there are many uncertainties regarding the parameters that affect the results, a Global Sensitivity Analysis (GSA) has been implemented to quantify the relation of the variance of the model output (impedance cardiogram − |dZ/dt|) and the variability of its input. The sensitivity analysis of the models’ output shows that the highest difference between the impedance cardiograms of the healthy and dissected conditions occurs when the velocity is highest in the aorta. Also, the highest difference in the maximum value of the impedance cardiograms of the healthy and dissected conditions
It has been shown that the size of the false lumen has a tremendous effect on the impedance cardiogram of the dissected condition. This effect shows that in some cases, the pathological changes caused by false lumen might end up in different calculated haemodynamic parameters by impedance cardiography in comparison to other methods. Furthermore, the development of the aortic dissection disease such as false lumen expansion and false lumen thrombosis cause some pathological changes which will alter the measured impedance cardiogram. Thus, applying impedance cardiography to track these changes can be helpful for the medical management of AD.
For future works, the electrical conductivity changes of the blood in case of disturbed aortic flow by setting up simulation models and experiments, and also the possibility of tracking false lumen thrombosis by impedance cardiography, will be investigated.
This work is supported by Graz University of Technology through the LEAD Project “Mechanics, Modelling, and Simulation of Aortic Dissection”. Also, the authors would like to acknowledge the use of high power computing resources provided by the ZID of Graz University of Technology.
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Altamirano-Diaz L, Welisch E, Dempsey AA, Park TS, Grattan M, Norozi K. Non-invasive measurement of cardiac output in children with repaired coarctation of the aorta using electrical cardiometry compared to transthoracic Doppler echocardiography. Physiol Meas. 2018; 17;39(5): 055003.)| false https://doi.org/10.1088/1361-6579/aac02b 10.1088/1361-6579/aac02b 29695645
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