The cardiovascular system consists of the heart, which is the central unit, the vessels, and blood. Its main functions are transporting blood, which carry essential substances, throughout the body. It also removes products and wastes produced from cellular metabolism. The cardiovascular system is closed, as the blood flows from the heart to every region of the body and back again [1].

Blood vessels are classified mainly as veins, arteries, and capillaries. All blood vessels basically consist of lumen (where the blood can flow) and wall [2]. In atherosclerosis, the lumen area is narrowed due to the formation of deposits on the vessel wall. These deposits may be made of cholesterol, macrophages, calcium, or other substances from the blood. Sometimes, the deposits grow to a certain size and stop growing, causing no problems. However, sometimes, the deposits grow and block the lumen, leading to many health problems [3].

The weakness in vessel wall causes the wall to be swollen and form aneurysms. These aneurysms may be formed in any blood vessel. Small ones are not dangerous but at their severe stages, some aneurysms can rupture, leading to internal bleeding [4]. In diagnosing vascular diseases, imaging techniques and ankle-brachial index are the most commonly used techniques which require complex instruments and, thus, the continuous presence of a trained operator [5].

Bioimpedance analysis (BIA) has been recently used for diagnosing several diseases since it is non-invasive, simple for users, and does not require a trained operator [6]. In BIA, weak AC current (I) is applied by two electrodes and the produced voltage (V) is measured by the same electrodes or by another two electrodes to calculate the impedance. In tetrapolar electrodes, two electrodes which are called “current electrodes” supply the current, while the two other electrodes are called “voltage sensing electrodes”, and are used for voltage measurements, as shown in Figure 1 [7].

The main objective of this research study was to investigate the effects of vascular diseases on vessel impedance. Herein, we present mathematical models to describe different vascular conditions. These conditions are normal vessels without any disorders, vessels with mural plaques, and vessels with wall aneurysms.

Moreover, a finite element model of the blood vessel and surrounding tissues was constructed and simulated. The blood vessel was modeled in the case of normal; thereafter, the mural plaques and aneurysms of different sizes and locations were added to the vessel. At each case, the model was simulated and the impedance was measured across the sensing electrodes. These measurements were carried out using three different frequencies (50 KHz, 100 MHz, and 1GHz), to determine the best frequency capable of detecting the vascular diseases.

Finally, the results were extracted and discussed to demonstrate:

- The ability of mathematical models in describing different vascular disorders, and their impact on the measured bioimpedance.

- The ability of BIA in detecting vascular diseases.

In this part, mathematical models were suggested to describe different cases of vascular disorders. Firstly, the general impedance law was described by equations (1) to (3) below [8],

Where:

Y is the admittance.

Z is the impedance (reciprocal of admittance).

G and C are conductance and capacitance.

A and L are area and length.

σ(f ), ∈ (f ) and ∈0 are the electrical conductivity, relative permittivity, and vacuum permittivity.

Secondly, mathematical models were suggested to describe the vessel impedance in three medical cases: normal case, stenosis case, and swelling (aneurysm) case.

In this section, 3D modeling of a blood vessel was simulated by COMSOL MULTIPHYSICS.5, and the impedance was measured at different vascular cases.

In model geometry, the simplified forearm model was constructed. The forearm was constructed as structures containing five components (i.e. skin, fat, muscle, bone and blood vessel). The taken length was 100 mm, as shown in Figure 7 . The thicknesses of skin, fat and muscles were 3 mm, 10 mm and 25 mm, respectively, and the bone diameter was 2 cm. The radiuses of blood and wall were 1.5 mm and 1.7 mm, respectively [10][11].

A tetrapolar configuration of copper electrodes was used; the outer two electrodes supplied current with values (1 mA) and the inner pair (A & B) were used as voltage sensing electrodes. The inter-electrode spacing was 25-10-25 mm. The dimensions of the sensing electrodes were

5x10x1 mm3 and cover 20 mm from the forearm length, as illustrated in Figure 7 .

Figure 7 shows the blood vessel in the normal case when the vessel is located under the skin.

Two different vascular disorders were studied using this model, as shown in Figure 8 .

Figure 8a displays the vessel with mural plaque; this plaque shape was studied at different heights (from 0.5 mm to 3 mm) and lengths (from 2 mm to 20 mm). In Figure Figure 8b , wall swelling was presented with different heights (from 0.5 mm to 4 mm) and lengths (from 2 mm to 20 mm). The center of the plaque or aneurysm was located at Z=50 mm.

In “single impedance measurement”, the voltage electrodes ‘A&B’ were positioned from Z=40 mm to Z=60 mm.

In “multi-segment impedance measurements”, the voltage electrodes ‘A&B’ were positioned firstly from Z=15 mm to Z=35 mm, and then moved with each 5 mm step until they reached Z= 65 mm & Z=85 mm; the impedance was measured with each movement.

Three different frequencies (50 KHz, 100 MHz, and 1GHz) have been used in modeling simulations. The dielectric properties of tissues have been defined by Gabriel et al. (1996), as shown in Table 1 [12] . The blood flow is assumed to be laminar and blood density is 1060 and viscosity is 0.005 Pa.s [13].

In meshing, the ‘finer’ element size is used to increase the model effectiveness in detecting the vascular disorders, as shown in Figure 9 .

Finally, the simulation was executed and the impedance was measured across the voltage electrodes at different vascular cases using different frequencies. In the next section, the simulation results will be represented and discussed.

In detecting vascular diseases, the imaging techniques and ankle-brachial index are the most used techniques. These techniques require complex and expensive instruments, and the continuous presence of a trained operator. In this study, the possibility of using the bioimpedance analysis (BIA) in detecting different vascular diseases was discussed. As it is a non-invasive, low cost, and simple analysis, and one that does not need a trained operator, it presents many benefits.

In recent years, BIA has become a popular modality in detecting various body diseases (e.g. blood analysis and cancer detection), in chest measurements (e.g. impedance cardiography), and in imaging [14]. There are a wide range of bioimpedance utilizations to estimate and evaluate the vascular states. In a study by Al-Harosh and Shchukin (2017) [15], the electrical impedance method was used in peripheral vein detection by applying a known alternating current of frequency (100 KHz) and measuring the resulting potential. Anand et al. (2016) [10] also applied measurements on human forearm within a frequency range of 1 KHz to 2 MHz; these measurements were done at three instances of radial artery diameter to mimic the condition of blood flow.

In invasive measurement, Stiles et al. (2003) [16] had constructed a finite element model to determine the feasibility of characterizing various types of atherosclerotic lesions in vivo. This was accomplished by simulating two electrodes which were attached to an angioplasty balloon in the coronary artery and measuring the impedance of type III, IV, Va, and Vb lesions. The simulation results showed that the lesions near the electrodes had a greater effect on the measured impedance; also, the four lesions types could be distinguished.

In non-invasive measurements, Ben Salah et al. (2015) [17] had performed several investigations in the field of diagnosis of cardiovascular diseases using the bioimpedance method. New techniques have been suggested in evaluating several hemodynamic parameters, such as cardiac output, stroke volume, aortic compliance, and cardiac index. Moreover, Anderson et al. (1984) [18] had presented a review of a theoretical basis and clinical application of bioimpedance plethysmography in the evaluation of peripheral vascular diseases. However, the accuracy of bioimpedance plethysmography was poor, especially in detecting small plaque sizes since the signals might contain motion artifacts and the flow could not be completely blocked, resulting in incorrect analysis. Moreover, the limb was modelled as a uniform cylinder and the vascular expansion was assumed to be uniform. This assumption was probably true only for healthy vessels, but not for diseased vessels. Also, the diagnosis procedures included long occlusion time (>2 min), causing a change in blood properties due to the decrement in oxygen saturation.

In other previously published studies, some authors studied vascular diseases and their effects on the mechanical properties of blood vessels, e.g. Vallez et al. (2015) [19]. These effects or changes would be reflected as a change in the electrical properties of the blood vessels.

Herein, we conducted a study to evaluate the possibility of detecting vascular diseases using BIA. In this study, mathematical models and finite elements models were developed to describe the vessel impedance for different medical conditions. Furthermore, this study was developed to estimate different vascular disorders non-invasively as a novel method, based on the bioimpedance phenomena. Various studies were applied to find the suitable frequency and electrode configurations for detecting the vascular diseases. To verify the passage of current within the blood vessel and guarantee the current uniformity, ‘distance current electrodes’ were recommended to be used. In the case of voltage electrodes, their length must be appropriate for the vessel diameter, whereas electrodes with small length cannot cover the full vessel diameter and, thus, their abilities to detect vascular disorders are reduced. Moreover, the results were improved by decreasing the internal distance between the voltage electrodes.

The simulation results demonstrate that the preferred frequency is 50 KHz, as it can detect the vascular diseases better than the other frequencies in the MHz or GHz range. This frequency (50 KHz) is characterized by high penetration depth and high efficiency to pass through intracellular and extracellular fluids. Moreover, it is high enough compared with the frequency of the heartbeat to consider the blood particles as stationary objects [14].

From the measured impedances, the presence of mural plaques impedes the blood flow and the increase in deposit height increases the probability of blocking the vessel. Also, it should be noted that the presence of deposits with any size increases vessel impedance and, thus, the measured impedance by electrodes. This what has been concluded before based on equation 9 to equation 12, whereas the presence of deposit decreases the blood area and increases vessel impedance. On the contrary, the presence of wall swelling (aneurysm) decreases the measured impedance, as it increases the blood volume (in the area of interest) and decreases its impedance, thus, vessel impedance. This is what has been deduced by equations 17 to 18.

The location of the aneurysm or plaque, relative to the position of the sensing electrodes, has a greater effect on the impedance. In the case of plaque, the maximum impedance is obtained only when all plaque sizes are located between the two sensing electrodes. Thereafter, the impedance begins to be decreased when a segment of plaque is situated outside the sensing electrodes. When all plaques are located away from the sensing electrodes, the presence of the plaque has an ignored effect on the measured impedance and, thus the value Z_S/Z_N is approximately equal to 1. These conclusions correspond to equations 13 to 16.

In vessels with mural aneurysms, the impedance has the smallest values when aneurysms of all sizes are located between the two voltage electrodes. Thereafter, the impedance is increased when the electrodes are moved further from the aneurysm, but the value ZA/ZN remains less than 1. When the aneurysm is located near the outside of the covered length by electrodes, the presence of the aneurysm has an ignored effect on the measured impedance and, thus, the value Z_A/Z_N is approximately equal to 1. These deductions correspond to equations 19 to 22.

This research presents the usage of bioimpedance technique in detecting vascular diseases. Mathematical models were suggested to describe the vessel impedance under different medical conditions. Also, the finite element model of the blood vessel with different disorders was constructed and simulated, and the results were extracted and compared with the mathematical models.

The mathematical models were successful at representing the vascular diseases, as the presence of any disorders (plaque, block, or aneurysm) could change the vessel impedance and, therefore, the measured impedance by the electrodes. This clarifies the ability of BIA in detecting vascular disorders and estimating their kinds and sizes.

Using frequencies in KHz range is preferable to be used in detecting the changes in vessel impedance because it has the ability to differentiate between normal and diseased blood vessels.

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BIA: bio impedance analysis

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The authors declare that they have no conflicts of interest.

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All Authors participated in drafting the article and revising it, also they all gave a final approval of the version to be submitted.