**Abstract**

The fetal blood flow in two arteries and one vein of the human umbilical cord could be influenced by the conditions of the fetal growth and placenta that the evaluation of the blood flow pattern by ultrasound Doppler velocimetry is important. That is, the mechanical environment in the umbilical cord should be kept to maintain the blood flow suitable for good fetus growth. In this chapter, a human umbilical cord model for finite analysis, based on the mechanical and histological characteristics is proposed. Considering that the active force production by hyaluronan, proteoglycan, smooth muscle cells, and myofibroblasts could influence the mechanical environment in the umbilical cord, the computation with the proposed model was carried out in order to evaluate the influence. The changes in the mechanical environment caused by the active force production and their influences on the fetal blood flow through the pressure rise and drop in the arteries of the umbilical cord are introduced.

**Keywords:** human umbilical cord, artery, vein, Wharton's jelly, hyaluronan, proteoglycan, smooth muscle cell, myofibroblast, active force production

## **1. Introduction**

The human umbilical cord, connecting the fetus and placenta, conveys fetus blood through two arteries and one vein [1]. While the umbilical cord itself spirals, the artery is accompanying the vein with spirals. The blood flow in the umbilical artery and vein, which ultrasound Doppler velocimetry is generally used to measure, shows different characteristics in pulsation: the artery has pulsation, but the vein, except the regions around the ductus venosus and portal vein, does not [2–6]. The difference in pulsation between the artery and vein has been also indicated by pressure [7]. The blood flow pattern could be influenced by the condition of fetal and placenta so that analyses of the flow pattern would be helpful for diagnosis. For example, the pulsation pattern of the blood flow in the artery and vein around the ductus venous and portal vein have been analyzed in pathological aspects [2–6]. In the meantime, the sizes of the umbilical cord, artery, and vein have been reported with some variation [8–14].

Considering the size of the umbilical cord at 40 weeks of gestational age, the diameters of cord, artery, and vein in Ref. [8], whose number of cases was larger than those in [9–14], was 17.8 mm, 4.6 mm, and 9.1 mm at average, respectively, while the length of the cord was more than 50 cm [1]. That is, the blood should travel a long way more than 50 times of the diameter. The distension of the artery and vein of the umbilical cord [15] would contribute to keeping the proper blood flow patterns in the artery and vein, necessary for fetal growth. In numerical analyses, the influence of the umbilical cord's coiling [16] and pressure drop [17] on the blood flow in the artery and vein, and that of arterial pressure pulse on the flow in the vein [18] have been reported. The mechanical environment in the umbilical cord would be arranged as keeping the shapes of the blood vessels, but the investigation has been barely carried out.

The human umbilical artery and vein show the characteristics different from major blood vessels in the body, including large and active endothelial layers [1, 19]. Moreover, the artery has shown two distinctive smooth muscle layers: inner layer, loosely arranged cells with abundant ground substance; outer layer, circularly arranged cells [20]. While myosin and vimentin have been found in both two layers, only the outer layer has indicated desmin [21]. These two blood vessels are covered with Wharton's jelly, whose college fiber has shown the network with canalicular-like structure, and cavernous and perivascular spaces (stromal cleft), which would be helpful for diffusion [22, 23]. Considering the control of the mechanical environment in the umbilical cord, it has been pointed out the contribution of hyaluronan and proteoglycan to pressure control [24], and myofibroblasts, which show active contraction [25, 26].

The mechanical properties of the human umbilical cord and blood vessels have been reported [27–29]. Considering the limitation of the range of the blood pressure in the arteries and vein [7], and nonlinearity in the stress-strain relationship [27–29], the elasticity in the limited range would be proper to estimate the mechanical environment in the umbilical cord. Given that the stress distribution at blood vessel walls would be homogeneous at physiological pressures [30], the assumption that the stress distributions at the umbilical artery and vein would be homogeneous at the average pressure could be made.

In this chapter, a computational model of the human umbilical cord is proposed in order to estimate the mechanical environment. Influence of the change in blood pressure, active force production in smooth muscle cells, myofibroblasts, hyaluronan, and proteoglycan on the mechanical environment and its relation to the fetus's blood flow is shown.

#### **2. Computational model of the human umbilical cord**

The computational model of the human umbilical cord has been developed on the finite element analysis (FEA) software, COMSOL Multiphysics® ver.6 (COMSOL, Inc. MA). The length of the umbilical cord is longer than 50 times of its diameter so that the cross section, whose normal vector was the axial direction of the umbilical cord, was modeled and analyzed as the representative of the umbilical cord. The analysis has been carried out two-dimensionally, as a plane strain problem.

#### **2.1 Spatial arrangement and constraint**

The spatial arrangement and size of each component are shown in **Figure 1** and **Table 1**, respectively. The blood vessel is composed of three layers: *Tunica Intima*,

*Mechanical Environment in the Human Umbilical Cord and Its Contribution to the Fetal… DOI: http://dx.doi.org/10.5772/intechopen.106565*

#### **Figure 1.**

*Components and spatial arrangement in the computational model of the human umbilical cord. A, the entire image of the model; B and C, the vicinities of the vein and artery, respectively.*


#### **Table 1.**

*Sizes of the human umbilical cord model.*

*Tunica Media*, and *Tunica Adventitia*. The intima is composed of a single layer of endothelial cells, and the adventitia shows a sparse distribution of smooth muscle cells without a clear boundary [19] so that the distinctive two layers of smooth muscle, an inner layer with loosely arranged smooth muscle cells and abundant ground substance, and an outer layer with circularly arranged smooth muscle cells [20], was categorized into *Tunica Media* in this study. The diameter of the umbilical cord, and the inner diameters of the artery and vein were the average values *in vivo* at 40 weeks of gestational age in [8], whose number of cases was larger than other reports [9–14].

**Figure 2.** *Finite element model of the human umbilical cord with triangular mesh (the number of elements, 29, 786).*

"Surroundings" and "Boundary" in the model are corresponding to the surroundings of the umbilical cord and limitations in the displacement of the umbilical cord, whose size was about 1.3 times of the umbilical cord's diameter. The thickness values of the amniotic epithelium, subamniotic zone, and blood vessels have been hardly measured *in vivo* so that each thickness was decided as follows: amniotic epithelium and subamniotic zone, less than 5% of the umbilical cord diameter (0.5 mm); artery and vein, only the layers of smooth muscle cells were modeled, and the thickness in each layer was set as 0.1 mm. The center positions of these blood vessels formed an equilateral triangle, whose sides were equally 8.84 mm. All the blood vessels were equally apart from the umbilical cord at the same distance: at the outer diameter, 0.4 mm. **Figure 2** shows the finite element model of the umbilical cord with triangular meshes (the number of elements, 29, 786). The fixation was set on the region, "boundary."

## **2.2 Mechanical properties**

The arterial and venous pressure *in vivo* has been reported as 88 mmHg and 41 mmHg on average, respectively [7]. While the venous pressure has hardly shown pulsation, the pressure pulsation in the artery, 12 mmHg, has been observed [7]. The size of each component of the model was based on the measurement *in vivo* so that all the shapes would be at the average pressures in the blood vessels. Given that the stress distribution would be homogeneous at the average pressure [30], the change in stress distribution, caused by the pressure rise and drop (12 mmHg (1.6 kPa)), was evaluated. The pressure in the vein was maintained at 0 Pa because of no pulsation.

Reference [28] has shown the mechanical properties of the umbilical vein and Wharton's jelly with two elastic moduli at high and low stress. Because the removal of the amniotic epithelium has not been mentioned, the results of Wharton's jelly would be those of the umbilical cord without the blood vessel. Also, the description of *Tunica adventitia*, which is difficult to separate properly from the surrounding tissue in general, has not been found so that the sample would include *Tunica media* at least. The stress (σ) at a thin-walled cylinder caused by the load of internal pressure (P) is calculated by the following equation:

$$
\sigma = \frac{Pr}{t} \tag{1}
$$

#### *Mechanical Environment in the Human Umbilical Cord and Its Contribution to the Fetal… DOI: http://dx.doi.org/10.5772/intechopen.106565*

where r and t are the radius and thickness of the cylinder, respectively. When the radius, thickness (*Tunica Media* only), and average pressure at the vein were applied to Eq. (1), the stress was more than 0.1 MPa, which was corresponding to the elastic modulus at high stress [28]. Considering that the surrounding tissues would be also at the high stress, the elastic moduli in *Tunica Media* of the vein and amniotic epithelium were set as 2.36 MPa and 11.1 MPa, respectively as **Table 2**, the list of the mechanical property in each component, shows. While the inner layer of the media in the artery was half of that in the media in the vein (1.18 MPa), the other layers around the blood vessels were set as 2.36 MPa. Considering that blood vessel walls would be incompressible in general, Poisson's ratio was set as 0.4999. Because vimentin, desmin, and α- and γ- actin have been found in the subamniotic zone as well as the layers of the blood vessel, the mechanical properties were set as those in the layers. The components of Wharton's jelly, hyaluronan and proteoglycan, contributing to diffusion, would show its compressibility so that Poisson's ratio was set as 0.1. Supposing that the elastic modulus of Wharton's jelly would be between those of the vein and amniotic epithelium, the lower one, corresponding to that of the vein, was set as that of Wharton's jelly. Plenty of fibers observed in amniotic epithelium [23] would hardly show incompressibility so Poisson's ratio was set as 0.3. Given that the umbilical cord could move freely, the elastic modulus of "Surroundings," corresponding to the surroundings of the umbilical cord, was set as 0.1% of that in the layer of the blood vessel. Also, Poisson's ratio of "Surroundings" was set as 0.4.

The active force productions in the layers of the blood vessels, Wharton's jelly, and subamniotic zone, in each model, are shown in **Table 3**. The magnitude of the force generated by the layers of the blood vessels were two types: one was twice of the other. The force direction was set as the opposite to the pressure in the artery. For evaluating the influence of the layers of the blood vessels and others, the force was 0 Pa (negative) at part of the models: Model A, all the force productions were active; Model B, all the force productions were active and the magnitude was twice of those in Model A; Model C, the force productions at the layers of the blood vessels were active; Model D, the force productions at Wharton's jelly and subamniotic zone were active; Model E, no force production.


#### **Table 2.**

*Mechanical properties of the human umbilical cord model.*


#### **Table 3.**

*Active force produced in the components. The plus and negative signs are corresponding to the directions of the pressure, outer and inner directions, respectively.*

#### **2.3 Mechanical environment**

The mechanical environment in the human umbilical cord model was evaluated by von Mises stress (σvm) (equivalent stress), described as below:

$$\sigma\_{\rm VM} = \frac{1}{\sqrt{2}} \left\{ (\sigma\_1 - \sigma\_2)^2 + (\sigma\_2 - \sigma\_3)^2 + (\sigma\_3 - \sigma\_1)^2 \right\}^{1/2} \tag{2}$$

where σ1, σ2, and σ<sup>3</sup> are three principal stresses. When σvm is equal to the yield stress of a material, the elastic deformation is changed to plastic one. σvm was used as a parameter that estimates how severe the mechanical environment is.

#### **3. Mechanical environment in the human umbilical cord**

#### **3.1 Pressure rise and drop in the arteries**

**Figure 3** shows the distribution of von Mises stress (σvm) in the umbilical cord model at the pressure rise in the arteries (1.6 kPa). In Models A, and B, where all the components, the layers of the blood vessels, Wharton's jelly, and subamniotic zone, produced active forces, the shapes of the arteries and veins were kept circular. Comparing Model B with Model A, the region for higher σvm around the vein became larger while the region around the arteries reduced. In Models C, D, and E, whose shapes of the blood vessels seemed not circular, higher σvm in the region around the

*Mechanical Environment in the Human Umbilical Cord and Its Contribution to the Fetal… DOI: http://dx.doi.org/10.5772/intechopen.106565*

#### **Figure 3.**

*von Mises stress distribution at the pressure rise in the arteries (1.6 kPa). The black solid line shows the shape before pressure rise. Active force production in each model, indicated in Table 3, is as follows: Model A, in all the components; Model B, in all the components, twice as large as that in A; Model C, in all the layers around the blood vessels; Model D, in Wharton's jelly and subamniotic zone; E, in no component.*

arteries became larger. **Figure 4** shows the distribution of σvm in detail: σvm along the lines in the umbilical cord, depicted in **Figure 3**, is indicated. All the models indicated that the peak of σvm at the artery and vein sides appeared around the amniotic epithelium in all the lines except Line b-b' at the artery side in the models except Model B. Considering that Line b-b' was passing through the region close to the artery and vein, which would be influenced largely by the pressure rise and active force productions, increase in σvm would be reasonable and could be prevented by an increase in the magnitude of the active force. In the meantime, the peak at the artery side of Line b-b' was much smaller than other peaks so that the regions around the blood vessels would keep smaller than those around the amniotic epithelium. The peak of σvm at Models B (vein side), and D and E (artery side) was larger than that in

#### **Figure 4.**

*von Mises stress distribution in detail at pressure rise in the arteries (1.6 kPa). The 6 lines, parallel to each other at the space interval of 0.2 mm from the center to the surroundings (the top of the right side.) were set to examine the stress distribution. The stress along the line in each model is indicated with the circles, corresponding to the peak at the artery and vein sides.*

Models A and C. The magnitude of the active force production at Model B would be too large. And the force produced by the layers of the blood vessels would be necessary to modulate the stress distribution in the umbilical cord.

**Figure 5** shows the distribution of von Mises stress (σvm) in the umbilical cord model at the pressure drop in the arteries (1.6 kPa). Models A and B, where the active forces were produced at all the components, kept the shape of the blood vessels circular. While Model B had the larger region around the vein for higher σvm than

*Mechanical Environment in the Human Umbilical Cord and Its Contribution to the Fetal… DOI: http://dx.doi.org/10.5772/intechopen.106565*

#### **Figure 5.**

*von Mises stress distribution at the pressure drop in the arteries (1.6 kPa). The black solid line shows the shape before pressure drop. Active force production in each model, indicated in Table 3, is as follows: Model A, in all the components; Model B, in all the components, twice as large as that in A; Model C, in all the layers around the blood vessels; Model D, in Wharton's jelly and subamniotic zone; E, in no component.*

Model A, the regions around the arteries for higher σvm in Model B were smaller than those in Model A. The circular shapes of the blood vessels have been hardly kept in Models C, D, and E, with the larger regions around the arteries for higher σvm. These characteristics of the shapes of the blood vessels and distribution pattern of σvm were similar to those in pressure rise as **Figures 3** and **5** show although Models C, D, and E caused the collapses of the arteries and umbilical cord, which were not observed in those models at the pressure rise. **Figure 6** shows the σvm along the lines, which were set as the case of pressure rise in **Figure 4**. The characteristics of all the graphs in **Figure 6** agreed with those in **Figure 4** because the pressures rise and drop had the same magnitude with different directions, and σvm is the function of the difference between the principal stresses as Eq. (2) shows. Hence, through the change in the arterial pressure, the stress around the blood vessels would be kept much lower than that around the amniotic epithelium.

**Figure 6.**

*von Mises stress distribution in detail at pressure drop in the arteries (1.6 kPa). The 6 lines, parallel to each other at the space interval of 0.2 mm from the center to the surroundings (the top of the right side.) were set to examine the stress distribution. The stress along the line in each model is indicated with the circles, corresponding to the peak at the artery and vein sides.*

#### **3.2 Mechanical environment and blood flow in the umbilical cord**

The σvm around the blood vessels (< 6 kPa) was kept much lower than that around the amniotic epithelium (> 12 kPa) when the pressure in the arteries and the active force production was changed. Considering that the adaptations of the blood vessels

*Mechanical Environment in the Human Umbilical Cord and Its Contribution to the Fetal… DOI: http://dx.doi.org/10.5772/intechopen.106565*

and the surrounding tissue to change in the mechanical environment could cause changes in their shapes and structures, the control of the mechanical environment would be effective to keep the shapes and structures of these components in the umbilical cord. Also, considering the adaptation and damage in the amniotic epithelium might be necessary. In the meantime, the shape of the blood vessels would be hardly kept unless the active forces were produced by all the components. The shape of the blood vessel would directly influence the blood flow pattern and cause change in wall shear stress, which could damage the blood vessel function and structure. Also, the collapse of the umbilical cord at the pressure drop with the active forces, which were produced partially, could damage its tissue structure. Hence, the active force productions at all the components would be necessary to maintain the proper blood flow in the umbilical blood vessels and avoid the damage to the tissue of the umbilical cord.

#### **4. Conclusion**

The human umbilical cord model for finite element analysis, based on the mechanical and histological characteristics, has been developed. The influence of the active force productions by hyaluronan, proteoglycan, smooth muscle cells, and myofibroblasts on the stress distribution in the umbilical cord was evaluated. As a result, the active forces produced by all the components would be necessary to maintain the blood flow properly.

#### **Acknowledgements**

This work was supported by Grant for Individual Research of Tohoku-Gakuin in 2021.

#### **Conflict of interest**

The author declares no conflict of interest.

#### **Author details**

Yoko Kato Faculty of Engineering, Tohoku Gakuin University, Miyagi, Japan

\*Address all correspondence to: ykato@mail.tohoku-gakuin.ac.jp

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
