The stem villi are categorized into the three groups: truncus chorii, next to the chorionic plate; rami chorii, next to the truncus chorii; ramuli chorii, between the rami chorii and the basal plate [1, 25]. The diameter gradually becomes smaller from the chorionic plate to the basal plate, and branches are found in all the groups except the truncus chorii. The branching pattern of the ramuli chorii is not equally dichotomous. The generations of branches in the rami chorii and ramuli chorii are up to 4, and from 1 to 30, respectively. Table 1 shows the size and branching pattern in the stem villi model. In this model, the chorionic and basal plates, and the boundaries between the categories (truncus chorii, rami chorii and ramuli chorii) were parallel to each other. Firstly, how to use the centripetal and centrifugal orders at the branch to describe the branching pattern in Table 1 is explained. Figs. (1a and 1b) show a simple branching pattern designated by the centripetal order and centrifugal order: the tip and trunk are designated as 1 and maximum, respectively, at the centripetal order, and vice versa at the centrifugal order. The centripetal order is used to evaluate dichotomy [26, 27]. The bifurcation ratio, R_b, is defined as the following equation:

Fig. (1). Villous tree model developed in this study. The branching pattern in the stem villi of the model was indicated by the centripetal and centrifugal orders at the branches, whose example is indicated by (a) and (b), respectively. (c-e) show the side views of the model: the stem villi, white; the villi except the stem villi, blue; the intervillous space, which the maternal blood passes through, blue. As (d) and (e) show, this model is between the chorionic and basal plates. The z axis of the Cartesian coordinate system is perpendicular to the chorionic plate. The size of this model is 34.8 × 34.8 × 24.5 [mm] (1200 × 1200 × 847 [pixels], 29 μm/pixel).

(1)

where N_u is the number of branches at the centripetal order u. If R_b is 2, the branching is equally dichotomous. Assuming that R_b is constant, N_u is a geometrical series given as:

(2)

where u_max is the maximum centripetal order. Hence,

(3)

R_b of the branches was calculated by the method of least squares. Table 1 shows that R_b in the rami chorii was 2, but that in the ramuli chorii was not equal to 2. These values indicate that the branching patterns in the rami chorii and ramuli chorii are equally and unequally dichotomous, respectively. The centrifugal order at the tip, C_f, in Table (1), corresponds to the generation of the branches. While C_f in the rami chorii was constant, that in the ramuli chorii was varied. That is because the branches in the ramuli chorii were not symmetric.

The diameter range in each category was as follows: truncus chorii, 900 – 3000 μm; rami chorii, 300 – 1000 μm; ramuli chorii, 50 – 500 μm [25]. Table 1 shows that the diameter ranges agree with the aforementioned one. The change of the diameter at the truncus chorii was much larger than those at the rami chorii and ramuli chorii. For the smooth connection between the truncus chorii and rami chorii, the derivative of the radius with respect to the z coordinate should be zero at the boundary. Hence, the following equation was used to determine the radius of the branch at the truncus chorii:

(4)

where r_max and r_min are the maximum and minimum radii in the truncus chorii, and z_tr is the boundary between the truncus chorii and rami chorii (z_tr = 2.9 mm). In the rami chorii, the radius was decreased as the distance along the axis was longer. The radius of the branch in the ramuli chorii became larger as the centripetal order increased. The radii of the branches at the connecting point were modulated as all the branches showed the same radius.

The branches in the rami chorii were equally dichotomous as well as symmetric, and 16 branches were connected to those in the ramuli chorii at the boundary. The branches in the ramuli chorii should be unequally dichotomous. For making unequally dichotomous branches, the 2D diffusion-limited aggregation (DLA) models [28] were made by the free-software, dla-nd, which was developed by the Dr. Mark J. Stock (http://markjstock.org/dla-nd/). Because the path between the branching points was not smooth in this model, the line between the branching points was set as an axis of a branch.

Fig. (2). Contraction directions at the branches of the rami chorii. Arrows show the directions of the contractions. Each branch in the stem villi contracts along its axis and toward the junction.

As Table (1) shows, the longest distance from the chorionic plate in the model was 24.5 mm. According to the previous reports, the thickness of the human placenta and chorioamniotic membrane, the surface of the placenta, were 25 mm [1] and 243 μm [29] on average, respectively. It is calculated that the distance between the chorionic and basal plates is 24.8 mm. The longest distance from the chorionic plate in the model was close to the calculated distance between the chorionic and basal plates, based on the previous reports [1, 29]. In the meantime, the cross section of the model, parallel to the chorionic plate, had the bounding rectangle, whose size was 23.8 mm × 22.6 mm (width × height). The previous report [1] indicated that the placenta at term, whose diameter was 220 mm, had 60 – 70 villous stems. The area based on this diameter is 3.80 × 10⁴ mm² so that the average cross section of the villous trees is 5.42 × 10² - 6.33 × 10² mm², whose corresponding diameter is 26 – 28 mm. The size of the cross section in the model was close to that based on the previous report [1]. Figs. (1c-1e) show the villous tree model developed in this research. The size was 34.8 × 34.8 × 24.5 [mm] (1200 × 1200 × 847 [pixels], 29 μm/pixel).The Cartesian coordinate system, whose z axis was perpendicular to the chorionic plate, was used to describe the position in the model. Its origin was also on the chorionic plate.

The contractile cells run along the longitudinal axis of the branch [4-11]. Each point at the boundary surface between the stem villi and the surroundings has two tangential directions as Fig. (2) shows. The tangential direction, closer to the axis of the branch than the other, was decided as the contraction direction.

At the truncus chorii, the axis of the branch was parallel to the z axis and its radius was largely changed as Equation (4) shows. The angle between the z axis and tangential direction (φ_o) and the differentiation of Equation (4) by z are as follows:

(5)

(6)

Considering that the direction of the contraction at the axis of the branch was (0, 0, -1),

(7)

When the angle between the point at the surface and the x axis in the xy plane is θ_o, the tangential direction is (sinφ_ocosθ_o, sinφ_osinθ_o, cosφ_o). Comparing the rami chorii with the truncus chorii at Table (1), the diameter change at the rami chorii was 25% of that at the truncus chorii. In addition, the rami chorii showed the range of the z coordinate, which was about 3.7 times larger than that at the truncus chorii. The change of the diameter at the rami chorii was much smaller than that at the truncus chorii. Hence, the tangential direction at the surface was parallel to the axis of the branch at the rami chorii. Because the change of the diameter was also small at the ramuli chorii, the tangential direction was determined in the same way.

As Figs. (1c-1e) show, assuming that the surroundings of the stem villi were one continuum in this model, the propagation of the displacement in the placenta was evaluated by the model. A wave equation is generally described as below:

(8)

where u is displacement vector, ρ is density, λ_l and μ are Lamé’s constant. μ also shows a shear elastic modulus. Generally, biological tissue is incompressible, so that the second term in Equation (8) is zero. Hence,

(9)

Hence, the shear wave (transverse wave), whose propagation is normal to the vibration and carried out in solid, was evaluated in this computation. The displacement caused by the shear wave is described as follows [30]:

(10)

where ξ_o is the amplitude, k is the wave number (k = 2π/λ, λ is wave length), r is the distance from the surface of the stem villi, t is time, and ω is angular frequency. ξ_o was 0.1 μm in all the computational conditions. The shear elastic modulus, μ, is described as follows [31]:

(11)

Considering that k = 2π/λ and ω=2πν (ν, frequency),

(12)

ρ was 1.0 × 10³ kg/m³ because of biological tissues [32]. ν was 1.0 Hz, and λ was 0.29, 0.58 or 1.45 mm (10, 20 or 50 pixels) in the computation so that the shear moduli were 8.41 × 10^-5, 3.36 × 10^-4 and 2.10 × 10^-3 Pa. The displacement is attenuated by viscoelastic properties so that the maximum distance for the propagation was 1.45, 2.9 and 4.35 mm (50, 100 and 150 pixels). In order to simplify the problem, t was set for zero. That is, the computation did not consider the time effect on the displacement. The maximum distance for the propagation was not dependent on the position. λ was kept constant in the surroundings, or became longer as the distance from the surface of the stem villi was longer. In the latter case, λ was increased from 0.29 mm to 1.45 mm every one-third of the maximum distance from the surface. Hence, there were 12 conditions in this computation.

Fig. (3). Displacement caused by the contraction of the stem villi. λ = 1.45 mm, and the maximum distance for the propagation = 4.35 mm. All the parameters at each z coordinate are shown. While (a) shows the displaced area, the magnitude (b) and direction (φ, θ) (c, d) of the displacement were described by the representative values, the mean and standard deviation (SD) for the displaced area. The solid and dotted lines in (c) and (d) are the mean and SD, respectively. The triangles show the characteristic positions, which were observed at all the computations. zd, zφ1, zφ2, and zθ are the z coordinates for these characteristic positions.

The displacement of the surroundings of the stem villi is described by the polar coordinate system (magnitude, φ (0° ≤ φ ≤ 180°) and θ (-180° ≤ θ < 180°)) because the coordinate system was useful to separate the displacement into its magnitude and direction. Fig. (3) shows the results when λ and the maximum distance for the propagation were 1.45 mm and 4.35 mm, respectively. Fig. (3a) shows that the displaced area was gradually increased as the z coordinate became larger. The long and steep slope was observed from the truncus chorii to the rami chorii. The same features were observed in all the computational conditions. Figs. (3b, 3c and 3d) show the mean and standard deviation (SD) of each parameter: magnitude, φ and θ. The mean and SD were calculated for all the points in the displaced area, whose magnitude was more than zero. When the displacement is perpendicular to the chorionic plate, the value of θ cannot be determined. Hence, such a critical point was not included for the calculation of the mean and SD in θ. Fig. (3b) shows that the SD normalized by the mean was calculated in order to evaluate the magnitude range of the displacement in each z coordinate. The peak of the normalized SD was observed around the boundary between the truncus chorii and rami chorii, which is named as z_d. The mean and SD about the direction of the displacement in each z coordinate are shown in Figs. (4c and 4d). As Fig. (4c) shows, the mean of φ was kept around 90 degrees, but the SD of φ indicated two peaks, whose positions are named as z_φ1 and z_φ2, respectively. Fig. (3d) shows that the mean of θ slightly decreased around the boundary between the rami chorii and ramuli chorii, which is named as z_θ, while the SD of θ was kept around 90°. These characteristic z coordinates, z_d, z_φ1, z_φ2, and z_θ, were observed at all the computations. Fig. (4) shows the images which visualizes the magnitude, φ, and θ in each characteristic z coordinate and the middle z coordinates of the truncus chorii (z_t), rami chorii (z_r), and ramuli chorii (z_rl) under the same computational condition as Fig. (3) shows. The displaced area became larger as the z coordinate was increased. The magnitude of the displacement was almost kept constant at every z coordinate although the magnitude in the limited area at z_d was high. The distributions of φ and θ were largely changed along the z coordinate as shown in Figs. (3c and 3d) The visualization of the displacement and direction like Fig. (4) is useful to find out critical points and important points for analysis.

As Fig. (4) shows, the magnitude of the displacement was kept almost constant except the magnitude was high near the stem villi at z_d. 90% of the displaced area showed that the magnitude relative to the maximum one was less than 0.17 for the maximum distance for the propagation = 1.45 mm and λ increasing as the distance from the surface of the stem villi, and 0.06 for other results. The SD normalized by the mean was large at z_d, but the high magnitude was in the limited area. Hence, the maternal and fetal blood circulations in the villous tree and intervillous space would be influenced by the constant distribution of the displacement in the placenta. Because the shape of the stem villi directly influences the displacement pattern, the model whose shape near the high magnitude is changed will be developed and used for the computation to examine whether or not such a high magnitude is inevitable.

Fig. (4). Displacement patterns in the characteristic positions. λ = 1.45 mm, and the maximum distance for the propagation = 4.35 mm. In addition to the characteristic positions in each parameter, zd, zφ1, zφ2, and zθ, the representative positions of the truncus chorii, rami chorii and ramuli chorii appear: zt, zr and zrl, which are the middle z coordinates of the truncus chorii, rami chorii, and ramuli chorii, respectively.

Fig. (5). Distribution of φ in the characteristic positions for the displacement at λ = 1.45 mm and the maximum distance for the propagation = 4.35 mm. (a-e) show the distributions of φ at the following characteristic positions, zt, zφ1, zr, zφ2 and zrl. The area fraction of φ, from 45° to 90°, for all the computation results is indicated in (f).

Figs. (5a-5e) show the distribution of φ at the characteristic z coordinates (z_φ1, z_φ2) and the middle z coordinate in each category (z_t, z_r, z_rl) for the same condition as shown in Fig. (3). The area fraction, which is the area for each φ normalized by all the displaced area, was used to evaluate the distribution of φ. The area fraction around φ = 90° (φ = 45° - 135°) was largest at z_φ1 (Fig. 5b) and smallest at z_φ2 (Fig. 5d). The same characteristics were observed in all the computations. For all the computational conditions, the mean and SD of the area fraction at φ = 45° - 135° were calculated at each z coordinate. As the average values in Fig. (5f) show, more than 90% of the displaced area showed φ from 45° to 135° at z_φ1. The area with the same range of φ was around 10% at z_φ2. Considering that the φ = 90° means the direction parallel to the xy plane, most of the displacement was parallel to the xy plane at z_φ1, and parallel to the z axis at z_φ2. Fig. (5f) shows that the mean of the area fraction at the other positions was around 0.3. Hence, φ did not have a preferred direction there. The displacement could help the maternal blood to go to the chorionic plate at z_φ2, spread parallel to the chorionic plate and the basal plate at z_φ1, and go toward the basal plate at z_φ2. In the meantime, the branches in the stem villi around z_φ1 and z_φ2 had the axes largely different from the directions of the displacement there: the axis of the branch in the truncus chorii is perpendicular to the chorionic plate, and that in the rami chorii was largely changed because of its curvature. The displacements at z_φ1 and z_φ2 could assist the fetal blood to pass through the vessels in the stem villi. Moreover, the maternal and fetal bloods could be homogenized by the displacement at z_t, z_r and z_rl. The SD was largest at z_t, and smallest at z_φ1. The SD at z_rl was larger than that at z_φ1, but much smaller than those at the other characteristic positions. Considering that λ and the maximum distance for the propagation are corresponding to the mechanical properties of the surroundings of the stem villi, φ at z_φ1 and z_rl would be hardly influenced by the mechanical property of the surroundings, but φ at the truncus chorii would be vulnerable to it.

Fig. (6). Distribution of θ for the displacement at λ = 1.45 mm and the maximum distance for the propagation = 4.35 mm. (a-d) show the distribution of θ in the characteristic positions, zt, zr, zθ and zrl. The SD of the area fraction in each class interval for all the computations (SDin) is indicated in (e).

Figs. (6a-6d) show the distribution of θ at the characteristic z coordinates (z_θ) and the middle z coordinate in each category (z_t, z_r, z_rl) for the same condition as Fig. (3) shows. The area fraction, the area for each θ normalized by all the displaced area, was used for describing the distribution. Figs. (6a and 6b) show that the similar distribution pattern was observed every 90° at z_t and z_r. The result agreed with the mean of θ at z_t and z_r, around zero. Fig. (6c) shows the distribution of θ at z_θ, where the area fraction at θ = -180° - 90° (the third quadrant) and θ = 0° - 90° (the first quadrant) was larger than that at θ = -90° - 0° (the fourth quadrant) and θ = 90° - 180° (the second quadrant). This result agreed with the decrease of the mean at z_θ in Fig. (3d). As Table (1) shows, the branches in the ramuli chorii were unequally dichotomous as well as asymmetric while those in the rami chorii were equally dichotomous as well as symmetric. Because the rami chorii had 16 branches connecting to those in the ramuli chorii, 16 types of the branching pattern were used in the ramuli chorii. z_θ was located around the boundary between the rami chorii and ramuli chorii. The branches of the ramuli chorii at z_θ were almost parallel to the z axis in the second and fourth quadrant, but those in the first and third quadrants were not. The angle of the branch at z_θ would cause the characteristic distribution of θ at z_θ. The similar distribution pattern was observed at z_rl, but each peak was much smaller than that at z_θ as Fig. (6d) shows. Because z_rl was placed on the middle of the ramuli chorii, the feature caused at z_θ would be weakened by the branches which showed various angles. All the computational results showed the same characteristics as Figs. (6a-6d) show. The SD of the area fraction in each interval (each value of θ) (SD_in) was calculated in order to evaluate the uniformity of the distribution in θ for all the computations. Fig. (6e) shows that the average value of SD_in at z_θ was largest and that at z_rl was smallest, among all the positions. The distribution of θ was most uniform at z_rl, and most fluctuated at z_θ in all the positions. The distribution of θ at z_θ would be influenced by the angle of the branches in the ramuli chorii. Because θ at z_rl did not have a preferred value, the maternal and fetal bloods could be homogenized. The SD values of SD_in at z_t and z_θ was much larger than those of other two positions, and that at z_rl was smallest in all the positions. The results show that the mechanical properties of the villus tree would strongly influence θ at z_t and z_θ, but hardly influence that at z_rl. Even if the mechanical properties of the villous tree are changed, the homogenization of the bloods around the ramuli chorii would be kept.