**2. Computational models**

#### **2.1. Villous tree model**

and vein. The fetal and maternal blood circulates separately so that the substances and gases are exchanged through the surface of the villous tree, without mixing these two types of blood [1]. The influence of the blood circulations on the exchange of the substances has been investigated [2, 3]. Also, the fetal and maternal blood circulations in the placenta have been evaluated by ultrasound Doppler and MRI so that the influences of these circulations on fetal growth have been indicated [4–9]. However, the direction of the blood flow in the placenta has been hardly determined yet. The villous tree is classified into the following villous types: stem villi, intermediate villi, terminal villi, and mesenchymal villi [1, 10]. At term, the villous tree is mainly composed of stem villi, mature intermediate villi, and terminal villi [1, 11, 12]. The stem villi, the main support of the villous tree, are connected to the mature intermediate villi, which is accompanied by the terminal villi. In the stem villi, arteries, veins, arterioles, venules, and capillaries are observed [1, 11]. The mature intermediate villi have arteriole, postcapillary venules, and capillaries, part of which are linked to the capillary loop in the terminal villi [1, 12]. The artery and vein in the stem villi are surrounded by contractile cells, which are axially aligned [13–20]. Also,

Since the contraction of the stem villi would contribute to the fetal and maternal blood circulations in the placenta, the computational model of the villous tree, which actively contracts, has been developed [24]. In this model, the contraction of the stem villi causes the displacement of the surroundings: the displacement propagates from the surface of the stem villi. The result of the computation based on this model indicated that the magnitude of the displacement was almost kept in the placenta, and the direction was helpful for the fetal and maternal circulations although several positions were vulnerable to the mechanical properties of the placenta [24]. In addition, the comparison between the displacement pattern estimated by this model and flow velocity measured by the aforementioned methods will provide the direction of the blood flow in the placenta [24]. However, how the parameters in this computation, including the shear elastic modulus of the placenta and the maximum distance for the propagation,

Hypoxia is classified into the following groups: preplacental hypoxia, less oxygen content in the maternal blood; uteroplacental hypoxia, normal oxygen content in the maternal blood but less content in the uteroplacental tissues; postplacental hypoxia, normal oxygen content in the maternal blood but less content in the fetus because of the problem in the fetoplacental perfusion [1, 25, 26]. In preplacental hypoxia and uteroplacental hypoxia, capillaries are highly branched and the terminal villi are clustered so that the terminal villi are predominantly observed [1, 25, 26]. The terminal villi in postplacental hypoxia are filiform and scarcely observed [1, 25, 26]. The villous tree model, which changes its shape based on the oxygen content, has also developed [27]. Assuming that an increase in the terminal villi is corresponding to the higher shear elastic modulus of the placenta and longer propagation distance of the stem villi contraction, the stem villi model [24] is useful to estimate the influence of hypoxia on the blood circulation in the placenta. However, such an analysis has not been done yet.

In this chapter, how three kinds of hypoxia influence the blood circulations in the placenta was evaluated. The computational model of the stem villi has been developed and used to estimate the distribution of the displacement in the placenta previously [24]. The distribution of the displacement was analyzed not for hypoxia, but for general characteristics of the

the contraction of the stem villi has been observed [14, 21–23].

28 Highlights on Hemodynamics

influence the displacement has not been investigated yet.

The computational model and parameters have been reported previously [24]. **Figure 1** shows the computational model, where the chorionic and basal plates are the boundaries for the fetal and maternal sites, respectively. Truncus chorii, rami chorii, and ramuli chorii are parts of the stem villi. The position in the model is indicated by the Cartesian coordinate: the z axis, whose origin is placed on the chorionic plate, perpendicular to the chorionic and basal plates. That is, the z coordinate is corresponding to the distance from the chorionic plate. The surroundings of the stem villi are composed of the intervillous space, where the maternal blood circulates, and all the villi except the stem villi. The size and branching patterns based on the histological reports [1, 10] were as follows: diameter, 3–0.3 mm; branching pattern, no bifurcation in the truncus chorii, equally dichotomous as well as symmetric in the rami chorii, and unequally dichotomous as well as asymmetric in the ramuli chorii.

**Figure 1.** Side view of the villous tree model. White region, stem villi; black region, other types of villi and intervillous space; size (x, y, z), 1200 × 1200 × 847 pixels (1 pixel = 29 μm) (source: modified from Figure 1 in Kato et al. [24]).

The displacement, *u*, is described by the following equation:

$$\mathbf{u} = \xi\_{\mathbf{e}} \cos kr \tag{1}$$

the z coordinate at the peak of the standard deviation normalized by the mean in the displaced area; zφ1 and zφ2, the z coordinates at the lower and higher peaks of the standard deviation of φ, respectively; zθ, the z coordinate at the lower peak of the mean from the rami chorii to the ramuli chorii. These positions have been used previously [24], but how the parameters influ-

**Figure 2** shows that the displaced volume normalized by the volume of the entire model except the stem villi region became larger as the maximum distance for the propagation, s, grew longer. However, the wavelength scarcely altered the size of the displaced region.

**Figure 2.** Displaced volume normalized by the volume of the model except the stem villi (2.96 × 104

, in the regression line, and the number of the samples, *n*,

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<sup>1</sup> <sup>−</sup> *<sup>R</sup>*<sup>2</sup> (3)

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 mm3 ).

ence the positions have not been investigated yet.

*<sup>F</sup>* <sup>=</sup> *<sup>n</sup>−<sup>2</sup> <sup>R</sup>*<sup>2</sup> \_\_\_\_

provide F-value, *F*, whose *df1* and *df2* were *1* and *n−2*, respectively:

**2.5. Statistical analysis**

**3. Results**

**3.1. Displaced region**

The coefficient of determination, *R2*

The level of significance was 0.05.

where *ξ<sup>o</sup>* is the amplitude (0.1 *μ*m), *k* is the wavenumber and *r* is the distance from the surface of the stem villi. The maximum distance for the propagation, s, was 1.45, 2.9, or 4.35 mm. Also, the shear elastic modulus, *μ*, is described as follows:

$$
\mu = \rho \lambda^2 \nu^2 \tag{2}
$$

where ρ is the density (1.0 × 10<sup>3</sup> kg/m3 ), λ is the wavelength (0.29, 0.58, or 1.45 mm), and ν is the frequency (1 Hz). Hence, the shear elastic modulus of the placenta was 8.41 × 10−5 Pa, 3.36 × 10−4 Pa, or 2.10 × 10−3 Pa. The displacement caused by the contraction was indicated by the polar coordinate: magnitude, φ (0 to 180°) and θ (−180 to 180°).

#### **2.2. Hypoxia model**

The villous tree in preplacental hypoxia and uteroplacental hypoxia has plenty of terminal villi while that in postplacental hypoxia has few filiform ones [1, 25, 26]. It was assumed that an increase in the number of the terminal villi makes the shear elastic modulus of the placenta higher and the maximum distance for the propagation of the displacement longer. Because the model, whose shear elastic modulus was 3.36 × 10−4 Pa, was the control model, the shear elastic moduli in the preplacental hypoxia and uteroplacental hypoxia model and the postplacental hypoxia model were 2.10 × 10−3 and 8.41 × 10−5 Pa, respectively. Also, the maximum distances for the displacement propagation of the preplacental hypoxia and uteroplacental hypoxia model and the postplacental model were 4.35 and 1.45 mm, respectively, since that of the control model was 2.9 mm. Because the shear elastic moduli and the maximum distance for the propagation had three types, respectively, nine different models were made. The models except the control model, the preplacental hypoxia and uteroplacental hypoxia model, and the postplacental model represent the conditions toward these types of hypoxia.

#### **2.3. Contraction**

Three types of hypoxia in the placenta alter the terminal villi but rarely the stem villi. How the hypoxia influences the contractile cells in the stem villi has been barely investigated. The previous report has indicated that fetal growth restriction enhances α-smooth muscle actin of the stem villi [28]. Hence, the amplitude, ξ<sup>o</sup> , in Eq. (1) could be maintained in all the models but also be changed as 0.4 μm for 8.41 × 10−5 Pa, 0.016 μm for 2.10 × 10−3 Pa because of the change in the elastic moduli.

#### **2.4. Characteristic positions**

In order to evaluate the influence of the terminal villi on the displacement distribution in the placenta, the mean and standard deviation of the magnitude, φ and θ in each z coordinate were used as the previous report indicated [24]. The middle positions of each part and several z coordinates which show the characteristic distributions of the magnitude, φ, and θ are used: zt , zr, and zrl, the middle positions of the truncus chorii, rami chorii, and ramuli chorii; zsd/mean, the z coordinate at the peak of the standard deviation normalized by the mean in the displaced area; zφ1 and zφ2, the z coordinates at the lower and higher peaks of the standard deviation of φ, respectively; zθ, the z coordinate at the lower peak of the mean from the rami chorii to the ramuli chorii. These positions have been used previously [24], but how the parameters influence the positions have not been investigated yet.

#### **2.5. Statistical analysis**

The displacement, *u*, is described by the following equation:

the polar coordinate: magnitude, φ (0 to 180°) and θ (−180 to 180°).

the shear elastic modulus, *μ*, is described as follows:

where ρ is the density (1.0 × 10<sup>3</sup> kg/m3

where *ξ<sup>o</sup>*

30 Highlights on Hemodynamics

**2.2. Hypoxia model**

**2.3. Contraction**

zt

the stem villi [28]. Hence, the amplitude, ξ<sup>o</sup>

change in the elastic moduli.

**2.4. Characteristic positions**

u = *ξ<sup>o</sup>* cos*kr* (1)

of the stem villi. The maximum distance for the propagation, s, was 1.45, 2.9, or 4.35 mm. Also,

μ = ρ *λ*<sup>2</sup> *υ*<sup>2</sup> (2)

is the frequency (1 Hz). Hence, the shear elastic modulus of the placenta was 8.41 × 10−5 Pa, 3.36 × 10−4 Pa, or 2.10 × 10−3 Pa. The displacement caused by the contraction was indicated by

The villous tree in preplacental hypoxia and uteroplacental hypoxia has plenty of terminal villi while that in postplacental hypoxia has few filiform ones [1, 25, 26]. It was assumed that an increase in the number of the terminal villi makes the shear elastic modulus of the placenta higher and the maximum distance for the propagation of the displacement longer. Because the model, whose shear elastic modulus was 3.36 × 10−4 Pa, was the control model, the shear elastic moduli in the preplacental hypoxia and uteroplacental hypoxia model and the postplacental hypoxia model were 2.10 × 10−3 and 8.41 × 10−5 Pa, respectively. Also, the maximum distances for the displacement propagation of the preplacental hypoxia and uteroplacental hypoxia model and the postplacental model were 4.35 and 1.45 mm, respectively, since that of the control model was 2.9 mm. Because the shear elastic moduli and the maximum distance for the propagation had three types, respectively, nine different models were made. The models except the control model, the preplacental hypoxia and uteroplacental hypoxia model, and

the postplacental model represent the conditions toward these types of hypoxia.

Three types of hypoxia in the placenta alter the terminal villi but rarely the stem villi. How the hypoxia influences the contractile cells in the stem villi has been barely investigated. The previous report has indicated that fetal growth restriction enhances α-smooth muscle actin of

but also be changed as 0.4 μm for 8.41 × 10−5 Pa, 0.016 μm for 2.10 × 10−3 Pa because of the

In order to evaluate the influence of the terminal villi on the displacement distribution in the placenta, the mean and standard deviation of the magnitude, φ and θ in each z coordinate were used as the previous report indicated [24]. The middle positions of each part and several z coordinates which show the characteristic distributions of the magnitude, φ, and θ are used:

, zr, and zrl, the middle positions of the truncus chorii, rami chorii, and ramuli chorii; zsd/mean,

is the amplitude (0.1 *μ*m), *k* is the wavenumber and *r* is the distance from the surface

), λ is the wavelength (0.29, 0.58, or 1.45 mm), and ν

, in Eq. (1) could be maintained in all the models

The coefficient of determination, *R2* , in the regression line, and the number of the samples, *n*, provide F-value, *F*, whose *df1* and *df2* were *1* and *n−2*, respectively:

$$F = n - 2\frac{R^2}{1 - R^2} \tag{3}$$

The level of significance was 0.05.

#### **3. Results**

#### **3.1. Displaced region**

**Figure 2** shows that the displaced volume normalized by the volume of the entire model except the stem villi region became larger as the maximum distance for the propagation, s, grew longer. However, the wavelength scarcely altered the size of the displaced region.

**Figure 2.** Displaced volume normalized by the volume of the model except the stem villi (2.96 × 104 mm3 ).

These results indicate that the displaced area in the placenta under preplacental hypoxia and uteroplacental hypoxia would be larger than that under postplacental hypoxia.

**Figure 3** shows the displaced area at the control model. Since the wavelength weakly influences the size of the displaced region, the mean of the displaced area at the same maximum distance for the propagation in each characteristic z coordinate was calculated. **Figure 4** show the relationship between the displaced area normalized by that at the maximum distance for the propagation of the control model (2.9 mm) and the characteristic z coordinates. The maximum distance for the propagation almost equally altered the size of the displaced area at all the characteristic position except zt and zsd/mean at both of the maximum distances. The influence of the maximum distance on the size of the displaced area at zt and zsd/mean were much larger than that on the other positions. Considering that zt and zsd/mean are close to the chorionic plate, all the types of hypoxia would strongly influence the blood circulations near the chorionic plate.

#### **3.2. Characteristic positions**

**Figure 5** shows that the maximum distance for the propagation scarcely influenced the characteristic z positions. The influence of the wavelength was also little. Hence, the characteristic z positions would be kept in all the types of hypoxia.

#### **3.3. Displacement: magnitude**

**Figures 6** and **7** show the mean of the displacement in each model when the amplitude ξo was maintained. As the wavelength and maximum distance for the propagation grew **Figure 4.** Displaced area normalized by that of the model for s = 2.9 mm at the characteristic z coordinates. The displaced

*]W ]U ]UO ]VGPHDQ ]˳ ]˳*

*]U ]˳* *]UO ]˳*

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**Maximum distance for the propagation [mm]**

**Figure 5.** Distance from the characteristic positions to the chorionic plate normalized by that between the chorionic and

area in each model (s = 1.45, 2.9, and 4.35 mm) was averaged.

**Normalized**

basal plates.

**z coordinate**

*]˥*

*]W ]VGPHDQ ]˥*

**Figure 3.** Displaced area normalized by the area of the model at each z coordinate. λ = 0.58 mm and s = 2.9 mm.

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These results indicate that the displaced area in the placenta under preplacental hypoxia and

**Figure 3** shows the displaced area at the control model. Since the wavelength weakly influences the size of the displaced region, the mean of the displaced area at the same maximum distance for the propagation in each characteristic z coordinate was calculated. **Figure 4** show the relationship between the displaced area normalized by that at the maximum distance for the propagation of the control model (2.9 mm) and the characteristic z coordinates. The maximum distance for the propagation almost equally altered the size of the displaced area at all the

the types of hypoxia would strongly influence the blood circulations near the chorionic plate.

**Figure 5** shows that the maximum distance for the propagation scarcely influenced the characteristic z positions. The influence of the wavelength was also little. Hence, the characteristic

**Figures 6** and **7** show the mean of the displacement in each model when the amplitude

**Figure 3.** Displaced area normalized by the area of the model at each z coordinate. λ = 0.58 mm and s = 2.9 mm.

was maintained. As the wavelength and maximum distance for the propagation grew

and zsd/mean at both of the maximum distances. The influence of

and zsd/mean were much larger than

and zsd/mean are close to the chorionic plate, all

uteroplacental hypoxia would be larger than that under postplacental hypoxia.

characteristic position except zt

32 Highlights on Hemodynamics

**3.2. Characteristic positions**

**3.3. Displacement: magnitude**

ξo

the maximum distance on the size of the displaced area at zt

that on the other positions. Considering that zt

z positions would be kept in all the types of hypoxia.

**Figure 4.** Displaced area normalized by that of the model for s = 2.9 mm at the characteristic z coordinates. The displaced area in each model (s = 1.45, 2.9, and 4.35 mm) was averaged.

**Figure 5.** Distance from the characteristic positions to the chorionic plate normalized by that between the chorionic and basal plates.

longer, the displacement became larger. When ξ<sup>o</sup>

**3.4. Displacement: direction (φ and θ)**

at zrl indicated the same tendency, but those at zt

**Figure 8.** Influence of the wavelength on the mean of the modified displacement.

**Figures 6**–**9**.

z coordinate on φ.

lus of the placenta grew larger, the mean of the displacement was largely decreased by the longer wavelength as **Figure 8** shows. In **Figure 9**, the longer distance for the propagation induced the larger displacement as shown in **Figure 7**. However, the longer wavelength induced the larger displacement in **Figure 7** but vice versa in **Figure 9**. Considering that the preplacental hypoxia and uteroplacental hypoxia model, and the postplacental hypoxia model showed the smallest elastic moduli and maximum distance, and the largest ones, the mean of the displacement would be different from the control model as shown in

While **Figure 10** shows that the influence of the wavelength on the range of the area fraction for φ = 45–135° among those at the characteristic z coordinates was barely consistent, the longer maximum distance for the propagation reduced the range as shown in **Figure 11**. Hence, the longer maximum distance for the propagation would weaken the influence of the

Both the wavelength and the maximum distance for the propagation influenced the standard deviation of the area fraction in each class interval of θ (SDin), but not consistently. SDin at zr was decreased by the longer wavelength and maximum distance as shown in **Figure 12**. SDin

and zθ did not.

became smaller as the shear elastic modu-

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35

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**Figure 6.** Influence of the wavelength on the mean of the displacement in each model.

**Figure 7.** Influence of the maximum distance for the propagation on the mean of the displacement in each model.

longer, the displacement became larger. When ξ<sup>o</sup> became smaller as the shear elastic modulus of the placenta grew larger, the mean of the displacement was largely decreased by the longer wavelength as **Figure 8** shows. In **Figure 9**, the longer distance for the propagation induced the larger displacement as shown in **Figure 7**. However, the longer wavelength induced the larger displacement in **Figure 7** but vice versa in **Figure 9**. Considering that the preplacental hypoxia and uteroplacental hypoxia model, and the postplacental hypoxia model showed the smallest elastic moduli and maximum distance, and the largest ones, the mean of the displacement would be different from the control model as shown in **Figures 6**–**9**.

#### **3.4. Displacement: direction (φ and θ)**

**Figure 6.** Influence of the wavelength on the mean of the displacement in each model.

34 Highlights on Hemodynamics

**Figure 7.** Influence of the maximum distance for the propagation on the mean of the displacement in each model.

While **Figure 10** shows that the influence of the wavelength on the range of the area fraction for φ = 45–135° among those at the characteristic z coordinates was barely consistent, the longer maximum distance for the propagation reduced the range as shown in **Figure 11**. Hence, the longer maximum distance for the propagation would weaken the influence of the z coordinate on φ.

Both the wavelength and the maximum distance for the propagation influenced the standard deviation of the area fraction in each class interval of θ (SDin), but not consistently. SDin at zr was decreased by the longer wavelength and maximum distance as shown in **Figure 12**. SDin at zrl indicated the same tendency, but those at zt and zθ did not.

**Figure 8.** Influence of the wavelength on the mean of the modified displacement.

**Figure 9.** Influence of the maximum distance for the propagation on the mean of the modified displacement.

**Figure 11.** Influence of the Maximum distance for the propagation on the range of the area fraction for the displaced

*/ PP / PP / PP ˨ PP ˨ PP ˨ PP*

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**Maximum distance for the propagation [mm]**

**Figure 12.** The standard deviation of the area fraction in each class interval of θ (SDin) at the rami chorii.

area at φ = 45–135°.

**SDin**

**Figure 10.** Influence of the wavelength on the range of the area fraction for the displaced area at φ = 45–135°.

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**Figure 11.** Influence of the Maximum distance for the propagation on the range of the area fraction for the displaced area at φ = 45–135°.

**Figure 9.** Influence of the maximum distance for the propagation on the mean of the modified displacement.

36 Highlights on Hemodynamics

**Figure 10.** Influence of the wavelength on the range of the area fraction for the displaced area at φ = 45–135°.

**Figure 12.** The standard deviation of the area fraction in each class interval of θ (SDin) at the rami chorii.

previously [24], and it has been shown that the magnitude and direction of the displacement would be helpful for the blood circulation in the placenta [24]. Because the general aspects of the displacement caused by the contraction were investigated in the previous study [24], the influence of each parameter on the displacement and how to modulate the parameter and model for representing the dysfunction of the placenta, including hypoxia, has been barely discussed. Based on the previous computation [24], the analysis for the preplacental hypoxia

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The maximum propagation distance influenced the displaced region, especially near the chorionic plate. However, the characteristic z coordinates, was not influenced by the shear elastic modulus and the maximum propagation distance. When increase in the shear elastic modulus of the placenta reduced the displacement caused by the contraction of the stem villi, the placenta in preplacental hypoxia or uteroplacental hypoxia caused smaller displacement than that in postplacental hypoxia. Also, changes in the magnitude and direction of the displacement would occur at the characteristic positions, kept in all the models. The range of the area fraction for the displaced area at φ = 45–135° got smaller as the maximum distance for the propagation became longer. The influence of the wavelength and the maximum distance for the propagation on the standard deviation of the area fraction in each class interval of θ (SDin) was dependent on the z coordinates. The SDin was more influenced by the z coordinates than the wavelength and the maximum distance so that SDin would be dependent on the shape of the stem villi. The ranges of the area fraction at φ = 45–135°and SDin in the postplacental hypoxia model was larger than those in the preplacental hypoxia and uteroplacental hypoxia model. Postplacental hypoxia would cause more varied displacement pattern than

According to the aforementioned results, postplacental hypoxia would have the small region displaced with various directions, and preplacental hypoxia and uteroplacental hypoxia would have the large region displaced with similar directions. In the intervillous space, the maternal blood in postplacental hypoxia might hardly circulate around the villous tree because of the small displacement regions with various directions while that in preplacental hypoxia and uteroplacental hypoxia might experience difficulties in circulation because of the large region displaced with similar directions. In the terminal villi, the fetal blood in postplacental hypoxia might have difficulty in circulation because the small amount of the capillaries would be received the displacement with various directions while that preplacental hypoxia and uteroplacental hypoxia might flow in the capillaries with difficulties because the large

amount of the fetal blood would receive the displacement with similar directions.

The villous tree model enables us to depict the displacement pattern in the placenta, which is linked to the mechanical environment. It is possible to apply the mechanical environment to the computation about the model of the terminal villi, which is composed of the capillaries [29]. In the meantime, telocytes in the placenta have been found [30–33]. The distribution of the telocytes should be considered because the placenta is an innervated organ. Moreover, a novel medical imaging method available for the placenta [34] is also developed. If the villus tree model and the aforementioned findings and methods are combined together, dysfunc-

and uteroplacental hypoxia model and the postplacental hypoxia model was done.

preplacental hypoxia and uteroplacental hypoxia.

tions in the placenta can be estimated more precisely.

**Figure 13.** The range of the area fraction for 45–135°, and range of SDin (θ). Post, postplacental hypoxia; pre&up, preplacental hypoxia and uteroplacental hypoxia.

**Figure 13** shows the ranges of the area fraction for φ = 45–135° and SDin at the control model (s = 2.9 mm, μ = 3.36 × 10−4 Pa), postplacental hypoxia model (s = 1.45 mm, μ = 8.41 × 10−5 Pa), and preplacental hypoxia and uteroplacental hypoxia model (s = 4.35 mm, μ = 2.10× 10−3 Pa). In both ranges, the postplacental hypoxia model was the largest while the preplacental and uteroplacental hypoxia model was the smallest. The result described that the direction of the displacement in the postplacental hypoxia model was more varied than that in the preplacental hypoxia and uteroplacental hypoxia model.

#### **4. Discussion**

In this chapter, the displacement caused by the contraction of the stem villi in the three types of hypoxia in the placenta was evaluated. While the villous tree is rich in terminal villi at preplacental hypoxia and uteroplacental hypoxia, it has few terminal villi at postplacental hypoxia. Assuming that increase in the terminal villi in the placenta induces higher shear elastic moduli of the placenta, postplacental hypoxia would show lower shear elastic moduli while preplacental hypoxia and uteroplacental hypoxia would show higher ones. In the meantime, the computational model of the villous tree with active contractions has been developed previously [24], and it has been shown that the magnitude and direction of the displacement would be helpful for the blood circulation in the placenta [24]. Because the general aspects of the displacement caused by the contraction were investigated in the previous study [24], the influence of each parameter on the displacement and how to modulate the parameter and model for representing the dysfunction of the placenta, including hypoxia, has been barely discussed. Based on the previous computation [24], the analysis for the preplacental hypoxia and uteroplacental hypoxia model and the postplacental hypoxia model was done.

The maximum propagation distance influenced the displaced region, especially near the chorionic plate. However, the characteristic z coordinates, was not influenced by the shear elastic modulus and the maximum propagation distance. When increase in the shear elastic modulus of the placenta reduced the displacement caused by the contraction of the stem villi, the placenta in preplacental hypoxia or uteroplacental hypoxia caused smaller displacement than that in postplacental hypoxia. Also, changes in the magnitude and direction of the displacement would occur at the characteristic positions, kept in all the models. The range of the area fraction for the displaced area at φ = 45–135° got smaller as the maximum distance for the propagation became longer. The influence of the wavelength and the maximum distance for the propagation on the standard deviation of the area fraction in each class interval of θ (SDin) was dependent on the z coordinates. The SDin was more influenced by the z coordinates than the wavelength and the maximum distance so that SDin would be dependent on the shape of the stem villi. The ranges of the area fraction at φ = 45–135°and SDin in the postplacental hypoxia model was larger than those in the preplacental hypoxia and uteroplacental hypoxia model. Postplacental hypoxia would cause more varied displacement pattern than preplacental hypoxia and uteroplacental hypoxia.

According to the aforementioned results, postplacental hypoxia would have the small region displaced with various directions, and preplacental hypoxia and uteroplacental hypoxia would have the large region displaced with similar directions. In the intervillous space, the maternal blood in postplacental hypoxia might hardly circulate around the villous tree because of the small displacement regions with various directions while that in preplacental hypoxia and uteroplacental hypoxia might experience difficulties in circulation because of the large region displaced with similar directions. In the terminal villi, the fetal blood in postplacental hypoxia might have difficulty in circulation because the small amount of the capillaries would be received the displacement with various directions while that preplacental hypoxia and uteroplacental hypoxia might flow in the capillaries with difficulties because the large amount of the fetal blood would receive the displacement with similar directions.

**Figure 13** shows the ranges of the area fraction for φ = 45–135° and SDin at the control model (s = 2.9 mm, μ = 3.36 × 10−4 Pa), postplacental hypoxia model (s = 1.45 mm, μ = 8.41 × 10−5 Pa), and preplacental hypoxia and uteroplacental hypoxia model (s = 4.35 mm, μ = 2.10× 10−3 Pa). In both ranges, the postplacental hypoxia model was the largest while the preplacental and uteroplacental hypoxia model was the smallest. The result described that the direction of the displacement in the postplacental hypoxia model was more varied than that in the preplacen-

**Figure 13.** The range of the area fraction for 45–135°, and range of SDin (θ). Post, postplacental hypoxia; pre&up,

In this chapter, the displacement caused by the contraction of the stem villi in the three types of hypoxia in the placenta was evaluated. While the villous tree is rich in terminal villi at preplacental hypoxia and uteroplacental hypoxia, it has few terminal villi at postplacental hypoxia. Assuming that increase in the terminal villi in the placenta induces higher shear elastic moduli of the placenta, postplacental hypoxia would show lower shear elastic moduli while preplacental hypoxia and uteroplacental hypoxia would show higher ones. In the meantime, the computational model of the villous tree with active contractions has been developed

tal hypoxia and uteroplacental hypoxia model.

preplacental hypoxia and uteroplacental hypoxia.

**4. Discussion**

38 Highlights on Hemodynamics

The villous tree model enables us to depict the displacement pattern in the placenta, which is linked to the mechanical environment. It is possible to apply the mechanical environment to the computation about the model of the terminal villi, which is composed of the capillaries [29]. In the meantime, telocytes in the placenta have been found [30–33]. The distribution of the telocytes should be considered because the placenta is an innervated organ. Moreover, a novel medical imaging method available for the placenta [34] is also developed. If the villus tree model and the aforementioned findings and methods are combined together, dysfunctions in the placenta can be estimated more precisely.
