**3.2.2 Photodegradation of pollutants**

During the same period, Frank and Bard applied TiO2 photocatalysis in the destruction of pollutants (Frank & Bard, 1997). Their preliminary studies on the oxidation of cyanide to cyanate, by using four different forms of TiO2 under xenon lamp, opened the way for the photodegradation of environmental pollutants.

In the last three decades, TiO2 powders have known wide applications into the recovery of water of industrial, agricultural or civil origin, as well as the decontamination of atmosphere and soil, through the mineralization of the pollutants, or at least their transformation into non-toxic compounds (Fox & Dulay, 1993; Gaya & Abdullaha, 2008; Hoffmann et al., 1995; Ravelli et al., 2011). Of course, the great advantage of these photocatalytic systems depends on the fact that they do not need the use of stoichiometric quantities of chemicals, potentially themselves polluting, but they act under light (often sunlight represents the best choice) in the presence of oxygen (Malato et al., 2007).

Moreover, the high band-gap value of TiO2 allow to oxidize various organic substrates, such as hydrocarbons and their derivatives to lower molecular weight ozidized species and CO2, volatile organic compounds (VOCs) and nitrogen oxides (NOx) present in urban atmosphere (Carp et al., 2004).


**2. Methods**

#### **2.1 Cell geometry**

In our model, a cell in 2D Ω ⊂ 2 is represented as an oriented polygon including a number of boundary vertices *V*∂Ω � *vi* ∈ ∂Ω ⊂ 2 , where the location of the vertex *vi* is denoted as *xi*. The set of boundary vertices *V*∂Ω, together with a set of internal vertices *VInt* and a set of triangular elements *T*Ω � *τi*,*j*,*k* : *vi*, *v j*, *vk* ∈ *V*∂Ω ∪*V*Intg define the geometry of cell Ω (**Figure 1a**). If two cells are closely in contact, a set of adhesive springs are generated between them (**Figure 1a**, red bars in the dashed blue box). There are several interior vertice on each cell boundary edge. They are evenly distributed along that edge. These interior vertice are the potential locations for newly generated adhesive spring to attach on. Any force applied on that interior vertex through the attached adhesive spring will be mapped onto its nearest end-node vertex of the corresponding boundary edge.

#### **2.2 Viscoelasticity of the cell**

Previous researches have demonstrated that the cell cytoskeleton exhibits viscoelastic characteristic [14, 15]. Following the studies of [16, 17], we assume that, during cell deformation and cell migration, linear viscoelasticity is adequate to describe the mechanical properties of the cell.

*A Dynamic Finite Element Cellular Model and Its Application on Cell Migration DOI: http://dx.doi.org/10.5772/intechopen.94181*

#### **Figure 1.**

*The cell geometry and the chemical pathway between cell-substrate and intercellular adhesion. (a) the cell in our model is represented as following: The cell boundary is defined by an oriented polygon including a number of boundary vertices. A triangular mesh tiling up a cell is generated based on the method of farthest sampling [11]. The E-cadhesion type of intercellular adhesions between two neighboring cells are represented as elastic springs (red bars in the blue box, the dashed blue box is for a closer view). (b) each triangular element exhibits viscoelastic characteristic using a generalized Maxwell model following [5, 12, 13]. (c) the positive feedback loop between focal adhesion and cell protrusion is built up in each vertex of the triangular mesh following [8]. Such network includes the proteins of integrin, Paxillin, Rac, and PAK. The protein Merlin on the cadherin is also included to count the effects of intercellular adhesion on cell migration [9].*

#### *2.2.1 Strain and stress tensors*

We use the strain tensor *ε*ð Þ *x*, *t* to describe the local cell deformation at *x* at time *t*. *ε*ð Þ *x*, *t* takes the form of *ε*1,1 ¼ *∂u*1*=∂x*1, *ε*2,2 ¼ *∂u*2*=∂x*2, and *ε*1,2 ¼ *ε*2,1 ¼ 1 2 ð Þ *∂u*1*=∂x*2 þ *∂u*2*=∂x*1 , where *u x*ð Þ , *t* defined as ð Þ *u*1ð Þ *x*, *t* , *u*2ð Þ *x*, *t T* ⊂ 2 is the displacement of *x* at time *t*. We use the stress tensor *σ* ð Þ *x*, *t* to describe the local forces at *x* at time *t*. Here *σ* is correlated with *ε* by a generalized Maxwell model: *σ* ð Þ¼ *x*, *t σ*∞ð Þþ *x*, *t σm*ð Þ *x*, *t* [5, 12, 13], where *σ*∞ð Þ *x*, *t* is the stress of the long-term elastic element and *σm*ð Þ *x*, *t* is the stress of the Maxwell elastic element. *E*∞, *Em*, and *ηm* denote the long-term elastic modulus, elastic modulus of the Maxwell elastic element, and viscous coefficient of the Maxwell viscous element, respectively (**Figure 1b**). The strain of the Maxwell elastic element *ε*1ð Þ *x*, *t* and the strain of the viscous element *ε*2ð Þ *x*, *t* sum up to the strain tensor *ε*ð Þ *x*, *t* : *ε*1ð Þþ *x*, *t ε*2ð Þ¼ *x*, *t ε*ð Þ *x*, *t* .

We assume that the total free energy of a cell is the summation of its elastic energy, its adhesion energy due to the contact with the substrate, its elastic energy due to the intercellular adhesions with neighboring cells, and its energy due to the forces exerting on the boundary.

#### *2.2.2 Cell elastic energy*

The elastic energy due to the deformation of the cell Ω is given by

$$E\_{\Omega}(t) = \frac{1}{2} \int\_{\Omega} \left(\sigma\_{\Leftrightarrow}(\mathfrak{x}, t) + \sigma\_{d}\mathfrak{d}\_{\vec{\eta}}(\mathfrak{x})\right)^{T} \mathfrak{e}\left(\mathfrak{x}, t\right) d\mathfrak{x} + \frac{1}{2} \int\_{\Omega} \sigma\_{m}(\mathfrak{x}, t)^{T} \mathfrak{e}\_{1}(\mathfrak{x}, t) d\mathfrak{x},\tag{1}$$

where *σa* is a homogeneous contractile pressure following [6].

#### *2.2.3 Cell adhesion energy due to the contact with the substrate*

The energy due to the adhesion between the cell and the substrate is given by [6].

$$\frac{Y(\mathbf{x},t)}{2}\int\_{\Omega} \mathfrak{u}\left(\mathbf{x},t\right)^{2}d\mathbf{x},\tag{2}$$

where *Y*ð Þ *x*, *t* is the adhesion coefficient at time *t* and is proportional to the strength of local focal adhesions [18]: *Y*ð Þ¼ *x*, *t nx*,*t n*0 *EstYa*, where *nx*,*t* is the number of bound integrins at location *x* at *t* (more details of calculating *nx*,*t* in **Model of focal adhesion**), *n*0 is a normalized constant number, *Est* is the stiffness of the substrate and *Ya* is the basic adhesion constant following [18].

#### *2.2.4 Cell adhesion energy due to intercellular adhesion*

The energy due to the intercellular adhesions, which are modeled as elastic springs, is given by 1 2 P *l kl ul*ð Þ*t* 2 , where *kl* is the spring constant of the spring *l*. Its orientation angle at time *t* is denoted as *θl*ð Þ*t* . Its transformation vector *T*ð Þ*θ* is denoted as cosð Þ *θl* , sin ð Þ *θl* , � cosð Þ *θl* ð Þ , � sin ð Þ *θl* . So *ul*ð Þ*t* can be written as *ul*ðÞ¼ *t T*ð Þ *θl* ð Þ *u*11ð Þ*t* , *ul*2ð Þ*t* , where *ul*1ð Þ*t* and *ul*2ð Þ*t* are the displacements of the two endnode vertice *x*1 and *x*2 of *l* at time *t*. The elastic force of *l* due to displacement of Δ*l* is applied on *xi* and *x j* as *fl* ¼ *f*ð Þ Δ*l el* and -*f*ð Þ Δ*l el*, respectively, where *f*ð Þ Δ*l* is the magnitude of *fl* , *el* is the unit vector of the orientation of *l*.

#### *2.2.5 Boundary and protrusion forces*

Furthermore, the local forces applied on the cell boundary also contribute to the energy. Following [6], the tension force along the cell boundary is considered. In addition, we also incorporate the protrusion force on the leading edge of migrating cell. The contribution of these two forces can be written as

$$\int\_{\partial\Omega} (\lambda(\mathfrak{x},t) + f(\mathfrak{x},t)) \mathfrak{u}(\mathfrak{x},t) d\mathfrak{x},\tag{3}$$

where *λ*ð Þ *x*, *t* is the line tension force and *f x*ð Þ , *t* is the protrusion force. Line tension force is written as *λ*ð Þ¼� *x*, *t f mκ*ð Þ *x*, *t n x*ð Þ , *t* , where *f m* is a contractile force per unit length, *κ*ð Þ *x*, *t* is the curvature, and *n x*ð Þ , *t* is the outward unit normal at *x* at time *t* [6]. Protrusion force is denoted as *f* ð Þ¼� *x*, *t f an x*ð Þ , *t* , where *f a* is the protrusion force per unit length.

#### *2.2.6 The total free energy and its dissipation*

In summary, the total free energy of cell Ω at time *t* is given by

$$E\_{\Omega}(t) = \frac{1}{2} \int\_{\Omega} \left( \left( \sigma\_{\alpha}(\mathbf{x}, t) + \sigma\_{a} \delta\_{\overline{\eta}} \right)^{T} \mathbf{e} \left( \mathbf{x}, t \right) + \sigma\_{m}^{T}(\mathbf{x}, t) \mathbf{e}\_{1} (\mathbf{x}, t) \right) d\mathbf{x} + \frac{1}{2} \int\_{\Omega} \left( \sigma\_{a}, \sigma\_{a}, 0 \right) \mathbf{e} \left( \mathbf{x}, t \right) d\mathbf{x} \tag{4}$$

$$+ \frac{Y(\mathbf{x}, t)}{2} \int\_{\Omega} \mathbf{u} \left( \mathbf{x}, t \right)^{2} d\mathbf{x} + \frac{1}{2} \sum\_{l} k\_{l} \mathbf{u}\_{l}(t)^{2} + \int\_{\partial \Omega} \left( \lambda(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_{l}(\mathbf{x}, t) \right) \mathbf{u}(\mathbf{x}, t) d\mathbf{x} \right). \tag{4}$$

*A Dynamic Finite Element Cellular Model and Its Application on Cell Migration DOI: http://dx.doi.org/10.5772/intechopen.94181*

The energy dissipation of *E*Ωð Þ*t* due to cell viscosity is determined by the viscous coefficient *ηm* and the strain of the viscous element *ε*2ð Þ *x*, *t* [19]: � Ð Ω*ηm ∂ ε*2 *∂t* � �2 *dx*. The dissipation of the total free energy of the cell can be written as

$$\begin{split} \frac{\partial}{\partial t} E\_{\Omega}(t) &= \frac{\partial}{\partial t} \Big[ \frac{1}{2} \Big]\_{\Omega} \Big( \left( \sigma\_{\text{ov}}(\mathbf{x}, t) + \sigma\_{d} \delta\_{\overline{\eta}} \right)^{T} \mathbf{e} \left( \mathbf{x}, t \right) + \sigma\_{m}^{T}(\mathbf{x}, t) \mathbf{e}\_{1} (\mathbf{x}, t) \Big) d\mathbf{x} + \frac{1}{2} \Big[ \int\_{\Omega} \left( \sigma\_{\text{av}}, \sigma\_{\text{a}} \right) \mathbf{e} \left( \mathbf{x}, t \right) d\mathbf{x} \Big] \\ &+ \frac{Y(\mathbf{x}, t)}{2} \int\_{\Omega} \mathbf{u} \left( \mathbf{x}, t \right)^{2} d\mathbf{x} + \frac{1}{2} \sum\_{l} k\_{l} \mathbf{u}\_{l} (t)^{2} + \int\_{\partial \Omega} \left( \lambda(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_{l}(\mathbf{x}, t) \right) \mathbf{u} \left( \mathbf{x}, t \right) d\mathbf{x} \Big] \\ &= - \int\_{\Omega} \eta\_{m} \Big( \frac{\partial \mathbf{e}\_{2}(\mathbf{x}, t)}{\partial t} \Big)^{2} d\mathbf{x}. \end{split} \tag{5}$$

Since *ηm ∂ ε*2 *∂t* ¼ *Emε*1 and *σ*∞ ¼ *E*∞*ε*, and *ul*ðÞ¼ *t T*ð Þ *θl ul*1 ð Þ , *ul*2 . Eq. (5) can be rewritten as

$$\begin{split} & \int\_{\Omega} E\_{\omega} \mathbf{e}(\mathbf{x}, t) \frac{\partial \mathbf{e}(\mathbf{x}, t)}{\partial t} d\mathbf{x} + \int\_{\Omega} \boldsymbol{\sigma}\_{m}^{T}(\mathbf{x}, t) \frac{\partial \mathbf{e}(\mathbf{x}, t)}{\partial t} d\mathbf{x} + \int\_{\Omega} (\boldsymbol{\sigma}\_{a}, \boldsymbol{\sigma}\_{a}, \mathbf{0}) \frac{\partial \mathbf{e}(\mathbf{x}, t)}{\partial t} d\mathbf{x} \\ & \quad + Y \int\_{\Omega} \mathbf{u}(\mathbf{x}, t) \frac{\partial \mathbf{u}(\mathbf{x}, t)}{\partial t} d\mathbf{x} + \sum\_{l} T(\theta\_{l})^{T} T(\theta\_{l}) k\_{l}(\mathbf{u}\_{l1}(t), \mathbf{u}\_{l2}(t)) \frac{\partial \mathbf{u}(\mathbf{x}, t)}{\partial t} \\ & \quad + \int\_{\partial \Omega} \left( \lambda(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_{l}(\mathbf{x}, t) \right) \frac{\partial \mathbf{u}(\mathbf{x}, t)}{\partial t} d\mathbf{x} = \mathbf{0}. \end{split} \tag{6}$$

$$\text{Denoting } \mathbf{B} = \begin{pmatrix} \partial / \partial \mathbf{x}\_{1} & \mathbf{0} \\ \mathbf{0} & \partial / \partial \mathbf{x}\_{2} \end{pmatrix}, \text{then } \mathbf{e}(\mathbf{x}, t) = \mathbf{B} \mathbf{u}(\mathbf{x}, t). \text{ According to Gauss's}$$

*∂=∂x*2 *∂=∂x*1 divergence theorem, we rewrote Ð Ωð Þ *σa*, *σa*, 0 *B ∂u x*ð Þ , *t ∂t* as Ð Ω*σa*∇ � *∂u x*ð Þ , *t ∂t dx*, which leads to Ð ∂Ω*σan x*ð Þ , *t ∂ u x*ð Þ , *t ∂t dx*.

$$\text{Denoting } \mathbf{A}\_{l} = \begin{pmatrix} c\_{l}^{2} & c\_{l}s\_{l} & -c\_{l}^{2} & -c\_{l}s\_{l} \\ c\_{l}s\_{l} & s\_{l}^{2} & -c\_{l}s\_{l} & -s\_{l}^{2} \\ -c\_{l}^{2} & -c\_{l}s\_{l} & c\_{l}^{2} & c\_{l}s\_{l} \\ -c\_{l}s\_{l} & -s\_{l}^{2} & c\_{l}s\_{l} & s\_{l}^{2} \end{pmatrix}, \text{ where } c\_{l} = \cos\left(\theta\_{l}\right) \text{ and }$$

$$\text{i.e. } (0) \text{ E- (6) are the constraints in.}$$

*sl* ¼ *sin* ð Þ *θl* . Eq. (6) can be rewritten as

$$\begin{aligned} \int\_{\Omega} \mathbf{B}^T \boldsymbol{\sigma}(\mathbf{x}, t)^T d\mathbf{x} + Y \int\_{\Omega} \boldsymbol{\mathfrak{u}}(\mathbf{x}, t) d\mathbf{x} + \sum\_{l} \mathbf{A}\_l k\_l(\boldsymbol{\mathfrak{u}}\_{l1}(t), \boldsymbol{\mathfrak{u}}\_{l2}(t)) &= \\ - \int\_{\partial\Omega} \boldsymbol{\sigma}\_d \boldsymbol{\mathfrak{u}}(\mathbf{x}, t) + \boldsymbol{\lambda}(\mathbf{x}, t) + f(\mathbf{x}, t) + f\_l(\mathbf{x}, t) d\mathbf{x} \end{aligned} \tag{7}$$

#### *2.2.7 Stress of viscoelastic cell and its update*

By using general Maxwell model, the stress *σ* ð Þ *x*, *t* can be written as [12, 13, 19]:

$$E\_{\approx} \mathfrak{e}(\mathfrak{x}, t) + \int\_{0}^{t} E\_{m} \mathfrak{e}\_{m \stackrel{t \to}{\mathbb{R}}} \frac{\partial \mathfrak{e}(\mathfrak{x}, s)}{\partial s} ds = \mathfrak{e}\_{\approx}(\mathfrak{x}, t) + \mathfrak{e}\_{m}(\mathfrak{x}, t). \tag{8}$$

During the time interval Δ*t* ¼ *tn*þ1 � *tn*, where *tn* is the *n*-th time step, *σn*þ1 *m* ð Þ *x* can be written as [19]:

$$\left[e^{-\frac{\Delta t}{\eta\_m/E\_{\infty}}}\sigma\_m^n(\mathfrak{x}) + \frac{E\_m}{E\_{\infty}}\right]\_{t\_n}^{t\_{n+1}} e^{-\frac{t\_{n+1} - \varepsilon}{\eta\_m/E\_{\infty}}} ds \frac{\sigma\_{\infty}^{n+1}(\mathfrak{x}) - \sigma\_{\infty}^n(\mathfrak{x})}{\Delta t}.\tag{9}$$

Therefore, the stress *σn*ð Þ *x* at *tn* can be written as

$$
\sigma^n(\mathfrak{x}) = \sigma^n\_{\infty}(\mathfrak{x}) + \sigma^n\_m(\mathfrak{x}) \tag{10}
$$

#### *2.2.8 Force balance equation for discretized time step*

For each triangular element *τi*,*j*,*k*, Eq. (7) at time step *tn*þ1 can be rewritten using Eq. (10) as

$$\begin{split} & \int\_{\mathfrak{r}\_{ij,k}} \mathbf{B}^T \left( E\_{\alpha} \mathbf{B} \mathbf{u}^{n+1} (\boldsymbol{\omega}) + e^{-\frac{\Delta t}{\mathfrak{r}\_m \boxplus \boldsymbol{\omega}\_m}} \ \sigma\_m^n + \gamma\_m A\_m \left( E\_{\alpha} \mathbf{B} \mathbf{u}^{n+1} (\boldsymbol{\omega}) - E\_{\alpha} \mathbf{B} \mathbf{u}^n (\boldsymbol{\omega}) \right) \right) d\mathbf{x} \\ & + Y \Big[\_{\mathfrak{r}\_{ijk}} \mathbf{u}^{n+1} d\mathbf{x} + \sum\_{l \in \mathfrak{r}\_{ij,k}} A\_l k\_l \left( \mathfrak{u}\_{l1}^{n+1}, \mathfrak{u}\_{l2}^{n+1} \right) = F^{n+1} (\boldsymbol{\omega}), \end{split} \tag{11}$$

where *γm* ¼ *Em=E*∞, *Am* ¼ 1�*e* � Δ*t ηm=E*∞ Δ*t ηm=E*∞ , and *Fn*þ1 ¼ �Ð ∂Ω*σa n x*ð Þþ , *t λ*ð Þþ *x*, *t f x*ð Þþ , *t fl* ð Þ *x*, *t dx*. Eq. (11) leads to the following linear force-balance equation

$$\mathbf{K}^{n+1}\_{\mathbf{r}\_{i\neq k}} \mathbf{u}^{n+1}\_{\mathbf{r}\_{i\neq k}} = \mathbf{f}^{n+1}\_{\mathbf{r}\_{i\neq k}},\tag{12}$$

where *Kn*þ1 *τi*,*j*,*k* , *un*þ1 *τi*,*j*,*k* , and *f n*þ1 *τi*,*j*,*k* are the stiffness matrix, displacement vector, and integrated force vector of *τi*,*j*,*k* at time step *tn*þ1 (see more details of derivation of (Eq. 12) in [11]).

We can then assemble the element stiffness matrices of all triangular elements into one big global stiffness matrix *Kn*þ1 . Therefore, the linear relationship between the concatenated displacement vector *un*þ1 of all cell vertice and the force vector *f n*þ1 on them is given by

$$\mathbf{K}^{n+1} \mathbf{u}^{n+1} = \mathbf{f}^{n+1}.\tag{13}$$

Changes in the cell shape at time step *tn*þ1 can be obtained by solving Eq. (13). For vertex *vi* at *xi*, its new location at next time step is then updated as *xn*þ1 *i* ¼ *xn i* þ *un*þ1ð Þ *vi* .

#### **2.3 Mechano-chemical pathway in the cell**

Upon contact with the environment, cells can transfer the mechanical cues into biochemical signals, which can trigger the initiation of further cellular behaviors [20]. In our model, we considered a mechano-chemical pathway consisting of two parts, where one is to regulate the feedback loop between focal adhesion and cell protrusion and the other is to regulate the transmission of mechanical signal between adjacent cells through intercellular adhesions.

#### *2.3.1 Model of focal adhesion*

For each vertex *vi* in cell Ω, we assign a constant number of integrin ligand on it. These integrin molecules can bind or unbind with fibronectin molecules on the

*A Dynamic Finite Element Cellular Model and Its Application on Cell Migration DOI: http://dx.doi.org/10.5772/intechopen.94181*

substrate underneath. Following [21], the numbers of bound and unbound integrin ligand molecules are determined by

$$\frac{dR\_b}{dt} = k\_f n\_s R\_u - k\_r R\_b,\tag{14}$$

where *Ru* and *Rb* are the numbers of unbound and bound integrin ligand, respectively; *k f* is the binding rate coefficient; *ns* is the concentration of fibronectin per cell vertex; *kr* is the unbinding rate coefficient. *kr* depends on the magnitude of the traction force *fr* applied on *vi*. Traction force *fr*ð Þ *x*, *t* on *x* at time *t* is given by *Y*ð Þ *x*, *t u x*ð Þ , *t* following [6]. *kr* is determined by *kr* ¼ *kr*0 *e*�0*:*04 *fr* þ 4*e* � 7*e*0*:*2 *fr* � � following [22], where *kr*0 is a constant. *k f* is related with the substrate stiffness by *k f* ¼ *kf*0*E*2 *st= E*2 *st* þ *E*2 *st*0 � � [16, 23], where *kf*0 and *Est*0 are constants (see Appendix for choosing *Est*0).

#### *2.3.2 Model of feedback loop between focal adhesion and cell protrusion*

We introduced a simplified model of a positive feedback loop to control the spatial distribution of the focal adhesions, which governs the direction of cell protrusion [8, 24]. In our model, this feedback loop involves proteins of Paxillin, Rac, and PAK (**Figure 1c**). Upon formation of focal adhesion, Paxillin is activated by active PAK. The active Paxillin then activates Rac, which in turn triggers the activation of PAK. The activated Rac is responsible for protruding cells [25]. Since the protein Merlin on the intercellular cadherin complex also plays a role in activating Rac [9], we include Merlin in our feedback loop. The protein concentration over time is updated through a set of differential equations following [8]:

$$k\frac{dr}{dt} = k\_{\mathbf{x},r} \left( k\_m \frac{C\_r^2}{C\_r^2 + m^2} \mathbf{x} - r \right) \tag{15}$$

$$\frac{dp}{dt} = k\_{r,p}(r-p) \tag{16}$$

$$\frac{d\mathbf{x}}{dt} = k\_{p,\mathbf{x}} \left( k\_{\mathbf{x}} \frac{p^2}{p^2 + C\_{\mathbf{x}}^2} n - \mathbf{x} \right) \tag{17}$$

where *x*, *r*, *p*, *m* and *n* are the concentrations of activated Paxillin, activated Rac, activated PAK, Merlin and bound integrins, respectively. *kx*,*r*, *kr*,*p*, *kp*,*x*, *km*, *kx*, *Cr*, *Cx* are the parameters of corresponding rates. The level of activated Rac was used to determine the protrusion force on the leading edge of the migrating cell (see details of cell protrusion model in Appendix).

#### *2.3.3 Model of mechanosensing through intercellular adhesion*

We added Merlin in the feedback loop (**Figure 1c**) following a previous study reporting that Merlin on the intercellular cadherin complex regulates the Rac activity [26]. As illustrated in **Figure 2a**, for two adjacent cells *C*1 and *C*2 where *C*1 is the leader cell and *C*2 is the follower cell, if both cells are at static state, Merlin molecules only locate on the cadherin spring (**Figure 2b** top). As reported by [9], Merlin suppressed the binding of integrin. Due to such suppression, Rac turns to inactivated on the Merlin-expressed site. Once cell *C*1 starts to migrate, tension force is generated on the cadherin spring between *C*1 and *C*2. Merlin is therefore

#### **Figure 2.**

*Mechanosensing through intercellular adhesion and tissue model for collective cell migration. (a) the intercellular adhesion of the cadherin spring (red springs in the blue box) are responsible for transmitting mechanical stimulus from leader cell (C*1*) to its follower cells (C*2 *and C*3*). (b) When cells are static, Merlin which inhibits the Rac activation is bound on the cadherin spring. Once leader cell migrates, stretch is generated on the cadherin spring, Merlin on the follower cell is delocated. Therefore, Rac is activated on the follower cell. (c) the size of the wound epithelial tissue is 720 μm* � *240 μm. The right boundary is set as the wound edge (yellow line). Cells can migrate towards the open space on the right. Three measurements are introduced to measure the collective cell migration: (1) migration persistence p t*ð Þ*n , the ratio of the distance from the current position at time tn to its initial position (green line), over the length of the traversed path (red curve); (2) normalized pair separation distance di*,*j*ð Þ *tn is the separation distance between a pair of cells at time tn (green lines) divided by the average length of the two cells' traversed path (red curves); (3) migration direction angle α*ð Þ *tn is the angle between the migration direction (red arrow) and the direction towards the wound (green arrow).*

delocalized from cadherin-attached site in response to the generated tension force. As a consequence, Rac is activated there to generate protrusion force to follow the leader cell (**Figure 2b** bottom [26]). For simplicity, we introduced the inactive Merlin phenotype along with the active Merlin phenotype on the two end vertice of one intercellular cadherin adhesion. The negative feedback loop of Merlin-Rac is modeled through a set of differential equations following [9, 26], where active Merlin and inactive Merlin can switch their phenotype, but only active Merlin can suppress the Rac activity (**Figure 1c**). The delocalization of Merlin was simply modeled as Merlin switching to inactive phenotype:

*A Dynamic Finite Element Cellular Model and Its Application on Cell Migration DOI: http://dx.doi.org/10.5772/intechopen.94181*

$$\frac{dm}{dt} = e - \delta\left(f\_t > f\_{t-thr}\right)k\_{m,\epsilon}\left(k\_p\left(\frac{p^2}{p^2 + C\_\epsilon^2} + k\_\epsilon\right)m\right) \tag{18}$$

$$\frac{de}{dt} = \delta(\left.f\_t > f\_{t-thr}\right)k\_{m,\epsilon}\left(k\_p\left(\frac{p^2}{p^2 + C\_\epsilon^2} + k\_\epsilon\right)m\right) - e\tag{19}$$

where *m*, *e* and *p* are the concentrations of Merlin, inactive Merlin, and PAK, respectively. *km*,*e*, *kp*, *ke*, *Ce* are the corresponding rate parameters. *δ*ð Þ *x* is a kronecker function that *δ*ð Þ¼ *TRUE* 1 and *δ*ð Þ¼ *FALSE* 0. *ft* is the tension force through the cadherin spring and *ft*�*thr* is a force threshold.

#### **2.4 Cellular tissue for collective cell migration**

In our model, the collective cell migration was modeled using a wound tissue of epithelial cells. The tissue size is 720 *μm* � 240 *μm*. The epithelial cell type is set to MCF-10A, which is used in the *in vitro* study [10]. The corresponding epithelialspecific parameters can be found in **Table 1**. We arbitrarily set the right boundary of the tissue as the wound edge, and cells can migrate towards the open space to the right of the wound edge (**Figure 2c**). The mechano-chemical pathway was initiated first in the cells on the wound edge after they migrate. We followed a previous study [10] to divide the location of cells into four sub-regions according to their distance to the wound edge: Regions I, II, III, and IV whose distance to the wound edge is 0–160 *μm*, 160–320 *μm*, 320–480 *μm*, and 480–640 *μm*, respectively (**Figure 2c**). We ran the simulation for 12 biological hours, the same experimental duration time in the *in vitro* study [10].

#### **2.5 Measurements of the collective cell migration**

In our model, the collective cell migration are measured using four measurements following [10]:

The migration persistence. At time step *tn*, the length of the straight line between cell positions at *tn* and initial time step *t*0 over the length of the migrating trajectory:

$$p(t\_n) = \frac{|\mathbf{x}(t\_n) - \mathbf{x}(t\_0)|}{\sum\_{k=0}^{n-1} |\mathbf{x}(t\_{k+1}) - \mathbf{x}(t\_k)|},\tag{20}$$

where *t*0 is the initial time, *x*ð Þ*ti* is the position at time step *ti* (**Figure 2c.1**).

The normalized separation distance. At time step *tn*, the separation distance of a pair of two adjacent cells 1-2, divided by the average length of their migrating trajectories:

$$d\_{1,2}(t\_i) = \frac{||\mathbf{x}\_1(t\_i) - \mathbf{x}\_2(t\_i)| - |\mathbf{x}\_1(t\_0) - \mathbf{x}\_2(t\_0)||}{\frac{1}{2} \left(\sum\_{k=0}^{i-1} |\mathbf{x}\_1(t\_{k+1}) - \mathbf{x}\_1(t\_k)| + \sum\_{k=0}^{i-1} |\mathbf{x}\_2(t\_{k+1}) - \mathbf{x}\_2(t\_k)|\right)},\tag{21}$$

where the numerator is the separation distance between cells *i* and *j* at time *tn*, and the denominator is the average path length of cells *i* and *j* at time *tn* (**Figure 2c.2**).

The direction angle. The angle between the cell migration direction and the direction towards the wound.


**1. Introduction**

Leptospirosis is a worldwide bacterial zoonosis showing greater occurrence in tropical and subtropical countries. It is transmitted mainly through direct contact with animals or urine but can also be acquired indirectly by ingesting contaminated water or food. The disease is typically occupational affecting particularly farmers, slaughterhouse workers, veterinarians and their co-workers [1].

Leptospirosis brings economic loss to cattle raisers as it causes reproductive disturbances like abortion or infertility. It is considered a reproductive system disease [2].

Clinical signs can be chronic such as abortion, mainly at the pregnancy's middle third, around the fifth month, estrus repetition and stillbirths as well as placental retention not always occuring. It can cause agalactia or decrease in milk production as well as infection of young calves [3].

Cattle are the main reservoirs of serovars Hardjo [4], and others such as Pomona and Grippotyphosa [5]. They are the preferential hosts of serovar Hardjo. Serovar Hardjoprajitino is responsible for decreases in cattle milk production and conception rates [6], and serovar Hardjobovis is associated with reproductive failures [2–7].

The main serovars found in Brazil are Hardjo, Wolffi, Pomona, Grippotyphosa, Icterohaemorrhagiae [3]. There is a prevalence of serovar Hardjoprajitino, also present in commercial vaccines. However, as with other domestic mammals, cattle can be infected by any pathogenic serovar [8]. Despite some degree of agent species selectivity, the disease is not serovar-specific.

Considering the impact of leptospirosis on cattle breeding as well as its effects on human and animal health, the present study was proposed with the goal of evaluating the consequences of leptospiral infection on the pregnancy and milk production rates of a confined dairy cattle herd with respect to the serological response to 16 serovars of Leptospira spp., of importance for herbivores, during 9 months, having it associated with productive, referring to milk production, and reproductive, referring to pregnancy rate, indexes as well as monitoring leptospiral infection evolution in two groups set up and kept under similar conditions, one with animals serologically positive for at least one of the evaluated serovars and another, the control group, with animals serologically negative at study onset, the results thereof compared vis-à-vis the studied variables.

## **2. Material and methods**

With owner's consent secured, the study took place in a dairy property the authors were familiar with. These premises were selected due to the permanent availability of veterinary assistance and the authors' good understanding of its zoosanitary practices. The dairy is a fenced property capable of animal self-replacement, located in the central region of the State of São Paulo, Brazil. Its stock counts 750 animals of which about 400 are of high genetic lineage, pure origin Black and White Dutch lactating cows kept in a semi-confinement system and milked three times a day. The production system is completely computerized, allowing data to be obtained on a monthly basis to evaluate the individual and herd productive and reproductive indexes. Milking is carried out with the help of a carousel-type parlor.

Animals are vaccinated against IBR, BVD, brucellosis and leptospirosis one week before dry-off, approximately 60 days before parturition, and receive a second shot 30 days later. Lactating cows are vaccinated between 120 to 128, 270 to 278 and 420 to 428 days of the lactation cycle.

The experimental groups were formed spliting 202 lactating animals in two groups. One group had 50 animals with non-reactive results to anti-leptospiral antibodies (G-1) while the other has 50 sera reactive animals with a microscopic agglutination test (MAT) titer ≥100 IU for at least one Leptospira serovar (G-2). G-2 was reduced to 39 animals by the end of the study as 11 were discarded by the owner during the experimental period. Both groups were set up with animals picked at the beginning of their lactation cycles affording longer monitoring times within their milk production periods.

G-1 and G-2 Blood samples were collected monthly for 9 months by mammary vein puncture to assess herd infection dynamics. In order to diagnose infection,

### *Serological Monitoring for* Leptospira *Spp. and Monitoring of Productive and Reproductive… DOI: http://dx.doi.org/10.5772/intechopen.98983*

MAT were performed employing live antigens from 16 serovars belonging to 10 serogroups: serovar Bratislava (serogroup Australis); Castellonis (Ballum); Canicola (Canicola); Djasiman (Djasiman); Grippotyphosa (Grippotyphosa); Copenhageni, Icterohaemorrhagiae (Icterohaemorrhagiae); Pomona (Pomona); Pyrogenes (Pyrogenes); Tarassovi (Shermani); Guaricura, Hardjobovis, Hardjo CTG, Hardjoprajitono, Mini, Wolffi (Sejroe). The titer cut-off point was 100 IU.

Production indexes, such as the monthly average of milk production in liters, as well as reproductive factors such as interval between birth and conception, conception rate at the first service, conception rate in all services, services per conception, age at first delivery and number of lactations were evaluated from each animal's history obtained from the dairy's database.

The results of this longitudinal observational study were analyzed by evaluating the relationship between infection by *Leptospira* spp. and the productive and reproductive parameters of cows. Regarding milk production, every month the corresponding milk output log10 was computed and used first to compute the area under the curve (AUC) for each animal and then the mean and standard deviation.

MAT serovar positivity at the diverse moments was established by means of descriptive statistics where positivity percentages represented the frequency distribution of occurrences. Association of pregnancy to reactive serovar for the two groups in the different moments was carried out with the Goodman association test for contrast between and within multinominial populations [8, 9], whose significance was designated with the help of lowercase and uppercase Latin letters. With significance indicated by lowercase and uppercase Latin letters. Milk production comparison for reacting serovar for each moment was done by independent samples Student's t-test [10]. Statistical results were discussed at the 5% significance level.


**3. Corporate Innovation and Sustainability Practices**

Governance, corporate social responsibility (CSR), and innovation are interconnected aspects of corporate strategy and performance. Governance refers to the systems and processes through which companies are directed, controlled, and managed [3,6]. It encompasses the roles and responsibilities of the board of directors, management, and other stakeholders in decision-making and accountability. While CSR involves a company's commitment to conducting business ethically and responsibly, considering the impact of its operations on society, the environment, and stakeholders. It goes beyond legal requirements and aims to create a positive impact on society. Effective governance practices can provide the structure and framework for companies to prioritize and integrate CSR and innovation into their strategies. Moreover, CSR initiatives can also foster innovation by encouraging companies to identify and respond to emerging social and environmental trends. CSR practices can inspire creativity, collaboration, and the exploration of new business models that generate both social and economic value [2–4]. Moreover, innovation can contribute to CSR by enabling companies to develop sustainable solutions, reduce their environmental footprint, or create products and services that meet societal needs. Innovation-driven CSR initiatives can enhance a company's reputation, competitiveness, and long-term sustainability. The following papers address the interrelations between governance, CSR, and innovation arguing that, when effectively integrated, can drive

responsible and sustainable business practices while fostering creativity, competitiveness, and value creation for companies.

For instance, the seventeenth paper (Contribution 17) examines whether transformational leadership influences ESG performance in SMEs, whether organizational innovation mediates the relationship between transformational leadership and ESG performance, and the moderating effect of external social capital on transformational leadership and organizational innovation. Based on higher-order theory, resource-based theory, stakeholder theory, the results of the study indicate that transformational leadership has a positive effect on ESG performance, and that organizational innovation partially mediates the relationship between transformational leadership and corporate ESG performance. Furthermore, external social capital moderates the direct relationship between transformational leadership and organizational innovation and moderates the role of organizational innovation as a mediator between transformational leadership and ESG performance. This study adds to our further understanding of the relationship between transformational leadership and ESG performance in SMEs, expanding the antecedent research on ESG performance and providing a basis for sustainable SME development.

Linked to sustainable development and leadership, the eighteen paper (Contribution 18) investigates the association between executives' environmental protection background and corporate green innovation, as well as the mechanisms that influence this relationship. Drawing on the upper echelons theory, the study reveals a positive correlation between executives' environmental protection background and corporate green innovation. This positive relationship remains robust even when using alternative regression models and accounting for different measures of green innovation. Moreover, the findings indicate that media attention and board independence have a positive moderating influence on the relationship between executives' environmental protection background and green innovation.

Theoretically, the nineteenth article (Contribution 19) introduces a framework that examines the potential influence of board independence and the utilization of digital technology on a corporation's environmental performance. They used a sample of 53 publicly listed Italian companies is selected, and data on board composition, greenhouse gas emissions, and expenditures for Enterprise Resource Planning (ERP) digital technologies are collected over a five-year period. The results of the analysis partially support the predictions made in the framework. Specifically, a higher degree of board independence is associated with improved environmental performance. Their further analysis reveals that the environmental performance of companies is positively influenced by the use of digital technologies when these companies have a higher proportion of independent directors on their boards. This research contributes to our understanding of the determinants of Corporate Digital Responsibility (CDR), indicating that a greater presence of independent directors on a board has a positive impact on CDR.

Looking at the overborrowing issue in China, the twentieth article (Contribution 20) explores the impact of overborrowing in China's state-owned enterprises (SOEs) on their innovation spending. The study benefits from a theoretical model within the unique institutional context of China's banks, specifically focusing on the administrative-economic governance. By analyzing a longitudinal panel dataset of Chinese listed companies from 2012 to 2018, the study confirms that overborrowing acts as a mediator between state ownership and innovation expenditure, emphasizing the importance of improving the monitoring of banks to foster innovation in transitional economies. Furthermore, the study investigates the influence of political connections and managers' R&D experience in leveraging the innovation resources available to SOEs. The findings of this study reveal a negative impact of government intervention on the allocation of innovation resources and contribute to our understanding of the role of debt governance in promoting innovation in transition economies.

Focusing on the corporate strategy broadly, the twenty-first article (Contribution 21) employs propensity score matching, ordinary least squares, and quantile regression techniques to examine the relationship between voluntary disclosure of social responsibility and

innovation investment in enterprises. The findings reveal that when enterprises engage in voluntary disclosure of social responsibility, it leads to an increase in innovation investment. In other words, corporate social responsibility has a significant positive impact on both innovation and investment. However, as the level of innovation investment in enterprises increases, the impact of corporate social responsibility on innovation gradually diminishes. This research highlights the complex dynamics between social responsibility and innovation investment, providing insights into their interplay within corporate strategies.

While the twenty-second article (Contribution 22) focuses on the importance of environmental corporate social responsibility (CSR) for achieving economic benefits and sustainable development is a subject of great interest among theorists and practitioners. However, the specific relationship between environmental CSR and green innovation performance remains unclear. To address this research gap, this paper proposed a research model, incorporating the mediating effect of shared vision capability and the moderating effect of resource slack. The aim is to investigate the impact of environmental CSR on green innovation performance and to determine the conditions under which this relationship is most significant. The results of the study confirm a positive association between environmental CSR and green innovation performance. Additionally, shared vision capability was found to mediate the link between environmental CSR and green innovation performance. Furthermore, resource slack was found to have a statistically significant moderating effect on the relationship between environmental CSR and green innovation performance. These findings provide valuable insights for managers in formulating management policies related to environmental CSR, shared vision capability, and green innovation performance. By leveraging these insights, enterprises can work towards sustainable development and contribute to environmental friendliness in society as a whole.

The impact of innovation quality has become a growing concern in the academic industry. In previous studies, the impact of TMT experience heterogeneity on enterprise innovation quality has not been well explored. In the context of enterprise technologies and innovation, the twenty-third article (Contribution 23) argues that high-quality innovation can solve the "bottleneck" problem of key enterprise technologies and drive the high-quality development of enterprises. Based on the panel data of Chinese A-share listed companies, the following results were found (1) TMT functional experience heterogeneity positively affects partner diversity to promote innovation quality, while industrial experience heterogeneity shows the opposite result. (2) Enterprise partner diversity partially mediates the relationship between TMT experience heterogeneity and innovation quality. (3) TMT technological participation positively regulates the relationship between TMT experience heterogeneity and enterprise partner diversity.

In China, the Shanghai and Shenzhen Stock Exchanges implemented regulations in 2008 that require certain public firms to disclose their social and environmental governance information in annual reports. Therefore, the twenty-fourth article (Contribution 24) examines the impact of mandatory social and environmental regulations on firm innovation. Using a difference-in-differences approach with propensity score matching, the study finds that the firms subject to the regulations experience a significant increase in innovation, as indicated by a higher number of total patents and invention patents. Furthermore, the study reveals that the positive association between MSER and firm innovation is primarily driven by the CSR-improving effect and market-reaction effect. Specifically, the treatment firms demonstrate an enhancement in CSR performance and a decrease in transient institutional investors. These results highlight the role of MSER in fostering firm innovation and emphasize the importance of CSR and market dynamics in driving this relationship.

Green innovation plays a vital role in driving sustainable development and promoting green circular economic practices within businesses. It involves organizations considering environmental, social, and governance (ESG) aspects, and the resulting ESG advantages can serve as a catalyst for enterprises to undergo a green transformation. Focusing on that, the twenty-fifth article (Contribution 25) focuses on Chinese A-share listed companies from 2009 to 2020 to investigate the relationship between ESG rating performance and corporate

green innovation, as well as the boundary mechanisms that influence this relationship. The findings demonstrate that higher ESG ratings are associated with increased levels of green innovation among listed enterprises. Furthermore, the relationship between ESG ratings and green innovation is strengthened by the institutional environment and the availability of redundant organizational resources. This study provides empirical evidence supporting the positive impact of ESG ratings on green innovation within enterprises.


*Article* **Food from the Depths of the Mediterranean: The Role of Habitats, Changes in the Sea-Bottom Temperature and Fishing Pressure**

**Porzia Maiorano 1,2, Francesca Capezzuto 1,2,\*, Angela Carluccio 1,2, Crescenza Calculli 3, Giulia Cipriano 1,2, Roberto Carlucci 1,2, Pasquale Ricci 1,2, Letizia Sion 1,2, Angelo Tursi 1,2 and Gianfranco D'Onghia 1,2**

**Abstract:** As part of the "*Innovations in the Food System: Exploring the Future of Food*" Special Issue, this paper briefly reviews studies that highlight a link between deep-sea fishery resources (deep-sea food resources) and vulnerable marine ecosystems (VME), species, and habitats in the Mediterranean Sea, providing new insights into changes in commercial and experimental catches of the deep-sea fishery resources in the central Mediterranean over the last 30 years. About 40% of the total landing of Mediterranean deep-water species is caught in the central basin. Significant changes in the abundance of some of these resources with time, sea-bottom temperature (SBT), and fishing effort (FE) have been detected, as well as an effect of the Santa Maria di Leuca cold-water coral province on the abundance of the deep-sea commercial crustaceans and fishes. The implications of these findings and the presence of several geomorphological features, sensitive habitats, and VMEs in the central Mediterranean are discussed with respect to the objectives of biodiversity conservation combined with those of management of fishery resources.

**Keywords:** seafood; fisheries resources; vulnerable marine ecosystem; environmental change; conservation; Mediterranean