# Non-Linear Modeling of Immune System Activation and Lymph Flow Dynamics
## 2 Materials and Models
### Lymph Flow  
The _general fluid flow equations_ are given by the mass conservation equation and the equations of motion. The _mass balance_/continuity equation is as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{x})}{\partial x}+\frac {\partial(\rho v_{y})}{\partial y}+\frac{\partial(\rho v_{z})}{\partial z}=0, \tag{1}\]  
The continuity Equation (1), expresses that the mass of a system does not change due to motion, with \(\rho\) density of the fluid and \(\mathbf{v}=(v_{x},v_{y},v_{z})\) fluid velocity. The _Navier-Stokes_ equations are obtained from [PERSON]'s equations:  
\[a_{i}=F_{i}+\frac{1}{\rho}\frac{\partial T_{ij}}{\partial x_{j}} \tag{2}\]  
where \(\mathbf{T}\) is the Cauchy stress tensor. For Newtonian fluids, the form of the stress tensor is  
\[T_{ij} = (-p+\lambda\theta)\delta_{ij}+2\mu D_{ij},\text{where }\theta=\text{D}_{ii}, \text{D}_{ij}=\frac{1}{2}\left(\frac{\partial v_{i}}{\partial x_{j}}+\frac{ \partial v_{j}}{\partial x_{i}}\right) \tag{3}\]  
and, therefore, \(a_{i}=F_{i}+\frac{1}{\rho}\left(-\frac{\partial p}{\partial x_{i}}+(\lambda+ \mu)\frac{\partial\theta}{\partial x_{i}}+\mu\Delta v_{i}\right)\)[19].  
_Lymph flow dynamics_ corresponding to (1) and (2) are \[\begin{split}&\frac{\partial A}{\partial t}+\frac{\partial(Au)}{ \partial x}=0\\ &\rho\bigg{(}\frac{\partial u}{\partial t}+u\frac{\partial u}{ \partial x}\bigg{)}=-\frac{\partial p}{\partial x}+\mu\frac{\partial^{2}u}{ \partial x^{2}}-R(u)\end{split} \tag{4}\]  
where \(A(x,t)\) is the cross-sectional area of the vessel, \(u(x,t)\) is the lymph flow velocity, \(p(x,t)\) is the pressure within the vessel, \(x\) is the axial position along the vessel, \(t\) is the time, \(\rho\) is the density of lymph, \(\mu\) is the viscosity of lymph, and \(R(u)\) is the resistance term accounting for friction and vessel properties [19; 20].  
The dynamics of the vessel wall intervene with the relationship between the pressure and the cross-sectional area: \(p-P_{\text{ext}}=K\Big{(}\sqrt{\frac{A}{A_{0}}}-1\Big{)}\), where \(P_{\text{ext}}\) is the external pressure acting on the vessel, \(K\) is the vessel wall stiffness coefficient, and \(A_{0}\) is the resting cross-sectional area of the vessel.  
The valve dynamics in a lymph flow are expressed by a unidirectional flow ensured by the lymphatic valves. The flow rate across a valve, \(Q\), can be modeled as follows:  
\[Q=\begin{cases}C_{v}(P_{2}-P_{1}),&\text{if }P_{2}>P_{1}\\ 0,&\text{otherwise}\end{cases}\]  
where \(P_{2}\) is the upstream pressure, \(P_{1}\) is the downstream pressure, and \(C_{v}\) expresses the conductance of the valve. The flow rate equation expresses the relation between the volumetric flow rate to velocity and the cross-sectional area: \(Q=Au\).  
Lymph flow is influenced by external forces such as muscle contractions, respiration, and arterial pulsations. The external force applied to the vessel can be modeled as follows: \(F_{\text{ext}}=\alpha\cos(\omega t)\), with \(\alpha\) the amplitude of the external force and \(\omega\) the frequency of the force.  
The energy conservation law for the lymph flow dynamics that includes contributions from flow energy, dissipation due to viscosity, and work performed by external forces is represented as follows:  
\[\frac{\partial}{\partial t}\bigg{(}\frac{1}{2}\rho u^{2}+\frac{p}{\rho}\bigg{)} +\text{ Dissipation }=\text{ Work from external forces.}\]