# Non-Linear Modeling of Immune System Activation and Lymph Flow Dynamics
## 2 Materials and Models
### Hagen-Poiseuille movement  
Unidirectional flow takes place in the presence of a pressure gradient, alongside a velocity distribution and a stress distribution (wall shear stress). The pressure is associated with the tension in the capillaries. Initially, the flow is similar to that of blood through arteries, capillaries, and veins, in shape. \[\Delta p=\frac{8\eta LQ}{\pi R^{4}},\tau=\frac{4\eta Q}{\pi R^{3}} \tag{5}\]  
where \(\Delta p=\) the pressure difference between two ends of one tube, \(\eta=\) viscosity, \(L=\) length of tube, \(Q=\) flow, \(R=\) radius of tube section, \(A=\) surface of tube section, and \(\tau=\) wall shear stress [21]. Although this approach accurately explains the movement of blood in arteries and veins, it needs adjustments to account for the movement of lymph in lymphatic channels, which is influenced by valves and contractions of the vessel walls, Figure 4.  
In [22], the Hagen-Poiseuille equation is also used, but in the context of graph/subgraph theory to compute the direction of lymph flow. In addition, the balance of the flow through vertices due to the mass conservation leads to a symmetric weighted Laplacian matrix. Pulsatile flow occurs throughout the circulatory system, regulating pressure through water exchange between the arterial, venous, and lymphatic systems. Fluid transport relies on pulsatile mechanisms, with external pressure playing a crucial role, and cells arranged to ensure unidirectional lymph movement. One key feature is how initial lymphatic capillaries regulate lymph entry into the lymphatic system via processes that maintain pressure homeostasis. Primary valves, or flap valves, are one-way gates formed by overlapping endothelial cells. With the help of the anchoring filaments in the extracellular matrix, these valves open when the interstitial hydrostatic pressure exceeds the pressure within the lymphatic tube, enabling fluid, cells and chemicals to enter. When the pressure within the vessel exceeds the interstitial pressure, the valves close, preventing backflow. This procedure ensures that lymph flows in one direction and that tissues are thoroughly drained. Figure 1B, from [17] contains additional information as well as a visual representation. Inside the lymphatic capillary (LC), the lymph flow is modeled as a porous medium using the Darcy-Brinkman equation, while outside the LC, the Navier-Stokes equation is applied. Equation (4)\({}_{2}\) becomes  
\[\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+\frac{1}{\rho} \frac{\partial p}{\partial x}=-8\pi\eta\frac{u}{A}.\]  
The second equation explains the fluid flow in relation to pressure forces in a capillary, forces that control fluid passage across capillary walls. The capillary filtration coefficient (K) measures how quickly fluid goes through the capillary wall, which reflects its permeability. Capillary hydrostatic pressure (\(P_{2}\)) forces fluid out of the capillary and into the interstitial space, while interstitial hydrostatic pressure (\(P_{1}\)) pulls fluid back into the capillary. The reflection coefficient (\(\sigma\)) measures how well proteins are maintained within the capillary--larger values mean better retention. This parameter takes up values between 0 and 1, where \(\sigma=0\) for a perfectly permeable membrane, and \(\sigma=1\) for a membrane that is perfectly selective. Oncotic pressures also play an important role. Capillary on-cotic pressure, created by plasma proteins such as albumin, pushes water back into the capillary, whereas interstitial oncotic pressure, caused by proteins in the interstitial space, pushes water out. These forces work together to determine the optimal balance of filtration and reabsorption. The viscosity of lymph is primarily influenced by its protein composition, which includes albumin, globulins, fibrinogen, and lipoproteins, as well as immune-related molecules such as cytokines and antibodies [23].