Some systems are in parallel. For instance, the resistance of lungs $R_{\text{lungs}}$. Assume here for simplicity that these parallel systems can be handled linearly. We also consider only Total Vascular Resistance for simplification. So we have now only systems in series under consideration.
Linear system for Systemic Vascular Resistance
Formula for the total resistance as mechanical definition as assuming linear connections between those systems:
$$R_{\text{total}}=R_A+R_a+R_c+R_V+R_v$$
where appropriate resistances are
- $R_c$ capillaries,
- $R_a$ arterioles,
- $R_A$ Small artery,
- $R_v$ venules and
- $R_V$ vein.
Nonlinear system for Systemic Vascular Resistance
There are many factors involved in these locations:
- permeability
- vasoconstriction
- ...
- growth factors (FGF) for renewal of the vessel wall
- different channels (activated, deactivated - mostly potassium channels)
where I skip the consideration of vasoconstriction, growth factors and different channels.
Permeability is one which applies to all types of vessels. Change in permeability of one location does not mean a linear change in permeability in another. A change in the permeability of small arterioles does not lead to a similar change in the arteries or venules.
The vessel is permeable to gas and liquid. Let's consider only these components. I want to include these facts into the definition because, in the pathological situations, different state of inflammation for instance can apply to a vessel. Let's call total permeability $S$ which is dependent on gas and liquid components, first on liquid component (so inner integral).
I could give you the nonlinear formula of the total resistance but I think it goes over the skills most here. Let's call the nonlinear resistance dependent on permeability of the vessel, $W(x,y)(t,f)$ in time and space. So consider only the nonlinear component of the permeability formula after parametrisation i.e. permeability, let's call it $T$, which is dependent on changes in water (w) and gas (g) components over a path integral. The path covers all components in the vascular system that are in series. So total resistance defined nonlinearly
$$W(x,y)(t,f) = \int_{\gamma} T(u) \frac{1}{u} \int_{t-u/2}^{t+u/2} T(w) \, S(x,y)(t,f) dw \, du$$
where the inner integral corresponds to the water component while the outer integral to the gaseous component. There is only an averaging in the inner integral when the width of the window is $u$. However, we do not any windowing here actually, just because of this averaging, so we lose no data and can handle the total resistance as a single process.
Notice that change in $T(u)$ leads to a change in $T(w)$ nonlinearly. I skip the nonlinear definition of $S(x,y)(t,f)$ which stands for the energy in time in each instance - you can think it as the amount of permeability in each position of the vessel. Actually, I also skipped the explicit definitions of the water and gaseous components too.
I have not found any work done on the nonlinear development of the resistance. I think this is necessary for defining inflammation explicitly.
How can you describe the total resistance as an nonlinear operation better?
