Simplification as a starting point for blood hydrodynamics modeling.
When a mathematical model is defined by its assumptions and limitations, the domain of application and the ability to describe or forecast is well established. Is in that stage when problems are detected and solved or the model is discarded as it becomes too narrow or too specific. A common problem when working with mathematical models is the decision to make a model that describes a phenomenon in the spatial domain as well as the temporal domain.
This decision can impact the ability to solve the model, as transient spatial models need high computational resources to be solved. However, simplification of the model to rather study the temporal process of the phenomenon instead of the spatial specifics of it can yield useful information.
Models that describe the process without spatial information of a phenomenon are often called 0-dimensional models. Such models can be successfully applied to complex problems where the spatial domain is too complex to be easily generated. An example of such models is the Windkesel model that describes the blood flow through the aorta. Proposed in 1800 by the German physiologist Otto Frank, the Windkessel model describes the arterial flow as an electronic circuit. There are different variations of the model but first wee need a way to approximate the volumetric flow outputted by the heart.
This will result in a pulsatory volumetric flow similar to what the heart outputs. Also, the first and second derivative of that input is calculated for the Windkessel model.
The simplest kind of Windkessel model is the two-element Windkessel model that is composed of two elements the arterial compliance and flow resistance as the blood passes through the aorta. Using the volumetric flow input as well as the Windkessel equations the blood pressure can be obtained with the following.
The three-element Windkessel model also known as the Broemser model adds a new element to the model, the resistance encountered by blood as it passes through the pulmonary aortic valve. For this model, the first derivative of the volumetric input is also used.
Finally the four elements Windkessel model adds the effect of inertia as blood travels through the aorta. That can be done by adding another parameter to the model called inertance.
Under the different Windkessel models the blood pressure follows a similar pattern as the volumetric input flow. Also after some time, the blood pressure reaches a dynamic equilibrium.
Taking the four elements Windkessel model and turning off the inertial elements return to a behavior similar to the three-element model. Also increasing the inertial load returns a pattern similar to a classical arterial pressure plot profile.
The Windkessel effect helps to damp the change in blood pressure during the cardiac cycle, the understanding of this simple effect aided the design of devices to aid the heart function. As always the complete code for this post can be found on my GitHub by clicking here. Talk to you soon.
Suggested reading.
Modeling a Heart Pump, Michelle Vellekoop
