Tuning Vascular Network Boundary Conditions

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Today, we will tuning 1D vascular network boundary conditions to achieve a desired flow distribution and inlet pressure in Flownex. Flownex allows users to easily specify outflow using the designer, by tuning and optimizing network boundaries to meet a desired set of input criteria. I will be using a characteristic model of the aortic arch through the iliac bifurcation with a prescribed transient inlet flowrate. I will walk through how to use the designer to specify outlet flow distributions at the outlet and the time averaged pressure at the inlet.

This method may be used to mimic patient data, determine the desired resistance or compliance chamber air volume when designing a benchtop model. The designer iterates specified network variables until the desired conditions are met. The designer can be used with individual components, or to set global parameters, dictating multipliers for subsets of boundary condition groups (such as downstream perfusion resistance).

Network Overview

This network will use a specified flowrate boundary condition and a three element windkessel (lumped model) boundary condition. This is a commonly utilized method to model the downstream compliance (from larger vessels) and resistance (primarily from smaller vessels and capillary beds). We will also be using a custom fluid to model blood. This network uses a script defined waveform for the inlet flow. Coefficients were used from this article, but the script can be applied to any Fourier series. This may also be accomplished with the Distributed Control System toolbox by summing the outputs of several waveform generators or by using a function generator component with a flow data set. This simplified model consists of 11 compliant pipes with three element windkessel boundary conditions (accumulators with proximal and distal resistance elements, exiting at atmospheric pressure).

This network also uses a global parameter to specify the period in scripts and Digital Control Task components, so we can easily adjust the period of the cardiac cycle.

Determining Flow Distribution

We will initially set our flowrate to the time averaged flowrate of 5.02L/min. We will start with an equivalent distal resistance at each branch. The steady state flow distribution for an equal distal resistance model is shown below.

We can specify our desired time averaged outflow for each branch. For this network, I will use 0.6L/min for the subclavian arteries, and 0.5L/min for the common carotids and set the Iliac artery flows to 1.41L/min. Our independent variables will be five of the six distal resistances (leaving one arbitrary resistance constant), and our dependent variables will be their resultant flowrates. The designer configuration is shown below.

Running the designer modifies the admittance of the flow resistance components to provide our desired flow distribution.

We will use these settings as an approximation of the time averaged flow distribution in transient runs.

Calculating Periodic Averages

Now that our inlet flow profile and outlet flow distribution are specified, we can use the statistics component to determine the time averaged pressure and flow for the inlets and outlets of our model. The statistics component highlighted below shows the time averaged flowrate of the ascending aorta (our inlet).

We can also create a new page to keep track of our distal resistances. This is very useful in large arterial networks. We can use views of distal resistance components to quickly check properties and determine periodic averages (shown below).

We can also use direct references in quick scripts calculate the total outflow of our system. This can be useful for tracking the mass flows in elastic pipes.

Using the Designer in Transient Simulations

The designer may also be used with properties obtained (at a specified time) from transient simulation. We will use the designer to set the mean arterial pressure (average pressure over 1 cardiac cycle) of the aorta to 100mmHg (gauge). We can easily scale our resistance to achieve our inlet pressure by running the designer in a transient simulation. I have created a global parameter to scale the distal resistances.

We can create a new designer configuration to set the scaling factor for our distal resistances. Our system should have a liner relationship between pressure and flow. This allows us to directly raise our inlet mean arterial pressure without modifying our flow distribution. We will use our global parameter (scaling distal resistances) as an independent variable to achieve our desired mean arterial pressure. We will also run the simulation for 30 seconds to allow our waveform to develop after changing our downstream resistance.

Results

After running the designer, we have achieved our target mean arterial pressure.

We can also check that our transient flow distribution matches our specifications by plotting the average (over one cardiac cycle) pressure value of each outflow.

Finally, we can also plot the resulting pressures of our inlet and outlets (upstream of the windkessel boundaries).

We can also use the designer to tune additional properties of waveforms by modifying properties of boundary conditions or pipes (vessels) using this workflow. Thanks for following along, make sure to check in for future posts.

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