Cardiovascular models: towards personalised digital health
Scientists have always been intrigued by the complexity of the heart, a sophisticated pump capable of beating three billion times over a lifetime. This longtime interest has proven to be well founded, as computational models of the circulatory system have turned out to be valuable tools in predicting cardiovascular diseases and possibly enabling personalised therapies.
Can we make patient-specific models more specific?
In the last few years, researchers have focused on the development of the so called patient-specific cardiovascular models, which enable a personalised representation of the patient’s condition through a computer model. The idea is simple: we know how blood flows in our veins and arteries, thanks to bio-mechanics, and we know that we can get a pretty good approximation of this flow by solving a couple of partial differential equations (Navier-Stokes equations). This gives us quite a solid knowledge of what happens inside our cardiovascular system.
But what if we want to know exactly what happens inside a specific patient? Can we combine the generic mathematical model with specific information pertaining our particular patient (the shape of their arteries, for example, or the velocity at which their blood is ejected from the heart)? In short, can we make our model patient-specific? The answer is yes: we need to collect some clinical data from our patient, usually through advanced imaging techniques, such as Computed Tomography and 4D-Flow MRI, and incorporate them into a mathematical model. In this way our virtual model will closely mimic what is happening inside the patient.
A model can be made patient-specific in a number of ways:
- the exact geometry of the patient’s anatomy can be reconstructed in the form of a 3D mesh, usually extrapolating this information from clinical imaging with a process called image segmentation;
- some parameters, such as the stiffness of the vessels or the electrical diffusivity can be computed indirectly to match patient’s data;
- the boundary conditions of the mathematical model can be properly tuned to match patient’s clinical measurements.
Let’s focus on the last point.
Why do we need boundary conditions?
If you ever tried to solve a differential equation (like those describing how blood flows in our veins), you know how crucial it is to choose appropriate boundary conditions. Basically, boundary conditions are in charge of modelling what happens outside our domain of interest. Imagine we want to model how blood flows in the aorta depicted above: we would first need to know what happens before blood enters the aorta, which means the conditions at the inlet (how fast is blood there? How much is the pressure?), and then what happens when blood leaves the aorta, so the conditions at the outlets. This choice will have a huge influence on the resulting modelled blood flow: a bad choice of boundary conditions may ruin the “specificity” of our model, or worse lead to totally wrong results. So how can we choose boundary conditions that don’t screw up our simulations? Is there a smart way to do that?
If we have some clinical data available (e.g., measurements of blood flow rates of out patient, or pressure measurements), we can use them to guide our choice of boundary conditions. We must be careful though, as such clinical data are noisy and scattered, and simply plugging them in our model can do more harm than good.
The solution? Optimal Control
A powerful solution to this problem is called optimal control. This advanced numerical framework gives the possibility to incorporate measurements data into a physical model, in such a way that the model will comply to the measurements, while still retaining all the good properties of being a mathematical model (accurate, physically consistent, and so on). This process is called data assimilation, and it is relevant in all those fields where we want to enrich a physical description of nature with experimental observations (weather forecasting is an example).
To make it simple, you have your model with some experimental data, and you let optimal control choose the solution to model equations that comes closer to those data, which does so by tuning a parameter, the so-called control.
If you want to know more about optimal control and its various applications, I suggest you to read this article, where optimal control is applied to the study of marine environment.
Now back to our cardiovascular models: what do they have to do with all of this? Well, researchers found out that patient-specific cardiovascular models could greatly benefit from optimal control: we have Navier-Stokes equations to describe our system, and we have some patient-specific measurements to assimilate into our model. That’s a good starting point. But what if we use optimal control to choose boundary conditions for us? This can be achieved by setting the boundary conditions as the control, and it is convenient for two reasons:
- we get rid of the responsibility of choosing boundary conditions (it’s hard, complicated, we could destroy our model → it’s better if someone else does that for us).
- we are able to incorporate data coming from the patient →our patient-specific model becomes even more patient-specific.
Now you are ready to see how we applied optimal control to a real clinical case.
Application of optimal control to a patient-specific aortic arch
For a bit of context, we worked on a patient-specific aorta, where we wanted to use optimal control to set outlet boundary conditions. We had some patient-specific measurements coming from 4D-Flow MRI, an advanced imaging technique which is able to measure the time evolution of blood inside vessels. We used this data to set the inlet boundary condition, but we also wanted to assimilate it to select the outlet ones.
We ended up with a mathematical model of the aortic arch, which is physically correct but also matches 4D-Flow MRI data. We used it to get some clinically relevant parameters, e.g., Wall Shear Stress, which are used by clinicians to make surgical plans and predictions.
In conclusion
Patient-specific cardiovascular models are truly a powerful tool, and their role is becoming even more critical as we are moving to a new personalised, digital approach to medicine and health sciences. Handling them, however, is anything but straightforward. In this short article we have introduced some of these difficulties, and a possible way to overcome them by means of optimal control.
