Biomedical Engineering Reference
In-Depth Information
that it is reasonable to think to a further step, moving from a “sequential” to an “inte-
grated” use of data and simulations. The DA approach entails measures and images
to be used not just for providing initial and boundary conditions, but to drive the
results by a sophisticated integration with the mathematical models. The outcome of
this process is an assimilated result where not only numerical computation is strictly
consistent with the individual data, but the noise affecting the data has been filtered
out by the mathematical modelling.
The “integrated” paradigm, which is well developed in other contexts such as the
weather forecasting, opens many challenging problems at the methodological and
practical level. The quality of the data in terms of their size, location in space and
frequency in time plays obviously a major role in the mathematical properties (well
posedness) of the assimilation problem (see [7]). Moreover, the noise that invariably
affects the data has an impact on the reliability of the entire process. A precise eval-
uation of this aspect is strictly related to both the type of data and the methods used
for the assimilation procedure. This leads to analyze and solve partial differential
equations with stochastic terms (see e.g. [61, 86, 87, 88]).
When the assimilation problem is solved with variational methods, the mathemat-
ical structure almost invariably can be represented as a feedback control loop. The
effective numerical solution of inverse problems in this form presents many open
concerns to be properly addressed (see [6]).
In this chapter we have presented three basic examples sharing this control-loop
structure, motivated by ongoing collaborations with medical doctors. The first pre-
liminary results enlighten the great potential of DA as a way for improving both
the reliability of numerical results and the quality of measures. As we have pointed
out, a certified reliability is crucial since bioengineering and medical communities
are increasingly resorting to scientific computing for taking decisions (see [49]).
The accomplishment of the new integrated paradigm - requiring new advanced and
increasingly interdisciplinary research - represents an exciting challenge of cardio-
vascular mathematics for the years to come.
Acknowledgements.
Marina Piccinelli and Alessandro Veneziani thank Emory University Re-
search Committee for the support of the Project “Image based numerical fluid structure interactions
simulations in computational hemodynamics”. Tiziano Passerini is supported by the NIH Grant
5R01HL070531-08 “Biology, Biomechanics and Atherosclerosis”. The research of C. Vergara
has been (partially) supported by the ERC Advanced Grant N.227058 MATHCARD. The authors
wish to thank Marijn Brummer (Emory Children's Healthcare of Atlanta), Eldad Haber (Univer-
sity of British Columbia, Canada), Robert Taylor (Emory School of Medicine), Michelle Consolini
(Emory School of Medicine), Michele Benzi (Emory University), Max Gunzburger (Florida State
University), George E. Karniadakis (Brown University).
References
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