Biomedical Engineering Reference
In-Depth Information
Control Science and Dynamical Systems
The second method comes from signal processing and control science. It is applied
to the representation of dynamic systems by state-space models and it relies on
the fact that the system is given by a set of ODEs, which may be converted by
Laplace transform to the study of transfer functions, i.e., the system in the frequency
domain. Such systems may be studied by their responses to input excitations to
better characterise them. Presentations of such identification methods may be found
in [ 93 , 137 ].
Inverse Problems
The third method, inverse problem solving, belongs to the domain of PDEs. The
models under consideration are close to the physical world and the method can
comprise almost all situations, but requires specific studies for each case and
nontrivial mathematics. The general principle is that observations of the real system
represented by a PDE model correspond to an ill-posed problem, i.e., that the system
of PDEs as it is given cannot be identified in a unique manner from the observations.
Nevertheless, small regularisations (such as Tikhonov's), i.e., small modifications of
the underlying differential operator, make the problem well-posed, i.e., amenable to
the identification of its parameters in a unique manner. For a general presentation,
see [ 76 ]. Recent developments on physiologically structured models may also be
found in [ 50 - 52 , 64 , 120 ].
5.2
Parameters in Macroscopic Models of Tumour Growth
In macroscopic models of tumour growth, parameter identification most often relies
on imagery techniques, mainly radiological or MRI, as in [ 131 ] for brain tumour
growth. But it is also possible to obtain tumour growth curves representing three-
dimensional growth by using a method which may seem very coarse, but which
has not found any really better competitor so far. It consists in growing a tumour
(homograft or xenograft, i.e., of the same animal species, or of another) under the
skin or on the skin of a laboratory rodent and measuring everyday by using a caliper
diameters in three dimensions (one longitudinal and two orthogonal transverse) of
the tumour, which is protected by the skin coating when the tumour is subcutaneous.
It is possible only when the tumour is already palpable under the skin (or visible
when it is on the skin), which excludes avascular tumours and generally involves
histologically heterogeneous, but physically (i.e., in density with respect to water)
homogeneous tumours. This allows an approximate estimation of the tumour mass,
assumed to be proportional to the number of tumour cells, from its approximate
volume and keeps the animal alive (until tumour weight reaches 10 % of the animal
body weight, at which point the animal is sacrificed for obvious ethical reasons).
Coarse as it may seem, this method is still widely used because of its simplicity, see
e.g., [ 35 , 61 , 124 ]. In a macroscopic (whole body) perspective, it should in principle
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