PipnP

Here, P represents an admissible set of infarction parameters.

16 ECG recording devices typically measure the difference between the potential at two leads (or the difference between the potential at one specific lead and the average of the potential at the remaining leads). In Sections 7.2 and 7.3 we assumed, for the sake of simplicity, that the observation data was of a simpler form; more precisely, that the electrical potential itself at the leads were measured. This is, however, not usually the case. In the present situation, we want to show that we are able to handle data of the form (7.131). The theory presented in Sections 7.2 and 7.3 can also, in a straightforward manner, be modified to incorporate such cases.

17 The purpose of the present section is to investigate the possibilities for characterizing an ischemia by this approach!

7.4.3 Differentiation of the Objective Function

In order to solve the problem (7.133), we must apply some sort of optimization procedure. As mentioned in Section 7.3.5, such minimization algorithms often require the partial derivatives of the involved objective function with respect to all of its variables. That is, we need to compute dJ dJ dJ

dpi dp2 dpn

This can, as already discussed in Section 7.3.5, be accomplished by a straightforward finite difference approach, i.e. by applying the approximations dJ _ J (pi, ...,Pi + Api,... ,pn) - J (pi,. ..,pi, . ..,Pn)

dpi Api for i = l,...,n. Note that, to compute the fraction (7.135) we must solve our model problem (7.121)—(7.129), not only in the case of the parameter set pi,p2,... ,pn, but also for the perturbated "infarction" parameters pi,... ,pi + Api,... ,pn. Thus, n +1 coupled problems, of the form (7.121)-(7.129), must be solved in order to compute all of the partial derivatives of J at a single point. This leads to an extremely CPU-demanding procedure. For a fairly large number n of infarction parameters, this cannot, within reasonable time limits, be done by even the fastest computers available today.

The Adjoint Problem Approach

Our aim is to develop an efficient technique suitable for differentiating J with respect to the infarction parameters. It turns out that all of the n partial derivatives (7.134) of J can be computed by solving a single auxiliary problem. Magic? No, it follows by a mathematical trick involving the so-called adjoint problem of (7.121)— (7.129). In fact, the derivation presented below, is similar to the argument presented in Section 7.3.5.

In the following, we will write I(v) instead of Iion(v), and we define the function ue in H, u0 in T.

With this notation at hand, we may write the model (7.121)—(7.129) on its variational form, also integrating in time, t* t* t* / vt0dxdt+ / MiVv ■ V0dxdt+ / MiVu ■ V(pdxdt

+ 1 I MoVu ■ Vipdxdt = 0 for all ^ e V (H U T), (7.137) Jo Jt where V(H) and V(H U T) represent suitable spaces of functions defined on H and H U T, respectively. The derivation of weak forms of systems of partial differential equations was discussed in detail in Chapter 3. Hence, we will not dwell any further upon this issue in the present context.

As mentioned above, since the function g = g(x, t; pi,... ,pn) depends on the infarction parameters, the solution (u,v) of the system (7.136)-(7.137) will also depend on pi,... ,pn. Let j e {1,...,n} be arbitrary and define, for the sake of simple notation, p = pj. Our goal is to compute the partial derivative of the cost-functional J with respect to the variable p. From (7.132) it follows that dt, dj t m

— = £ [di(t)-{u(ri+i^ -u(ri,t)} uP(ri+2,t) -up(ri,t)

where

1 f up(ri,t) = t—r up(x,t) dx for i = 1,...,k, (7.139)

\ri\ Jr cf. (7.130). In (7.139), up(x, t) denotes the partial derivatives of u(x,t) with respect to the infarction parameter p, i.e.

We differentiate (7.136)-(7.137) with respect to p and find that

Jo Jh Jo Jh

+ / i gI'(v)vpfadxdt = 0 for all fa e V(H), (7.140)

Jo Jh

( i MiVvp ■ Vipdxdt + i ( (Mi + Me)Vup■ Vipdxdt Jo Jh Jo Jh t*

+ 1 I MoVup ■ Vip dx dt = 0 for all p e V(H U T) (7.141) Jo Jt

If we define the operator

A [(r,T), (^,p)] = [ [ rt^dxdt+ [ ( MVr ■ V(pdxdt

Jo Jh Jo Jh

+ [ ( M,Vt ■ V(pdxdt + [ [ gI'(v)r$dxdt Jo Jh Jo Jh rt* r ft*

+ i i MiVr ■ Vp dx dt + i i (M, + Mb)Vt ■ Vp dx dt Jo Jh Jo Jh

+ / M0Vt ■ Vp dx dt, Jo Jt then it follows from (7.140)-(7.141) that (vp, up) satisfies t*

0 0

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