Controlling the Kelvin force: basic strategies and applications to magnetic drug targeting

Abstract

Motivated by problems arising in magnetic drug targeting, we propose to generate an almost constant Kelvin (magnetic) force in a target subdomain, moving along a prescribed trajectory. This is carried out by solving a minimization problem with a tracking type cost functional. The magnetic sources are assumed to be dipoles and the control variables are the magnetic field intensity, the source location and the magnetic field direction. The resulting magnetic field is shown to effectively steer the drug concentration, governed by a drift-diffusion PDE, from an initial to a desired location with limited spreading.From step 3 there exists {(v τ , w τ )} τ >0 such that v τ v and w τ w in H 1 (0, T ) andwhere we have used thatᾱ τ is a minimizer for J τ together with (4.5).5.-Convergence: Since (ᾱ τ ,θ τ ) is a minimizer we deduce the inequality J τ (ᾱ τ ,θ τ ) ≤ J τ (Π τᾱ , Π τθ ), whence applying first step 3 and next step 2 we see that

A note on the application of the Guermond–Pasquetti mass lumping correction technique for convection–diffusion problems

“…Let us note that the case of using the first correction (i.e. the case with 1  n in Definition 1) is the most frequently used case of application of the Guermond-Pasquetti technique in literature (see [12][13][14][15][16][17]), mainly due to its greatest simplicity and observations in [11] that the first correction is usually sufficient to compensate the dominating dispersive effects of mass lumping in pure transport problems (in particular, it is rigorously proved in [11] that the first correction eliminates the leading terms in the consistency error of lumped scheme in the 1D pure transport case). The following proposition justifies this via Fourier analysis (note that we also consider the case with the presence of diffusion In the following proposition we use the standard concept of the (local) equivalence of functions in mathematical analysis (e.g., see [26] (obtained in Lemmas 1-2 and Remarks 1-2) are even with respect to z , so their power expansions contain only even powers of h (respectively, Proposition 9 provides the first terms of the corresponding expansions).…”

“…However, the authors did not make any detailed theoretical estimates of the quality and accuracy of the solution depending on the number of terms of the matrix series taken. Despite the computational attractiveness and wide application of this powerful technique (e.g., in constructing maximum principle preserving methods [12][13][14], level set methods for two-phase flows [15], various engineering applications [16,17] etc. ), and the overall high quotability of the paper [11], these important issues remain still unexplored.…”

External optimal control of fractional parabolic PDEs

“…In many realistic applications, the source/control is placed outside the domain where a PDE is fulfilled. Some examples of problems where this may be of relevance are: (a) Magnetic drug delivery: the drug with ferromagnetic particles is injected in the body and external magnetic field is used to steer it to a desired location [6,7,38]; (b) Acoustic testing: the aerospace structures are subjected to sound from the loudspeakers [35].…”

Model-based optimized steering and focusing of local magnetic particle concentrations for targeted drug delivery

“…These decisions are based on the minimization of a so-called cost function containing metrics of interest such as particle movement time, energy consumption, particle spreading, etc. Reported control strategies have treated the guiding or steering of single particles or droplets (Probst et al, 2011;Komaee & Shapiro, 2012;Khalil et al, 2016) and distributed ferrofluids by means of electromagnets (Shapiro, 2009;Komaee, 2017;Antil et al, 2018;Liu et al, 2020). An in-depth review of many prior magnetic drug targeting solutions with a focus on control systems for magnetic fluids was provided by Nacev et al in 2012(Nacev et al, 2012.…”

“…His cost function is formulated such that the particle dispersion and the required movement time be minimized, subject to the constraint that the center of mass remains on a desired trajectory. Another optimization-based approach was elaborated by Antil et al., who aimed at moving a domain of particles from an initial to a desired location by keeping the forces on particles in this domain almost constant, thereby minimizing particle spreading (Antil et al., 2018 ). We use the optimal control framework as well to ensure optimality over a period of time, but do not require a predefined trajectory between the beginning and end point of the ferrofluid, which is not always available beforehand or optimal.…”