Discussion by David FOLIO about magnetic microrobots

Commonly, the choice of the microrobots depends on the considered procedure, as well as the targeted environments (networks: vascular, urinary, nervous, or cavities: brain, eye, bladder, etc.). One most basic principle of magnetic actuation relies on the use of the magnetic forces, commonly referred as *bead pulling*. Different shapes of microrobots exhibit various modes of operations when magnetic fields and gradients are applied [2], as illustrated in Figure 1.

The arrow that represents \(\grad{\vb{B}}\) in Figure 1 (a) is there for the sake of clarity only. In no case \(\grad{\vb{B}}\) is a 2D or 3D vector. As \(\vb{B}\) is a vector with up to 3 components, \(\grad{\vb{B}}\) is then a Jacobian matrix.

## Magnetic manipulation

The basic principle of magnetic actuation is to manipulate microrobots with the use of electromagnetic fields to induce forces and/or torques on it. Basically, any magnetized objects can be subjected to forces and torques with an externally-imposed electromagnetic field. The magnetic forces and torques developed on a magnetized object are usually expressed as follows [1]–[4]: \[\begin{align}
\vb{f}_m &=V_m \ \left(\vb{M} \cdot \grad\right) \vb{B} \label{eq-Fm}\\
\vb{t}_m &=V_m \ \vb{M} \times \vb{B} \label{eq-Tm}
\end{align}\] where \(V_m=\tau_m V\) is the volume of magnetic materials, with \(V\) the total volume and \(\tau_m\) the magnetization rate of the microrobot.

The magnetization \(\vb{M}\) (A/m) denotes the magnetic moment per unit volume due to any individual *magnetic moments* \(\vb{m}\) existing in the material, that is: \[
\vb{M} = \frac{1}{V_m}\sum_{V_m} \vb{m}
\] According to Ampere’s law, a magnet is equivalent to a circulating electric current. The elementary magnetic moment \(\vb{m}\) can be represented by a tiny current loop. If the area of the loop is \(S\), and the circulating current is \(i\), then: \(\vb{m}= i S\).

When a \(\vb{B}\)-field (T) passes through a magnetic material there can be some ambiguity regarding which part of the field that origins from the external field and which part that origins from the sample itself. It is then motivated to introduce the auxiliary field^{1} \(\vb{H}\) (A/m) as: \[\begin{equation}
\vb{B}=\mu_0\left(\vb{H}+\vb{M}\right)=\mu\vb{H}\label{eq-BH}
\end{equation}\]

In free space \(\vb{M}=0\), that is outside a magnetic material \(\vb{B}\) and \(\vb{H}\) are almost identical, both fields point in the same direction but differ in size by the factor \(\mu\) which is the permeability for the specific medium. Inside any magnetic materials, there exists a relationship between \(\vb{H}\) and \(\vb{M}\), which depends on the magnetic properties of the material (see also this figure).

To actuate a microrobot, the magnetic field must either vary spatially (e.g. for bead pulling), or temporally (e.g. using a rotating or oscillating field), as illustrated in Figure 1. Bead pulling commonly used a strong magnetic gradient \(\grad{\vb{B}}\) to steer magnetic microrobots. Helical or flagellated microswimmers use a time varying magnetic fields to control the propulsion torque \(\eqref{eq-Tm}\). It has been shown that helical microswimmers are mainly efficient in arterioles or capillaries where the Reynolds number (\(\mathrm{R}_e<1\)) remains small [5]–[7]. In contrast, bead pulling remains a more appropriate propulsion scheme in the arterial system [6].

To maximize either the magnetic pulling force \(\eqref{eq-Fm}\) or the torque \(\eqref{eq-Tm}\), the microrobot should possess strong magnetization \(\vb{M}\), which depends on its magnetic properties.

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## Footnotes

Some authors call \(\vb{H}\) the magnetic field. In fact, Sommerfeld [8, p. p45] opines that “The unhappy term ‘magnetic field’ for \(\vb{H}\) should be avoided as far as possible” [3].↩︎