### Frequency mixing magnetic detection sensor

The FMMD method applies an AC magnetic field with two different frequencies to the SPIONs sample to improve the SNR of the received signal unlike other MPI studies^{1,7,8,10,25}. SPIONs with nonlinear magnetization as shown in Fig. 11a are exposed to magnetic fields consisting of two frequency components (f_1) and (f_2) as shown in Fig. 11c. Then, due to the non-linear magnetization characteristics, the response signal would be distorted as shown in Fig. 11b. The particles saturates at higher fields, leading to higher harmonics and frequency mixing components in the spectrum analysis such as fast fourier transformation (FFT) of response signal,**c**, as shown in Fig. 11d. The magnetization M of superparamagnetic magnetic particles may be approximated by the Eq. (1)

$$begin{aligned} M(upmu H) = M_s cdot {mathcal {L}}left( dfrac{m_0 upmu _0 H}{k_B T}right) . end{aligned}$$

(1)

where ({mathcal {L}}(x)) is the so-called Langevin function^{26}:

$$begin{aligned} {mathcal {L}}(x) = coth(x) – left( dfrac{1}{x}right) . end{aligned}$$

(2)

(m_0) is the magnetic moment in Am(^2) of a single magnetic particle, (upmu _0 = 4pi times 10^{-7}) Vs/Am denotes the vacuum permeability, (k_B) is the Boltzmann constant, *T* is the temperature in Kelvin, (M_s) is the saturation magnetization of the particles, and (x=m_0upmu _0H/k_BT) denotes the dimensionless (scaled) magnetic field. In case the magnetic particles are exposed to a magnetic field consisting of two distinct excitation frequencies (f_1) and (f_2)(with (f_1 > f_2)):

$$begin{aligned} upmu _0 H(t) = B_0[A_0 + A_1 sin (2 pi f_1 t) + A_2 sin(2 pi f_2 t)], end{aligned}$$

(3)

where, (B_0) is the scale factor of the magnetic field, (A_0) is the DC magnetic field, and (A_1) and (A_2) are the amplitudes of high frequency (f_1) and low frequency (f_2), respectively.

The third order mixing component (f_1 + 2f_2) of Langevin function’s Taylor expansion at a field *x* near 0 may be written as

$$begin{aligned} M_3(t) = M_s frac{A_1A^2_2}{8} cdot {mathcal {L}}^{(3)}(x) times cos[2pi (f_1 + 2f_2)t], end{aligned}$$

(4)

The approximated Eq. (4) is valid only in the limit of small excitation amplitudes. The physical meaning of Eq. (4) is the probe value indicating the strength of the SPIONs concentration. When using one frequency as in the traditional methods, the parameters (f_1) and (f_2) become identical, i.e. (f1 = f2). Since (M_3 (t)) is scaled at a ratio of (A_1 A^2_2) for the frequency ((f_1 + 2f_2)) component in the SPIONs concentration, the higher (A_1 A^2_2), the higher the SNR performance.

In summary, the scale factor of a single frequency is (A_1^3), the scale factor of the FMMD method is (A_1 A ^ 2_2). To increase (A_1), one can need high AC current amplifiers and coils made from stranded litz wire which is able to minimize skin effects^{28}.

On the other hand, for (A_2) at low frequencies, an impedance caused by a coil inductance is low, so that high SNR can be realized with normal AC amplifiers and single wire coils.

In terms of low power and portability for PoCT-MPI, it is important to transmit and receive signals in a single control device, because many common semiconductor chips for a device, such as regulators, processors and memories can be shared.

We proposed FMMD sensor device capable of transmitting, amplifying, receiving and processing two-channel sinusoidal analog signals in one device as shown in Fig. 12a. The measurement head (blue cube box) has a small coil with 8 mm diameter to measure the magnetization properties of small sample with SPIONs as shown in Fig. 12a. A new coil with 40 mm diameter was designed to measure samples of rodent size as shown in Fig. 12b. In addition, if necessary, it can be linked with an external power amplifier to generate a peak-to-peak 20 mT magnetic field at the center of our designed FFL generator. To measure a sample with a diameter of 40 mm, we use AE Techron’s AE7224 power amplifier, which amplifies the sinusodal signal at (pm 60) V at 1 A. The millivolt-level raw analog signal obtained from the receiving coil is input to the sensor and then it is filtered and amplified up to (pm 4) V.

Figure 12c is the result of frequency analysis of the output signal in zeroing state without a sample. The high and the low frequency are applied at (7,000,hbox{Hz},(f_1)) and (75,hbox{Hz},(f_2)) and for the amplitude in Tesla, (A_1) is about 2 mT for 7,000 Hz and (A_2) is about 15 mT for 75 Hz. Note that the frequency values used here are empirical hyperparameters selected by consideration of coil impedance matching and repeated experiment. When a sample containing SPIONs is not inserted, then no harmonic peaks occurs except fundamental frequencies as shown in Fig. 12c.

On the other hand, when the sample containing SPIONs is loaded into the measurement coil, we can observe harmonic peaks in the software of FMMD sensor. The voltage peaks of the harmonic frequencies such as 7,150 (hbox{Hz} (= f_1 + 2 f_2)) are proportional to the concentration of the SPIONs as shown in Fig. 12d. This value is also proportional to (M_{s} frac{A_{1} A_{2}^{2}}{8} cdot {mathcal {L}}^{(3)}(x)), the magnitude term of Equation 4. If the frequency and amplitude of the input signal are constant, it is a voltage proportional to (M_{s} cdot {mathcal {L}}^{(3)}(x)), so it can be used as a measurement signal for the distribution of SPIONs samples.

### Hybrid FFL generator

Signals obtained from the FMMD sensor are the integral of harmonic signals from all the SPIONs existed in the sample. Therefore, the combined signals need to be separated into spatial signals by introducing a selection field in the magnetic field excitation process^{1,7,25}. Since FFL can achieve a speedup by the number of pixels corresponding to the line length over FFP, we used FFL as the selection field^{8}.

Traditional FFL generators can be implemented with quadrupole magnets^{27} as shown in Fig. 13a. In the quadrupole magnet, two magnets face each other with the same polarity, **S**, in X direction. The other pair of magnets faces each other with **N** polarity in Z direction. Then, this combination of magnets produces a long FFL in the Y direction at the center of XZ plane, as shown in Fig. 14a. However, this structure has the disadvantage that the sample is situated in a closed space. Note that it is difficult to scan in the Z direction when the sample is inserted in the Y direction, since the direction FFL and the insertion of samples are parallel to each other. To make the sample inlet in the middle of the magnet, a torus-shaped magnet could be consider, but it has a problem that the magnetic field is not homogeneous in the gradient field region. In detail the magnetic field of the inner surface exists regardless of whether it is square or circular, so that the direction of the magnetic field is difficult to be uniform at the sample position in a donut-shaped magnet. After all, a magnet in the form of a solenoid coil is the solution.

Figure 13b shows the hybrid generator proposed to solve this problem in which the permanent magnet at one side of the Z-axis is replaced with a coil-based electromagnet, so that the sample can be fed along the Z direction. Figure 13c is the computer-aided design of **b**.

Note that due to the difference in characteristics between the magnet and the coil, XZ plane may be asymmetric in the Z direction when (|G_x| ne |G_z|) as shown in Fig. 14b,c, where (G_{x}:=frac{partial H_{x}}{partial x},G_{z}:=frac{partial H_{z}}{partial z}) are the gradients of each direction. Figure 14a shows the simulation results of the magnetic field of a quadrupole magnet based FFL generator. To imagine the shape of the FFL, you can think of a pencil with a long circular cross section that stands in the Y direction. On the other hand, Fig. 14b shows a long elliptic cross section on the XZ plane because the gradient field in the Z direction is smaller than that in the X direction. The value of (G_x) and (G_z) is 2.9 and (-2.9) for quadrupole magnet and 2.9 and (-1.0) for hybrid structure, respectively, where (G_x) and (G_z) are from Equation 6. Figure 14b,c show almost identical results of the simulated magnetic fields (**b**) and that of measured magnetic fields (**c**). To measure the internal magnetic field of the FFL generator, 150 (or (5 times 5 times 6)) positions are set, then the magnetic intensity are measured at each position (intermediate values are interpolated with bilinear method).

The frame of the FFL generator is made of monomer casting (MC) nylon material. measuring (20 times 33 times 45,hbox{cm}^3) and weighing less than 40 kg. The maximum size of sample (cylindrical shape) can be loaded is 40 mm in diameter and 60 mm in length.

### Shifting FFL for sinogram

We need to mathematically model the selection field before moving it. In modeling the rotation and translation of the FFL generator, we introduce homogeneous coordinates^{29,30} in which the last element increases by one dimension for ease of computation. Assuming that the gradient field of the selection field is linear, the gradient matrix ({{varvec{H}}}_0^{{rm S}}({{varvec{r}}})) represents the magnetic field of the selection field in Fig. 13a. Accordingly, Equation (5) in Euclidean space can be transformed into homogeneous coordinates.

$$begin{aligned} {{varvec{H}}}_0^{{rm S}}({{varvec{r}}}_0)&= left( begin{array}{ccc}{G_{x}}&{} {0} &{} {0} \ {0} &{} {G_{y}} &{} {0} \ {0}&{} {0} &{} {G_{z}}end{array}right){{varvec{r}}}_0 \&= left( begin{array}{ccc}{G_x}&{} {0} &{} {0} \ {0} &{} {0} &{} {0} \ {0}&{} {0} &{} {-G_x}end{array}right) {{varvec{r}}}_0\&= left( begin{array}{ccc}{2.5} &{} {0} &{} {0} \ {0} &{} {0} &{} {0} \ {0} &{} {0} &{} {-2.5}end{array}right) {{varvec{r}}}_0 = G_0 r_0, end{aligned}$$

(5)

where ({{varvec{r}}}_0 = [x,y,z]^{T}) is the position vector and (G_{x}:=frac{partial H_{x}}{partial x},G_{y}:=frac{partial H_{y}}{partial y}), and (G_{z}:=frac{partial H_{z}}{partial z}) are the gradients of each direction.

Since the new *G* matrix represents the selection field in the FFL generator, one can multiply position vector (underline{mathbf{r }}) by *G* to get the H field vector (H ^ S (r)) of the point. For the simulation prior to prototyping, we define the rotation matrix (R_z) to rotate the FFL generator by (theta) around the Z axis as Eq. (7).

Then, a translation matrix T is defined to shift (t_x), (t_y), and (t_z) in X, Y, and Z directions. Next, the H field matrix ({{varvec{H}}} ^ {{rm S}} _ {R,T} (varvec{{bar{r}}})) is calculated by Eq. (9).

We can see that the rotation of a selection field consisting of gradient fields by Eq. (9) has the same effect as the FFL scan as shown in Fig. 15. It shows the result of FFL scanning simulation over ((0 le theta _z le 180)^circ), ((-40,hbox{mm} le t_x le 40,hbox{mm})) sections. Note that an FFL scan is performed on the 2D slice of the sample in the XY plane perpendicular to the Z-direction.

$$begin{aligned} {{varvec{H}}}^{ {rm S}}({{varvec{r}}})&= left( begin{array}{cccc}{G_{x}} &{} {0} &{} {0} &{} {G_{x_{os}}} \ {0} &{} {G_{y}} &{} {0} &{} {G_{y_{os}}}\ {0} &{} {0} &{} {G_{z}} &{} {G_{z_{os}}} \ {0} &{} {0} &{} {0} &{} {1}\ end{array}right) {{varvec{r}}}\&= left( begin{array}{cccc}{2.5} &{} {0} &{} {0} &{} {0} \ {0} &{} {0} &{} {0} &{} {0}\ {0} &{} {0} &{} {-1} &{} {0} \ {0} &{} {0} &{} {0} &{} {1}\ end{array}right) {{varvec{r}}} = G r, end{aligned}$$

(6)

where (underline{mathbf{r }}), ([x,y,z,1]^{T}), is a position vector in 3D space, and (G_{x_{os}}), (G_{y_{os}}), and (G_ {z_{os}}) are constant offset fields in the corresponding directions, respectively. The offset fields are introduced to indicate how far the center of the each gradient field is from the origin of the FFL generator.

$$begin{aligned} varvec{R_z}({theta _{z}})= & {} left( begin{array}{cccc}{cos theta _z} &{} {-sin theta _z} &{} {0} &{} {0} \ sin theta _z &{} cos theta _z &{} {0} &{} {0}\ {0} &{} {0} &{} 1 &{} {0} \ {0} &{} {0} &{} {0} &{} {1}\ end{array}right) . end{aligned}$$

(7)

$$begin{aligned} {{varvec{T}}}({{t_x,t_y,t_z}})= & {} left( begin{array}{cccc}{1} &{} {0} &{} {0} &{} {t_x} \ {0} &{} {1} &{} {0} &{}{t_y}\ {0} &{} {0} &{} 1 &{} {t_z} \ {0} &{} {0} &{} {0} &{} {1}\ end{array}right) . end{aligned}$$

(8)

$$begin{aligned} G_{rot}= & {} R times Gnonumber \ T_{rot}= & {} R times Tnonumber \ G_{R,T}= & {} G_{rot} times T_{rot}^{T}nonumber \ {{varvec{H}}}^{ {rm S}}_{R,T}(varvec{{bar{r}}})= & {} G_{R,T} times {bar{r}} , end{aligned}$$

(9)

where ({bar{r}}) is ([x,y,z,1]^T) and (times) is a matrix multiplication. Figure 16 shows that the three-dimensional selection field can be rotated and translated by Eq. (9). For the sake of understanding, nine combinations consisting of Z-axis rotation ((-45^circ), 0(^circ), 45(^circ)) and translation ((-40,0,40)) mm in the rotated X-axis direction are also displayed. The combination in the middle shows the default selection field as ((t_x, theta _z) = (0,0)). As the value of translation (t_x) changes, the FFL of the XZ plane and the XY plane can be observed to move linearly. Also, as (theta _z) changes, we can see that the FFL standing in the XY plane rotates to an angle.