A new approach for highly accurate, remote temperature
probing using magnetic nanoparticles
Authors: Jing Zhong, Wenzhong Liu, Li Kong, and Paulo Cesar Morais
Supplementary
Model Construction and Inverse Calculation
According to the first-order Langevin function and its Taylor’s expansion (see
Equations (1) and (2) in the Methods section), the applied magnetic fields (Hi) plus
the corresponding recorded magnetisations (Mi) allow the construction of the
following sets of equations to assess the temperature (T):

 H1

H13 3 2 H15
H17 7
M

x
y

y


y  
 1

45
945 4725
 3



H

H3
2 H 25
H7
 M 2  x  2 y  2 y 3 
 2 y7  
45
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
 3




H

H 3 3 2 H n5
H7
 n y7  
M n  x  n y  n y 
45
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 3


,
(1)
and


1 
 M 1  x coth yH 1 

yH1 





 M 2  x coth  yH 2   1 
yH 2 






1 
 M n  x coth  yH n  

yH n 




.
(2)
Equation (1) represents the fundamental model for temperature probing using a
Taylor’s expansion of Langevin’s function at lower magnetic fields. Equation (2)
represents the model for temperature probing over the full range of magnetic fields.
Solving Equations (1) and (2) facilitates temperature probing using magnetic
nanoparticles.
Two methods of inverse calculation, based on the matrix solution and the least
square error, are used to solve Equations (1) and (2). Because the inverse calculation
method based on the matrix solution is suitable for equations that can be transformed
into the matrix form, it can be used only to solve Eq. (1). Then, Eq. (2) can be
rewritten into matrix form as:
 H1

3
 M1  
M   H2
 2   3
  
  
M n  
 Hn
 3
Given
 M1 
M 
Y   2 ,
 
 
M n 
 H1

 3
 H2
A 3


H
 n
 3
H13
45
H3
 2
45
2 H15
945
2 H 25
945

H n3
45
2 H n5
945



H n7
4725
H13

45
H3
 2
45
2 H15
945
2 H 25
945
H17

4725
H7
 2
4725
H n3
45
2 H n5
945



  xy 
 3
  xy 
  xy 5 
 7
  xy 
 
 

H17
4725
H7
 2
4725










H n7
4725
and
.
(3)
 xy 
 xy 3 
 
X   xy 5  ,
 7
 xy 
 
Equation (3) can be simplified as:
Y  AX
.
(4)
In this study, the point number n equals 20, which is greater than the Taylor’s
expansion order of 5, indicating that system matrix A is of full column rank. Therefore,
the solutions to Eq. (4) can be described by:
X   AT A AT Y
1
.
(5)
The vector X in Eq. (5) contains different polynomials, comprised of variables x and y.
To access temperature probing independent of the concentration of the magnetic
nanoparticles, a combination of the 1st and 2nd items (xy and xy3) yields the optimal
solution y*.
In contrast, the inverse calculation method based on the least square error mainly
uses an iteration method to determine the optimal solution, which provides the least
square error between the measured and theoretical magnetisations. Therefore, this
approach is able to solve both Equations (1) and (2). For example, Equation (1) is
transformed into:

 H1

H13 3 2 H15 H17 7


x
y

y 

y    M1
 1

45
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 3


3
5
7

H

H
2H 2
H
 2  x  2 y  2 y 3 
 2 y7    M 2
45
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
 3




H

H 3 3 2 H n5
H7
 n y7    M n
 n  x  n y  n y 
45
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 3


.
(6)
Similarly, Equation (2) is transformed into:


1 
1  x coth yH 1 
  M1
yH

1





 2  x coth  yH 2   1   M 2
yH 2 






1 
 n  x coth  yH n  
  Mn
yH n 




.
(7)
n
When the square error     i2 reaches its minimum value, the values of variables
i 1
x and y indicate the optimal solutions (x* and y*). In the present report, the iteration
method of Levenberg-Marquardt will be used to solve Equations (6) and (7).
Simulation of Temperature Probing
A. Model and inverse calculation
The model construction and the inverse calculation method are key aspects of
temperature probing using magnetic nanoparticles because both significantly affect
the temperature probing accuracy. When probing temperature using magnetic
nanoparticles, the truncation error, caused by the Taylor’s expansion approach, will
generate a significant temperature probing error, as long as the applied magnetic field
does not fulfil the low magnetic field requirement. In contrast, the inverse calculation
method based on the least square error provides more accurate temperature probing
because of the stronger rejection of noise. In this study, a 5th-order Taylor’s expansion
and an iteration method using the Levenberg-Marquardt algorithm are employed for
model construction and determining the least square error, respectively.
Figure S1. Temperature probing using different combinations of models and inverse calculation
methods. (a) shows the measured temperature, and (b) shows the measured temperature error.
ET_TM, ET_TL and ET_LL represent temperature probing using a combination of Taylor’s
expansion and the matrix solution, Taylor’s expansion and the least square error, and Langevin’s
function and the least square error. The maximum applied magnetic field was 400 Oe with a step
size of 20 Oe. SNR was set to 80 dB. The effective magnetic moment MsV was set to 1.6×10-19
Oe.
Considering the model construction and the inverse calculation method stated
above, three model-method combinations are worth examining, namely, Taylor’s
expansion and the inverse calculation method based on the matrix solution, Taylor’s
expansion and the inverse calculation method based on the least square error, and
Langevin’s function and the inverse calculation method based on the least square error.
Employing these three model-method combinations, we used a simulation to assess
the temperature and the corresponding temperature error. Figure S1 shows the results
of temperature probing using the different combinations of models and methods, and
Table S1 presents the maximum values and the standard deviations of the temperature
probing errors. Figure S1 and Table S1 indicate that the model and the selected
method significantly affect the temperature probing accuracy. The inverse calculation
method based on the least square error improves the temperature standard deviation
by a factor of approximately 15, compared with the inverse calculation method based
on the matrix solution. Additionally, Langevin’s function reduces the maximum
temperature probing error from 0.77 K to 0.34 K. However, the impact on temperature
probing of selecting the theoretical model and the inverse calculation method differs.
Thus, to investigate these two aspects in greater detail, the present study reports on
temperature probing using different combinations of models and methods with
different signal-to-noise ratios (SNR) and values for the maximum applied magnetic
fields.
Table S1. The maximum values and standard deviations for the temperature probing error using
the different models and inverse calculation methods
Model and Inverse Calculation Method
Maximum (K) Standard deviation (K)
Taylor’s expansion and matrix
3.00
2.45
Taylor’s expansion and least square error
0.77
0.16
Langevin’s function and least square error
0.34
0.16
When employed directly, the inverse calculation method determines the impact
of noise on the temperature probing accuracy. Using the Taylor’s expansion, the
inverse calculation methods based on the matrix solution and the least square error
allow different noise rejection ratios that predict the differences in temperature
probing accuracy. Figure S2 indicates that the inverse calculation method based on the
least square error provides greater temperature probing accuracy in comparison with
the matrix solution, particularly at a lower SNR. With a SNR of 80 dB, the inverse
calculation method based on the matrix solution provides a maximum temperature
probing error of approximately 11.2 K with a standard deviation of 8.11 K, whereas
the least square error approach provides a maximum error of approximately 1.11 K
with a standard deviation of 0.55 K. These findings indicate that the temperature
probing accuracy improves by a factor of approximately 10. Additionally, Figure S3
shows the curve of the standard deviation of the temperature probing errors and SNR
using different inverse calculation methods. Comparing the two methods, we found
that the standard deviation of the temperature probing errors using the inverse
calculation method based on the least square error is significantly lower than that
using the inverse calculation method based on the matrix solution. Moreover, with an
increase in SNR from 80 dB to 120 dB, the temperature probing accuracy using the
inverse calculation method based on the matrix solution improves from 8.11 K to
0.092 K (a factor of approximately 90), compared with an improvement from 0.55 K
to 0.0055 K (a factor of 10) using the inverse calculation method based on the least
square error. This finding indicates that temperature probing using the inverse
calculation method based on the least square error supports a higher rejection of the
impact of noise on temperature probing.
Figure S2. The temperature errors at different signal-to-noise ratios (80, 90, 100, 110, and 120
dB). (a) The temperature error for the combination of the Taylor’s expansion and the inverse
calculation based on the matrix solution; (b) the temperature error for the combination of the
Taylor’s expansion and the inverse calculation based on the least square error. TT represents the
theoretical value. The maximum applied magnetic field was 200 Oe with a point number of 20.
Figure S3. The standard deviations of the temperature errors at different SNR values using the
inverse calculation methods based on the matrix and the least square error.
The truncation error caused by Taylor’s expansion mainly affects the truncation
error of temperature probing. At low magnetic fields, the magnetisation curve can be
sufficiently described by the Taylor’s expansion of Langevin’s function. However, as
the applied magnetic field is increased, the truncation error increases, thus increasing
the temperature probing error. Figure S4 shows the temperature probing at different
maximum values of the applied magnetic field. Figure S4a shows that as the applied
magnetic field is increased, the maximum error of temperature probing using the
Taylor’s expansion increases significantly due to the truncation error of the Taylor’s
expansion when describing the magnetisation curve. With the maximum applied
magnetic field set to 200 Oe, the maximum error of temperature probing is 0.66 K,
whereas this value is approximately 7.58 K with a maximum applied magnetic field of
600 Oe. The variation in the temperature probing error shown in Figure S4b differs
from that shown in Figure S4a, as a result of increasing the applied magnetic field.
Moreover, Figure S5 shows the standard deviation of the temperature probing errors
using two models. We found (see Fig. S5) that at applied magnetic fields from 200 to
400 Oe, the standard deviation is approximately equal for the two models (i.e.,
Taylor’s expansion and Langevin’s function). However, as the applied magnetic field
is increased, a significant variation in the truncation error affects the standard
deviation of the temperature error. Nevertheless, the relative ratio of the standard
deviation and the truncation error in Figure S4b does not vary significantly. Therefore,
the truncation error for the magnetisation curve as described by Taylor’s expansion is
important for determining the truncation error of temperature probing.
Figure S4. The temperature errors at different maximum values of the applied magnetic field (200,
250, 300, 350, and 400 Oe) with the same point number of 20 (the steps of the applied magnetic
field are 10, 12.5, 15 and 20 Oe, respectively). (a) The temperature errors using a combination of
the Taylor’s expansion and the inverse calculation based on the least square error; (b) the
temperature errors using a combination of Langevin’s function and the inverse calculation based
on the least square error. TT represents theoretical values. The SNR equals 80 dB.
Figure S5. The standard deviation of the temperature error as a function of the applied magnetic
field using different models (Taylor’s expansion versus Langevin’s function).
Both the theoretical model and the inverse calculation method affect the
temperature probing accuracy. The theoretical model mainly affects the truncation
error of temperature probing, whereas the inverse calculation method significantly
affects the impact of noise on the temperature probing accuracy. Additionally, the
theoretical model based on Langevin’s function does not affect the truncation error of
temperature probing, but the inverse calculation method allows a significant rejection
of the impact of noise on temperature probing. Thus, a combination of the model
based on Langevin’s function and the inverse calculation method based on the least
square error is highly promising for yielding more accurate temperature probing using
magnetic nanoparticles.
B. Applied magnetic field
With respect to magnetic nanothermometry, a hysteresis curve describing
magnetic nanoparticles is crucial for assessing the temperature and the accuracy of the
measurements. To assess the temperature, a discrete magnetisation curve, that is, a set
of data for M versus H, should be obtained for a given temperature. However, a
particular set of applied magnetic fields may provide a particular value for the mean
square error from the discrete magnetisation curve, as well as the Jacobian condition
when performing the inverse calculation, thus affecting the temperature probing
accuracy. Therefore, to improve the temperature probing accuracy while applying
magnetic fields, a combination of the model based on Langevin’s function and the
inverse calculation method based on the least square error is employed. Thus, we
evaluated the impact of the applied magnetic field on temperature probing using
different sets of applied magnetic fields.
Figure S6 presents the temperature probing errors and the standard deviation for
different maximum values for the applied magnetic field. As the maximum value of
the applied magnetic field is increased from 200 to 800 Oe, the maximum error of
temperature probing decreases from 0.8 to 0.084 K. Moreover, the inset shows the
standard deviation of the temperature probing error, which decreases from 0.46 to
0.054 K. This finding indicates that as the maximum applied magnetic field is
increased by a factor of 4, the accuracy of temperature probing improves by a factor
of 10. Therefore, an increase in the maximum value of the applied magnetic field
improves the temperature probing accuracy.
Figure S6. The temperature error using Langevin’s function and the inverse calculation method
based on the least square error. Using the same point number (20), ET1, ET2, ET3 and ET4
represent the temperature probing errors at the maximum values of the applied magnetic fields of
200, 400, 600 and 800 Oe, respectively. The inset shows the standard deviation (STD) of the
temperature probing error.