Characterization of magnetic relaxation when biofunctionalized magnetic nano-particles are associated with biomarkers in the liquid state in biomedical applications

This study determined the characteristics of the time-dependent effective relaxation time seff and magnetization M when biofunctionalized magnetic nanoparticles (BMNPs) associated with biomarkers in a liquid immunoassay. Carcinoembryonic antigen (CEA) was used as the biomarker. BMNP is an anti-CEA that was coated onto dextran-coated Fe3O4 and labeled as Fe3O4-anti-CEA. The phase lag q of M with respect to the applied field H was measured using a sensitive homemade alternative-current susceptometer, and q was measured using lock-in detection. The results were used to estimate seff using the relationship tan q(t) 1⁄4 useff(t), where 2pf 1⁄4 u and f is the excitation frequency. Dseff increased with FCEA, where Dseff was the increment of seff after Fe3O4-anti-CEA associated with CEA, and FCEA was the concentration of CEA. Additionally, M enhanced when FCEA increased. We attributed these enhancements to magnetic dipole–dipole interactions among MNPs that contributed extra M and in turn enhanced seff. Magnetic clusters after the association were verified using a transmission electron microscope. This study established the relationship between Dseff/seff,0 and FCEA, where seff,0 1⁄4 seff (FCEA 1⁄4 0), allowing the assay of an unknown amount of CEA molecules.


I. Introduction
Immunoassays use a variety of labels for assaying antibodies or antigens. These labels are typically chemically conjugated or linked to desired analyses. The most common methods are enzyme-linked immunosorbent assay, 1 radioimmunoassay, 2,3 and real-time quantitative polymerase chain reaction. 4 Recently, magnetic immunoassays (MIAs) 5,6 have gained considerable attention because there is negligible magnetic background in a biomolecular sample during magnetic detection; thus, MIAs can reach a high detection sensitivity. In MIAs, biofunctionalized magnetic nanoparticles (BMNPs) are used as labels. 7 The assays involve the specic binding of a bioprobe to its analyte, and a magnetic label is conjugated to one element of the pair to form magnetic clusters. Because of molecular interactions between MNPs, their magnetic properties change aer the association. These changes in magnetic characteristics provide a sensing mechanism. The signal for magnetic clusters is then detected using magnetic sensors, which measure the change of magnetic signals that is induced by magnetic clusters. Changes in magnetic signals are proportional to the concentrations of analytes (e.g., viruses, bacteria, tumor markers, and cardiac markers) in samples. Different sensing mechanisms have been used for MIAs, such as Brownian relaxation time, 8 Néel relaxation time, 9-12 remanent magnetization, 13,14 saturation magnetization, 15,16 spin-spin relaxation time, 17 and alternative-current (AC) susceptibility. [18][19][20][21][22][23][24][25][26] Effective relaxation time s eff is one of the magnetic properties that are affected by the magnetic clustering of BMNPs when they associate with biomarkers. The value of s eff reects the effect of clustered MNPs on Brownian and Néel relaxations. 27 To study s eff , Liao et al. used a homemade AC susceptometer. 28 In the current study, a sensitive homemade AC susceptometer was used to characterize s eff when BMNPs that are composed of Fe 3 O 4 -anti-carcinoembryonic antigen (CEA) associated with CEA in biomedical applications. This study found that Ds eff increased with the concentration of CEA, F eff , where Ds eff was the increment of s eff . This increase was attributed to dipole-dipole interactions between MNPs, which contributed extra magnetization and in turn increased the value of s eff . A relationship between Ds eff /s eff,0 and F CEA was established, where s eff,0 ¼ s eff (F CEA ¼ 0). This relationship is based on a (Ds eff /s eff,0 )-versus-F CEA curve and can be used to determine an unknown amount. This paper also discusses the merits of using the Ds eff /s eff,0 ratio for biosensing.

II. Experiments
BMNPs are anti-CEAs that were coated onto dextran-coated MNPs and labeled as Fe 3 O 4 -anti-CEA (MF-CEA-0060, MagQu Co., Taiwan). The biotarget was CEA (AbD). A feasibility study for clinical use reported the development of antibodyfunctionalized magnetic nanoparticles for the immunoassay of CEA. 29 The mean diameter of BMNPs was 53.8 AE 17.8 nm, 23 as detected through dynamic light scattering (Nanotrac-150, Microtrac) and shown in Fig. 1a. The magnetization reagent was characterized using a vibrating sample magnetometer (Hystermag MagQu Co. Taiwan). The magnetic clustering aer association was measured using a eld-emission transmission electron microscope (Philips Tecnai F20 G2). Before being examined, the reagent consisted of magnetic clusters that were dispersed in an extremely diluted solvent (1 : 50). The samples were then dried under vacuum at room temperature for 48 h.

III. AC susceptibility
The AC susceptibility c ac (u) of the samples is expressed as follows: 28,30 where i ¼ (À1) 1/2 , q ¼ tan À1 us eff (t) ¼ c 00 /c 0 is the phase lag for the time-varying magnetization, M(t), with respect to the applied AC magnetic eld, H(t), and s eff is the effective relaxation time. In this study, q(t) was detected using a homemade AC susceptometer and the lock-in detection technique at 18 kHz. The time evolution of s eff (t) was measured by monitoring q(t).
The value of s eff (t) was estimated using the relationship q ¼ tan À1 us eff (t). The value of given as a function of the concentration of biomarkers to characterize their abundance.

IV. Results and discussion
To explore and conrm the combination of Fe 3 O 4 -anti-CEA and CEA, a TEM was used to verify the presence of magnetic clustering when CEA associated with Fe 3 O 4 -anti-CEA. Fig. 2a shows a TEM image of CEA molecules; single CEA molecules are marked with small dashed circles. A CEA molecule has a mean length of approximately 29 nm, 31 which is close to the value given in Fig. 2b. Fig. 2b shows a TEM image of magnetic clusters when Fe 3 O 4 -anti-CEA associated with CEA molecules; some magnetic clusters are marked with a large dashed circle. Each magnetic cluster was composed of several small magnetic clusters. Also marked is the independent Fe 3 O 4 -anti-CEA and MNPs inside the magnetic clusters; the formation of magnetic clusters was veried using a TEM. The size of associations with CEA molecules was changeable; if the concentration of CEA molecules was higher, then a magnetic cluster was formed. Thus, the size of Fe 3 O 4 -anti-CEA associated with CEA molecules increased.  reagent with F CEA ¼ 1, 5, and 10 mg mL À1 associated with CEA, respectively. This increase is attributable to dipole-dipole interactions between MNPs in the magnetic clusters. Thus, the magnetization increased with the CEA concentration.
For a homogeneous MNP with a total number of particles, N, a mean magnetic moment, hmi, and an effective mean diameter, hdi eff , the magnetization M(H) at H is expressed as follows: and   Fig. 4b shows the relationship between s eff and time for the reaction process when the reagent conjugated with different concentrations of F CEA . The value of s eff for the reagent was 0.93 ms when t ¼ 0 and remained constant until t ¼ 7200 s, which demonstrated that the AC susceptometer had stable performance. In the reaction with 10 mg mL À1 CEA, the value of s eff was approximately 0.93 ms when t ¼ 0 and the value of s eff increased to 1.43 ms when t ¼ 7200 s. The s eff -versus-t curve showed saturation at approximately t ¼ 7200 s.
Yang et al. studied the effect of temperature on immunoreaction kinetics for a BMNP assay of biomarkers for colorectal cancer. 33 The activation energy that was required for the reaction in immunoassay was determined to be 10 À19 J per molecule. At higher temperatures, a signicant increase was observed in the number of molecules with energy that was higher than the activation energy, which is the principal reason for the increase in the reaction rate. The current study indicated that it takes approximately 2 h to complete the association at 298 K. An increase in the reaction temperature decreased the reaction time. Fig. 5 illustrates the value of s eff as a function of F CEA when the reagent associated with various concentrations of CEA; the red line is a guide. It is seen that s eff ¼ 0.93 ms for a reagent that consisted of Fe 3 O 4 -anti-CEA. The value of s eff increased to 1.43 ms when F CEA ¼ 10 mg mL À1 associated with the reagent. The increase in the value of s eff aer association with Fe 3 O 4 -anti-CEA is due to magnetic interactions between MNPs in the associated magnetic clusters. The value of M increased; therefore, the value of s eff increased.
Two simultaneous relaxations occur during the reaction process: Néel and Brownian relaxation. In Brownian relaxation,  the magnetic moment is locked to the crystal axis and the particle rotates. The corresponding relaxation time is s B when the eld is turned off. In Néel relaxation, the magnetic moment rotates within the crystal, and the corresponding relaxation time is s N . The Brownian relaxation time s B is expressed as 29 where h is the viscosity coefficient, k B ¼ 1.38 Â 10 À23 J K À1 is the Boltzmann constant, T is the absolute temperature (K), and V H is the hydrodynamic volume of BMNPs. The Néel relaxation time, s N , is given by the expression, s N ¼ [(p) 1/2 /2]s 0 exp[r]/(r) 1/2 , where r ¼ KV M /k B T, K is the anisotropy constant, s 0 is the relaxation time, and V M ¼ (4p/3)R 3 is the magnetic volume of MNPs with radius R. The approximate mean core value was hRi ¼ 37 nm in this study. If hRi ¼ 37 nm, then s Brownian is in the order of microseconds and s Néel is in the order of milliseconds or longer. 27 The mean hydrodynamic diameter of BMNPs in this work was approximately 47 nm, which equates to a core size of approximately 38 nm. Because 1/s eff ¼ 1/s Brownian,eff + 1/s Néel,eff and 1/s Brownian,eff >> 1/s Néel,eff , the values of 1/s eff,eff $ 1/s Brownian,eff can be approximated.
Brownian relaxation dominates the time dependency of s eff in the association process that is shown in Fig. 4b. Fig. 6 illustrates Ds eff /s eff,0 as a function of F CEA when Fe 3 O 4 -anti-CEAs conjugated with CEAs, where Ds eff ¼ s eff (F CEA ) À s eff (F CEA ¼ 0) and s eff,0 ¼ s eff (F CEA ¼ 0) at t ¼ 7200 s. In an assay of 10 mg mL À1 CEA molecules, Ds eff /s eff,0 ¼ 0.05 as F CEA . Fig. 6 Ds eff /s eff,0 as a function of F CEA ¼ 0.01 mg mL À1 and Ds eff /s eff,0 increases to 0.54 when F CEA ¼ 0.01 mg mL À1 ; Ds eff /s eff,0 increases when F CEA increases.
The value of Ds eff /s eff,0 as a function of F CEA is described by a logistic function: 34 where Ds eff ¼ s eff (F CEA ¼ 0) and s eff,0 ¼ s eff (F CEA ¼ 0). The solid line is a tting curve to eqn (6), with tting parameters A ¼ 0.03, B ¼ 13.9, F 0 ¼ 7243 mg mL À1 , and g ¼ 0.49. The relationship between Ds eff /s eff,0 and F CEA was established. This relationship, (Ds eff /s eff,0 )-versus-F CEA , allows the assay of an unknown amount  of CEA molecules for biosensing. Colorectal tumors, which are caused by uncontrolled cell growth in the colon or rectum, 35 are the most commonly diagnosed cancer, especially in developed countries. 36 Quantitative measures of CEA molecules are used as a tumor marker to monitor colorectal carcinoma treatment, to identify recurrences aer surgical resection, and to stage or localize the spread of cancer. The detection sensitivity in this study can be determined from the variation of Ds eff /s eff,0 . The standard deviation of Ds eff /s eff,0 was 0.033 as F CEA ¼ 0.001 mg mL À1 . The estimated detection limit of 0.0037 mg mL À1 was obtained by the highest value of Ds eff /s eff,0 and transferred using a logistic function. The detection limit in our system reached the detection criteria for colorectal carcinoma diagnosis.

V. Conclusions
This study determined the relationship between time and s eff when a reagent that consisted of Fe 3 O 4 -anti-CEA associated with CEA molecules. Because of magnetic interactions between MNPs in magnetic clusters, which contributed extra magnetization, Néel relaxation and Brownian relaxation increased, which resulted in an increase in the value of s eff . Magnetic clustering was veried using a TEM. A relationship between Ds eff /s eff,0 and F CEA was established, which obeyed a logistic function, allowing the assay of an unknown amount of CEA molecules. The c ac detection platform for sensing biomarkers is highly sensitive, easy to use, and can be used to assay a wide variety of analytes, such as proteins, DNA, viruses, and tumor markers.

Conflicts of interest
There are no conicts to declare.