Abstract:
The invention relates to an integrated measurement system to detect a quantity of magnetic particles in a sample. The measurement system includes a substrate. An electromagnetic (EM) structure disposed on the surface of the substrate is configured to receive a sample including the magnetic particles in proximity thereof. The integrated measurement system also includes an electrical current generator disposed on the surface of the substrate which is electro-magnetically coupled to the EM structure. The electrical current generator is configured to cause an electrical current to flow in the EM structure. The integrated measurement system also includes an effective inductance sensor disposed on the surface of the substrate which is configured to measure a selected one of an effective inductance and a change in effective inductance. The invention also relates to a method to determine the number of and/or the locations of the magnetic particles in a sample.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to and the benefit of U.S. provisional patent application Ser. No. 61/068,513, filed Mar. 7, 2008, which application is incorporated herein by reference in its entirety. 
    
    
     STATEMENT REGARDING FEDERALLY FUNDED RESEARCH OR DEVELOPMENT 
     The U.S. Government has certain rights in this invention pursuant to Grant No. N00014-04-C-0588 awarded by ONR—Space and Naval Warfare Systems Center. 
    
    
     FIELD OF THE INVENTION 
     The invention relates in general to bio-sensors and more particularly to micro/nano bead based cell/bio-molecule sensors. 
     BACKGROUND OF THE INVENTION 
     Non-optical methods for bioassay are of interest to the interdisciplinary fields of biology, applied physics, and microelectronics. Magnetic micro/nano beads have been studied for use in cell/bio-molecule sensing as one non-optical method. While the magnetic behavior of magnetic micro/nano beads can be detected without using costly imaging systems, sensing magnetic micro/nano particles remains a challenging task. Various detection methods have been proposed to address this sensing challenge. Traditionally, superconducting quantum interference devices (SQUIDs), giant magnetoresistance (GMR) arrays and atomic force microscopy have been used for their high sensitivity. However, sensing methods such as those based on SQUIDs or GMR arrays typically cannot be fabricated with standard integrated processes as CMOS. Also, both SQUID and GMR array generally require relatively costly postprocessing steps. Moreover, GMR techniques need externally generated biasing magnetic fields. Such biasing magnetic fields can be provided using either permanent or electro-magnets, which leads to extra power-consumption, large form-factor and high cost. Moreover, the biasing magnetic fields need to be calibrated to set a correct orientation, which increases the difficulty of use. 
     Another prior art sensor technology, the Hall sensor, is available in CMOS processes. Hall sensors, however, need a relatively high power external biasing field, thus precluding most low power portable battery powered applications. The biasing magnetic fields also have all of the impediments mentioned before for GMR sensors. Hall sensor based systems are also generally unsuitable for use in micro-fluidic systems. For example, for optimum sensitivity, Hall sensors should have dimensions (sensor size and passivation layer thickness) on the order of the dimensions of the magnetic beads to be sensed. Such Hall dimensions typically limit the hall sensor to a small sensing area and preclude sensor compatibility over a range of sizes of magnetic particles of interest. Moreover, to achieve a close proximity between the Hall sensor sensing part and the magnetic samples, expensive post processes, such as etching, are needed. 
     Therefore, what is needed is a more efficient and flexible system and method for making micro/nano magnetic bead based cell/bio-molecule measurements. 
     SUMMARY OF THE INVENTION 
     In one aspect, the invention relates to an integrated measurement system to detect a quantity of magnetic particles in a sample. The measurement system includes a substrate having a surface. The integrated measurement system also includes an electromagnetic (EM) structure disposed on the surface of the substrate which is configured to receive a sample including the magnetic particles in proximity thereof. The integrated measurement system also includes an electrical current generator disposed on the surface of the substrate which is electro-magnetically coupled to the EM structure. The electrical current generator is configured to cause an electrical current to flow in the EM structure. The integrated measurement system also includes an effective inductance sensor which is disposed on the surface of the substrate and configured to measure a selected one of an effective inductance and a change in effective inductance; thereby to detect the quantity of magnetic particles. 
     In one embodiment, the magnetic particles include magnetic beads. 
     In another embodiment, the integrated measurement system comprises a CMOS structure. 
     In yet another embodiment, the electrical current generator comprises a quasi-static electrical current generator. 
     In yet another embodiment, the magnetic particles include magnetic micro/nano beads. 
     In yet another embodiment, the measurement system includes a cell/bio-molecule sensing system. 
     In yet another embodiment, the measurement system includes an impedance based sensing system. 
     In yet another embodiment, the measurement system includes a transmission line based sensing system. 
     In yet another embodiment, the measurement system includes an oscillator based sensing system. 
     In yet another embodiment, the oscillator based measurement further includes a mixer and wherein a difference frequency is measured to detect the quantity of magnetic particles. 
     In yet another embodiment, the oscillator based measurement includes a low noise oscillator. 
     In yet another embodiment, the integrated measurement system further includes a temperature regulator. 
     In yet another embodiment, the temperature regulator is configured to set a temperature of the integrated measurement system. 
     In yet another embodiment, the temperature regulator is configured to set a temperature of the quantity of magnetic particles in the sample. 
     In yet another embodiment, the measurement system further includes a sample delivery structure. 
     In yet another embodiment, the sample delivery structure includes a microfluidic delivery structure. 
     In yet another embodiment, the measurement system further includes a second EM structure disposed on the surface of the substrate, wherein the first EM structure is configured to receive a sample including a plurality of target particles and the second EM structure is configured to receive a control solution. 
     In yet another embodiment, the control solution includes a sample lacking magnetic particles. 
     In yet another embodiment, the control solution includes a sample having magnetic particles. 
     In yet another embodiment, the control solution includes a sample lacking magnetic properties. 
     In yet another embodiment, the control solution includes a sample having magnetic properties. 
     In yet another embodiment, the measurement system further includes a sensor configured to make a differential sensing measurement. 
     In yet another embodiment, an oscillator based measurement system includes a sensing oscillator and a reference oscillator configured to obtain a correlation on a 1/f 3  phase noise and wherein the sensing oscillator and the reference oscillator are configured to measure a difference between a first set of frequency counts of the sensing oscillator and a second set of frequency counts of the reference oscillator. 
     In yet another embodiment, a differential sensing system is configured to suppress a common mode noise. 
     In yet another embodiment, the measurement system further includes an M×N array of EM structures disposed on the surface of the substrate, wherein each element of the array of EM structures is configured for a sensitivity to a particular type of target particle and the measurement system is configured to measure a plurality of quantities of the particular types target particles for each of a plurality of sample volumes, each sample volume of the plurality of sample volumes including a portion of a common sample. 
     In yet another embodiment, the measurement system further includes an M×N array of EM structures disposed on the surface of the substrate, wherein each element of the array of EM structures is configured to measure a quantity of target particles of a respective one of a plurality of different samples. 
     In another aspect, the invention relates to method for determining the number of magnetic particles in a sample including the steps of: providing an integrated magnetic particle sensor having a sensor sample volume, an electromagnetic (EM) structure, an electrical current generator, and an effective inductance sensor; providing a sample including a plurality of magnetic particles; delivering the sample to the sensor sample volume; generating an electrical current in the EM structure to establish a magnetic field in the sensor sample volume; measuring parameter of the sensor sample volume; and determining the number of magnetic particles in the sample based on the parameter. 
     In one embodiment, the step of measuring a parameter comprises the step of measuring an electrical parameter of said sensor sample volume and said step of determining the number of magnetic particles comprises the step of determining the number of magnetic particles in said sample based on said electrical parameter. 
     In another embodiment, the step of measuring a parameter comprises the step of measuring a magnetic parameter of said sensor sample volume and said step of determining the number of magnetic particles comprises the step of determining the number of magnetic particles in said sample based on said magnetic parameter. 
     In one embodiment, the step of providing an integrated magnetic particle sensor includes providing an impedance measurement based magnetic particle sensor having a sensor sample volume. 
     In another embodiment, the step of providing an integrated magnetic particle sensor includes providing a transmission line based magnetic particle sensor having a sensor sample volume. 
     In yet another embodiment, the step of providing an integrated magnetic particle sensor includes providing an oscillator based magnetic particle sensor having a sensor sample volume. 
     In yet another embodiment, the step of measuring an parameter includes measuring an parameter of said sensor sample volume by averaging a set of measurements to lower the noise floor and to improve a measurement sensitivity. 
     In yet another embodiment, averaging a set measurements includes averaging a set of frequency counting measurements. 
     In yet another embodiment, the method further comprises the following additional steps: generating an electrical current in said EM structure to establish a magnetic field in a sensor sample volume lacking a sample; measuring a parameter of said sensor sample volume lacking a sample; determining an empty sensor sample volume measurement based on said parameter; and determining the number of magnetic particles in said sample based on said parameter of said empty sensor volume and said sensor volume having a delivered sample. 
     In yet another aspect, the invention relates to an integrated measurement system to detect a location of magnetic particles in a sample. The measurement system includes a substrate having a surface. The integrated measurement system also includes an electromagnetic (EM) structure disposed on the surface of the substrate which is configured to receive a sample including the magnetic particles in proximity thereof. The integrated measurement system also includes an electrical current generator disposed on the surface of the substrate which is electro-magnetically coupled to the EM structure. The electrical current generator is configured to cause an electrical current to flow in the EM structure. The integrated measurement system also includes an effective inductance sensor which is disposed on the surface of the substrate and configured to measure a selected one of an effective inductance and a change in effective inductance; thereby to detect the location of magnetic particles. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The objects and features of the invention can be better understood with reference to the drawings described below, and the claims. The drawings are not necessarily to scale, emphasis instead generally being placed upon illustrating the principles of the invention. In the drawings, like numerals are used to indicate like parts throughout the various views. 
         FIG. 1  shows a block diagram of one embodiment of a general form of a magnetic particle detection system. 
         FIG. 2  shows a flow chart of one exemplary method for magnetic particle sensing. 
         FIG. 3  shows a block diagram of one exemplary impedance measurement system. 
         FIG. 4  shows one embodiment of a micro/nano magnetic particle detection system based on a transmission line S-parameter measurement. 
         FIG. 5  shows one embodiment of a micro/nano magnetic particle detection system using an inductance sensor based on an oscillator. 
         FIG. 6  shows one embodiment of an improved oscillator based sensor. 
         FIG. 7  shows a comparison of phase noise linewidth for an impedance based sensor (left side) compared to an oscillator based sensor (right side). 
         FIG. 8A  shows a graph of noise floor σ 2   Δf/f0  plotted against time (T) illustrating a different relative power between 2k 2 /T noise and 1/f 0   2 T 2  for an exemplary sensor system. 
         FIG. 8B  shows a graph of noise floor σ 2   Δf/f0  plotted against time (T) illustrating a different relative power between 2k 2 /T noise and 1/f 0   2 T 2  for another exemplary sensor system. 
         FIG. 9  shows a block diagram of one exemplary differential sensor system. 
         FIG. 10  shows a schematic diagram and time line illustrating how common mode noise can be suppressed through differential sensing. 
         FIG. 11  shows a block diagram of a two dimensional (2D) M×N dimensional sensor array. 
         FIG. 12  shows an illustration of an exemplary sensor circuitry. 
         FIG. 13  shows a schematic diagram of the magnetic particle sensor based on a Colpitts LC oscillator. 
         FIG. 14  shows a flowchart of the experimental procedures useful to perform experiment 1. 
         FIG. 15A  shows an illustration of a sensor inductor with delivered magnetic beads. 
         FIG. 15B  shows an illustration of a sensor inductor with higher concentration of delivered magnetic beads than shown in  FIG. 15A . 
         FIG. 16  is a graph showing frequency in Hz versus time in seconds for a measurement cycle. 
         FIG. 17  is a graph showing Δf/f per bead versus measurement number for 22 measurements. 
         FIG. 18  shows a flowchart of experimental procedures useful to perform experiment 2. 
         FIG. 19A  shows an illustration of an overall experimental sensor setup. 
         FIG. 19B  shows an illustration emphasizing the sensor of  FIG. 19A . 
         FIG. 20A  is an illustration showing the channel state of the microfluidic channel when the valve is open. 
         FIG. 20B  is an illustration showing the channel state of the microfluidic channel when the valve is closed. 
         FIG. 20C  is an illustration showing a more detailed view of the PDMS structure of  FIG. 20A  and  FIG. 20B . 
         FIG. 21  shows a sensor inductor with delivered magnetic beads. 
         FIG. 22  shows measurement results plotted as Δf/f per bead versus measurement number for 8 measurements. 
         FIG. 23  shows an exemplary inductor structure that can generate a magnetic field to polarize the illustrated symbolic magnetic particle. 
         FIG. 24  is a graph showing the results of an exemplary three-dimensional (3D) EM (electromagnetic simulation). 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The description which follows is organized into four sections. Part I generally introduces the inventive sensor system. Part II describes the inventive method using a flow diagram. Part III describes three embodiments of a micro/nano magnetic particle detection system, an impedance based sensor, a transmission line based sensor, and an oscillator based sensor. Part IV describes several exemplary implementations of the sensor system including arrays and two implementation examples. Theoretical underpinnings are described for the sensing methods, and for an approximate close-form solution to quantify the inductance change in the magnetic particle sensors described herein. 
     Part I, Introduction 
     Bio-systems, on their own, typically do not generate significant magnetic signals. Therefore, cell/bio-molecule sensing systems, and methods, using magnetic bead based sensors (e.g. to detect magnetic micro/nano beads) offer several advantages for magnetic detection based cell/bio-molecule measurements, including a relatively quiet sensing background. Fluorescent label optical techniques offer a widely used alternative cell/bio-molecule measurement technology. However, as compared with fluorescent label optical techniques, magnetic beads do not exhibit signal quenching or decaying problems. Magnetic bead-based measurements are largely immune to such problems since their characteristic signal typically exhibits a stable relationship with respect to external excitation for a long measurement time. The stable nature of magnetic bead-based measurement signals also lends itself to filtering techniques that can provide an improved signal to noise ratio (SNR) by signal averaging. 
     Another advantage is the ability of magnetic beads to manipulate attached cells/macromolecules. The ability to manipulate attached cells/macromolecules can lead to several desirable features including, bio-sample delivery, concentration/separation, and affinity binding facilitation both with and without valve/channel based conventional micro-fluidic systems. Also, micro/nano magnetic beads can be engineered to be biocompatible and can be made available for most commonly used bio-probe coatings. Therefore, micro/nano magnetic beads are particularly well suited for bio sensing platforms. 
     As described above, the design of efficient and flexible systems and methods for sensing magnetic micro/nano particles remains a challenging task. The challenge is due in part to their magnetic property (superparamagnetic for most of the off-the-shelf magnetic micro/nano particles) which offers a relatively low effective relative permeability value (μ r ). Such a low μ r , typically around 2 to 3, can lead to small magnetic measurement signals. Traditional magnetic toroid shapes are generally not suitable for use with planar sensors, where magnetic excitation/sensing is generally carried out in an open-magnetic-loop fashion. Also, demagnetization effects can further degrade the sensitivity. 
     In some of the embodiments of micro/nano magnetic particle detection systems and methods described herein, the magnetic material properties of micro/nano magnetic particles are used to measure an effective inductance change of a sensor structure. The sensor structure can be viewed as an optimized electromagnetic (EM) structure. In contrast with prior art methods, the inventive structures described herein are generally compatible with standard integrated circuit processes, i.e. free of any external biasing magnetic field setups, capable of being adapted for magnetic manipulation, and providing a relatively large sensing area which can accommodate magnetic particles of virtually any size. Sensors as described herein-below can be made fully portable and battery-powered and can be better integrated with sample delivery structures, such as micro-fluidic systems. Such sensors systems can also be fabricated as a hybrid lab-on-chip (LOC) for point-of-care (POC) medical diagnostic support applications. Such sensor systems can also be used for parallel sensing for many same/different samples in an array fashion. The sensor schemes described herein can be implemented in planar format, and are therefore compatible with standard integrated circuit processes. 
     Part II, General Description of the New Sensor 
     In this section, a new class of sensors that uses a new detection method for sensing magnetic particles is described. Various embodiments of these sensors generally include the following functional blocks: 
     1) One or more electromagnetic (EM) structures for sensing a cell/bio-molecule sample; 
     2) One or more circuits that can generate an electrical current conducting through the EM structures of block  1 ; 
     3) One or more circuits to sense the effective inductance and/or a change in effective inductance; and optionally; 
     4) A structure to deliver the test samples to the one or more optimized electromagnetic (EM) structures. 
       FIG. 1  shows a block diagram of one embodiment of a general form of a magnetic particle detection system as described above. An electromagnetic (EM) structure for sensing a cell/bio-molecule sample is configured to evaluate a sample volume in or near the EM structure  201 . EM structure  201  can operate without a sample, or with a sample having insignificant magnetic properties, or with certain magnetic properties such as control samples, to make control measurements. EM structure  201  can also operate to evaluate a sample having magnetic particles such as bio-molecules and cells having attached magnetic particles. A sample can be manually placed in, or in the vicinity of the EM structure for a measurement of the sample. More typically, a test sample delivery structure  204  can be used to deliver a sample for measurement to a sample volume, such as by use of a liquid or suspended liquid. An electrical current generator  202  generates an electrical current flow through the sensing EM structure  201 . An effective inductance sensor  203  measures an effective inductance, or a change in effective inductance, such as a change in effective inductance from an empty sample volume to a sample volume occupied by a sample with magnetic particles. 
     The electrical current generator  202  can also be a quasi-static current generator. A quasi-static current generator can generate a DC (direct current) and/or one or more AC currents (alternating electrical currents). If an AC current is generated, the frequency of the AC current should be low enough, so that the dimension of the circuits and/or the EM structure where this AC current will flow through will be much smaller than the electromagnetic wavelength at this frequency in a media in which the circuit and/or the EM structure is placed. 
     As described in more detail below, the various functional blocks ( 201 ,  202 , and  203 ) need not necessarily be implemented as separate structural blocks. For example, in some embodiments, a given structural block can include both the functions of blocks  201  and  202  simultaneously. Also, note that the electrical current conducting through the EM structures can be described interchangeably as a current, or as a voltage or power level that causes the current. 
     The working mechanism is described as follows: First, a sample is delivered onto the sensor through the delivery structure. Magnetic particles can be included directly in the sample where the magnetic particles are the sensing target. Micro- or nano-meter level non-magnetic particles, such as bio-molecules and cells, can also be attached with magnetic particles to serve as a target sample. In this case, detecting the presence of magnetic particles infers the existence of target nonmagnetic particles. Samples with no magnetic particles can also be used to characterize sensor response in a control case. Next, the current generation circuitry generates an electrical current which is conducted through the EM structure. Based on the strength of the current and the shape of the EM structure, a magnetic field is established throughout a sample space where a target sample or control sample can also be present. The magnetic field polarizes magnetic particles present in the space and induces magnetization of the particles, which increases the total magnetic energy in the space. The total number of magnetic particles and their locations present in a sample volume determines the total magnetic energy change, which is related to the effective inductance (both/either self and/or mutual inductance) change of the EM structure. Theoretical details are presented hereinbelow. Then the inductance sensing circuit measures the inductance (both/either self and/or mutual inductance) value and/or its change for the EM structure, which infers the presence and the number of magnetic particles and their location information present in a sample. Since the EM structure can be designed with a location-dependent sensitivity, the sensor systems described herein can also sense information related to the location and distribution of the magnetic particles. 
     The flow diagram of  FIG. 2  shows one embodiment of the inventive method. The method shown in  FIG. 2  can determine the number of magnetic particles in a sample by following the steps of: providing a magnetic particle sensor having a sensor sample volume; providing an electromagnetic (EM) generating structure; providing a sample comprising a plurality of magnetic particles; delivering the sample to the sensor sample volume; generating an electrical current in the electromagnetic generating structure to establish a magnetic field in the sensor sample volume; measuring a magnetic and/or electrical parameter of the sensor sample volume, typically effective inductance or a change in effective inductance; and determining the number of magnetic particles in the sample based on the magnetic and/or electrical parameter. 
     A baseline measurement of the sensor response can be made before delivering the samples (e.g. after cleaning and/or washing away samples) by measuring some specific control samples (magnetic or non-magnetic), and/or by measuring a reference sensor. The signals measured, for example for the samples, can be processed with the baseline measurement(s), e.g. by addition or subtraction, to yield the desired information. 
     Part III, Three Embodiments of a Magnetic Particle Detection System 
     Part III describes three embodiments of a micro/nano magnetic particle detection system, including as alternatives an impedance based sensor, a transmission line based sensor, and an oscillator based sensor. The functional blocks described as a general structure in Part II do not necessarily correspond one-to-one with structural block of each embodiment described in this section. For example, in some embodiments, one structural block can perform shared functions. For each embodiment, a correlation is given to the general embodiment described in Part II and illustrated by the block diagram of  FIG. 1 . 
     Impedance Measurement Based Sensor 
     In one embodiment of a micro/nano magnetic particle detection system, an impedance based sensor can be used to directly measure the impedance of a sample volume.  FIG. 3  shows a block diagram of one exemplary impedance measurement system. A signal (e.g. a voltage, current or power) is generated by source  301  and measured by sense  302  (a sensor). The bandwidth of source  301  and sense  302  can be narrowband, tunable narrowband, or broadband. Block  303  is a sensing block which interacts with the samples. Block  303  can include one or several lumped inductors or circuits formed by lumped inductors together with other components, such as capacitors. In particular, parallel or series or multi-resonance LC resonators can be formed for use in block  303 . The impedance of the resonator of block  303 , both in amplitude and phase, can be highly dependent on an inductance value at or near a resonant frequency of the resonator. 
     Now, comparing the exemplary impedance measurement based sensor shown in  FIG. 3  to the general embodiment described in Part II and the block diagram of  FIG. 1 , it can be seen that, source  301  corresponds to the one or more circuits that can generate an electrical current conducting through the EM structures (block  202 ,  FIG. 1 ). Sense  302  corresponds to the one or more circuits to sense the effective inductance and/or a change in effective inductance (block  203 ,  FIG. 1 ), and block  303 , the sensing block which interacts with the magnetic particles, corresponds to the one or more electromagnetic (EM) structures for sensing a cell/bio-molecule sample (block  201 ,  FIG. 1 ). Note that in some embodiments, block  303  can include and/or replace some of the sourcing and sensing functions of blocks  301  and  302 . 
     Transmission Line Based Sensor 
     Transmission lines can be generally viewed as including a having distributed or lumped component parameters. Transmission lines have characteristics that are highly dependent on the parasitic inductance per unit length (distributed model) or the inductance per synthetic section (lumped component model). Two such characteristics of a transmission line, the characteristic impedance Z 0  and the delay per section τ are shown below in Eq. 1 and Eq. 2: 
     
       
         
           
             
               
                 
                   
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     One system and method to measure the change in inductance L of a transmission line is to measure the S-parameters for the transmission line under study.  FIG. 4  shows one embodiment of a micro/nano magnetic particle detection system using a transmission line S-parameter based sensor structure. As can be seen in  FIG. 4 , circuit block  404  can function as port  1  and circuit block  406  can function as port  2  for a two-port S-parameter measurement. Block  405  functions as the sensing block which interacts with the samples. Block  405  can include a single ended transmission line, a differential transmission lines or any suitable microwave circuit having a transmission line. Suitable exemplary microwave circuits include, but are not limited to, filters, couplers, and resonators. 
     Now, comparing the exemplary transmission line based sensor shown in  FIG. 4  to the general embodiment described in the block diagram of  FIG. 1  of part II, it can be seen that Block  405 , the one or more circuits that can generate an electrical current conducting through the EM structures (block  201 ,  FIG. 1 ), and blocks  404  and  406  can correspond to the one or more circuits that can generate an electrical current conducting through the EM structures (block  202 ,  FIG. 1 ), and the one or more circuits to sense the effective inductance and/or a change in effective inductance (block  203 ,  FIG. 1 ). 
     Oscillator Based Sensor 
     Resonant structures can be made by combining inductors and capacitors and/or a microwave resonator. Resonant structures include parallel, series, and multi-mode resonators. An oscillator can be based on such resonant structures. An oscillator based on one or more resonators has an oscillation frequency. The oscillation frequency of the oscillator can be measured directly to indicate an inductance(s) (and/or the equivalent inductive part(s)) or change of inductance(s) (and/or change of the equivalent inductive part(s)) of the resonator(s), for example, using the relationship shown by Eq. 3 below. 
     
       
         
           
             
               
                 
                   
                     
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     One/multiple oscillator(s) can be made based on one/multiple resonator(s) to have one/multiple oscillation frequency/frequencies. The oscillation frequency (frequencies) can also be used to sense both/either the self/mutual inductance(s) (and/or the self/mutual inductive part(s)). 
       FIG. 5  shows one embodiment of a micro/nano magnetic particle detection system using an inductance sensor based on an oscillator. Block  507  includes the sensing structure whose (self/mutual) inductor (or equivalent self/mutual inductive part) will change its value when magnetic particles are present, and together with capacitors (or equivalent capacitive part), block  507  forms a resonance tank for the oscillator. Block  508 , the circuitry for the oscillator core, pumps that power to the lossy tank to maintain a steady oscillation. Block  508  can include cross-coupled transistor pairs, such as have been used in negative-gm oscillator designs, or other suitable feedback structures such as are used in a Colpitts oscillator design. Block  509 , a frequency counter, can be an off-the shelf type unit, such as a commercial frequency counter, or can be integrated counter such as an integrated synchronous or asynchronous adder. 
     Now, comparing the exemplary oscillator based sensor shown in  FIG. 5  to the general embodiment described in Part II and the block diagram of  FIG. 1 , it can be seen that block  507  corresponds to EM structures (block  201 ,  FIG. 1 ), and block  508  corresponds to the one or more circuits that can generate an electrical current conducting through the EM structures (block  202 ,  FIG. 1 ) as well as the one or more circuits to sense the effective inductance and/or a change in effective inductance (block  203 ,  FIG. 1 ), while counter block  509  serves as a read-out device. 
       FIG. 6  shows one embodiment of an improved oscillator based sensor. Note that blocks  507 ,  508  and  509  of  FIG. 6  are equivalent to the same numbered blocks in  FIG. 5 . In  FIG. 6 , an additional mixer  612  mixes f sense , the output of oscillator core  508  with a local oscillator frequency, such as an external frequency f LO . The oscillation tone f sense  is translated by mixing. The mixer  612  output includes the sum and difference frequencies: f sense +f LO  f sense −f LO . There are at least two advantages to counting the downconverted tone f sense −f LO . First, the sensitivity of the signal frequency sensed by the counter is increased from Δf/f sense  to Δf/(f sense −f LO ). Second, counting at a lower frequency of f sense −f LO , instead of a higher frequency of f sense  makes the counter design both more reliable as well as saving electrical power (more energy efficient). Block  613  is used to filter out undesired frequencies, such as the unwanted tone f sense +f LO . Thus, it can seen that block  612  and block  613  can be used to reduce the requirements of counter  509  (by lowering the frequency to f sense −f LO ) as well as to improve the resolution of the frequency read-out. Moreover, although not shown in  FIG. 6 , multiple mixers and multiple filters can be used for multi-step downconversion. 
     Using the impedance sensing method described above on a resonator structure, the impedance function linewidth can be fundamentally limited by the quality factor of the EM sensing structure. By contrast, when using an oscillator based measurement as the EM sensing structure, the phase noise linewidth is significantly reduced. Reduction in phase noise linewidth leads to an ultra-high sensor sensitivity which can easily detect a small frequency (inductance) change.  FIG. 7  shows a comparison of phase noise linewidth for an impedance based sensor (left side) compared to an oscillator based sensor (right side). 
     By averaging the measured data, such as the frequency counting results for the oscillator based measurement scheme, the sensor system can achieve an improved noise floor (i.e. improved sensitivity). Also, the sensor can achieve a high sensitivity by use of a low noise oscillator, differential sensing scheme and/or a temperature regulator structure. By choosing an appropriate measurement time T (frequency counting time) for an oscillator-based measurement implementation, a low sensor noise-floor σ 2   Δf/f0  (i.e. improved sensor sensitivity) can be achieved.  FIG. 8A  shows a graph of noise floor σ 2   Δf/f0  plotted against time (T). The 2k 2 /T noise is from the 1/f 2  phase noise, 1/f 0   2 T 2  is the relative frequency counting error due to the principle of uncertainty, and the 2ζ 2  is due to the 1/f 3  phase noise. Therefore, by choosing a large enough T, one can achieve the minimum achievable noise floor of 2ζ 2 .  FIG. 8A  and  FIG. 8B  show graphs of noise floor σ 2   Δf/f0  plotted against time (T) illustrating a different relative power between 2k 2 /T noise and 1/f 0   2 T 2  for two different sensor systems. Note, although not plotted here, if 1/f n (n&gt;3) exists for the oscillator phase noise, the σ 2   Δf/f0  plot with respect to time T will start to increase after some T max . Then the optimum sampling time T should not exceed T max , but needs to be large enough to be in the 2ζ 2  flat range. For the oscillator based measurement implementation, using the same principle of differential sensing, if the 1/f 3  phase noise can be made correlated between the sensing oscillator and the reference oscillator, by taking the frequency counting difference on the two, this 1/f 3  phase noise can be suppressed which leads to a smaller 2ζ 2 , and therefore a lower noise floor and a better sensor sensitivity. 
     Delivering Structures 
     Samples compatible with sensors described herein can be in any physical state, such as gas, liquid or solid, the physical state often dependent on or related to a particular application. Therefore, there are many possible implementations or configuration of a sample delivery system. Several exemplary delivery systems are described herein, each of which can be made compatible with any of the sensor designs described herein. For example, samples can be delivered via a sub-μL volume pipette controlled by fine step motor. Also, a microfluidic channel can be designed to deliver a sample in the fluid or gas state. The microfluidic channel approach also can provide an enclosed environment for the sample. In another sample delivery approach, optical tweezers can be used to deliver individual magnetic particles. Optical tweezers are well suited for deliver where a very small amount of sample needs to be delivered with high accuracy. 
     Part IV, Exemplary Implementations of the Sensor System 
     In this part, exemplary embodiments of system level sensor implementations are described. 
     Differential Sensing System: 
       FIG. 9  shows a block diagram of one exemplary differential sensor system. Block  715  and block  717  of  FIG. 9  represent two sensors of any suitable type, including those sensor types described above. Preferably, block  715  (“sensor A”) and block  717  (“sensor B”) are the same type of sensor. In general, they should share the same operation environment if needed and possible. For example the environment can include the electrical environment such as the supply, bias, and ground. Another example of the environment can be the thermal environment, such as the temperature. Also, in general, the sensors of block  715  and block  717  should be situated physically close together, preferably as close together as practical to improve matching between the two sensors for similar sensor response. Black arrows  716  and  718  represent the corresponding delivery systems of the samples to bock  715  and block  717 . 
     In one embodiment, sensor A (block  715 ) can be used as a main sensor while sensor B (block  717 ) can be used as a reference sensor. Using a main sensor and a reference sensor, as an example, the differential sensing can be performed as follows: Structure  716  delivers the sample_ 1  as the target sample, while structure  18  delivers the sample_ 2  (or empty sample) as the control sample. The response of sensor A and sensor B can be recorded separately. Then the differences between the two sensor response signals can be calculated to produce the differential sensing results. 
     By having differential sensing, any common-mode noise/offset for the differential sensing sensor pair can be suppressed, as long as there is good matching between the two sensors. Exemplary non-ideal common-mode effects and offsets that can be removed by differential sensing include, by way of example, drift as a function of temperature, power supply noise, and other common-mode artifacts. Note that, in a generalized differential scheme, there can be multiple main sensors and/or multiple reference sensors. Also, the roles of main sensor and reference sensor can be interchanged. This means, for example, take sensor A as the main sensor and sensor B as the reference and do the sensing procedures described above to produce a differential signal result  1 . Then, take sensor B as the main sensor and sensor A as the reference to get differential signal result  2 . The two results can be processed, such as by averaging to further suppress noise and/or offsets in sensor response. The controls can also have samples with/without magnetic properties or with/without magnetic particles. The differential sensing scheme senses the difference between the target samples and the control (reference) samples. 
       FIG. 10  shows a schematic diagram of a differential sensing scheme implemented as an oscillator based measurement and a time line illustrating how common mode noise can be suppressed through differential sensing. This differential sensing method (taking the difference on the output from a sensing sensor and a reference sensor) can be used to remove the common-mode noise/drifting for the sensor system, thus yielding an overall lower noise floor for better sensor sensitivity. 
     Sensor Arrays 
     Two or more of any of the sensors as described above can be extended into a sensor array structure.  FIG. 11  shows a block diagram of a two dimensional (2D) M×N dimensional sensor array. Each block of  FIG. 11  represents a sensor. The 2D array depicted in  FIG. 11  can be either reduced to a 1D array or extended to a 3D array, possibly limited only by particular fabrication and packaging technologies. A sensor array can be fabricated either on a single chip, multiple chips, in a complete discrete basis, or by any combination thereof. One advantage of a sensor array is that it can improve sensing throughput by a significant factor. The following examples illustrate two exemplary sensor array applications: 
     In a first example application, an incoming sample can have multiple targets labeled with magnetic particles. If Sensor ij , for example, has a specific sensitivity to target T ij , which means Sensor ij  only works when T ij  is in the solution, the M×N sensor array can detect M×N targets simultaneously. Sensor specificity can be achieved, for example, through standard affinity binding-washing procedures. 
     In a second example application, multiple samples can be input to a sensor array. In this exemplary case, the delivery structure is designed to access individual sensor elements independently. Therefore, with an M×N sensor array, M×N types of sample can be sensed simultaneously. 
     Also, using arrays having a combination of elements, some of which operate according to the first example application, and some of which operate according to the second example application, as described above, a hybrid array of the two array types can be made that can sense multiple samples with multiple targets at the same time. Thus, it can be seen that the aforementioned variations in sensor implementation are not mutually exclusive of each other. Based on a specific application, various sensor array types can be combined to form an optimized sensor system. Such array sensor systems are well suited for use in a low cost fully integrated portable battery powered lab-on-a-chip (LOC) type system. 
     IMPLEMENTATION EXAMPLE 
     Two experiments using laboratory test setups of two embodiments of a magnetic particle sensor based on the sensing mechanisms as described above are now described. An LC resonator (as described in section III above) was used as the sensor core. A low noise Colpitts oscillator was built based on a resonator, and the sensor was powered at 4.5V by 3 AA batteries. Thin-film technology was adopted to fabricate both the circuit board and the inductors.  FIG. 12  shows an illustration of the sensor circuitry.  FIG. 13  shows a schematic diagram of the magnetic particle sensor based on Colpitts LC oscillator. Inductor L s  is used as the sensing inductor. Together with C 1  and C 2 , inductor L s  forms the resonator of block  507  in  FIG. 5 . Transistor T 1  forms the oscillator core as block  508  in  FIG. 5 . R 1 , R 2 , R e , Lc are present for biasing purposes, while C 3  and C 4  are used for coupling and bypass purposes. An off-the shelf model 53150A HP frequency counter was used for frequency counting as block  509  in  FIG. 5 . The magnetic particles used in this experiment were DynaBeads® MyOne particles having different concentrations were used as the magnetic particles for the experiment. DynaBeads® Beads are available from the Invitrogen Corporation of Carlsbad, Calif. De-ionized (“DI”) water was used to dilute the magnetic particle solution and to wash the surface of the sensor. Two sets of experiments were performed as described below. 
     Experiment 1 
     Experiment 1 demonstrated magnetic particle sensing without a micro-fluidic channel. The objective of this experiment was to test the functionality of the sensor in an open environment condition (without a fluidic channel).  FIG. 14  shows a flowchart of the experimental procedures used to perform experiment 1. The surface temperature of the chip rises during operation of the sensor. This thermal effect, together with the open environment, induces fast vaporization of the DI water in the magnetic particle solution. Note that the baseline measurement (frequency counting on f 2 ) and the target measurement (frequency counting on f 1 ) together with its sample delivery can be interchanged in time. Therefore, the recorded frequency f 1  and f 2  corresponds to an oscillation frequency of dried beads in the inductors and an oscillation frequency with a dried inductor surface, after achieving the thermal steady-state. The sensitivity can be further defined by Eq. 4: 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       f 
                     
                     f 
                   
                   = 
                   
                     
                       
                         f 
                         2 
                       
                       - 
                       
                         f 
                         1 
                       
                     
                     
                       f 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
       FIG. 15A  and  FIG. 15B  show illustrations of the sensor inductor with delivered magnetic beads, which appeared in the experiment as a brown layer (color not shown in  FIG. 15A  and  FIG. 15B ). Note that the inductor was coated with a parylene layer (thickness of about 3 μm) for electrical isolation purposes.  FIG. 16  shows a graph of frequency in Hz versus time in seconds for a measurement cycle. Note that the DI water vaporized within about 200 s.  FIG. 17  shows the corresponding frequency measurement results respectively plotted as Δf/f per bead versus measurement number for 22 measurements. The average Δf/f per bead is 3.7*10 −3  ppm or 3.7 ppb. In comparison, a Maxwell simulation of Δf/f per bead for this setup was 4.0*10 −3  ppm. Therefore, it can be seen that the measurement results were in close agreement with the simulated value. 
     Experiment 2 
     Experiment 2 demonstrated magnetic particle sensing using a micro-fluidic channel. The objective of this experiment was to test the functionality of the bio-sensor in an enclosed aqueous condition. A microfluidic channel together with pneumatic control valves were fabricated in a poly-dimethylsiloxane (PDMS) material. The microfluidic channel and pneumatic control valves were used to deliver magnetic particle samples to the sensor, as well as to form a sensing chamber. The sensing chamber substantially prevented vaporization of the DI water during detection.  FIG. 18  shows a flowchart of the experimental procedures used to perform experiment 2. Note that the baseline measurement (frequency counting on f 4 ) and the target measurement (frequency counting on f 3 ) together with its sample delivery can be interchanged in time. As in experiment 1, the surface temperature of the chip rises during operation of the sensor. 
     During an operational mode of the sensor, the DI water was preserved in the magnetic particle solution. Therefore, the recorded frequencies f 3  and f 4  correspond respectively to an oscillation frequency with the bead solution on the inductors and an oscillation frequency with only DI water. Then the sensitivity can be defined as follows: 
     
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       f 
                     
                     f 
                   
                   = 
                   
                     
                       
                         f 
                         4 
                       
                       - 
                       
                         f 
                         3 
                       
                     
                     
                       f 
                       4 
                     
                   
                 
               
               
                 
                   ( 
                   
                     Eq 
                     . 
                     
                         
                     
                     ⁢ 
                     5 
                   
                   ) 
                 
               
             
           
         
       
     
       FIG. 19A  shows an illustration of the overall sensor setup.  FIG. 19B  shows an illustration emphasizing the sensor. Circle  710  highlights the PDMS structure which includes the microfluidic channel and pneumatic control valves. Circle  711  highlights a pressure sensor used to monitor the air pressure in the pneumatic control valves.  FIG. 20A  through  FIG. 20C  illustrate the operation of the microfluidic channel and the pneumatic control valves in more detail.  FIG. 20A  shows the channel state when the valve is open.  FIG. 20B  shows the channel state when the valve is closed. The solution contained a green dye in both of the aforementioned two cases. In  FIG. 20C , which shows a more detailed view of the PDMS structure, arrows  810  indicate the fluidic channel and the arrows  811  indicate the control path. 
       FIG. 21  shows a sensor inductor with delivered magnetic beads. Here, spiral  910  is the sensing inductor and darkened area  912  centered near the middle shows an aggregation of the magnetic beads. 
       FIG. 22  shows the measurement results plotted as Δf/f per bead versus measurement number as sensitivity data for sensing inside the microfluidic channel (for 8 measurements). The average Δf/f per bead was found to be 5.2*10 −3  ppm. The simulated Δf/f per bead was calculated as 4.8*10 −3  ppm. The slightly higher average Δf/f from the measurement is thought to be caused primarily by the non-equal distribution of the magnetic beads in the chamber on top of the sensing inductor. However, the two results still match relatively well. 
     Theoretical Discussion 
     Although the theoretical description given herein is thought to be correct, the operation of the systems and devices described and claimed herein does not depend upon the accuracy or validity of the theoretical description, but rather on the ability to make and use the systems and devices according to the methods and procedures described. That is, later theoretical developments that may explain the observed results on a basis different from the theory presented herein will not detract from the inventions described herein. 
     Theoretical Analysis of Sensing Methods 
     We now provide a theoretical basis for the sensing methods described herein. With a quasi-static assumption, electrical current I conduction in the sensor structure generates a magnetic field H ext  at coordinates (x,y,z) according to Biot-Savart Law of Eq. 6: 
     
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         ⇀ 
                       
                       ext 
                     
                     ⁡ 
                     
                       ( 
                       
                         x 
                         , 
                         y 
                         , 
                         z 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       I 
                       
                         4 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       
                         ∮ 
                         
                           C 
                           ′ 
                         
                       
                       ⁢ 
                       
                         
                           
                             ⅆ 
                             
                               l 
                               ′ 
                             
                           
                           × 
                           
                             R 
                             ⇀ 
                           
                         
                         
                           R 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   6 
                 
               
             
           
         
       
     
     Other electromagnetic formula can be used to calculate the H ext  at coordinates (x,y,z) if the quasi-static assumption is not valid. 
     This magnetic field polarizes one or more magnetic particles present in the magnetic field.  FIG. 23  shows an exemplary structure that generates an induced magnetization M causing a polarization of a magnetic particle. The exemplary magnetic field generating sensor structure of  FIG. 23  uses a spiral 6-turn symmetric inductor. The black arrows show a current I. The sphere of  FIG. 23  represents the magnetic particle. 
     Most commercially available magnetic particles, such as micro/nano magnetic beads, include magnetic nanoparticles dispersed in a nonmagnetic matrix. The magnetization M of such micro/nano magnetic beads can be expressed in a Langevin function form as shown in Eq. 7: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       ⇀ 
                     
                     ⁡ 
                     
                       ( 
                       
                         H 
                         ⇀ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         M 
                         sat 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             coth 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   
                                     μ 
                                     0 
                                   
                                   ⁢ 
                                   
                                     m 
                                     p 
                                   
                                   ⁢ 
                                   H 
                                 
                                 kT 
                               
                               ) 
                             
                           
                           - 
                           
                             ( 
                             
                               kT 
                               
                                 
                                   μ 
                                   0 
                                 
                                 ⁢ 
                                 
                                   m 
                                   p 
                                 
                                 ⁢ 
                                 H 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                     
                     · 
                     
                       
                         H 
                         ⇀ 
                       
                       H 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
           
         
       
     
     {right arrow over (H)} is the total magnetic field inside of the bead, instead of the external excitation magnetic field {right arrow over (H)} ext . At high temperature or low excitation magnetic fields (Curie regime), the Langevin function can be approximated and reduced to a classical formula for magnetization. This classical formula for magnetization, as shown in Eq. 8, can be used to determine an effective susceptibility (χeff) of the magnetic particle from experimental data. 
     
       
         
           
             
               
                 
                   
                     
                       
                         M 
                         ⇀ 
                       
                       ⁡ 
                       
                         ( 
                         
                           H 
                           ⇀ 
                         
                         ) 
                       
                     
                     ≈ 
                     
                       
                         
                           
                             M 
                             sat 
                           
                           ⁢ 
                           
                             μ 
                             0 
                           
                           ⁢ 
                           
                             m 
                             p 
                           
                         
                         
                           3 
                           ⁢ 
                           k 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           T 
                         
                       
                       ⁢ 
                       
                         H 
                         ⇀ 
                       
                     
                   
                   = 
                   
                     
                       χ 
                       eff 
                     
                     ⁢ 
                     
                       H 
                       ⇀ 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     For DynaBead® MyOne, one type of magnetic bead often used in immunoassay, the χeff is around 1.4, which by use of Eq. 9, results an effective permeability (μeff) given by:
 
μ eff =χ eff +1  Eq. 9
 
     In excitation/sensing schemes that use an open magnetic loop, demagnetization effects should also be taken into consideration. By applying the demagnetization factor {right arrow over (D)}, which is often in a 3×3 tensor format, the magnetic field inside of the bead and the externally applied magnetic field, can be related as shown in Eq. 10:
 
 {right arrow over (H)}={right arrow over (H)}   ext   −{right arrow over (D)}·{right arrow over (M)}   Eq. 10
 
     In general, the demagnetization factor {right arrow over (D)} depends on the geometry of the magnetic material and the position at which the magnetic field is evaluated. With an assumption of spherical shape of the magnetic bead, and taking the magnetic field {right arrow over (H)} at the center of the sphere as the average magnetic field inside of the bead, {right arrow over (D)} can be reduced to the following diagonal matrix: 
               [           1   /   3         0       0           0         1   /   3         0           0       0         1   /   3           ]     .         
Therefore, the coordinate system can be chosen such that the X axis is aligned with the external magnetic field {right arrow over (H)} ext . The apparent magnetic permeability μapp can be defined as χapp+1. Combining Eq. 3 and Eq. 5 yields the apparent magnetic susceptibility χapp as shown below in Eq. 11:
 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       ⇀ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           H 
                           ⇀ 
                         
                         ext 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           χ 
                           eff 
                         
                         
                           1 
                           + 
                           
                             
                               D 
                               xx 
                             
                             ⁢ 
                             
                               χ 
                               eff 
                             
                           
                         
                       
                       ⁢ 
                       
                         
                           H 
                           ⇀ 
                         
                         ext 
                       
                     
                     = 
                     
                       
                         
                           
                             χ 
                             eff 
                           
                           
                             1 
                             + 
                             
                               
                                 1 
                                 3 
                               
                               ⁢ 
                               
                                 χ 
                                 eff 
                               
                             
                           
                         
                         ⁢ 
                         
                           
                             H 
                             ⇀ 
                           
                           ext 
                         
                       
                       = 
                       
                         
                           χ 
                           app 
                         
                         ⁢ 
                         
                           
                             H 
                             ⇀ 
                           
                           ext 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
           
         
       
     
     Eq. 11 yields two important results. First χ app  is always smaller than χ eff . Second, χ app  has its maximum value of 3 when χ eff  approaches infinity. These results show that that even if the magnetic bead is made of ferromagnetic material with high susceptibility (a factor of hundreds or thousands), χ app  still remains small, which leads to small magnetic signal. This is actually the fundamental reason why magnetic bead sensing is challenging. 
     The total magnetic energy in the space can be calculated with or without the presence of magnetic beads: 
     
       
         
           
             
               
                 
                   
                     W 
                     m 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             ∫ 
                             ∫ 
                           
                         
                         V 
                       
                       ⁢ 
                       
                         
                           H 
                           ⇀ 
                         
                         · 
                         
                           B 
                           ⇀ 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         v 
                       
                     
                     = 
                     
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               ∫ 
                               ∫ 
                             
                           
                           V 
                         
                         ⁢ 
                         μ 
                         ⁢ 
                         
                           
                              
                             H 
                              
                           
                           2 
                         
                         ⁢ 
                         
                           ⅆ 
                           v 
                         
                       
                       = 
                       
                         
                           1 
                           2 
                         
                         ⁢ 
                         
                           
                             ∫ 
                             
                               ∫ 
                               ∫ 
                             
                           
                           V 
                         
                         ⁢ 
                         
                           μ 
                           app 
                         
                         ⁢ 
                         
                           
                              
                             
                               H 
                               ext 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           ⅆ 
                           v 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     If the inductance value is defined to quantify the total magnetic energy in the space with a certain excitation electrical current (Eq. 13), the presence of magnetic beads can directly yield an effective inductance change ΔLeff (Eq. 14): 
     
       
         
           
             
               
                 
                   
                     W 
                     m 
                   
                   = 
                   
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         L 
                         eff 
                       
                       ⁢ 
                       
                         I 
                         2 
                       
                     
                     = 
                     
                       
                         1 
                         2 
                       
                       ⁢ 
                       
                         
                           ∫ 
                           
                             ∫ 
                             ∫ 
                           
                         
                         V 
                       
                       ⁢ 
                       
                         
                           H 
                           ⇀ 
                         
                         · 
                         
                           B 
                           ⇀ 
                         
                       
                       ⁢ 
                       
                         ⅆ 
                         v 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   13 
                 
               
             
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       L 
                       eff 
                       
                         S 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         21 
                       
                     
                   
                   = 
                   
                     
                       
                         L 
                         eff 
                         
                           S 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                       - 
                       
                         L 
                         eff 
                         
                           S 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                     = 
                     
                       
                         2 
                         ⁢ 
                         
                           ( 
                           
                             
                               W 
                               m 
                               
                                 S 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             - 
                             
                               W 
                               m 
                               
                                 S 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                           ) 
                         
                       
                       
                         I 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   14 
                 
               
             
           
         
       
     
     S 1  and S 2  above denote two states with different magnetic bead presences. Therefore, as shown above in appendix I, we have derived a way to quantitatively detect the presence of magnetic beads by effective inductance change. 
     Derivation of Approximate Closed-Form Solution 
     We now present an approximate closed-form solution to quantify the inductance change in magnetic particle sensors described above. 
     The intrinsic sensitivity of a magnetic particle sensor can be defined as ΔL/L per bead. Eq. 14 above showed than ΔL/L can be calculated by evaluating the total magnetic energy change in the space. However, direct application of Eq. 13 demands calculating B and H fields in space together with volume integration, which is less suitable for an analytical derivation. In the description which follows, it is shown that a close-form approximate solution can still be obtained by defining mutual inductance between the bead and the inductor coil. This analytical solution can serve as a guideline for further inductor optimization using EM software. 
     The setup is as follows: An arbitrary shaped inductor L ind  conducts a DC or AC current of I ind  placed at the origin. Assume there is only one magnetic bead existing in the space at position (x,y,z). The excitation field, H ext , at (x,y,z) can then be calculated based on Eq. 6 above. Also, assuming the magnetic bead is small enough to homogeneously experience the H ext  field, Eq. 11 (above) gives the magnetization inside of the bead as follows: 
     
       
         
           
             
               
                 
                   
                     
                       M 
                       ⇀ 
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           H 
                           ⇀ 
                         
                         ext 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         χ 
                         eff 
                       
                       
                         1 
                         + 
                         
                           
                             D 
                             xx 
                           
                           ⁢ 
                           
                             χ 
                             eff 
                           
                         
                       
                     
                     ⁢ 
                     
                       
                         H 
                         ⇀ 
                       
                       ext 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   15 
                 
               
             
           
         
       
     
     Now, approximate the magnetic particle to be a cylindrical shape with a cross-sectional area of S and height of h. The magnetic field distribution due to the induced magnetization in Eq. 15 can be viewed equivalently as generated by a volume current density J m  and a surface current density J ms  as shown in Eq. 16: 
     
       
         
           
             
               
                 
                   
                     A 
                     ⇀ 
                   
                   = 
                   
                     
                       
                         
                           
                             μ 
                             
                               0 
                               ⁢ 
                               
                                   
                               
                             
                           
                           
                             4 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             V 
                           
                           ⁢ 
                           
                             
                               
                                 ∇ 
                                 
                                   × 
                                   
                                     M 
                                     ⇀ 
                                   
                                 
                               
                               R 
                             
                             ⁢ 
                             
                               ⅆ 
                               v 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             μ 
                             0 
                           
                           
                             4 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             ∮ 
                             s 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   M 
                                   ⇀ 
                                 
                                 × 
                                 
                                   a 
                                   n 
                                 
                               
                               R 
                             
                             ⁢ 
                             
                               ⅆ 
                               s 
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             μ 
                             0 
                           
                           
                             4 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             ∫ 
                             V 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   J 
                                   ⇀ 
                                 
                                 m 
                               
                               R 
                             
                             ⁢ 
                             
                               ⅆ 
                               v 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             μ 
                             0 
                           
                           
                             4 
                             ⁢ 
                             π 
                           
                         
                         ⁢ 
                         
                           
                             ∮ 
                             s 
                           
                           ⁢ 
                           
                             
                               
                                 
                                   J 
                                   ⇀ 
                                 
                                 
                                   m 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   s 
                                 
                               
                               R 
                             
                             ⁢ 
                             
                               ⅆ 
                               s 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   16 
                 
               
             
           
         
       
     
     With the homogeneous assumption of M, the volume current density J m  goes to zero, and the surface current density J ms  is M circulating on the lateral surface of the cylinder. Therefore, the magnetic field induced by the magnetization is equivalent to a small coil conducting a current I particle  of Mh. Therefore, we have a magnetic system with two coils, the original excitation coil C 1  of the inductor and the artificial coil of the magnetic particle C 2 . 
     The magnetic flux increase for the excitation coil is given by
 
Δφ= M   C1,C2   I   particle   =M   C2,C1   I   particle ,  Eq. 17
 
where M C2,C1  is the mutual inductance from coil C 1  to coil C 2  and M C1,C2  is the mutual inductance from coil C 2  to coil C 1 . Due to the reciprocity, M C2,C1  should equal M C1,C2 .
 
     This mutual inductance can be directly calculated as 
                       M       C   ⁢           ⁢   1     ,     C   ⁢           ⁢   2         =       M       C   ⁢           ⁢   2     ,     C   ⁢           ⁢   1         =         ∫     ∫         B   ext     →     ·     ⅆ     S   →             I     =              B   ext          ⁢   S     I           ,           Eq   .           ⁢   18               
Where {right arrow over (B ext )} is the B field generated from the excitation coil C 1  and S the cross-sectional area of the magnetic particle and I is the I ind . By considering all the factors, the relative inductance increase due to the presence of one magnetic particle can be expressed as
 
     
       
         
           
             
               
                 
                   sensitivity 
                   = 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           L 
                           eff 
                         
                       
                       
                         L 
                         ind 
                       
                     
                     = 
                     
                       
                         
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           φ 
                         
                         
                           
                             I 
                             ind 
                           
                           ⁢ 
                           
                             L 
                             ind 
                           
                         
                       
                       = 
                       
                         
                           
                             χ 
                             m 
                           
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     D 
                                     xx 
                                   
                                   ⁢ 
                                   
                                     χ 
                                     m 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               μ 
                               0 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                                
                               
                                 B 
                                 ext 
                               
                                
                             
                             2 
                           
                           
                             
                               I 
                               ind 
                               2 
                             
                             ⁢ 
                             
                               L 
                               ind 
                             
                           
                         
                         ⁢ 
                         
                           Volume 
                           particle 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   19 
                 
               
             
           
         
       
     
     The result show in Eq. 19 can be extended to a case where multiple particles are present in the space, where index i indicates the i th  magnetic particle. In Eq. 20 which follows below, it is assumed that the particles are relatively sparsely spaced so that the induced magnetization of any one particle does not affect the polarization of other particles. 
     
       
         
           
             
               
                 
                   sensitivity 
                   = 
                   
                     
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           L 
                           eff 
                           
                             S 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             21 
                           
                         
                       
                       
                         L 
                         ind 
                       
                     
                     = 
                     
                       
                         ∑ 
                         i 
                       
                       ⁢ 
                       
                         
                           
                             χ 
                             m 
                           
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 
                                   
                                     D 
                                     xx 
                                   
                                   ⁢ 
                                   
                                     χ 
                                     m 
                                   
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               μ 
                               0 
                             
                           
                         
                         ⁢ 
                         
                           
                             
                                
                               
                                 B 
                                 
                                   ext 
                                   , 
                                   i 
                                 
                               
                                
                             
                             2 
                           
                           
                             
                               I 
                               ind 
                               2 
                             
                             ⁢ 
                             
                               L 
                               ind 
                             
                           
                         
                         ⁢ 
                         
                           Volume 
                           
                             particle 
                             , 
                             i 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   20 
                 
               
             
           
         
       
     
     Eq. 19 indicates that optimizing the sensitivity of an inductor can be achieved by maximizing 
                      B   ext          2       I   ind   2           
at the particle and minimizing the self inductance L ind . This optimization of sensitivity can also be addressed as maximizing the ratio between the increased magnetic energy due to the magnetic particle and the magnetic energy of the inductor itself.
 
     Ansoft Maxwell V11, a 3D EM simulator (available from Ansoft, LLC, 225 West Station Square Drive, Suite 200, Pittsburgh, Pa. 15219) was used to numerically simulate the magnetic bead sensing/excitation process through calculation. The results of an exemplary simulation are shown in  FIG. 24 . The curve marked UL shows the upper limit of the sensitivity, while the curve marked LL shows the lower limit of the sensitivity. Two curves marked TL were used to indicate trendlines. It can be seen from the graph of  FIG. 24 , that the quantity ΔL/L per bead has an inverse relationship with the cube of the inductor radius, indicating that the smaller the inductor size is, the higher the sensitivity. This inverse-cubic relationship can lead to a tradeoff of sensitivity and quality factor of the inductor at a certain frequency. 
     While the present invention has been particularly shown and described with reference to the structure and methods disclosed herein and as illustrated in the drawings, it is not confined to the details set forth and this invention is intended to cover any modifications and changes as may come within the scope and spirit of the following claims.