{'fraction_non_alphanumeric': 0.03229453748851671, 'fraction_numerical': 0.012013285280192212, 'mean_word_length': 4.631516116195782, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Magnetophoretic separation of blood cells at the microscale

We present a method and model for the direct and continuous separation of red and white blood cells in plasma. The method is implemented at the microscale using a microfluidic system that consists of an array of integrated soft-magnetic elements embedded beneath a microfluidic channel. The microsystem is passive, and is activated via application of a bias field that magnetizes the elements. Once magnetized, the elements produce a nonuniform magnetic field distribution in the microchannel, which gives rise to a force on blood cells as they pass through the microsystem. In whole blood, white blood cells behave as diamagnetic microparticles while red blood cells exhibit diamagnetic or paramagnetic behavior depending on the oxygenation of their hemoglobin. We develop a mathematical model for predicting the motion of blood cells in the microsystem that takes into account the dominant magnetic, fluidic and buoyant forces on the cells. We use the model to study red/white blood cell transport, and our analysis indicates that the microsystem is capable of rapid and efficient red/white blood cell separation.


I. Introduction
Microfluidic systems with magnetic functionality are finding increasing use in bioapplications that involve the immobilization and separation of biomaterials such as cells, proteins, antigens and DNA. 1 In most such applications, the biomaterial needs to be magnetically "labeled" with magnetic micro/nano-particles in order to provide sufficient coupling to an applied field to enable manipulation. Direct manipulation (without labeling) is not usually practical as the intrinsic magnetic susceptibility of most biomaterials is so small that they do not efficiently couple to a field. However, there are exceptions such as blood cells, which can be manipulated with practical magnetic fields. [2][3][4][5][6][7][8] Furthermore, in venous (deoxygenated) blood, white blood cells (WBCs) behave as parametric microparticles, while deoxygenated red blood cells (RBCs) exhibit diamagnetic behavior. [9][10] Thus, these two types of cells move in opposite directions in an applied field, and therefore in principle they can be magnetically separated from one another.
To date, relatively few systems have been developed that magnetically separate unlabeled blood cells, despite a substantial need for fast, accurate, and inexpensive blood cell analysis. 11 One such system has been developed by Takayasu et al. 10 In this system, the blood flows continuously through a flexible rectangular conduit (3.5 mm x 1 mm) that is wrapped in the form of a helix with an overall diameter and length of 80 mm and 100 mm, respectively. The magnetic force is provided by a magnetic wire (1mm diameter) that is wrapped along the length of the conduit in close proximity to it. The wire is magnetized by placing the conduit/wire assembly along the axis of a superconducting solenoidal electromagnet. This system separates deoxygenated blood into three outlets that contain WBC-rich plasma, deoxygenated RBC-rich plasma and cell-depleted plasma, respectively.
A much smaller microscale system has been developed by Han and Frazier. 12 In this system there are two identical and parallel flow channels, which are 30 mm long and have a height and width of 50 µm and 150 µm, respectively. The magnetic force is provided by an integrated rectangular (50 µm x 120 µm) soft-magnetic bar that runs the length of the channels (between them), and is magnetized by a permanent magnet. This system separates deoxygenated blood into three outlets; two that contain deoxygenated RBC-rich plasma, and a third that contains WBC-rich plasma.
In this paper we introduce a magnetically functional microfluidic system for the continuous and direct separation of blood cells from plasma. The fluidic structure in the microsystem consists of an inlet that feeds a single microfluidic channel with three outlets (Fig. 1). The magnetic functionality is provided by an array of integrated softmagnetic elements, which are embedded transverse and adjacent to the microchannel. A bias field is used to magnetize the elements, and once magnetized these elements produce a magnetic force on blood cells as they flow through the microchannel. Specifically, the magnetized elements repel WBCs and attract deoxygenated RBCs (Fig. 1c). The microsystem is oriented with the microchannel (blood flow) aligned with the gravitational force, and the cells are separated perpendicular to the flow. The reason for this is because the fluidic and gravitational forces are greater than the magnetic force over extended regions of the flow channel, and therefore they need to be orthogonal to the direction of cell separation so as not to resist it. In this paper we study the transport of blood cells through the microsystem. We model blood cell separation taking into account magnetic, fluidic and gravitational forces. Our analysis indicates that the microsystem is capable of separating WBC-rich plasma, deoxygenated RBC-rich plasma and celldepleted plasma into respective outlets.
The cell/plasma separation method presented here has advantages over competing techniques such as centrifuging, or magnetic activated cell sorting (MACS), which involves magnetic labeling. Specifically, small samples of blood can be processed within minutes with the cells in their native state. Furthermore, the microsystem is compact and efficient; i.e., energy is needed to sustain blood flow through the microchannel, but the cell/plasma separation itself consumes no energy. Thus, the microsystem is highly portable and holds potential for low cost point-of-service medical diagnostic applications.
It is also ideal for filtering weakly magnetic micro/nano-particles from a transport fluid.
Lastly, we note that this paper represents an extension of earlier work. 13 Specifically, the microsystem described here is an enhancement of one introduced in a previous paper, which was limited to the separation of WBCs from deoxygenated RBCs, with no ability to extract cell-depleted plasma. Similarly, a model for predicting cell transport through the microsystem was developed earlier, but its application was limited to a proof-of-concept study. Here, we use the model for parametric analysis to study the cell separation performance of the microsystem as a function of its magnetic structure.

II. Theory
In this section we briefly review a model for predicting blood cell transport through the microsystem; a detailed description of the model can be found in a previous paper. 13 The model takes into account the dominant forces on the cells and is based on the following simplifying assumptions: the blood cells are treated as noninteracting rigid microspheres; the blood flow in the microchannel is laminar; the impact of particle motion on the flow field is ignored; a fixed value is used for the blood viscosity (complex blood rheology is ignored); and wall effects are ignored when determining the fluidic force. While this model is not suitable for a rigorous analysis of cell transport, we use it to assess the feasibility of cell separation based on the interplay of the dominant forces on the cells. A rigorous analysis of cell transport would require a complicated, fully-coupled structural/fluidic numerical analysis that would account for the interactions of numerous structurally deformable cells, and this is beyond the scope of the present work.
We predict the motion of blood cells using Newton's law, (1) where c m and c v are the mass and velocity of the cell, and m F , f F , and g F are the magnetic, fluidic and gravitational force (including buoyancy), respectively. The magnetic force is obtained using an "effective" dipole moment approach and is given by [14][15][16] where c,hyd R is the effective hydrodynamic radius of the cell, and η and f v are the viscosity and the velocity of the fluid, respectively. The gravitational force is given by where c ρ and f ρ are the densities of the cell and fluid, respectively ( 2 g 9.8 m/s = ). Note that gravity acts in the + x direction, parallel to the flow (see Figs. 1c and 2b).

A The Magnetic force
The magnetic force depends on the field in the microchannel and the coupling of the cells to this field. The magnetic field is a superposition of the bias field bias H and the field e H due to the array of magnetized elements, These fields are not independent as bias H induces e H . Thus, to predict the total field in the microchannel we need models for the bias field, the magnetization of the elements, and the field of the magnetized elements. Our model for the bias field is based on an analytical analysis of the field due to a rectangular block magnet 17 where d N is the demagnetization factor of the element, which is geometry dependent, and es M is its saturation magnetization. The demagnetization factor for a highly permeable ( e χ ≈ ∞ ) long rectangular element of width 2w and height 2h that is magnetized parallel to its height can be obtained using analytical formulas (see Fig. 2b).
Specifically, both the demagnetization factor N d and the aspect ratio of the element h p w = can be defined parametrically as a function of a variable k over the domain 0 1 k < < as follows: 19

and ( )
K k and E( ) k are the complete elliptic integrals of the first and second kind, respectively, To determine the magnetization of the elements, we first use Eqs. (7) and (8) and In these equations, e M is determined using Eq. (6).
The field due to the entire array of elements is obtained via superposition. Let N e denote the number of elements, and let the first element be centered with respect to the origin in the x-y plane. All other elements are positioned along the x-axis as shown in Fig.   2b. We identify the elements using the index n = (0,1,2,3,4, …, N e -1). The field components due to the first element (n = 0) are given by Eqs. (10), (11). The n'th element is centered at n x s = , and its field and force components can be written in terms of the 0'th components as follows: ( 1, 2,3, ) .
The total field of the array is obtained by summing the contributions from all the elements, The force components are given by 13,16,17 ( ) and ( ) In Eqs. (15) and (16) we have assume that the bias field is constant and in the y-direction.

B. Fluidic force
The fluidic force is predicted using Stokes' law for the drag on a sphere in a viscous fluid. The blood flow in the microchannel is assumed to be laminar, and since the diameters of WBCs and RBCs are small relative to the channel height, we assume that the fluid velocity is relatively constant across the cells. We use Eq. (3) to estimate the drag force at a given time using the particle velocity at that time, and the fluid velocity at the position of the cell at that time. Let c h and c w denote the half-height and half-width of its rectangular cross section (Fig. 2a). We assume fully developed laminar flow parallel to the x-axis and obtain where f v is the average flow velocity in the channel. 16,17 Note that the distance y in Eq.
(17) is taken with respect the center of the magnetic elements (Fig. 2b). We substitute Eq.

(17) into Eq. (3) and obtain the fluidic force components
and fy , c,y We use these in the equations of motion below.

C. Blood cell properties
The

D. Equations of motion
The equations of motion for blood cell transport through the microsystem can be written in component form by substituting Eqs. (15), (16), (18) and (19)

II. Results
We use Eqs. (20) - (22) to study blood cell transport through the microsystem.
We start with a specific configuration in which the microchannel is 120 µm high  10) and (11), and these elements have an aspect ratio / 1 p h w = = . We use Eqs. (7) and (8) to determine the demagnetization factor for these elements, which turns out to be N 0.456 d = (p 191, It instructive to examine the magnetic force provided by the elements. To this end, we use Eqs. (15) and (16) to predict the magnetic force on deoxygenated RBCs in the microchannel above an isolated array of three magnetized elements with the dimensions and spacing as defined above. The force components mx F and my F across the microchannel are shown in Figs 3a and 3b, respectively. These components are computed over the dotted rectangular region show in Fig. 3c, which extends from 20 µm to 100 µm above the elements, and from -100 µm to the left of the array to + 100 µm to the right of the array. Note that mx F , which is in the flow direction, changes sign across each element; while my F , which is responsible for cell separation, also changes sign across each element, but acts predominantly downward above each element. Thus, the magnetic elements will attract deoxygenated RBCs, but repel WBCs and oxygenated RBCs. Also not that my F is on the order of a pico-Newton near the elements, but decreases rapidly with distance from the elements. By comparison, the gravitational force  Fig. 4a and b, respectively. The trajectory profiles are slightly irregular due to the spatial variation of the magnetic force. 13,16,17 Note that the respectively. Notice that cell separation occurs over a shorter distance (faster) as the size and spacing of the magnetic elements increase relative to the dimensions of the channel.
Lastly, we study WBC separation as a function of WBC susceptibility. Note that in our previous simulations we used 6 9.

IV. Conclusion
We have presented a method and model for the direct and continuous separation of deoxygenated whole blood into three components: deoxygenated RBC-rich plasma, WBC-rich plasma, and cell-depleted plasma. The method can be implemented in a passive magnetophoretic microsystem that enables rapid processing of small blood samples. The microsystem is compact, efficient and can be fabricated using established methods. 12,[22][23][24] As such, it holds substantial potential for point-of-service medical diagnostic applications. The microsystem is also suitable for a broad range of applications that involve the manipulation, immobilization or filtering of weakly magnetic micro or nanoparticles.