Jason Z. Moore AnnArbor,MI48109 Modeling of the Plane ...wumrc.engin.umich.edu/wp-content/uploads/sites/51/... · The curved leading edge of the needle tip acts as the cutting edge,

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Jason Z. MooreDepartment of Mechanical Engineering,

University of Michigan,Ann Arbor, MI 48109

Qinhe ZhangDepartment of Mechanical Engineering,

Shandong University,Jinan, Shandong 250100, China

Carl S. McGillDepartment of Biomedical Engineering,


Haojun ZhengDepartment of Precision Instrument and

Mechanology,Tsinghua University,

Beijing 100084, China

Patrick W. McLaughlinDepartment of Radiation Oncology,


Albert J. ShihDepartment of Mechanical Engineering,

and Department of Biomedical Engineering,University of Michigan,

Ann Arbor, MI 48109

Modeling of the Plane NeedleCutting Edge Rake and InclinationAngles for BiopsyHollow needles are one of the most common medical devices, yet little study has focusedon the needle tip cutting geometry for biopsy, which is a tissue cutting process. Thisresearch develops mathematical models to calculate the inclination and rake anglesalong cutting edges on needle tips generated by planes. Three types of plane needle tips,the one-plane bias bevel, multi-plane symmetrical, and two-plane nonsymmetric needles,are investigated. The models show that the leading tip of a bias bevel needle has aninclination angle of 0 deg, the worst configuration for cutting. Symmetric multiplaneneedles on the other hand have very high inclination angles, 60, 56.3, and 50.8 deg,given a needle formed by two-, three-, and four-plane, respectively, for a bevel angle of30 deg and can assist more effective needle biopsy. The rake angle is at its greatest value(the best configuration for cutting), which equals the 90 deg minus the bevel angle, at theinitial cutting point for the bias bevel needle. Experiments are performed using three 11gauge two-plane symmetric needles with 20, 25, and 30 deg bevel angles on bovine liverand demonstrate that the needle tip geometry affects biopsy performance, where longerbiopsy samples are collected with needles of higher rake and inclination angle.�DOI: 10.1115/1.4002190�

Introduction

Hollow needles are one of the most commonly used medicalevices. A major medical procedure using hollow needles is bi-psy, in which a tissue sample is cut and removed for examinationo diagnose cancer or other diseases. There are a variety of differ-nt types of needle biopsy systems for cutting and retaining theissue samples. One of which being commonly used is the end-cuteedle biopsy, which is the focus of this paper. An end-cut biopsys in general performed in three steps, as shown in Fig. 1. Theeedle and stylet �inner rod of needle� are first advanced to theront of the location where the biopsy is to take place �Fig. 1�a��;he needle is then advanced forward, cutting and trapping theissue sample inside the needle �Fig. 1�b��; and lastly, the needle isemoved from the body with the biopsy sample inside �Fig. 1�c��.ll needle biopsy systems use the sharp edge of the needle tip toerform the primary cutting operation.

Current needle biopsy systems can be improved with longeriopsy sample lengths, decreases in sample fragmentation, andncreases in biopsy performance consistency �1–3�. Studies havehown that the ability of doctors to make an accurate diagnosis ismproved by biopsy samples that are longer and less fragmented1,2�. Current end-cut biopsy needles have been shown to produceery inconsistent performance results, failing to capture any tissue7% of the time �3�.

The needle geometry has a direct affect on biopsy sampleength, as will be shown by biopsy experiments in Sec. 6. Blade

Contributed by the Manufacturing Engineering Division of ASME for publicationn the JOURNAL OF MANUFACTURING SCIENCE AND ENGINEERING. Manuscript receivedugust 18, 2009; final manuscript received July 2, 2010; published online September
0, 2010. Assoc. Editor: Burak Ozdoganlar.
ournal of Manufacturing Science and EngineeringCopyright © 20

ded 20 Sep 2010 to 141.213.232.87. Redistribution subject to ASM

cutting experiments performed by Moore et al. �4� also showedthat cutting edge geometry of a blade plays an important role inthe ability of the tissue to be cut in blade insertion experiments.

Needle tips come in a wide variety of different designs, yetthere is a limited research on needle cutting geometry, the topicthis paper directly addresses. The effects of needle geometry oninsertion forces and needle deviation were analyzed by O’Leary etal. �5� and Podder et al. �6� for solid needles. Blade cutting forceand modeling on tissue have also been studied before �7�. Needlebending analysis, needle steering, and tissue modeling have beenperformed in many studies �8–13�. This survey found no study ondefining the cutting edge geometry of a biopsy needle. A bettergeometrical understanding of needle tips could lead to an opti-mized needle tip geometry, which would increase the biopsysample length, decrease the amount of sample fragmentation, andincrease the consistency of performance, thereby improving thedoctor’s ability to make an accurate diagnosis.

Commercial needle tips are generated by a special burr-freegrinding process. The simplest needle tip geometry is the one-plane bias bevel tip, as shown in Fig. 2. By indexing the needle tipin different orientations during grinding, various needle tip geom-etries can be generated by planes. Figure 3 shows examples oftwo-plane and three-plane needles. A multiplane needle containsplanes oriented symmetrically or nonsymmetrically around theneedle. A symmetric orientation means that the planes are tilted inequal angle relative to the needle axis and placed in equal inter-vals around the circumference of the cylindrical needle, while anonsymmetric design means the planes are oriented in any otherconfiguration. The two- and three-plane symmetric and nonsym-metric needle tips are illustrated in Fig. 3. Three parameters, the
needle radius, the number of planes, and the tilt angle, are re-
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uired to define the symmetric plane needles. Defining the orien-ation for nonsymmetrical needles is more complex due to thedded variables.

The curved leading edge of the needle tip acts as the cuttingdge, as marked in Figs. 2 and 3, during the needle insertion intoissue. For the bias bevel needle, the cutting edge that performshe tissue cutting action during needle insertion transitions fromhe outer edge to the inner edge across the needle tip, as shown inig. 2�b�. In the case of multiplane formed needles, the cuttingdge is usually only the inner edge, as shown in Fig. 3. Theerformance of the needle for biopsy is dependent on the orienta-ion and sharpness of cutting edge.

In machining, the tool geometry is critical to the cutting opera-ion �14�. Two critical parameters at the oblique cutting edge thatefine the cutting geometry are the rake angle � and inclinationngle � �15�. The rake and inclination angle affect the biopsyeedle performance. Understanding the values of � and � is im-ortant for selecting the needle geometry for efficient tissue cut-ing in biopsy. The goal of this research is to develop mathemati-al models to calculate � and � around the needle circumferenceor various needle tips formed by planes. Needle insertion experi-ents are conducted to demonstrate the effects of � and � on

iopsy cutting length. In this study, the model for inclination anglef a bias bevel needle tip is first developed, followed by symmet-ic multiplane needle tips, and then by nonsymmetric two-planeeedle tips. The model for � of the needle is then derived. Theool geometry parameters � and � are determined around theeedle circumference. Specific factors in needle geometry influ-

ig. 1 End-cut biopsy operation performed in three steps: „a…eedle positioned in front of biopsy area, „b… needle advancing,utting and trapping biopsy sample, and „c… needle with biopsyample removed from tissue

ig. 2 One-plane bias bevel needle tip and cutting edge: „a…eneration of needle tip by a plane and „b… cutting edge of biasevel needle

Fig. 3 Multiplane symmetric and nonsymmetric needle tips

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encing the � and � are also examined. Lastly, biopsy performanceexperiments are conducted, which measure the length of tissuebiopsy sample and relates to the needle cutting edge geometry.

2 Mathematical Model for Inclination Angle of PlaneNeedles

2.1 Bias Bevel One-Plane Needle. The bias bevel needle ismade by one-plane being ground at a specified bevel angle, �, asshown in Fig. 4. An xyz axis with the z axis coinciding with theneedle axis and the x axis passing through the lowest point of theneedle tip profile is defined. The outside radius of the needle is rand the radial position of a point A along the needle is defined as�.

When observing the bias bevel needle in the xz plane, as shownin Fig. 4�b�, the needle profile forms a straight line with a slope of−cot � and a z intercept of r cot �, thereby yielding the equation ofz= �r−x� cot �. Putting in terms of �, the parametric equations ofan ellipse to define the shape of the needle tip are

x = r cos �

y = r sin � �1�

z = r�1 − cos ��cot �

The normal vector of the xy plane v= �0,0 ,1� and the tangentvector s, shown at point A on Fig. 4�a�, can be expressed as s= �−r sin � ,r cos � ,r cot � sin ��.

The angle between the Pr plane �plane with normal vector v�and s is the inclination angle, �.

sin � =�s · v��s��v�

=�cot � sin ��

1 + cot2 � sin2 ��2�

The equation for � of one-plane bias bevel needle tip is

��,�� = arcsin�cot � sin ��

1 + cot2 � sin2 ��0 � � � 2�� 3�

This inclination angle model along with all the other inclinationand rake angle equations presented were validated by measuringthe inclination and rake angles directly from SolidWorks models.

2.2 Symmetric Multiplane Needle. A symmetrical multi-plane needle tip is formed by evenly spaced planes being posi-tioned at identical bevel angles ��. Two-plane and three-planeneedle tips are shown in Fig. 5. The xyz axis is defined using the

Fig. 4 Bias bevel needle model: „a… bias needle tip and „b… xzcross section

same rule as in the bias bevel needle. Commercially, the three-

Transactions of the ASME


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lane symmetric needle is called a Franseen needle.The geometry of a symmetric multiplane needle is based on the

ame principles of using the parametric equations for an ellipse,q. �1�. The inclination angle for a symmetrical multiplane needle

s based on Eq. �3� with a phase shift of �i−1��2� / P� applied to, where the P��2� is an integer representing the number oflanes and i�=1,2 , . . . , P� defines which segment of the plane iseing examined. This leads to the general inclination angle equa-ion for the symmetrical multiplane needle being

��,P,�� = arcsin

cot � sin�� − �i − 1�2�

P�

1 + cot2 � sin2�� − �i − 1�2�

P� ,

�

P�2i − 3� � � �

�

P�2i − 1� �4�

2.3 Nonsymmetric Two-Plane Needle. A nonsymmetriceedle formed by two planes, as shown in Fig. 6, can be fullyefined given the offset height between two planes �h�, offsetngle between planes ��, the bevel angle of the two planes rela-

ig. 5 Multiplane symmetrical needle tips: „a… two-plane andb… three-plane

Fig. 6 Nonsymmetric two-plane needle

ournal of Manufacturing Science and Engineering


tive to the z axis ��1 and �2�, and the radius of the needle �r�. Theneedle cutting edge is formed by two ellipses, marked as ellipsesI and II. These two ellipses intersect at two sharp tips. The non-symmetry of the planes makes determining the position of thesharp tip of the needle, i.e., location where the two planes meet,difficult. These two sharp points, marked as A and B in Fig. 6,have � equals �A and �B, respectively.

Based on Eq. �1�, the parametric equations for ellipse I withbevel angle �1 are

x1�� = r cos �

y1�� = r sin � �5�

z1�� = r�1 − cos ��cot �1

The parametric equations for ellipse II can be formed by addinga phase shift � and the change in height h.

x2�� = r cos �

y2�� = r sin � �6�

z2�� = r�1 − cos �� + ��cot �2 + h

The location of where the two planes meet is then found by settingz1��=z2��.

r�1 − cos ��cot �1 = r�1 − cos �� + ��cot �2 + h �7�The MAPLE software is used to solve this equation in terms of

�A and �B, the location where two ellipses meet. The solution of�A and �B is long and presented in Appendix.

The inclination angle can be found at any � along the radius ofthe needle for ellipses I,

��1,�� = arcsin�cot �1 sin ��

1 + cot2 �1 sin2 ��0 � � � �A�

and ��B � � � 2�� 8�and ellipse II,

��2,�,�� = arcsin�cot �2 sin�� + ��

1 + cot2 �2 sin2�� + ��A � � � �B�

�9�

3 Mathematical Model for Rake Angle of Bias BevelNeedles

The oblique cutting configuration is applied to find the rakeangle at a point on a bias bevel needle cutting edge. The rakeangle � is defined as the angle between two planes Pr and A�

measured in plane Pn, as shown in Fig. 7. Pr is the plane at cuttingpoint and parallel to the xy plane, A� is the face plane of theneedle tip surface, and Pn is a plane with a normal vector s, vectortangent to cutting edge �14�. These three planes can be found on abias bevel needle tip, as shown in Fig. 7. The normal vector tothese planes are defined using �, �, and r. The plane Pr has anormal vector v= �0,0 ,1�, the plane A� has a normal vector n�

= �cos � ,0 , sin ��, and the plane Pn has a normal vector s= �−r sin � ,r cos � ,r cot � sin ��.

Vectors a and b mark the intersection of planes PnA� and PnPr,respectively, as illustrated in Fig. 7. Vectors a and b are definedusing the cross products of the normal vectors:

a = s n� = �r cos � sin �,r cot � sin � cos � + r sin � sin �,

− r cos � cos �� 10�

b = s v = �r cos �,r sin �,0� �11�The rake angle for a bias bevel needle is the angle between vector
a and b:
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� = arcosa · b

�a��b�= arccoscos2 � sin2 � + sin2 � �12�

Equation �12� can also be applied to two-plane symmetriceedles because the elliptical cutting edge of a bias bevel needle isdentical to that of a two-plane symmetric needle.

Results of Inclination Angle and Discussions

4.1 Bias Bevel One-Plane Needle. Figure 8 shows the incli-ation angle for three bias bevel needle tips with �=15, 30, 45eg. At �=0 and 180 deg, � is equal to 0 deg, regardless of �. Theip of the bias bevel needle ��=180 deg�, where the needle firstontacts tissue during insertion, has �=0, which is the worst cut-ing configuration for needle insertion. This problem can beolved using the multiplane needle, as will be discussed in Sec..2.

Smaller � creates larger inclination angles where the maximum=� /2−� for a bias bevel needle. A larger � reduces �, therebyreating higher cutting forces �4�.

4.2 Symmetric Multiplane Needle. A multiplane symmetriceedle’s inclination angle can be solved using Eq. �4�. The per-pective view and � of a symmetric multiplane needle tip of P2, 3, 4, �=30 deg, and r=1 mm are shown in Fig. 9. The num-er of needle tip points �sharp points that occur where ellipseseet and equally space around the radius� on the needle tip is

qual to P.Similar to the bias bevel needle, as shown in Fig. 8, the mini-um � is also equal to 0 for all symmetric multiplane needles.nlike the bias bevel needle, the location of �=0 is the last point

nd not the first point on the needle tip cutting edge contacting the

Fig. 7 Definition of rake angle in the bias bevel needle

Fig. 8 Inclination angle of bias bevel needles

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tissue. The number of points where �=0 is equal to the number ofplanes used to construct the needle because �=0 is located at thebottom of the ellipse formed by each plane.

At a tip of the multiplane needle where the needle first contactstissue during insertion ��=90 deg for P=2, �=66.7 deg for P=3, and �=45 deg for P=4�, the � is maximum, which is the bestcutting configuration for needle insertion. This is opposite to thebias bevel needle with �=0 at the needle tip point. The �=60,56.3, 50.8 deg for P=2, 3, 4, respectively, for �=30 deg. For agiven �, increase the P beyond 2 reduces the maximum �, therebymaking the needle less effective at cutting.

The effect of � on � of multiplane needles with P=3 and 4 isshown in Fig. 10. The same trend as seen in the bias bevel needle�Fig. 8� is observed that the increase in � reduces the �.

4.3 Nonsymmetric Two-Plane Needle. The effects of � andh on � of nonsymmetric two-plane needles are discussed in thefollowing sections.

4.3.1 Effect of �. Figure 11 shows the shape of the needlecutting edge and � for �1=30 deg, �2=30 deg, r=1 mm, h=0 mm, and �=45, 90, 180 deg. Increasing � shifts the locationof the maximum and minimum �, which occur on ellipse II. El-lipse I remains fixed in place, therefore, the � from 0��90 deg remains the same as � varies from 45 to 180 deg.Changing � can lower the high � that occurs on ellipse II, thereby

Fig. 9 Cutting edge profile and � for multiplane needles withP=2, 3, and 4, �=30 deg, and r=1 mm

making it more difficult to cut the soft tissue.



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4.3.2 Effect of h. Figure 12 shows the shape of the needleutting edge and inclination angle for �1=30 deg, �2=30 deg, r1 mm, �=135 deg, and h=0, 1, and 2 mm. Increasing h causesllipses I and II to meet at � values closer to �. Again, ellipse Iemains fixed in place; therefore, the � remains the same for 0

��90 deg. Varying h can lower the maximum � reached byllipse II.

A unique effect of varying h is observed on �, which becomesiscontinuous as in the case of h=1.55 and 2 mm. This occursecause the � on one side of the needle tip is different than the �n the opposite side of the needle tip. The cutting characteristicshange greatly at this transition point. This interesting character-stic may greatly affect the cutting performance of the needle as ahole.

4.4 Comparison of the Inclination Angle. A bias beveleedle contains a full elliptical cutting edge profile that forms thease needle tip shape for all types of plane needles given theyontain the same � value. Adding more planes and varying therientation of the planes only changes how the basic ellipticalutting edge profile is sectioned together and does not change theange of � that are possible for a given � value.

The maximum � achieved by a plane needle is dictated by thelane with the smallest �, where �� /2−� for all plane needles.his occurs because the value � /2−� marks the steepest point on

he bias bevel ellipse and therefore the point of maximum incli-ation. As illustrated in Fig. 9, this high � is not reached in thease of symmetric multiplane needles where P2. Varying thelane orientation, such as h and �, can also determine if the maxi-um � is equal to � /2−�.The minimum � for all plane needles is 0 deg and the number

f times that this occurs around the needle circumference is equalo P given that P1. In the case of the one-plane bias beveleedle �P=1�, there are two points where �=0, at the top ��180 deg� and bottom ��=0� of the elliptical cutting profile.ultiplane needles �P1� do not contain top portions of the el-

iptical cutting edge profile and instead contain one lower portion

ig. 10 Inclination angle of a symmetrical multiplane needleP=3 and 4… with �=15, 30, and 45 deg

f the elliptical cutting edge profile for each plane that is used to



form the needle tip. Thereby, creating the situation on mutliplaneneedles where the number of points with �=0 equals the numberof planes.

5 Results of Rake Angle and DiscussionFigure 13 shows the results of the � for a bias bevel needle with

�=15, 30, and 45 deg. The range of � is between 0 and � /2−�.For other types of plane needles, this statement remains true be-cause all plane needles are based on the same basic ellipse geom-etry.

The maximum � values occur at �=0 and 180 deg, top andbottom of the ellipse, with a value equal to � /2−�. The higherrake angle represents a sharper cutting edge and makes it easierfor tissue to pass over the needle cutting surface. All plane needlescontain at least one relative minimum point of the ellipse; there-fore, the maximum � is � /2−� for all plane-needles.

The minimum � occurs at �=90 and 270 deg, midpoints of theellipse where the cutting edge changes from the inside to theoutside of the needle, with a value of 0. The � does not becomenegative because the cutting edge changes sides at the point �=0. Varying P, h, and � can potentially raise the minimum � ifthe midpoint of the ellipse is not revealed, thereby making the

Fig. 11 Cutting edge profile and � when �1=30 deg, �2=30 deg, r=1 mm, and h=0 for �=45, 90, and 180 deg

cutting edge always on the inside of the needle.

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Biopsy Needle Experiments and ResultsTo study the effect of needle geometry on biopsy performance,

iopsy experiments were performed on bovine liver tissue using1 gauge two-plane symmetric needles of 20, 25, and 30 degevel angles. These needles are shown in Fig. 14. Two-plane sym-etric needles are chosen because the symmetric tip design

venly directs tissue to the inside of the needle. A one-planeeedle does not have this symmetry and therefore could cause theissue to flow over the needle tip face rather than to the inside ofhe needle.

ig. 12 Cutting edge profile and � when �1=30 deg, �230 deg, r=1 mm, �=135 deg, and h=0, 1.5, and 2 mm

ig. 13 Rake angle of bias bevel needle for �=15, 30, and 45
eg
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The overall biopsy experimental setup is shown in Fig. 15�a�.The experimental setup uses a linear stage �Siskiyou Instrumentmodel 200cri� to drive the needle into the tissue. One solid pieceof fresh bovine liver is placed into a tissue holder with a crosssection of 5050 mm2 and a pneumatic cylinder applies a con-stant pressure of 15.5 kPa to the top of the tissue to ensure repeat-able tissue constraint conditions. The tissue holder has an openarea where the needle is inserted into the tissue, as shown in Fig.15�b�. The linear stage inserts the needle through the 50 mm sec-tion of tissue at a constant speed of 1.5 mm/s. Once through thetissue, a stylet was used to push the biopsy sample out of theneedle. The length of the biopsy sample out of the needle was thenmeasured and recorded. Each of these three needles was insertedinto the tissue ten times for a total of 30 trials.

Based on Eqs. �4� and �12�, the complete range of inclinationand rake angles of the three needles used in the experiments isshown in Fig. 16. The needles of higher bevel angles containlower rake and inclination angles. Lower rake and inclination

Fig. 14 Two-plane symmetric 11 gauge needles of �=20, 25,and 30 deg

Fig. 15 „a… Overall experimental setup for biopsy needle per-
formance testing and „b… close up view of needle and tissue


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ngles have a negative effect on performance of the needles, ashown by the biopsy length results in Fig. 17. The needles withigher bevel angles collected shorter length samples of tissue. Thexperimental results demonstrate that the needle cutting edge ge-metry plays an important role in biopsy performance and thetudy of � and � around a needle is crucial to optimizing biopsyeedle performance. It is worth noting that only a limited range ofevel angles, 20 deg��30 deg, were tested. Outside thisange, other factors may become dominated and affect the biopsyesult.

Concluding RemarksThe inclination and rake angles around the radius of the needle

ere studied for the three needle tip styles examined: the biasevel, symmetric multiplane, and nonsymmetric two-planeeedles. The rake angle was solved for the bias bevel needle. Theaximum inclination angle achievable is dictated by � �� /2�� and other variables such as h, �, and the number of planesould only decrease this maximum. For symmetric multiplaneeedles, increasing the number of planes lowered the inclinationngle. For multiplane needles, the number of points where �=0as equal to the number of planes used to construct the needle.arying h and � shifted the needle geometry, thereby changinghere the ellipses met. The rake angle for plane needles was

imited by � �0�� /2−��. Biopsy performance experimentshow that the needle tip geometry defined by inclination and rakengles directly affects biopsy performance. Longer biopsyamples are obtained for two-plane needles with higher rake andnclination angles �lower bevel angles�.

Future work will focus on two areas. One is to further study theeedle geometry by examining needle geometries beyond the flat

ig. 16 Rake and inclination angles for a two-plane symmetriceedle of �=20, 25, and 30 deg

ig. 17 Average biopsy sample length collected for 11 gaugewo-plane symmetric needles of varying bevel angles „error bar
epresenting the range of standard deviation…


plane and by examining the other needle geometry characteristics,such as contact length and exposed needle area upon insertion.The second area is to more explicitly correlate the cutting param-eters to the experimental performance of needle biopsy. Newneedle tip geometry will be manufactured and studied experimen-tally to determine the most efficient geometry for tissue biopsycutting.

AcknowledgmentThis research work is sponsored by the National Science Foun-

dation �NSF� Award CMMI No. 0825795, National Natural Sci-ence Foundation of China �Award No. 50775119�, and supportedby the University of Michigan Radiation Oncology Department.

Appendix: Solution of the �A and �B for the Two-PlaneNonsymmetric Needle

The solution of �A and �B are expressed related to two param-eters for E1 and E2 with 0��180 deg. In E1, there is a minussign in front of the square root while in E2 there is a positive sign.If �=180 deg, then the limit as � approaches 180 deg can betaken in order to find �A and �B.

�A = arctan� r tan �1 − r tan �2 + h tan �2 tan �1

− E1 tan �1 cos �

−tan �2

tan �1 cos �+ 1�

�B = arctan� r tan �1 − r tan �2 + h tan �2 tan �1

− E2 tan �1 cos �

−tan �2

tan �1 cos �+ 1�

�0 � � � 180 deg�where

�E1,E2� = − h tan2 �2 tan �1 + r tan2 �1 cos � − r tan �1 tan �2

+ r tan2 �2 − r tan �1 cos � tan �2

+ h tan2 �1 cos � tan �2 � �− h2 tan4 �1 tan2 �2

+ 2r2 tan3 �1 tan �2 − 2r2 tan3 �1 cos � tan �2

− 2r2 tan3 �1 cos2 � tan �2 + h2 tan4 �1 cos2 � tan2 �2

+ 2rh tan4 �1 cos2 � tan �2 − 2rh tan3 �1 cos2 � tan2 �2

+ 2r2 tan3 �1 cos3 � tan �2 − 2rh tan4 �1 tan �2

+ 2rh tan3 �1 tan2 �2�0.5/�tan2 �2 − 2 tan �1 cos � tan �2

+ tan2 �1�

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Outcome Using 29 mm Cutting Length,” Urol. Int., 75�3�, pp. 209–212.�2� Iczkowski, K., Casella, G., Seppala, R., Jones, G., Mishler, B., Qian, J., and

Bostwick, D., 2002, “Needle Core Length in Sextant Biopsy Influences Pros-tate Cancer Detection Rate,” Urology, 59�5�, pp. 698–703.

�3� Ubhayakar, G., Li, W., Corbishley, C., and Patel, U., 2002, “Improving Glan-dular Coverage During Prostate Biopsy Using a Long-Core Needle: TechnicalPerformance of an End-Cutting Needle,” BJU Int., 89�1�, pp. 40–43.

�4� Moore, J. Z., McLaughlin, P. W., McGill, C. S., Zhang, Q., Zheng, H., andShih, A. J., 2009, “Blade Oblique Cutting of Tissue for Investigation of BiopsyNeedle Insertion,” Trans. NAMRI/SME, 37, pp. 49–56.

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Jason Z. Moore AnnArbor,MI48109 Modeling of the Plane ...wumrc.engin.umich.edu/wp-content/uploads/sites/51/... · The curved leading edge of the needle tip acts as the cutting edge,