Flying Focal Spot (FFS) in ConeBeam CTMarc Kachelrie, Michael Knaup, Christian Penel, Willi A. Kalender

Fig. 1. A rotating envelope tube with focal spot deflection capability.

Abstract In the beginning of 2004 medical spiral-CTscanners that acquire up to 64 slices simultaneously becameavailable. Most manufacturers use a straightforward acqui-sition principle, namely an Xray focus rotating on a circularpath and an opposing cylindrical detector whose rotationalcenter coincides with the Xray focus. The 64slice scanneravailable to us, a Somatom Sensation 64 spiral conebeamCT scanner (Siemens, Medical Solutions, Forchheim, Ger-many), makes use of a flying focal spot that allows for viewbyview deflections of the focal spot in the rotation direc-tion (FFS) and in the zdirection (zFFS). The FFS featuredoubles the sampling density in the channel direction andin the longitudinal direction. Up to four detector readingscontribute to one view (projection). A significant reductionof inplane aliasing and of aliasing in the zdirection can beexpected. Especially the latter is of importance to spiral CTscans where aliasing is known to produce socalled windmillartifacts. We have derived and analyzed the optimal focalspot deflection values and z as they would ideally occurin our scanner. Based upon these we show how image recon-struction can be performed in general. A simulation studyshowing reconstructions of mathematical phantoms furtherprovides evidence that image quality can be significantlyimproved with the FFS. Aliasing artifacts, that manifest asstreaks emerging from highcontrast objects, and windmillartifacts are reduced by almost an order of magnitude withthe FFS compared to a simulation without FFS.

I. Introduction

CORRECT sampling requires to satisfy the Nyquistcondition: at least two sample points should be takenper spatial resolution element (Shannon sampling theo-rem). In many cases this situation is not easy to achieve.

In CT the spacing of the detector samples is slightly

Institute of Medical Physics (IMP), University of ErlangenNurnberg, Henkestr. 91, 91052 Erlangen. Corresponding author: PDDr. Marc Kachelrie, Email: marc.kachelriess@imp.unierlangen.de.

larger than the active width of the detector pixels far from sampling the active detector area twice. Oneworkaround is the quarter detector offset. During a 360

rotation each ray is measured twice and the data are re-dundant. Shifting the detector array (channel direction)by one quarter of the detector sampling distance pays outsince opposing rays interlace and, by combining opposingviews, one effectively doubles the sampling [1]. However,for conebeam scans and for spiral scans this kind of dataredundancy is not really available since opposing rays donot exist; they rather differ by their tiltangle wrt the ro-tation axis and by their zposition. Further, the quartershift does not improve the sampling in the detector rowdirection (zdirection).

An alternative is deflecting the focal spot between ad-jacent detector readouts (figure 1) as it is done in ourtube [2]. This flying focal spot can be used to double thesampling density in both directions regardless of the coneangle and the spiral trajectory. In this paper we specifythe FFS deflection values, we clarify the geometry, we in-troduce the subfan approximation, we present the way westore FFS data in a viewbased (opposed to a readingbased) format and we demonstrate the effect of the FFS ina simulation study.

II. Source and Detector Geometry

With RF being the distance of the undeflected focal spotto the isocenter, d being the table increment per rotation,d = d/2 and being the view angle let us define the spiralsource trajectory as

s(, , z) =

(RF + RF) sin( + )(RF + RF) cos( + )

d + z

.

The vector s() = s(, 0, 0) is the source trajectory vec-tor of a spiral CT scanner without FFS. The additionalparameters and z are the focal spot deflection angleand deflection length that are used to improve the inplaneand the axial sampling properties of CT. These deflectionparameters are of very small magnitude. Using the smallangle approximation (first order Taylor series in the two de-flection parameters) is justified and will be implicitly usedbelow when deriving the ideal values for and z. Theremaining parameter RF is not independent. It accountsfor small variations in the radius of the true focal spot tothe isocenter. Since the rotation axis of the Xray tubesis parallel to the scanners rotation axis a focal spot de-flection z in the zdirection will effectively change RF byRF = z/ tan where is the anode angle (in our case7). Note that the variation in RF due to is of secondorder in and therefore negligible.

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(a)FFS

(b)zFFS

(c)View encoding

Fig. 2. Geometry of the FFS deflection and view encoding diagram

The detector is a cylindrical detector whose symmetryaxis coincides with s(). We parameterize it as

r(, , b) = s() + RFD

sin( + )

cos( + )0

+ b

001

,

where is the angle within the fan and b is the detectorslongitudinal coordinate. Note that the deflection parame-ters do not enter the definition of the detector coordinates.

To treat sampling we must go into discretization. Weassume equidistant sampling in all three coordinates andlet phys, phys and bphys denote the sample spacing.The sampling distance bphys is the width of the detectorrow. It is related to the nominal slice thickness S by scalingto the isocenter: S = bphysRF/RFD.

III. Flying Focal Spot Deflection

A. FFS

The sampling distance in the fanbeam close to theisocenter is given as RFphys. The flying focal spot inthe direction aims at doubling the sampling density toyield a sampling distance of 12RF

phys (see figure 2a).Two adjacent readings place the x-ray focus at

s( 12phys,, 0).

To obtain the value for the deflection angle we scale thedesired sampling distance from the isocenter to the focalspot:

12RF

phys RFDRD

is the distance between the two focal spot positions. Thisdistance is equal to 2RF from which follows

=14

physRFDRD

0.043.

The numerical value given applies to our scanner wherephys = 2/4640, RFD = 1040 mm, RD = 470 mm. Thedistance 2RF between two focal spot positions yields0.85 mm approximately.

B. zFFS

The sampling distance in the zdirection scaled to theisocenter is given by S = bphysRF/RFD. The zFFS aimsat doubling the sampling density (see figure 2b). Two ad-jacent readings place the Xray focus at

s( 12phys, 0,z).

The rays shall be longitudinally separated by S/2 whenintersecting the isocenter. Neglecting the table incre-ment per reading d phys which is of orders of magnitudesmaller than z we find by simple geometric scaling that2zRD/RFD should be just the same distance S/2 whichmeans

z =14S

RFDRD

0.33 mm.

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The numerical value applies to our scanner where S =0.6 mm, RFD = 1040 mm, RD = 470 mm. Obviously,both focal spot positions are separated by about 0.66 mmin the zdirection. Due to the 7 anode angle we getRF 2.7 mm which means that the radial separationof both focal spots is 5.4 mm.

C. Both FFSs

If both the and the zFFS are switched on simulta-neously the deflection values and z remain the same.However, the focal spot itself will jump in a rectangularfashion relative to the anode. Four physical readings

s( 12phys phys,,z)make up one projection. The order in which the focal spotpositions are visited depends on the implementation andmust be stored together with the view definition (see sec-tion V).

IV. Subfan Approximation

A. Subfan Approximation

Since the focal spot deflection is only a tiny angle onecan approximate each of the two deflected subfans by anideal subfan whose focal spot coincides with the rotationalcenter of the detector cylinder. Thus, we try to replace thedeflected system

s(, , 0) r(, , b)

by the ideal fanbeam system (that is a system where thefocal spot is fixed relative to the detector)

s( + ) r( + , + , b).

Regarding figure 2a and using the small angle approxima-tion we find that RFD = RF yields the desired com-pensation, and

= RFRFD

= 14

phys

results. If the FFS is not used = 0 implies = 0.Note that the minus sign is introduced since a clockwisedeflection of the focal spot can be approximately compen-sated by a counterclockwise deflection of the detector chan-nels.

B. zSubfan Approximation

For the zFFS a similar approximation can be made. In-stead of assuming a shifted focal spot during reconstructionone may rather assume a shifted baxis, i.e. a detector arraythat is shifted by b. Thus, we try to replace the deflectedsystem

s(, 0, z) r(, , b)

by the ideal system

s() r(, , b + b).

Fig. 3. The penumbra cannot be used for imaging. Therefore the prepatient collimators must open wide enough. Two dose profilesoffset by a distance O result when using the zFFS.

Regarding figure 2b we find that z/RF = b/RD yieldsthe desired compensation and

b = RDRF

z = 14S

RFDRF

= 14

bphys

results. If the zFFS is not used z = 0 implies b = 0.The minus sign shows that a forward deflection of the focalspot can be approximately compensated by a backwarddeflection of the detector rows.

V. View Definition and Storage

We define the view or projection to be the collection ofadjacent readings that comprises a total FFS cycle. Thus,the view consists of two readings if either the FFS or thezFFS is switched on and it consists of four readings if bothflying focal spots are used.

Our view encoding diagram in figure 2c shows howwe combine one, two, or four readings in one projection.Switching off the FFS is encoded by setting or z tozero.

Technically, we assign one view angle n to up to fourphysical readings. The view angles are sampled as

n = 0 + n

with {1, 2, 4}phys, depending on the FFS settings.Each projection is parameterized as

m = 0 + mbl = b0 + lb

where and b can be equal to or half of the physicalvalues phys and bphys, respectively, depending on theFFS settings.

Since each virtual view n consists of up to four physicalreadings A, B, C, and D we store four correction anglesA, B, C, and D to be able to retrieve the physical

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gantry positions from each virtual view, channel, and sliceindex (n,m, l):

nml = n +

with being one of A, B, C, or D. The mapping is a func-tion of the channel number m and of the slice number l asfollows

m odd m evenl odd A Bl even C D

The physical rays start and end points of projection n,channel m and slice l are given as

snml = s(n + , ()m, ()lz

)

rnml = r(n + , m ()m, bl ()lb

).

The subfan approximation allows to approximate these co-ordinates with the following:

snml = s(n + + ()m

)

rnml = r(n + + ()m, m, bl

).

VI. Reconstruction

Since our reconstruction is based on rebinning to parallelgeometry we do not need to use the subfan approximation.The rebinning equations we use rather exactly account forthe (odd) geometry of the FFS scanner.

A. Azimuthal Rebinning

Azimuthal rebinning means resorting the rays into nonequidistant parallel geometry by performing a resamplingin the direction. From the source (fanbeam) datap(, , b) one obtains azimuthally rebinned destination(fanparbeam) data p(, , b) where = + standsfor the angle of a ray wrt a fixed coordinate axis. It is thisstep where one can ideally account for the four correctionangles .

B. Longitudinal Rebinning

Many popular reconstruction algorithms, such as the spi-ral zinterpolation (singleslice CT), the spiral zfiltering(up to 4slice CT) or the advanced singleslice rebinning(conebeam spiral CT) [3] perform a resampling of the datawhere only the longitudinal coordinate b is involved. Sincethis means taking into account the focal spots zpositionlongitudinal rebinning is the step to account for z.

C. Convolution and 3D Backprojection

The rebinned data undergo a standard parallelbeamconvolution (for example using the Shepp and Logan ker-nel). Then, the data are backprojected along exactly thesame line they have been measured.

D. A Comment on the Quarter Detector Offset

Most clinical CT systems double the sampling densityin the direction by shifting the detector by a quarter ofthe detector width. Data acquired from opposing views

(a)Slice sensitivity profiles SSP(z) with and without zFFS.

(b)Point spread functions PSF(r) with and without FFS.

Fig. 4. Quantification of spatial resolution. Without zFFS oneobtains the expected broadening of the SSP due to linear inter-polation between adjacent detector rows. The zFFS improvessampling and the effective slice width (FWHM) equals the nom-inal slice width. The same behaviour is observed inplane wherethe radial point spread function is plotted.

(about 180 apart) that would normally be redundant willnow interlace. Mathematically, the quarter shift impliesthat 0 (2Z + 1)/4.

To gain further advantage when using the FFS one de-mands that the combined subfans have an effective quar-ter offset: 0 (2Z + 1)/4. Physically, this corre-sponds to an offset of one eighth of an detector element:0 (2Z + 1)phys/8.

VII. Dose Issues

The zFFS requires to open the prepatient collimator abit wider than it is necessary without a flying focal spot:the collimators mechanical inertia prohibits to follow themovement of the flying focal spot with such high frequency(in the order of 10 kHz). Basic geometric considerationshelp to get an idea of the magnitude of this dose increase.

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A schematic illustration is given in figure 3. It basicallyshows the trapezoidal dose profile at the axis of rotation.The area of the dose profile that can be used for imaging isthe collimated slices area C. Other parts of the dose profileare relevant for the patient dose but irrelevant for imaging.Thus, the patient dose is proportional to C + P + O whichis a factor of (C +P +O)/C compared to an ideal scanner.

For scanners without zFFS there is only one trapezoidaldose profile and the offset O would be zero which meansa dose of C + P . Evidently, the dose ratio of scans withzFFS and those without is given by (C+P +O)/(C+P ) =1+O/(C+P ). For our scanners 2320.6 mm collimation(C = 19.5 mm) this evaluates to about 1.06 which impliesa dose increase of only 6% with the zFFS.

VIII. Simulations

For our simulation we use the Sensation 64 geometrywith a slice thickness of S = 0.6 mm. The number of read-ings per rotation is set to 4 1160, the number of physicaldetector channels to 672 and the number of physical detec-tor rows to 32. Further, phys = 2/4640, RF = 570 mmand RFD = 1040 mm. The resulting fan angle and coneangle yield 52 and 1 .9 , respectively. A spiralpitch of p = 1.3, that corresponds to a table increment of25 mm per rotation, was simulated.

To mimick a finite detector area, focal spot area and inte-gration time we used a 3 3 subsampling for the detectorand the focal spot and a threefold subsampling for theview angle. Alltogether 243 monochromatic needle beamswere computed and averaged to obtain one measured value.The simulation software used was ImpactSim (www.vamp-gmbh.de).

To assess spatial resolution a delta object was simu-lated and its response was evaluated by computing ra-dial and longitudinal profiles. To study the artifactbehavior the FORBILD head phantom (www.imp.uni-erlangen.de/phantoms.htm) was simulated.

The following FFS combinations were simulated:

FFS N360 M Lnone 4640 672 32

FFS 2320 1344 32zFFS 2320 672 64both 1160 1344 64

Image reconstruction was performed with the Feldkamptype extended parallel backprojection (EPBP) [4].

IX. Results

Figure 4 shows the axial and longitudinal profiles weobtained. Apparently, the double z and the double sampling improve spatial resolution by about 20 to 30%each. It should be noted that these improvements havebeen achieved without modifying the kind of interpolationused: in all cases standard linear interpolation was applied.We also evaluated image noise and found an increase ofnoise with increasing spatial resolution. Quantitatively, theincreased noise corresponds to the theoretically expectedcost for the increased resolution.

Fig. 5. Head phantom. Aliasing artifacts on the left side are greatlyreduced with the zFFS. The increase in noise is due to the higherzsharpness of the zFFSreconstruction.

The main purpose for introducing double sampling isthe expected reduction of aliasing artifacts. This reduc-tion is clearly visible in the reconstructions of figure 5.Streaks emerging from high contrast objects are stronglysuppressed by the Nyquistconform double sampling.

X. Discussion

Spatial resolution and image quality can be greatly im-proved with the focal spot deflection. Especially the recon-struction of small slice thicknesses becomes possible with-out using tricks such as interpolation kernels with under-shoots (negative values). Obviously it is worth to invest intechnology satisfying the Nyquist criterion rather than justheading for smaller detectors.

References

[1] W. A. Kalender, Computed Tomography. Wiley & Sons, 2004.ISBN 3895782165, 2nd Edition.

[2] P. Schardt, J. Deuringer, J. Freudenberger, E. Hell, W. Knupfer,D. Mattern, and M. Schild, New xray tube performance in com-puted tomography by introducing the rotating envelope tube tech-nology, MedPhys, vol. 31, no. 9, pp. 26992706, 2004.

[3] M. Kachelrie, S. Schaller, and W. A. Kalender, Advancedsingleslice rebinning in conebeam spiral CT, Med. Phys.,vol. 27, pp. 754772, Apr. 2000.

[4] M. Kachelrie, M. Knaup, and W. A. Kalender, Extended paral-lel backprojection for standard 3D and phasecorrelated 4D axialand spiral conebeam CT with arbitrary pitch and 100% doseusage, Med. Phys., vol. 31, pp. 16231641, June 2004.

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