ECG-correlated image reconstruction from subsecond multi ...

ECG-correlated image reconstruction from subsecond multi-slice spiral CTscans of the heart

Marc Kachelrieß,a) Stefan Ulzheimer, and Willi A. KalenderUniversity of Erlangen-Nurnberg, Institute of Medical Physics, Krankenhausstrasse 12,Erlangen D-91054, Germany

~Received 6 December 1999; accepted for publication 3 May 2000!

Subsecond spiral computed tomography~CT! offers great potential for improving heart imaging.The new multi-row detector technology adds significantly to this potential. We therefore developedand validated dedicated cardiac reconstruction algorithms for imaging the heart with subsecondmulti-slice spiral CT utilizing electrocardiogram~ECG! information. The single-slice cardiacz-interpolation algorithms 180°CI and 180°CD@Med. Phys.25, 2417–2431~1998!# were general-ized to allow imaging of the heart forM-slice scanners. Two classes of algorithms were investi-gated: 180°MCD~multi-slice cardio delta!, a partial scan reconstruction of 180°1d data withd,F ~fan angle! resulting in effective scan times of 250 ms~central ray! when a 0.5 s rotation modeis available, and 180°MCI~multi-slice cardio interpolation!, a piecewise weighted interpolationbetween successive spiral data segments belonging to the same heart phase, potentially providing arelative temporal resolution of 12.5% of the heart cycle when a four-slice scanner is used and thetable increment is chosen to be greater than or equal to the collimated slice thickness. Data seg-ments are selected by correlation with the simultaneously recorded ECG signal. Theoretical studies,computer simulations, as well as patient measurements were carried out for a multi-slice scannerproviding M54 slices to evaluate these new approaches and determine the optimal scan protocol.Both algorithms, 180°MCD and 180°MCI, provide significant improvements in image quality,including extremely arythmic cases. Artifacts in the reconstructed images as well as in 3D displayssuch as multiplanar reformations were largely reduced as compared to the standardz-interpolationalgorithm 180°MLI ~multi-slice linear interpolation!. Image quality appears adequate for precisecalcium scoring and CT angiography of the coronary arteries with conventional subsecond multi-slice spiral CT. It turned out that for heart ratesf H>70 min21 the partial scan approach 180°MCDyields unsatisfactory results as compared to 180°MCI. Our theoretical considerations show that afreely selectable scanner rotation time chosen as a function of the patient’s heart rate, would furtherimprove the relative temporal resolution and thus further reduce motion artifacts. In our case anadditional 0.6 s mode besides the available 0.5 s mode would be very helpful. Moreover, if tech-nically feasible, lower rotation times such as 0.3 s or even less would result in improved imagequality. The use of multi-slice techniques for cardiac CT together with the newz-interpolationmethods improves the quality of heart imaging significantly. The high temporal resolution of180°MCI is adequate for spatial and temporal tracking of anatomic structures of the heart~4Dreconstruction!. © 2000 American Association of Physicists in Medicine.@S0094-2405~00!00208-X#

Key words: computed tomography~CT!, multi-slice spiral CT, heart, 4D reconstruction

I. INTRODUCTION

Coronary artery disease is the most important cause of deathin western civilizations: For example 13.7% of all deaths arecaused by ischaemic heart disease.1 Therefore there is astrong need for early, preventive cardiac diagnosis, i.e., non-invasive imaging techniques.

Most of the techniques available today, such as cardiacultrasound, fluoroscopy, conventional computed tomography~CT!, spiral CT, and electron-beam computed tomography~EBT!, suffer from several drawbacks: they fail to detectsmall amounts of calcium and some of them are not readilyavailable~e.g., EBT!.2

Since spiral CT offers many features such as excellentvolume scanning capability, high rotation speed~subsecondscanning!, quantitative imaging, high resolution, and broadavailability3,4 there have been many attempts to use standard

spiral scanning for cardiac imaging.5–7 The only methodsknown to us that used dedicated cardiac reconstruction algo-rithms for heart imaging are the ones presented in a previousarticle: Electrocardiogram~ECG! data, recorded during thescan, may be used to divide projection data into ranges thatare allowed or forbidden to be utilized for reconstruction.8

The results are promising, although further work has to bedone to improve image quality and to establish means forquantification of coronary calcium.

In 1998 new CT systems have become available whichsimultaneously measureM parallel slices instead of a singleslice. We will refer to these systems as multi-slice CT. In thisarticle, we show that this new scanning scheme offers thepossibility to further improve the quality of cardiac imagingin computed tomography. The algorithms specified here are ageneralization of the cardiac algorithms 180°CI and

1881 1881Med. Phys. 27 „8…, August 2000 0094-2405 Õ2000Õ27„8…Õ1881Õ22Õ$17.00 © 2000 Am. Assoc. Phys. Med.

180°CD:8,9 180°MCI ~multi-slice cardio interpolation!, apiecewise linear interpolation between adjacent spiral datasegments and measured detector slices, is the multi-sliceequivalent of 180°CI whereas 180°MCD~multi-slice cardiodelta!, a partial scan reconstruction of 180°1d data withd,F ~fan angle!, is the corresponding adaption of 180°CD.They aim at overcoming the disadvantages of their single-slice equivalents: broadened slice sensitivity profile~SSP!for 180°CI and introduction of parallel streak artifacts due tothe selection of different data segments~180°CI and180°CD! case.10,11

During the annual meeting of the RSNA 1999, other car-diac imaging approaches were presented by some groups.Basically they can be divided into partial scan algorithmswith a temporal resolution oft rot/2 and segmented ap-proaches with a temporal resolution better thant rot/2. Someof them use simultaneously acquired ECG data for gatingpurposes. The first article to mention is a partial scan recon-struction with manual selection of the ‘‘timing shift’’ to se-lect a temporal reconstruction position corresponding tominimum cardiac motion given a reconstruction positionzR .12,13 Using a simultaneously acquired ECG to automati-cally detect the optimal reconstruction position for a partialscan has been investigated as well14 yielding a method quitesimilar to 180°MCD. Phase-coded reconstruction with ‘‘im-proved sector size’’ selection15 is using data from more thanone cardiac cycle for reconstruction. However, data seg-ments and the number of ‘‘sectors’’ are not chosen as dy-namically as compared to 180°MCI.

Our article presents the definitions of the algorithms180°MCD and 180°MCI as well as profound theoretical con-siderations concerning resolution, image noise, patient dose,and temporal resolution. The corresponding SSP predictionsas well as simulations of a virtual motion phantom will bepresented in comparison to a standard multi-slicez-interpolation algorithm. Moreover, the performance of thenew algorithms is demonstrated in patient studies and com-pared to the standardz-interpolation algorithm 180°MLI.

A benchmarking of the cardiac algorithms in comparisonto the multi-slice filtered interpolation~180°MFI! and themulti-slice linear interpolation~180°MLI! including cases ofarythmic cardiac motion and phantom measurements will bepresented elsewhere.16

This article is organized as follows: The materials sectionwill introduce the concept of relative and absolute temporalresolution, and present the virtual motion phantom used forsimulation studies. The algorithms section treats the standardand cardiac algorithms involved: 180°MLI, 180°MCI, and180°MCD. The definitions of the dedicated cardiac algo-rithms and theoretical considerations concerningz-resolutionand image noise are given there. The methods section intro-duces the experiments performed followed by the corre-sponding results section. Temporal and spatial resolution, re-constructions of the virtual motion phantom, noisemeasurements, coronary calcium measurements, the patientstudy, as well as dose considerations are presented there. Thearticle ends with the discussion section and list of nomencla-ture.

II. MATERIALS

A. Temporal resolution

The relative temporal resolutionw, describing the fractionof the cardiac cycle contributing to an image, is determinedas the full width at tenth maximum~FWTM! of the phasesensitivity profile~PSP!. The PSP contains the weighted con-tribution of the cardiac phases to the final image. It is thetemporal equivalent to the slice sensitivity profile~SSP!.Since the PSP is not a smooth function~in contrast to SSPswhich are the result of a convolution with the beam profile!

the full width at half maximum~FWHM! is not a usefuldescriptor for temporal resolution whereas the FWTMwmore adequately describes the width of the PSP. The abso-lute temporal resolutionteff , describing the period of timecontributing to an image, is given byteff5w/fH .

B. Virtual motion phantom

To evaluate our algorithms we have used the virtual car-diac motion phantom of Fig. 1~a! which has first been pre-sented in Ref. 10. It consists of several calcifications whichare located as a lattice of 335 points~spacing: 15 mm inyand 20 mm inx!. The three rows contain calcifications witha diameter of 3 mm, 2 mm, and 1 mm, respectively. The firstand the second column contain calcifications which are in-variant under translations inz ~i.e., cylindrical objects!whereas the third through the fifth column contain sphericalcalcifications. The calcifications of the first and third columnare not subjected to motion, the second and fourth columnobjects are moved horizontally and the fifth column calcifi-cations are moved inz-direction.

To simulate motion, the empirical motion function of Fig.1~b! was used. We are aware of the fact that this kind ofmotion is not physiologic and not necessarily realistic. Nev-ertheless, since our intention is to prove that our algorithmsare able to provide a reconstruction using only a small frac-tion w of the heart cycle, this kind of motion function iswell-suited for our purposes.

There are two advantages: During 20% of the cardiac in-terval there is no motion at all. Thus, if the algorithms areable to depict the object without motion artifacts, it can beconcluded that at most 20% of the cardiac cycle have beenused for reconstruction and that the algorithm’s temporalresolution lies below that value. Moreover, the remaining80% of R –R can be used to show how the algorithm dealswith cardiac motion and, in addition to this, it can be shownthat a temporal and spatial tracking of the calcifications ispossible.

The amplitudeA of the motion function was chosen asA55 mm since from the literature it is known that cardiacmotion ~e.g., for the ventricular wall! does not exceed lineardimensions of about 10 mm.17,18 Quantum noise was addedto draw attention away from those reconstruction artifactsthat have nothing to do with the cardiac motion and to givean impression of the noise characteristics of the differentalgorithms. Polychromacity and nonlinear partial volume ef-

1882 Kachelrieß, Ulzheimer, and Kalender: ECG-correlated image reconstruction 1882

Medical Physics, Vol. 27, No. 8, August 2000

fects were not simulated to reduce the computation time ofphantom data generation.

True cardiac motion is 3D in contrast to the 1D motionused by our phantom. Due to the enormous variety of pos-sible 3D trajectories we decided to break them down into 1Dtrajectories and assume the results to be representative forthe 3D case. In addition, measurements using a cardiac mo-tion and calibration phantom with a realistic 3D LAD~leftarterior descending coronary artery! motion function19 haveconfirmed our assumption quite well.

C. Measurements, simulations, and reconstruction

The reconstruction algorithms were implemented on astandard personal computer~PC! with dedicated reconstruc-tion and image evaluation software ImpactIR~VAMPGmbH, Mohrendorf, Germany!; the reconstruction time liesbelow 5 s per image on a 450 MHz Pentium CPU with 250MB of memory.

Simulations of the virtual heart phantom were performedto evaluate the scan protocolp51/M . The simulated scannergeometry is equivalent to the SOMATOM Volume Zoom~Siemens Medical Systems, Erlangen, Germany!: The dis-tance of the focal spot to the center of rotation isRF

5570 mm, the distance of the detector to the center of rota-tion is RD5435 mm, and the radius of the field of measure-ment is RM5250 mm. This corresponds to a fan angleF552°. We simulated 43672 detector elements with quarteroffset corresponding to a collimation ofM3S5431 mm.1160 views per rotation and a rotation time of 0.5 s weresimulated.

All measurements were performed on a subsecond spiralfour-slice CT scanner~SOMATOM Volume Zoom, SiemensMedical Systems, Erlangen, Germany! using a rotation timeof t rot50.5 s. The nominal slice thickness was selected to be

eitherS51.0 mm orS52.5 mm and the table increment waschosen betweend5S andd52S to allow a complete cover-age of the heart~typically 12 cm to 15 cm! during a singlebreathhold~less than 30 s!.

An ECG monitor~Siemens Medical Systems, Erlangen,Germany! was used to record the patient’s electrocardiogramand to automatically detect theR-peaks~these are the mostpronounced peaks of the ECG! from the ECG data. TheseR-peaks then were used to perform synchronization with theprojection numbers, yielding the cardiac phasec as a func-tion of a: c(a).

III. ALGORITHMS

Multi-slice CT systems offer not only the possibility toselect data points obtained at the same angle of rotation~360° algorithms, measured spiral! and points obtained fromopposite views~180° algorithms, calculated or rebinneddata! such as in single-slice spiral CT,20–22 but also to choosethe slice from which data points are obtained.

In general, a z-interpolated projection at position(b,a,zR) is a linear combination of suitable, measured pro-jections:

P~b,a,zR!5 (kPZ

m51,...,M

wkmP~bk ,ak ,m !,

where the weightswkm5wkm(zR) are real-valued and

bk5~2 !kb, ak5a12dodd,kb1kp,

with k running over integer values (kPZ) and 0<a,2p~sequence data!. Furthermore, conservation of mass demands

(kPZ

m51,...,M

wkm~zR!51 ;zRPR.

FIG. 1. ~a! The virtual cardiac motion phantom that is used throughout this article. Cylindrical as well as spherical calcifications of varying diameter areavailable. For this image a single-slice spiral scan withd5S51 mm was simulated without motion, 180°LI was used for reconstruction.~0/500! ~b! Themotion function that has been used to simulate cardiac motion. The function is composed of a sinusoidal motion with amplitudeA extending over 80% of thecardiac cycle and a constant part~20% of R –R! with no cardiac motion. In the article we refer to the slow motion phase, which is located at 80% ofR –Rin this plot, to the medium motion phase at 10% and to the high motion phase at 30%.



A. Standard algorithm

As a standard algorithm for comparison we chose to usethe linear interpolation algorithm 180°MLI~multi-slice lin-ear interpolation!.4 The more commonz-filtering approachessuch as the algorithm 180°MFI~multi-slice filtered inter-polation!4 make use of all available data to accumulate dosein case of overlapping data acquisition. For low pitch values,as used in this article, their temporal resolution is even worsethan 180°MLI and thus they are unsuitable for a fair com-parison to our cardiac algorithms. For the cased5S ourimplementation of 180°MLI is equivalent to the single-slicealgorithm 180°LI, and it allows to select one of theM mea-sured slices to be used for reconstruction. Moreover, we havethe possibility to select a certain slice for other pitches aswell. This enables us to reconstructM different images at agiven z-position using the standard algorithm. TheseM dif-ferent images correspond to a temporal offset oft rotS/dwhich is the time span between two adjacent slices to reachthe reconstruction positionzR .

B. Algorithm 180°MCI

The basic idea of 180°MCI remains the same as for itssingle-slice equivalent 180°CI:8 only data acquired in ac-cepted phases of cardiac motion are used for reconstruction.The fact that we are now dealing withM slices complicatesthe situation as compared to the single-slice case. True linearinterpolation would be only reasonable for table incrementsbeing a multiple of the slice thickness, i.e.,dPNS. Othercases potentially result in~more complicated! sampling pat-tern alongz that require abrupt switches of the interpolationfunction within one projection~constant a! which yieldstreaking artifacts in the images—a reason why the CTmanufacturers use weighting or filtering algorithms insteadof true linear interpolation to assemble the planar data. Hightemporal resolution, as required for cardiac imaging, de-mands overlapping scans and thus a low table increment perrotation. For a four-slice scanner, for example, the aboverequirement and the demand for overlapping data acquisitionwould result in the possibilitiesd5S, d52S, and poten-tially d53S.

An approximate upper limit ford follows from the requestfor high z-resolution. During the duration 1/f H of one heartcycle the detector array should not advance by more thanMslice thicknesses in order to have full interpolation possibili-ties for anyz and any cardiac phasec. Otherwise the algo-rithm would have to wait for the next cardiac cycle and thuswould use data further away from the reconstruction plane.This upper limit evaluates to

d<MSf Ht rot . ~1!

Thus, for example, the settingd53S would be rejected, as-suming a rotation time oft rot50.5 s and a heart rate of 60min21 ~i.e., f Ht rot51/2! for M54. Moreover, since typicalheart rates lie in the range of 50 to 120 min21 one finds thatS<d<2S is desirable where the lower bound is a self-

imposed limit concerning patient dose. On the other hand,d'S is not always feasible since volume coverage orz-resolution would be too low.

The above discussion shows that 180 °MCI must allowfor arbitrary table increments, i.e.,dPR. This is achieved byintroducing a dynamic weighting~filtering! approach. Thefiltering is done inz ~similarly to what is known already forstandard multi-slice algorithms such as 180 °MFI! as well asin the cardiac phasec. This requires to define two weightingfunctions, wdist(z) and wphase(c), the first responsible forz-filtering, the latter responsible for cardiac phase filtering.Then we simply evaluate the weighting equation

PMCI~b,a,zR ,cR!

5 (kPZ

m51,...,M

wkmP~bk ,ak ,m !Y (kPZ

m51,...,M

wkm , ~2a!

with the total weight chosen as the product of the distanceand the phase weight

wkm5wkm~zR ,cR!5wdist~z~ak ,m !2zR!wphase~c~ak!2cR!.~2b!

The reconstruction is centered aboutzR ~reconstruction posi-tion! and about the cardiac phasecR ~reconstruction phase!.It must be pointed out that this approach allows for arbitrarypitches. Nevertheless, only a solution that complies with Eq.~1! will yield optimal image quality in terms of bothz- andtemporal resolution.

The weight functions themselves can be chosen to be tri-angular:

wdist~z !5L~z/ z !, wphase~c !5L~c/ c !,

where z denotes the width of the filter inz-direction andc<1/2 the filter width for the cardiac phase. These widthsdetermine the spatial resolution inz-direction and the tempo-ral resolution of 180°MCI. Nevertheless, this correspondenceis only approximate since the normalization procedure@de-nominator of Eq.~2a!# introduces deformations of the weightfunction which will not allow for an analytic calculation ofthe corresponding figures of merit—the FWHM of the SSPdeterminesz-resolution and the FWTM of the phase sensi-tivity profile ~PSP! determines the temporal resolutionteff .To ensure at least one data point contributing for a givenview anglea on either side of the reconstruction plane wewill additionally demandz>S. This lower limit for z yieldsa linear interpolation between two neighboring measuredslices and consequently the optimal achievablez-resolutionS. Smaller settings for thez-filter width could potentiallyresult in cases with only one data point on one side ofzR

contributing to the image plane.The width selection has been implemented as follows.

Given zR and a filter width z the algorithm seeks for theminimal phase widthc that allows for a complete data set.The completeness condition demands for eachaP@0;2p)andbP@2

12F, 1

2F# to have at least one data point on eachside of the reconstruction planezR contributing to the image~from the definition ofak andbk it is obvious that opposingrays are taken into account by rebinning!. Thus, the phase



width c is a function ofzR to adaptively account for varyingheart rates. It does not vary as a function of~b, a!. Allreconstructions shown in this article were performed usingz5S.

The strategy for dynamic width selection ensures to ob-tain the optimal achievable temporal resolution regardless ofthe current~local! patient heart rate. Thus arythmic heartrates and other abnormal behavior are covered optimally bythe 180°MCI approach.

Although the parameterc is dynamically adapted by ourimplementation 180°MCI it pays to derive the optimal widthsettings theoretically under the assumption of a locally con-stant heart rate:

1. Width settings and relative temporal resolutionof 180°MCI

The triangular distance weight collects contributions fromzR2 z to zR1 z, i.e., over a distance ofz on either side of thereconstruction plane. The overlap between thisz-width andthe detector@interpolatable width (M21)S# corresponds to atable translation ofz1(M21)S and thus is available for anangular increment of

Damax52pz1~M21!S

d,

or, equivalently,

nmax5z1~M21!S

d>p21

rotations on either side ofzR .a. Optimal data filling. From Ref. 8 we know that the

width w of the allowed cardiac interval must exceedu12 f Ht rotu to allow for optimal data filling. This quantity, re-flected by the baseline widthw52c<1 of the phase weightsin our new filtering approach, must fulfill

w>wopt5u12 f Ht rotu.

b. Range restriction. Now care has to be taken to collect acomplete data set within the given rangeDamax, i.e., to col-lect at leastp1F contiguous data on either side of the re-construction plane. Following the derivation in Ref. 8 wecome to the result

w>w res5wopt1p1F2Damaxwopt

DaH.

c. Trivial case. Excluding the trivial case of collectingmore thanp1F data during one heart cycle gives

w<w triv5

p1F

DaH.

Summing up, the multi-slice equivalent of Eq.~6! in Ref.8 yields

w>wopt5U12

2p

DaHU5u12 f Ht rotu,

w>w res5wopt1p1F2Damaxwopt

DaH, ~3!

w<w triv5

p1F

DaH, w,1.

Here, as compared to 180°CI, we have dropped the relationwhich ensures that data gaps due to forbidden ranges will bemuch smaller than the slice thickness, since due to the multi-slice acquisition these gaps are no longer existent.

Equation~3! ensures optimal data scanning, i.e., data stillmissing after one rotation will be at the right position duringthe next rotation, and it excludes trivial cases, i.e., caseswhere all data are allowed or cases wherew is large enoughto collect complete data during less than one cardiac cycle.Of coursef H is assumed to be locally constant, i.e., during2nmax rotations.

The important parameter in cardiac imaging is the relativetemporal resolutionw of the algorithm and not the effectivescan timeteff , sinceteff does not describe the fraction of theheart cycle that is depicted in the image. The relative tempo-ral resolutionw is the portion ofR –R that contributes to acertain reconstruction. For examplew515% means that thealgorithm was able to depict the cardiac motion to within15% of R –R.

Figure 2 depicts the system of Eqs.~3! for nmax54 andF50 ~central ray!; we have introduced the abbreviationf5(p1F)/2p for the sake of convenience. The shaded areastogether with the bold lines represent the allowed and non-trivial settings forw. Obviously the minimal achievable rela-tive temporal resolution isf /nmax and it can be achieved onlyif the heart rate and the scanner’s rotation time satisfy thecondition f Ht rot516 f /nmax. For all other values off Ht rot

the temporal resolution is worse than this minimal value.As an example let us assume thatM54, t rot50.5 s, and

z5d5S, i.e., nmax54. Then only patients with a heart rateof 105 or 135 min21 would give the optimalw512.5% ~thecorresponding absolute temporal resolution regarding thecentral ray would beteff'71 ms andteff'56 ms, respec-tively!. Other heart rates~which are quite probable! would bereconstructed with a relative temporal resolution worse than12.5%. From this point of view a high rotation speed is notalways optimal. It would rather be desirable to have a freechoice oft rot , at least within a certain range. This, unfortu-nately, is not the case with scanners available today.

Nevertheless, high rotation speeds offer the chance to skipone rotation until the next allowed data can be acquired: Ifthe patient’s heart rate is relatively low, it might be the casethat during thenmax rotations there are onlynmax/2 alloweddata intervals available. This would correspond to substitut-ing

wopt5u12 f Ht rotu→wopt5u122 f Ht rotu.

Graphically, this substitution appears as scaling down theplot in Fig. 2 by a factor two. Similarly, there are situations



where it might be necessary to skip even two or three rota-tions ~up to nmax21!. Then the analogue to the equationabove are the substitutions

wopt5u12 f Ht rotu→wopt5u123 f Ht rotu

]

wopt5u12 f Ht rotu→wopt5u12~nmax21! f Ht rotu.

Figure 3 depicts the minimum of the original graph and thescaled versions. Moreover, we have included the case of veryhigh heart rates where it might be necessary to ‘‘reorder’’ theallowed intervals in order to minimize the distance to the

reconstruction plane. An analytic description of complicatedsituations like the one described would be very complex dueto the large number of different cases that have to be consid-ered. Thus we will not try to analyze each possible case. Aswill be shown below~Fig. 5 in the results section!, our ana-lytically derived minimal width settings and the correspond-ing graphs describe the real situation very well.

The meaning of Fig. 3 is the following: The graph depictsthe relative widthw as a function of heart rate and rotationtime ~to be more precise,w is a function off Ht rot!. As can beclearly seen, the absolute minimal widthw5 f /nmax can bereached only for a few settings off Ht rot . For other combi-

FIG. 2. Illustration of the conditions set in Eq.~3!. The shaded areas together with the bold lines are the allowed values forw depending onf Ht rot . The smallestpossible width settings~assuming thatf Ht rot.1/n! for fixed n5nmax arewmin5f/n. They occur wheneverf Ht rot516 f /n. The lower bounds correspond to theparabolic functionw resand thev-shaped functionwopt , respectively. Upper bounds arew triv and 1. The plot is drawn forF50 ~central ray!, d5S, andM54,i.e., p51/4. The gray bars at the top map the interval@50, 120# min21 of typical heart rates for five different rotations times ranging from 0.3 to 0.75 s ontothe abscissa.

FIG. 3. Illustration of the conditions set in Eq.~3! in combination with its scaled versions including the case of high heart rates. Only the minimal possiblewidths for a givenf Ht rot are plotted. The dashed lines correspond to thenmax54 different functionswopt that have to be taken into account~see text!. Theguiding lines~dotted lines! are the ones of Fig. 2. We left out the corresponding dotted lines of the scaled versions for the sake of clarity. The dashed-dottedline depicts the minimal achievable temporal resolutionw in the case ofnmax51, i.e., when the complete data has to be acquired during one rotation.



nations of heart rate and rotation time, the relative temporalresolutionw will not reach the optimal valuef /nmax. Thegraph of 180°MCI lies almost always below the dashed-dotted line which corresponds to a partial scanreconstruction—this is a reconstruction using only 180°1F of contiguous data such as 180°MCD~presented below!and the other partial scan approaches that have been re-viewed in the introduction section—and consequently thetemporal resolution of 180°MCI is superior to a partial scanreconstruction. Only for those cases where 180°MCI mustcollect all data during one heart beat the temporal resolutionis the same as for a partial scan approach and the two graphscoincide. Phase selectivity, in addition, is always inherentlygiven.

As has just been shown, the algorithm does not yield theminimal relative temporal resolution for allf H . When this isabout to occur the best workaround is to simply choose adifferent rotation time. Choosing a different rotation timecorresponds to a scaling of the plot along the abscissa. Ift rot

was freely adjustable one could, for any arbitrary heart rate,reach the optimal relative temporal resolutionf /nmax ~whichhas already been shown in Ref. 8!. Unfortunately only a fewdiscrete values oft rot are selectable on medical CT scanners.

On the SOMATOM Volume Zoom there is a 0.5 s and a0.75 s scan mode available; higher rotation times are not ofinterest here as they would not allow for sufficient volume

coverage. Figure 4~a! shows the consequences of using eitherthe 0.5 s or the 0.75 s rotation: only for heart rates around 90min21 the slower scan mode has an advantage over the highspeed mode. The reason why we hardly gain improvementsby additionally considering the 0.75 s mode is the following:since the greatest common divisor of 50 and 75 is quitelarge—to be more precise, gcd(50,75)525—and since theunwanted peaks of the graph represent resonance phenomenaof heart rate and rotation time, the probability that these phe-nomena occur in both, the 0.5 s and the 0.75 s scan mode isvery high.

To show that an alternative, slower scan mode can beadvantageous, if chosen properly, we have added Fig. 4~b!

which gives the relative temporal resolution of a hypotheticalCT scanner that possesses both a 0.5 s and a 0.6 s rotationmode. As compared to Fig. 4~a! the additional 0.6 s modeimproves the achievable relative temporal resolution consid-erably while still allowing for high volume coverage. Forexample, the resonance peak at 80 min21 has vanished com-pletely due to the use of the 0.6 s scan mode.

Moreover, the algorithm 180°MCI can also benefit fromlower rotation times such as 0.4 or 0.3 s. This can be easilyseen from Fig. 3 which contains five gray bars. These barsdepict the location of the interval@50, 120# min21 of typicalheart rates for the rotation timest rotP$0.3,0.4,0.5,0.6,0.75% s. Although for increasing rotation speed this interval

FIG. 4. Relative temporal resolution of 180°MCI as a function of the patient’s heart rate when~a! 0.5 and 0.75 s and~b! 0.5 and 0.6 s scan mode are available.The dashed line corresponds to the 0.75 and 0.6 s mode, respectively. The dotted line@in Fig. 4~a! only visible around 90 min21# represents the 0.5 s mode.The solid line is the minimum of both and thus represents the minimal achievable temporal resolution.



moves more and more to the left of the graph~since f Ht rot

decreases! the linear region, wherew triv dominates and thecomplete data must be acquired during one heart beat, is notreached yet. Thus 180°MCI will be still advantageous over asimple partial scan reconstruction even for shorter scantimes.

2. Prediction of 180°MCI slice sensitivity profiles

The analytical derivation of 180°MCI’s slice sensitivityprofile is far from trivial due to the normalization conditionin Eq. ~2a! imposed on the weight functionswdist(z) andwphase(c). Choosing the weights as triangular functions, asproposed above, does not simplify the situation significantly.The normalization can, in principle, only be donea poste-riori, i.e., after knowing how many data points really con-tribute to each triangle. These difficulties arise already innoncardiac spiral multi-slice algorithms, which might be areason why no analytically calculated SSPs for standard spi-ral multi-slice interpolation algorithms are found in the lit-erature~for example Ref. 23 ends up with a convolutionequation for the SSP which is not or cannot be further evalu-ated analytically!.

In principle, the projection data of a delta peak located atthe origin,

P~b,a,m !5d~b !dS a1mDaS2

1

2~M11!DaSD * II S a

DaSD

5d~b !II S a

DaS1m2

1

2~M11! D ,

must be inserted into Eq.~2! and integration with respect toa andb must be performed:

SSP~zR ,cR!5E da (kPZ

m51,...,M

wkmII S ak

DaS1m2

1

2~M11! D .

Here, we have assumed the original slice profile to be ad-equatly represented by a rectangle function. Since there is noway known to us to circumvent the normalizing problemanalytically we will present numerical results in the resultssection.

3. Image noise of 180°MCI

Due to the filtering technique the cardio interpolation usesat leastp1F data on either side of the reconstruction planefor reconstruction. The exact amount of data contributing toan image depends on both the width of the cardiac weightsand the width of thez-filtering weight. An analytic descrip-tion of this rather complicated situation cannot be given.Nevertheless, the number of data points cannot fall below thevalue used for the standard single-slice algorithm 180°LI andthe interpolation used by 180°MCI is a two or more pointinterpolation with equally distributed positive weights on theaverage whereas the 180°LI interpolation uses two pointsalways. Thus the image noise will be equal to or lower than180°LI which is thoroughly discussed in Refs. 3,4,22,24.

Moreover, as long as the minimal possible setting for thecardiac width is used—a situation that will frequently occursince the algorithm’s aim is to minimizew—the used amountof data will approach the 180°LI value. Thus the image noiseof 180°LI cannot only be regarded as an upper limit but alsoas a good approximation to the noise of multi-slice cardiointerpolation.

C. Algorithm 180°MCD

This algorithm~multi-slice cardio delta! is the multi-sliceequivalent of the 180°CD algorithm:8 It aims at reducing theeffective scan time by doing a partial scan reconstruction. Inthe spiral case this corresponds to a next-neighbor interpola-tion since the data are taken from the slice which is closest tozR . A range ofp1F contiguous data is needed to gain acomplete data set. Nevertheless, only 180°1d with d,Feffectively contribute to the reconstruction of the heartitself.8 Of course, the considerations on scan time will be thesame as in the single-slice case: since 180°1d data contrib-ute to the heart the effective scan time will be slightly largerthan 1

2t rot .The main advantage of the multi-slice technology con-

cerning the 180°MCD algorithm is that the next-neighborinterpolation can now become phase-selective. As long asthe detector array needs longer than the duration 1/f H of oneheart cycle to pass a givenz-position zR , a next-neighborz-interpolation can be centered about the given cardiac phasecR . We will denote the corresponding view angle aboutwhich the partial scan reconstruction shall be centered byaC

and we havec(aC)5cR . Since it might be the case that thiscenter-view exactly lies at the edge of the overlapping inter-val between the detector andzR it does not suffice to demandd/ f Ht rot<MS as it is the case for 180°MCI. An additionaldata range of12(p1F) must rather be available and thus thecorresponding restriction for the scan parameters is

d•S ~ f Ht rot!21

1

1

2

p1F

2pD<MS. ~4a!

However, our implementation of 180°MCD allows for ex-trapolation at the outermost slices to be able to do withoutthis additional data range. This feature yields the less restric-tive pitch restriction

d<MSf Ht rot . ~4b!

The partial scan approach 180°MCD requires a set ofp1F contiguous data. We will assume that the detector arraydoes not completely pass a givenz-position during thisp1F angular increment. This ensures that the maximum dis-tance of the center of a detector line to the reconstructionpositionzR does not exceedS/2. We get

d<MS2p

p1F. ~4c!

In most cases Eq.~4c! is less strict than Eq.~4a!. For ex-ample using our settingst rot50.5 s andF552° we find thatas long asf H<90/29•120 min21'372 min21, i.e., for virtu-ally any patient, Eq.~4a! is the dominating pitch restriction.



As an example assumef H560 min21, t rot50.5 s, andF552°. Then the table increment is restricted by Eq.~4a! tod&1.72S or, equivalently,p&0.43. Using Eq.~4b! yieldsd<2S and p<0.5. Increasing heart rates allow for higherpitch values.

The algorithm 180°MCD can be given as

PMCD~b,a,zR ,cR!

5 (kPZ

m51,...,M

wkmP~bk ,ak ,m !Y (kPZ

m51,...,M

wkm , ~5a!

with

wkm5wkm~zR ,cR!5II S ak1bk2aC

pD II S z~ak ,m !2zR

S D .

~5b!

The weights are designed to contribute the necessary 180°data range, i.e.,aC2

12p<ak1bk,aC1

12p and only the

nearest-neighboring detector lines, i.e.,zR212S<z(ak ,m)

,zR112S.

1. Prediction of 180°MCD slice sensitivity profiles

Inserting the projection data of the delta peak

P~b,a,m !5d~b !dS a1mDaS2

1

2~M11!DaSD * II S a

DaSD

5d~b !II S a

DaS1m2

1

2~M11! D

into Eq. ~5! and integrating overa andb yields

SSP~zR ,cR!5E da (m51,...,M

II S a2aC

pD

3II S z~a,m !2zR

S D II S a

DaS1m2

1

2~M11! D

5E da (m51,...,M

II S a2aC

pD II S a

DaS1m

2

1

2~M11!2

zR

S D II S a

DaS1m2

1

2~M11! D

5E da (m51,...,M

II S a2aC

pD

3II S a2aR~m !

DaSD II S aR

~m !1aR

DaSD ,

where we have used

aR52pzR

d

and

aR~m !

5aR2mDaS112~M11!DaS ;

aR is the angle under which the center of the detector array

reaches the reconstruction position, whereasaR(m) is the view

angle of the center of slicem reachingzR .The product of the second and third rectangle function

can be simplified and the slice sensitivity profile becomes

SSP~zR ,cR!5E da (m51,...,M

II S a2aC

pD

3II S a2aR~m !

112 aR

~DaS2uaRu!∨0D

5 (m51,...M

II S aC

pD * II S aC2aR

~m !1

12 aR

~DaS2uaRu!∨0D

5 (m51,...,M

IIp,~DaS2uaRu!∨0** ~aC2aR~m !

112 aR!.

Evidently, a further analytic description of the SSP in fullgenerality is not very instructive. Nevertheless, the integrandabove indicates that there are certain pitch values that allowfor simplification: for DaSPp/N the shifts mDaS of thesecond rectangle function@given byaR

(m)# exactly divide thewidth p of the first rectangle. Consequently, the rectangle ofwidth p overlaps@as long as Eq.~4! is fulfilled# exactlyp/DaS rectangle functions of width (DaS2uaRu)∨0 whichgives a total area of overlap of

SSP~zR ,cR!5

p

DaS„~DaS2uaRu!∨0…

5pLS aR

DaSD5pLS zR

S D .

This value is independent ofaC and thus independent of thecardiac phasecR . The slice quality descriptors—exact fordP2SN under the restriction Eq.~4!—can now be stated

FWHM5S51.00S,

FWTM595 S51.80S,

FWTA52~12A1/10!S'1.37S,

SPQI53/4575.0%.

For pitch settings other than those derived above, the slicesensitivity profile will depend oncR . The reason is that the~now nonintegral! overlap of the rectangles will be modu-lated by the value ofaC . Numeric results will be shown inthe results section.

D. Image noise of 180°MCD

180°MCD is a partial scan reconstruction which meansthat in the center of the image only half of the data contributeto each image as compared to 180°MLI. Thus, in comparisonto a conventional step and shoot 360° CT scan, the imagenoise is expected to rise by a factor of& ~360° data areredundant by a factor of 2 as compared to the 180° partialscan reconstruction!.



IV. METHODS


To confirm our ~quite complicated! theorectical resultsconcerning the temporal resolution of 180°MCI we havesimulated situations corresponding to heart rates rangingfrom 40 to 150 min21 and regarded the algorithm’s output ofthe relative temporal resolutionw. The simulated scan pa-rameters areM54 andd5S.

Relative temporal resolution of 180°MCD has been cal-culated asw5

12 f Ht rot since only half a rotation contributes to

each image, i.e.,teff512trot .

B. Slice sensitivity profiles

Since the profiles cannot be determined analytically wewill rather present numerical results. SSPs and their figuresof merit are taken directly from our implementation of180°MCI and 180°MCD where they are calculated automati-cally during eachz-interpolation process in a spinoff manner.

C. Virtual motion phantom

To compare the approaches presented in this article wehave simulated the heart phantom for various heart ratesranging from 50 to 135 min21. The simulation parameterswere d5S51 mm andM54. The dedicated cardiac algo-rithms are compared to the standardz-interpolation algorithm180°MLI which we modified to use only one of theM slicesfor reconstruction. Axial as well as multiplanar reformations~MPRs! have been investigated.

D. Image noise comparison

The reconstructed volumes of the simulated motion phan-tom have been used to study the noise behavior relative tothe standard algorithm 180°MLI. An ROI~region of interest!was placed in the area above the calcifications in the heart tomeasure the noise as the standard deviation of the CT values.Heart rates ranging from 40 to 135 min21 have been consid-ered for the image noise comparison.

E. Coronary calcium scoring

To quantify the performance of the three algorithms, wehave calculated a coronary calcium score~CCS! for the 3mm calcifications. This was done as follows: for a completereconstructed volume of the virtual motion phantom~corre-sponding to the volume shown in the results section below inFigs. 8 and 9! we placed ROIs around each calcification andintegrated over all pixels above 130 HU. The underlyingheart~50 HU! was taken into account by subtracting 50 HUfrom each pixel value prior to the integration. It must benoted that the object of this article is not to propose or testcalcium scoring algorithms; we simply chose the 130 HUthreshold since it is used for the well-known Agatstonscore.25 We are aware of the fact that this kind of threshold-ing is not optimal but the volume score used here suffices forour purposes. Moreover, we are not presenting the absolutescores achieved but rather the scores of the moving calcifi-

cations relative to their motionless counterparts. For eachheart rate, algorithm, and phase~slow, medium, and highmotion! three numbers result: the relative score of the mov-ing cylinder and the relative scores of the spheres moving inx- and inz-direction.

F. Patient study

A patient study consisting of 25 patients scheduled forcardiac chest exams was carried out to test the dedicatedcardiac algorithms using clinical data. Informed consent wasobtained. Heart rates ranged from 50 to 120 min21. Theclinical data were reconstructed with the dedicated recon-struction algorithms 180°MCD and 180°MCI and the stan-dard algorithm 180°MLI to evaluate the cardiac algorithms.

G. Dose considerations

To investigate patient dose and image noise values moreprecisely we have calculated the relative temporal resolutionw and the expected dose increase factors for both prospectiveand retrospective gating for a wide range of heart ratesf H

and possible table incrementsd. For the prospective trigger-ing we assume the reconstruction phasecR to be fixed priorto the scan and we assume the radiation to be switched ononly for those projections that are used for reconstruction.One would then be able reconstruct images at arbitraryzR

but fixedcR .The dose increase factors were calculated as follows. As a

reference algorithm we use 180°MFI~and not 180°MLI!since 180°MFI uses data redundancies to accumulate dose.This algorithm is the standard algorithm of multi-slice scan-ners and thus dose comparisons should be done relative to180°MFI. We simulated noisy rawdata which then were re-constructed using 180°MFI and 180°MCI for various tableincrements and heart rates. Image noisesMFI andsMCI wascorrected by multiplying with the square-root of the accor-dant effective slice thickness~FWHM of the SSP! Seff,MFI

andSeff,MCI . The relative dose factor for retrospective gatingwas then calculated as the square of the noise ratios

D rel5sMCI

2 Seff,MCI

sMFI2 Seff,MFI

.

Multiplying this value by the fraction of data used to recon-struct exactly one constant cardiac phasecR5const gives therelative dose value for prospective gating since, as men-tioned before, here we assume the radiation to be switchedon only for data needed for reconstruction. Although thisfeature is not readily available on commercial scanners thecorresponding values may be of interest since they quantifythe data utilization of 180°MCI.

V. RESULTS


To validate the theoretical results of Fig. 4 we have in-cluded a plot of the relative temporal resolution achieved byour implementation of 180°MCI in Fig. 5. These resultsagree well with the theoretical description. The steps in the



graph are due to the fact that our implementation allows onlyfor integer values of the cardiac phase. The other deviationsfrom the theoretical curve result from those cases where theselection of data requires a complicated reordering of al-lowed data ranges. Those situations were excluded from thetheoretical consideration as has been mentioned above.

The dashed-dotted line of Fig. 5 corresponds to the rela-tive temporal resolution achieved by 180°MCD. Its value isworse than 180°MCI’s value for almost every heart rate ex-cept for the resonance cases.

B. Slice sensitivity profiles

1. SSPs for 180°MCI

Some examples of slice sensitivity profiles are shown inFig. 6. Obviously, the profiles vary dependent on the recon-struction phasecR . Nevertheless, since these variations arenot too instructive we have picked out four cases: Fig. 6~a!

shows the casef H5135 min21 where the highest temporalresolution of 56 ms is achieved. The profile shows only littlevariations for varyingcR . Figures 6~b!, 6~c!, and 6~d! givean example of how the pitch influences the shape of the SSP,given a constant heart rate~here: 80 min21!. The plots givenare similar to plots of other typical heart rates and table in-crements which is the reason why only these four represen-tative figures are shown.

Obviously, the profiles are unsymmetrical and biased toone side for a given reconstruction phasecR . However, thesedistortions are of the order of one magnitude lower than theFWHM of the profiles and thus less than one-tenth of a reso-lution element inz-direction. Moreover, the bias is zero onaverage~regarding all reconstruction phases 0<cR,1 for aconstantz-positionzR or, equivalently, regarding all possiblez-positionszRPR for a given reconstruction phasecR!. Thusthe fact that the profiles are locally biased and unsymmetricalwill have no impact on image quality.

Evaluating the SSPs for all cardiac phasescR and a num-ber of typical table increments (S<d<2.5S) and a range oftypical heart rates~50 to 120 min21! has shown that the

FIG. 5. Relative temporal resolution achieved by our implementation of180°MCI ~bold line! plotted from 40 to 150 min21 for the 0.5 s scan mode~compare to Fig. 4!. As has been predicted, the minimal relative temporalresolution ofw512.5% can be reached only for certain heart rates.M54,d5S. The theoretical relative temporal resolution of 180°MCD has beenincluded for comparison purposes~dashed-dotted line!.

FIG. 6. 3D-plots of 180°MCI’s SSP for the casef H5135 min21 andp51/4 yielding the highest achievable temporal resolution of 56 ms~a! and for the typicalheart ratef H580 min21 at varying pitch values~b!, ~c!, ~d!. The z-resolution, measured by the FWHM of the SSP, lies in the order of 1.3S and the profile’sSPQI is about 83%.



individual profiles are quite similar, regarding their figures ofmerit. We found FWHM'1.3S, FWTM'2.3S, and SPQI'83% ~the slice profile quality index SPQI is the area withinFWHM divided by the total SSP area and describes howclose the profile’s shape is to the ideal rectangle!.3 The varia-tions with respect tocR , f H , andd lie in the order of 10% ofthese values.

2. SSPs for 180°MCD

Figure 7 gives an impression ford532S andd5

52S, pitch

settings which yield SSPs varying withaC . The 3D plot~afunction of both z and aC! shows that deviations from aperfect triangle function will occur. Nevertheless, these de-viations are negligible as compared to the single-slicealgorithms8 and the above figures of merit are a good ap-proximation. Moreover, SSP(0,cR)5p for any value ofcR

and arbitrary table incrementd. This means that the achiev-able contrast does not depend on the cardiac phase.

C. Virtual motion phantom

Figure 8 shows for each simulated heart rate six images~two rows, three columns!. The upper row of images corre-sponds to the slow motion phase. The lower row is located atthe high velocity phase of cardiac [email protected]. Fig. 1~b!#. Thethree columns correspond to thez-interpolation algorithms180°MLI, 180°MCI, and 180°MCD. The standard linear in-terpolation algorithm is not ECG correlated. Thus we havechosen the reconstruction positionzR such that the upper rowdepicts the best case regarding motion artifacts and the lowerrow shows the worst case. In general, this reconstructionposition does not coincide with the center (z50) of thespherical calcifications~the 333 rightmost calcifications!and consequently some of them may not be depicted.

Each image is annotated with the relative temporal reso-lution w and the absolute temporal resolutionteff . Thesenumbers arenot taken from the theoretical results given inthe previous sections but rather were calculated as theFWTM of the phase sensitivity profile, PSP,—the profilewhich contains the cardiac phase weighted according to itscontribution to the central ray of the image—directly by the

z-interpolation algorithm. Only for the standardz-inter-polation 180°MLI these numbers were calculated theoreti-cally: the PSP of 180°MLI is a triangle function of full widtht rot50.5 s and thus the FWTM is given byteff50.9t rot

5450 ms. The corresponding relative temporal resolutionwas calculated asw5teff fH .

As can be easily seen, for low heart rates all algorithmsare able to depict the slowly moving calcifications withoutapparent motion artifacts~upper row!. Nevertheless, the re-construction in the high motion phase~lower row! showsdifferences between the algorithms even for low heart rates:180°MCI and 180°MCD are preferable over the standard re-construction.

For all simulated heart rates, ranging from 50 to 135min21, the cardiac algorithms are—not surprisingly—superior to the standard linear interpolation. Especially180°MCI shows good performance for high heart rates, evenfor tachycardic situations (f H.100 min21). It must bepointed out again that for the standard algorithm 180°MLIwe chose thez-position to yield either best or worst imagequality whereas for the dedicated cardiac algorithm the im-age quality remains the same throughout the complete vol-ume.

Figure 9 demonstrates the performance of the standardand the cardiac algorithms in thez-direction by means ofmultiplanar reformations~MPR!. It can be clearly seen that180°MLI does not display the objects continuously. Espe-cially the sphere moving alongz appears to be noncontigu-ous. The cardiac algorithms, in contrast, display the calcifi-cations as expected. Only in the high motion phase problemswith the sphere moving inz-direction become apparent. Nev-ertheless, 180°MCI displays the respective sphere better inthe tachycardic mode~105 min21! than it does in the 60min21 mode. This is a result of the said resonance phenom-enon: During one 60 min21 heart cycle the scanner rotatesexactly twice and thus cannot gain any new information. Theeffective scan times of 180°MCI and 180°MCD are the samein this case and the only major difference between these twoalgorithms is the image noise. This is confirmed by the 60min21 high motion phase images of 180°MCD and

FIG. 7. 3D-plots of 180°MCD’s SSP ford51.5S and 2.5S as a function of thez-position and the cardiac phasecR which corresponds to the reconstructionview centeraC . SSPs are normalized to@0, 1#. Sinced¹2SN the plots deviate from the perfect triangular SSP that has been derived in the text. Thesedeviations become negligible for larger table increments. The plot-range ofaC was chosen to be@2p, p# to show all features of the SSP and to comply withEq. ~4! ~assuming a heart rate of 60 min21! for both pitch settings,d51.5S, and 2.5S, andM54.



FIG. 8. Images of the cardiac motion phantom for a wide range of heart rates.~0/500!.



180°MCI. The only remedy for this situation would be tochose a slightly different rotation time, as has been men-tioned previously.

D. Image noise comparison

These resonance cases are also manifested in the imagenoise values of Table I. As has been predicted, 180°MCDshows slightly increased noise values as compared to180°MLI: 180°MCD increases noise by& and 180°MLI byA4/3 as compared to a step-and-shoot scan,22 thus the ex-pected value of 180°MCD relative to 180°MLI isA3/2'122%. 180°MCI’s image noise lies below 180°MLI. Nev-ertheless, for certain heart rates 180°MCI’s image noise isvery close to 100%, i.e., forf H5105 min21 and for f H

5135 min21, whereas for heart rates which are in resonancewith the scanner’s rotation time~here: t rot50.5 s! it is sig-nificantly lower ~60, 80, and 120 min21!. Here it becomes

clear that as soon as the resonance case occurs 180°MCI usesredundant data for averaging and thus for noise reduction.On the other hand, for heart rates which allow to reduce therelative temporal resolution by collecting data from differentcardiac phases during successive rotations no averagingtakes place, and image noise increases towards the imagenoise of 180°MLI.

Of course we have done the noise comparison using thesame spatial in-plane resolution for all algorithms. For theclinical images below, however, we decided to use the re-dundant data~opposing rays! available for 180°MLI and180°MCI combined with the quarter detector offset of ourscanner to increase the spatial resolution of the images in-stead of using it to reduce noise. For this reason the180°MCD images of the clinical cases presented below willappear smoother and less noisy than the 180°MLI and180°MCI images.

FIG. 9. Coronal MPRs through the 3 mm calcifications for 60 and 105 min21. Since 180°MLI is not phase-correlated the MPR depicts the movingcalcifications in a sinusoidal fashion. The cardiac algorithms perform much better. For the medium motion phase it can be seen that the moving calcificationsare displaced by the full amplitudeA55 mm of the motion function from their origin. For the high heart rate only 180°MCI shows a good image quality. Thisis due to its high temporal resolution.~0/500!.



E. Coronary calcium scoring

The results obtained by our volume scoring algorithm aregiven in Table II. For each algorithm~180°MLI, 180°MCI,and 180°MCD! each motion phase~slow, medium, and high!and each heart rate~from 40 to 135 min21! three numbers aregiven: the relative score of the 3 mm cylinder and the tworelative scores of the 3 mm spheres, one moving inx- andone moving inz-direction. The ideal score is 100% and thosevalues which are superior to the standard approach 180°MLIare underlined.

From these numerical results we can conclude that thecalcium scoring will yield better results~less deviation fromthe optimal 100%! for the cardiac algorithms than for thestandard algorithm. Especially in the slow motion phase theCCSs for 180°MCI and 180°MCD are clearly superior. How-ever, a general underestimation of the true~100%! score bythe standard algorithm 180°MLI must not be deduced fromTable II. Depending on the motion function used and de-pending on the scan start it may also be the case that thestandard algorithm overestimates the true calcium score.19

F. Patient study

Examples for patient data are presented in Figs. 10~lowheart rate! and 11~high heart rate!. The two dedicated car-diac algorithms are compared to the standardz-interpolationthere. We simply regard the four-slice scan as being foursingle-slice scans with a temporal offset oft rotS/d50.33 s.By picking out each one of the four slices separately andapplying 180°LI we can reconstruct images at four temporallocations, separated by 0.33 s each~left column!. The tem-poral spacing of 0.33 s corresponds to an offset in cardiacphase of 28% for the 51 min21 patient and to 52% for the 95min21 patient. For both patients the image quality of thestandard reconstruction is surprisingly good, which is due tothe low rotation speed of 0.5 s. Nevertheless, especially theleft ventricle shows double contours due to the cardiac mo-tion in Fig. 10 ~for m51, 2, and 4! and in Fig. 11 motionartifacts become apparent in the right ventricle~m52, 3, and4!. The most severe finding is that the display of the com-plete volume by 180°MLI is discontinuous and shows ex-treme stepping artifacts in the MPR~bottom!. 180°MLI isnot phase selective and thus good transaxial image quality isgiven only by chance; an acceptable display of completevolumes is not possible and the standardz-interpolationshould not be used for 3D displays.

The partial scan approach 180°MCD~middle column!,reconstructed atcR50%, 25%, 50%, and 75%, improvesthis situation significantly. Especially for the low heart ratepatient ~Fig. 10! the image quality is superior to 180°MLI:there are no double contours in the axial images and theMPR shows almost no stepping artifacts. In Fig. 11, how-ever, the heart rate~95 min21! is too high for 180°MCD andthe MPR, although much better than 180°MLI, shows dis-continuities. These discontinuities result from the fact thatthe tube positionsaC , around which the partial scan is cen-tered, vary from heart beat to heart beat. Consequently, mo-

TABLE I. Noise values obtained for the virtual cardiac phantom relative tothe value obtained in 180°MLI.p51/4.

f H 180°MCD 180°MCI

40 123% 69%50 125% 83%60 124% 57%70 127% 81%80 125% 56%90 126% 71%

105 126% 89%120 125% 42%135 126% 91%

TABLE II. Coronary calcification score for various heart rates. Motion phase ‘‘slow’’ corresponds tocR580%, ‘‘medium’’ is located atcR510%, and‘‘high’’ is centered aboutcR530%. The triples,A B C, are relative numbers~given in %! as follows:A is the relative CCS of the moving and the stationary3 mm calcium cylinder,B is the score of the 3 mm sphere moving inx relative to the score of the stationary 3 mm sphere, whereasC is the relative scorefor the sphere moving inz-direction. Values greater than 100 mean that too much calcium has been detected. Underlined values denote that the respectivecardiac algorithms score equal or better than the standard algorithm. The average and standard deviation values are given additionally.

f H

Undef. Slow Medium High

180°MLI 180°MCI 180°MCD 180°MCI 180°MCD 180°MCI 180°MCD

40 92 88 84 99 99 103 101 101 102 99 96 89 101 103 95 96 94 58 105 100 8650 80 82 67 100 102 97 101 102 99 99 100 92 100 96 97 101 76 62 117 102 6160 88 91 59 99 101 95 100 94 94 98 93 89 100 99 97 103 82 21 109 104 4570 88 85 49 99 93 103 97 87 90 100 94 91 107 106 93 94 79 65 119 100 3680 87 85 53 99 100 89 99 95 85 105 100 85 109 109 92 122 79 62 127 99 3890 90 84 45 100 100 96 101 95 73 95 103 101 120 104 84 77 55 45 121 97 23

105 9 1 91 38 101 97 100 110 87 60 99 99 93 130 115 72 83 68 77 115 104 41120 8 3 74 39 91 86 47 95 93 55 120 82 52 135 101 67 112 74 14 136 110 33135 8 3 75 41 101 104 97 114 97 52 99 96 90 132 96 62 84 65 67 114 89 40

Mean 87 84 53 99 98 92 102 95 79 102 96 87 115 103 84 97 75 52 118 101 45

Sigma 4 6 14 3 5 16 6 5 18 7 6 13 14 6 13 14 10 20 9 5 17



tion artifacts that depend on both, the cardiac motion and thecurrent view direction, will vary throughout the volume andresult in these stepping artifacts.

180°MCI ~right column of Figs. 10 and 11!, in contrast,achieves very good image quality for all heart rates. It is ableto correctly depict the anatomic structures of the heart, thecalcifications, and the coronary arteries. Motion artifacts aregreatly reduced, even for high heart rates, and the multipla-nar reformations are of high quality. Phase selectivity to-gether with high temporal resolution, which is especially im-

portant for patients with high heart rates, can be achieved by180°MCI.

Another example for phase selectivity is demonstrated byFig. 12 which depicts the heart throughout a complete cyclein steps of 15%. In the plane shown the cardiac motion caneasily be traced. Moreover, it becomes clear that there is amotion component perpendicular to the image plane. For ex-ample a calcification begins to appear at the second half ofthe cardiac cycle. Except for image noise there is hardly anydifference between 180°MCD and 180°MCI in this case,

FIG. 10. Patient with 51 min21 reconstructed at constantz-position. For the standardz-interpolation 180°MLI~left column! we have picked out the measuredslicesm51 throughm54 ~left column from top to bottom! and thus are able to reconstruct four images at the same position with an temporal offset oft rotS/d50.33 s. For the cardiac algorithms 180°MCD~middle column! and 180°MCI~right column! the reconstructions were centered about the cardiacphases 0%, 25%, 50%, and 75%. The MPRs at the bottom demonstrate the performance of the algorithms for the complete volume. Collimation 431 mm, d51.5 mm. ~0/500!.



since the heart rate is close to the resonance casef Ht rot

51/2, i.e., close to 60 min21 ~c.f. Figs. 3, 4, and 5!. We thushave simply omitted the MCI images here.

Phase constancy, as given in Fig. 13 for the same patient,is important for 3D displays and, especially, for the quanti-fication of coronary calcium. The images are reconstructed atfive adjacentz-positions separated by 1 mm. The standardz-interpolation shows the calcification only in one image. Incontrast, the calcification is depicted clearly and with only

few artifacts by 180°MCD. Although quantification of coro-nary calcium is beyond the scope of this article one caneasily tell that the 180°MLI reconstruction will in this spe-cific case yield a CCS that is too low. This has two reasons:~a! the contrast of the calcification is reduced due to themotion artifacts~e.g., atzR50 mm! and~b! the calcificationand the table probably move in opposite directions at thatmoment and the calcification is depicted in only one of theslices. Of course the opposite may happen as well. If the

FIG. 11. Patient with 95 min21 reconstructed at constantz-position. For the standardz-interpolation 180°MLI~left column! we have picked out the measuredslicesm51 throughm54 ~left column from top to bottom! and thus are able to reconstruct four images at the same position with an temporal offset oft rotS/d50.33 s. For the cardiac algorithms 180°MCD~middle column! and 180°MCI~right column! the reconstructions were centered about the cardiacphases 0%, 25%, 50%, and 75%. The MPRs at the bottom demonstrate the performance of the algorithms in the complete volume. Collimation 431 mm,d51.5 mm. ~0/700!.



calcification moved in scan direction it could possibly beimaged too often and would thus lead to an overestimation ofthe real calcium score. The real score can obviously only beobtained using a phase-selective image reconstruction algo-rithm such as 180°MCD or 180°MCI, a fact which is alsoconfirmed by Table II.

A high temporal resolution together with full phase selec-tivity is given especially by 180°MCI. Figure 14 again dem-onstrates this quite impressively: whereas the standard algo-rithm shows motion artifacts and thus overlapping structures180°MCI clearly depicts the aortic valve at arbitrary func-tional states. The valve is closed atcR50% and has opened

FIG. 12. Heart images as a function of the cardiac phasecR at fixedz-position. The images are centered about the cardiac phases 0%, 15%, 30%, 45%, 60%,and 75% with 180°MCD. The images are nearly artifact-free and the cardiac motion can be traced from image to image. The effective scan timeteff

5264 ms corresponds to a relative temporal resolution ofw529%. Parameters:d5S52.5 mm. Patient’s heart rate: 65 min21. ~0/500!.

FIG. 13. Heart images as a function of thez-position at fixed cardiac phasecR . The upper line shows the standardz-interpolation 180°MLI for five differentz-positions. Obviously the anatomy is depicted in different, undefined phases of cardiac motion. A coronary calcification measurement in these imageswouldyield the wrong score, since the calcification is depicted clearly in only one of the images. Further on, the score would be dependent on the absolute time scale,i.e., on the scan begin. In the lower row we show the corresponding images reconstructed at 60% ofR –R with 180°MCD. Since the cardiac phase iswell-defined throughout the volume, these images can be used for the assessment of coronary calcium. As one can see, the calcification has az-extend of atleast 5 mm. Thus the correct calcium score is much higher than achieved by the standard algorithms. Effective slice thickness: 2.7 mm.d5S52.5 mm.Patient’s heart rate: 65 min21. ~0/500!.



up atcR520% ofR –R. Apparently, these images are nearlyartifact-free. A movie of the same slice can be viewed anddownloaded from http://www.imp.uni-erlangen.de/e/research/cardio/. Another example showing an oblique MPRof the aortic valve for a patient with strongly varying heartrate throughout a complete cardiac cycle will be givenelsewhere.16

G. Dose considerations

The results of the dose calculations are given in Table III.The table gives for each heart rate~from 40 to 150 min21!

and some typical table increments three numbers: the relative

temporal resolution of 180°MCI, the dose increase factor forprospective triggering, and the dose increase factor for retro-spective triggering.

As an example let us look atf H570 min21 and d51.5S. 180°MCI will yield a relative temporal resolution ofw515% which corresponds toteff5w/fH5129 ms. Assum-ing retrospective gating we find a relative dose factor of 5.8.This means, if the image noise should remain the same as for180°MFI the operator had to multiply the tube current by5.8. On the other hand, if the tube current was left at itsoriginal value ~standard scan protocol! the image noise isexpected to increase by a factor ofA5.852.4.

FIG. 14. Patient with 90 min21, reconstructed at constantz. The standard reconstruction 180°MLI shows motion artifacts~overlapping structures in the leftventricle and the aortic valve!. Moreover, the cardiac phase is undefined. The dedicated cardiac algorithm 180°MCI depicts the heart in defined phases. At 0%,the end of the diastolic phase, the aortic valve is still closed whereas shortly after the start of the systolic phase, at 20% ofR –R, it has opened. Thesereconstructions are nearly free of artifacts except for the streak appearing in the valve at 20%. Collimation: 432.5 mm, table incrementd53.8 mm.~0/700!.

TABLE III. Relative temporal resolutionw ~in %!, the expected dose increase factor for prospective cardiactriggering, and the expected dose increase factor for retrospective triggering are given as triples,A B C, for thecardiac algorithm 180°MCI for a number of table increments and heart rates. Empty entries correspond to pitchvalues exceeding the allowed limit~assumingz5S!.

df H 1.0S 1.25S 1.50S 1.75S 2.00S 2.25S 2.50S

40 min21 16 1.2 4.8 16 1.2 5.145 min21 13 2.0 9.0 16 1.8 6.9 17 1.7 6.450 min21 16 1.4 6.6 17 1.6 5.8 19 1.4 5.155 min21 12 1.4 5.9 17 1.5 4.7 20 1.4 4.3 21 1.3 3.960 min21 25 1.1 3.4 24 1.1 3.2 24 1.1 3.1 25 1.1 3.3 25 1.1 3.265 min21 14 1.4 6.7 17 1.3 4.6 20 1.2 4.2 22 1.4 3.8 23 1.3 3.670 min21 13 1.1 6.2 14 1.1 5.1 15 1.3 5.8 19 2.1 5.3 22 1.7 4.3 26 1.3 3.375 min21 18 2.2 7.7 24 1.3 4.5 25 1.4 4.9 27 1.7 4.1 28 1.5 3.7 29 1.4 3.4 30 1.3 3.380 min21 33 1.0 2.7 33 1.0 2.7 33 1.0 2.7 33 1.0 2.8 33 1.0 2.8 33 1.2 2.7 33 1.1 2.685 min21 16 2.2 7.2 25 1.3 4.1 28 1.1 3.4 28 1.1 3.4 29 1.2 3.6 30 1.4 3.1 32 1.3 2.990 min21 15 1.1 4.6 15 1.3 4.8 19 1.3 4.4 23 1.2 4.1 24 1.2 4.4 27 1.7 3.4 31 1.4 3.095 min21 18 1.1 5.2 19 1.1 4.6 20 1.1 4.6 20 1.2 4.1 20 1.3 4.3 25 1.7 3.2 28 1.6 3.0100 min21 14 1.2 6.3 16 1.4 6.3 16 1.2 5.5 20 1.6 4.1 23 1.5 3.8 29 1.5 2.7 33 1.3 2.4105 min21 12 1.6 7.2 15 1.6 4.8 21 1.3 3.9 27 1.4 3.1 33 1.4 3.0 38 1.3 2.4 40 1.3 2.3110 min21 19 1.6 4.5 28 1.6 3.6 35 1.2 2.8 39 1.2 2.3 42 1.1 2.2 45 1.2 2.0 46 1.2 1.9115 min21 41 1.2 2.4 45 1.1 1.9 47 1.1 1.9 49 1.0 1.8 49 1.0 1.8 49 1.1 1.7 49 1.1 1.7120 min21 50 1.0 1.6 51 1.0 1.6 50 1.1 1.7 50 1.0 1.6 51 1.0 1.6 50 1.0 1.6 50 1.1 1.6125 min21 41 1.1 2.3 45 1.1 2.1 47 1.1 1.9 48 1.1 1.9 48 1.0 1.8 49 1.0 1.7 49 1.1 1.8130 min21 19 1.4 4.8 28 1.3 3.1 36 1.2 2.6 39 1.2 2.3 42 1.2 2.2 45 1.1 2.0 46 1.2 1.9135 min21 13 1.4 8.3 15 1.3 5.2 21 1.4 3.9 29 1.4 3.0 34 1.4 2.9 39 1.2 2.2 42 1.3 2.2140 min21 14 1.1 6.1 16 1.3 5.6 18 1.2 4.4 21 1.6 3.7 25 1.3 3.0 31 1.2 2.7 35 1.5 2.5145 min21 19 1.0 4.7 19 1.1 4.8 21 1.1 4.7 21 1.5 4.1 21 1.3 3.6 24 1.4 3.3 28 1.6 2.8150 min21 15 1.2 4.8 17 1.2 4.3 20 1.2 4.3 23 1.3 4.0 24 1.3 4.0 26 1.7 3.8 29 1.8 3.2



Although the resulting dose values seem to be quite highit should not be forgotten that this is only the case for retro-spective gating. As can be seen from the table a prospectivegating scan would yield dose values comparable to the stan-dard method but would be limited to one prospectively se-lected cardiac phasecR . In the example above the relativedose factor for prospective triggering would be 1.3 which isonly a 30% increase when compared to standard 180°MFI.

Using 180°MCI and 180°MCD together with prospectivecardiac triggering would not significantly increase the dosevalues as compared to a standard scan since radiation wouldbe switched on only for data contributing to the image. Ofcourse prospective triggering does not allow for a retrospec-tive choice of the reconstruction phasecR and thus no tem-poral tracking of anatomic structures would be possible. Inaddition, prospectively triggered scans cannot cope very wellwith arythmic patients, a fact that is also known fromEBT.26,27 Using retrospective triggering~especially in com-bination with 180°MCI! arythmic cases pose no severe prob-lem and yield high image quality.16

VI. DISCUSSION AND CONCLUSIONS

Cardiac imaging by~nonelectron beam! CT scanning hasbeen limited due to the fact that scan times achieved bymedical CT scanners have been too long to depict the heartwithout degradation of image quality due to motion. Evenfor relatively slow motion, i.e., in the diastolic phase, stan-dard z-interpolation algorithms suffer from motion artifacts.The cardiac algorithms presented in this article show prom-ising results. The respective reconstructions are nearlyartifact-free as compared to the standardz-interpolation al-gorithm.

The cardiac z-interpolation algorithms 180°MCD and180°MCI have been shown to be superior to standardz-interpolation algorithms. This is due to the better temporalresolution and due to the phase selectivity of the cardiacalgorithms. They can effectively reduce motion artifacts in-plane and throughout the volume. For higher heart rates,however, the interpolation approach 180°MCI clearly outper-forms the partial scan approach 180°MCD. This is mainlydue to its high temporal resolution~effective scan times of56 ms are potentially possible for a four-slice scanner withd>S and t rot50.5 s for 180°MCI! and due to the weightingapproach which helps to smooth away motion and data tran-sition artifacts. In contrast to 180°MCD which yields goodresults only for heart rates&70 min21 there is no such limitfor 180°MCI. This fact is clearly demonstrated in Fig. 15.Even for this kind of 3D display a temporal tracking is pos-sible and allows for impressive displays of the beating heart~http://www.imp.uni-erlangen.de/e/research/cardio/!. How-ever, in contrast to MPRs, tedious 3D editing is required toobtain shaded surface displays~SSD!.

The cardiac z-interpolation algorithms 180°MCI and180°MCD limit the table incrementd per 360° rotation: Ac-cording to Eqs.~1! and~4! the pitchp5d/MS is restricted tothe upper limit

p< f Ht rot .

This is the maximal pitch allowed to have fullz-interpolationpossibilities at any cardiac phasecR . For 180°MCI thisvalue corresponds toz'S. Of course higher pitch values arepossible for 180°MCI ifz is chosen to be greater. Neverthe-less the demand for high temporal resolution requires over-lapping data acquisition and thus setting the pitch as low asreasonably achievable, i.e.,p! f Ht rot is desirable. Moreover,choosing thez-filter width significantly larger than the slicethickness can be avoided by choosing a larger slice thicknessprior to the scan. In this case the tube current should bereduced to reduce dose. Thus our recommendation is tochoosep' f Ht rot for 180°MCD and to usep! f Ht rot and areduced tube current for 180°MCI. Since the latter require-ment is more restrictive it must be met for cases where onewants to compare between 180°MCI and 180°MCD.

Of course an overlapping data acquisition, i.e.,p,1,means an increase in patient dose which is not always ac-ceptable and might be regarded as a drawback of 180°MCI.Nevertheless, it has been shown that, assuming a constantscanner rotation time, for many heart rates image noise issignificantly lower for 180°MCI than for the standardz-interpolation algorithm 180°MLI and for 180°MCD. Re-garding Table I it might be concluded that the image noisereductions of 180°MCI lie in the order ofA1/2 and thus thetube current could be reduced by a factor of 2 to gain noisevalues equivalent to 180°MLI. This value, however, dependson the patient’s heart rate and thus cannot be used as a stan-dard for all f H . Moreover, in case of optimal sampling~interms of relative temporal resolution! Table I shows that theimage noise is not significantly lower since no data redun-dancies occur and thus the tube current should not be re-duced. Moreover, the true noise and dose values, given rela-tive to 180°MFI ~which accumulates dose but has a verypoor temporal resolution for low pitch values! can be takenfrom Table III.

180°MCD and 180°MCI have the potential to improvecoronary calcium scoring in multi-slice CT. It has beenshown that the achieved calcium scores are closer to the true

FIG. 15. SSD of the heart. The patient’s heart rate was 86 min21, reconstruc-tion was done using 180°MCI. Scan parameters: 431 mm collimation andd51.5 mm table feed per rotation.



value than for nonphase correlated algorithms~180°MLI!.However, future work is necessary to establish an exact cal-cium scoring method. The cardiac algorithms presented inthis article can be a good basis for this algorithm, but, inaddition, there is a strong need for a calibration standard anda calibration and motion phantom. This would enable theuser to calibrate the achieved calcium score to the true physi-cal value as a function of the heart rate, patient diameter,tube voltage, and scanner type.

The advantages of the new dedicated multi-slice cardiacalgorithms over standard methods are evident. Phase selec-tivity and high temporal resolution~relative toR –R and ab-solute in time! can be achieved for a wide range of heartrates. Artifacts in the images, not only for the slow motionphase, are significantly reduced as compared to the standardz-interpolation. Further on, the investigated cardiac algo-rithms may allow for a drastic improvement of calcium scor-ing using spiral CT instead of EBT. However, more flexibil-ity in choosing rotation times~such as an additional 0.6 srotation mode besides the available 0.5 s! would further im-prove the image quality for certain heart rates. Evidently, afurther increase in maximum rotation speed~to below 0.5 s!would add tremendously to the quality of cardiac imagingwith multi-slice spiral CT.

ACKNOWLEDGMENTS

This work was supported by Grants AZ 262/98 and AZ322/99 of ‘‘Bayerische Forschungsstiftung, D-80333Munchen, Germany.’’ We thank Dr. Stefan Achenbach andDr. Ulrich Baum who carried out the patient studies for avery efficient and pleasant collaboration.

NOMENCLATURE

Notations and definitions used throughout this article aregiven below. Where it is possible, they conform to the nota-tions and definitions of Ref. 8, at least in the limitM51. Wewill express some quantities by the corresponding rotationangle ~e.g., slice thickness will not only appear asS, butequivalently asDaS52pS/d! and switch, if convenient,from one to the other representation.

* convolution symbol•* n n-fold selfconvolution of function•d(•) Dirac’s delta functiondodd,k indicator function: 1 ifk odd, otherwise 0II( •) rectangle function with support@21/2, 1/2#

and area 1L(•) triangle function of characteristic width~5

half width! 1, L5II* IIII a,b** (•) II( •/a)* II( •/b), for explicit expressions cf.

Ref. 8, Eq.~B2!

II a,b,c*** (•) II( •/a)* II( •/b)* II( •/c), for explicit expres-sions cf. Ref. 8, Eq.~B1!

b•c floor function, yields greatest integer lower orequal•

d•e ceiling function, yields smallest integergreater or equal•

x∨y maximum function,x∨y5max(x,y)

x∧y minimum function,x∧y5min(x,y)iff if and only ifa projection angle,aP@0,2p) for a sequence

~360°! scan orz-interpolated data,aPR for aspiral scan

DaH angle of rotation during one heart beat,DaH

52p( f Ht rot)21

aR angle associated with reconstruction positionzR ,aR52pzR /d

aR(m) angle under which themth slice reacheszR

DaS angle of rotation while advancing the table byS, DaS52pS/d52p/pM

Damax to restrict the interpolation at positionaR tothe range @aR2Damax,aR1Damax#,Damax

52pnmax.p1F

ak ,bk angles suitable for interpolation ata, b withkPZ

b angle within fan,bP@212F, 1

2F#

c(a) cardiac phase~with respect toR –R! as afunction of the view angle,c(a)P@0,1)

cR cardiac phase about which to center the recon-struction

d table feed per 360° rotationF fan angle, in our caseF552°f abbreviationf 5(p1F)/2pf H patient’s heart rate, typically 50 min21< f H

<120 min21

f Ht rot ~fractional! number of heart beats per rotationm, M slice index and number of simultaneously

measured slices, 1<m<Mnmax maximal number of rotations on each of both

sides ofzR to be taken into account for inter-polation,nmaxPR

p pitch, p5d/SMP(b,a,m) projection data for systems withM simulta-

neously measured slices,m51,...,MPX(b,a,zR) sequence raw data obtained from

z-interpolation at positionz5zR for algorithmX

S nominal slice thicknessSSP~•! slice sensitivity profile, either as a function of

positionz or angleateff absolute temporal resolution5effective scan

time, teff5w/fH

t rot time for a 360° rotationw relative temporal resolution5relative width

~with respect toR –R! of the successive inter-vals in the cardiacR –R cycle. For measureddata, w is given as the full width at tenthmaximum ~FWTM! of the phase sensitivityprofile. wP(0,1#

wdist(z), z weighting function of characteristic widthzfor 180°MCI that weights the distance of ameasured point to the reconstruction plane,z>S

wphase(c), c weighting function of characteristic widthcfor 180°MCI that weights the cardiac phase



deviation of a measured point to the desiredreconstruction phase,c,50%

wopt lower limit for w for optimal data fillingw res lower limit for w to restrict the data range to

nmax rotations on either side ofzR

w triv trivial upper limit for w, for w>w triv the trivalcase occurs: only one heart cycle contributes

z z-position of detector array center,z5d•a/2p

z-axis axis of rotationzR arbitrary selectable reconstruction position,

zR5d•aR/2p

z(a,m) z-position of the center of themth measuredslice, z(a,m)5(a/2p)d1Sm2S(M11)/2

(C/W) notation used for the window setting,C is thewindow’s center,W its width

The freely selectable parametercR is used to select thedesired cardiac phase;cR determines the relative center ofthe time window with respect to eachR –R interval. Arith-metics using the cardiac phasec are meant to be modulo 1 totake into account its periodicity.

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