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X-ray energy optimisation in computed microtomography

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1989 Phys. Med. Biol. 34 679

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Phys. Med. Biol., 1989, Vol. 34, N o 6, 679-690. Printed in the UK

X-ray energy optimisation in computed microtomographyPer Spanne?Atomic and Applied Physics Division, Brookhaven National Laboratory, Upton, NY1973,USA

Received 6 July 1988, in final form 25 October 1988

Abstract. Expressions describing the absorbed dose and the num ber of incident photonsnecessary for the detection of a contrasting detail in x-ray transmission CT imaging of acircularphantomare derivedas unctionsof he inearattenuationcoefficients of thematerialscomprising heobjectand hedetail.Ashellofadifferentmaterialcanbeincluded to allow simulation of CT imaging of the skulls of small laboratory animals. Theequations are used to estimate the optimum photon energy in x-ray transmission computedmicrotomography.Theoptimumenergydependsonwhether henumberof ncidentphotons or the absorbed do se a t a po in t in the ob jec ts minimised. For a water object of300 mm diameter the two optimisation cri ter ia yield optimum photon energies differ ing byan order of magnitude.

1. IntroductionX-ray ransmissioncomputed omography is nowa outinediagnosticmethod nradiology and is oftenused nnon-medicalapplications ordetection of detailsextending over one or several millimetres. Only aew cases of high-resolution computedtomography, CT, with aspatialresolutionconsiderablybelowonemillimetreusingx-ray tubes have been reported (Burstein et a1 1984, Carlsson et a1 1985, 1987, Elliottand Dover 1982, 1984, Elliott et a1 1987, Sato et a1 1981, Seguin et a1 1985). Thespatial resolution in these cases, ranging from a few hundred micrometres down toabout 15pm,was generally limited by the properties of the radiation sources used intheexperiments, in particular heir imitedx-rayfluencerates.However, with theadvent of synchrotron x-ray sources, one can now conceive f high-resolution imagingat least of small objects with a spatial resolution in the micrometre range (Grodzins1983, Thomson et a1 1984, Bonse et a1 1986, Carlsson er a1 1987, Spanne and Rivers1987, 1988, Flannery et a1 1987).The very high x-rayfluenceratesavailablefromthese sources make it possible to tune the x-rays in energy and still keep the fluencerate high enough to allow rapid imaging. Energy tuning is an important advantagesince he absorbed dose at a point in the imaged object or the imaging time for aconstant absorbed dose exhibits a minimum as a function of the x-ray energy whendetecting a detail in an object with a fixed degree of confidence. The sharpness of thisminimum is more pronounced with decreasing object size both in conventional projec-tion radiography (e.g. Carlsson 1981, Carlsson et a1 1987) and computed tomography(Grodzins 1983). Monoenergetic x-rays are also an advantage in CT imaging of largeobjects, for example humans, since beam hardening artefacts can be avoided.t Present address: Radiation Physics Department, Linkoping U niversity, S-S81 85 Linkoping, Sweden,0031-91S5/89/060679+ 12S02.50 0 989 IO Publishingtd 679


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680 P SpanneIn hispaperanexpression or heabsorbeddoseat hecentre of an x-raytransmission CT imaged object is derived for the case when a detail is to be detectedwith a specified signal-to-noise ratio. The analysis is purely theoretical in the sensethat no consideration is paid to any effects due to a limited dynamic range for thedetector. The absorbed dose has been chosen as quality index since radiation damagemay set the ultimate limit on the spatial resolution achievable in imaging biological

specimens with spatial resolutionsn the micrometre range. The purpose f the analysisis to derive a formula to be used as an aid in selecting photon energies for computedmicrotomography.Grodzins (1983) studied how the numberof photons necessary to achieve a certainprecision in a CT image varied with photon energy. However, he did not include anycontrasting details in the signal-to-noise ratio, and his analysis is therefore valid onlyforadetermination of the inearattenuation coefficient in apictureelement in ahomogeneous object. The present analysis is done to derive a formula which explicitlycontains the linear attenuationoefficient of a contrasting detailn a materialof differentatomic composition. The formula therefore also covers high contrast details. Grodzinsoptimal energy for a water object of 30 cm diameter is almost an order of magnitudelarger than can be expected from other calculations (e.g. Brooks and Di Chiro 1976),a difference which the present analysis shows is due to the fact that he optimised thenumber of incident photons nstead of the absorbed dose. This may be of use nmaterials estingwhereabsorbeddosesare of noconcern.Anotherreasonforanextendedanalysis is that differences between attenuation coefficients of differentmaterials are maximised at energies where photoelectric absorption is the predominat-ing interaction process, which happens to be in the energy region suitable for computedmicrotomography. An analysisnot ncludingcontrastingdetails hereforedoesnotshow the full potential of this imaging method.2. Mathematical modelX-ray ransmission CT images are generallyobtained by reconstruction rom ineintegrals, butmathematicalmethods or hree-dimensional magingusingplanarintegrals are available (Shepp 1980). The planar integral method is useful in magneticresonance imaging and perhaps also in x-ray emission CT. Its application o rans-mission CT is not straightforward, but a two-step method using images reconstructedfrom line integrals to derive the planar integrals of the linear attenuation coefficienthas been proposed (Cho et al 1984). The photon energy optimisation problem willtherefore be related to that in the two-dimensional case. It is, however, not clear thatthe two-step method offers any advantage compared to slice-by-slice reconstructionssince the computational time increases and since Cho et a1 claim that the differencein signal-to-noise ratio for the two methods is small in imaging homogeneous objects.The situation may be different when the planar integral method is applied to x-rayemission CT because the planar integrals are hen in principle directly measurable.Since his work dealsonly with transmission CT the analysis is limited o wo-dimensional CT reconstructions from line integrals.A formula for the variance f the reconstructed mean l inear attenuation oefficientin a volume element in a homogenous circular object will be used as a basis for theanalysis (Goreand Tofts 1978, FaulknerandMoores 1984). It is valid when thedetector is totally absorbing and theoise in he number f detected photons s thereforethe only source of noise in the measured projection values. By a projection is meant


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X-raynergyptimisation in CMT 68 1here the line integral of the linear attenua tion coefficient along a ray path , ob taine din practice by taking the logarithm of the inverse value of the fraction of the numberof photo ns transm itted for the ray path . F or th e discrete case the reconstructed linearatten uat ion coefficient for an imageelementcan be expressed mathem atically as adouble sum over the sampling angles and the sampling points n the profiles (Fau lknerand Moo res 1984):

where ~ ( x ,) is the linear attenuation coefficient at the poin t (x , y ) , p ( R ,neAB) theprojection value at the point (R , oAB) in a polar coord inate ystem with B as angularand R as radial coordinate, AB the angular increment during data sampling, ne thenumb er of the sampling angle, A R the distance between the sampling points, nR thenumber of the sampling point in the profile, m the number of sampling angles, g thefilter function a nd L the diameter of the reconstruction region.Now , since the projection values are equal to the logarithms of the inverse valuesof the fraction of transmitted photons, application of Gausss approximation formulaegives the followingexpression or hevariance of the econstructedattenuationcoefficient in a volume elem ent:

provided the stochastic variables N ( R ,n e AB) , the numbers of detected photons, canbedescribedasuncorrelatedGaussianprocesses.Here E is used odenote heexpectation value and V the variance of a stochastic variable. In order to evaluatethis double sum for an arbitrarypicture element in ( x ,y ) t is necessary to find all raystraversing it. This problem is unsolved for the general case. However, the evaluationis simple if o nly the centra l pixel in a circular object is conside red (Fau lkner andMoores 1984). In this case R equals zero and the expectation value of the num berof transmitted photons is equal for all sampling angles:

when photo n counting is used as detection method . With photo n cou nting the single-event size distribution for the detector signal does not have to be considered (S panne1988). Note that equation 3) is valid only for mono energetic radiation. The evaluationof the noise can now be do ne by a summ ation of the squ are d filter terms an d anevaluation of the expectation values for the transmitted numb er f pho tons for all rayspassing throug h the central volume element.I n imaging, it is the possibility of detectingdetails with differe ntattenuationcoefficients which is of interest. Ho we ver, since we ca n only evalu ate the theore ticalnoise for he centra l pixel, it is notpossiblesimply to ntroduce a detail nto hecalculations and to comp are the noise inside and outside i t. Instead we either haveto com par e the noise in the centra l pixels of one image with a nd one image withoutthe detail or assume that the presence of the detail d oes n ot perturb the noise outsideit. Let us consider a circular detail concentric to the central volume element and defineou r signal as the difference between the reconstructed attenuation coefficient in the


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682 P Spannecentralpictureelement with and without he detail. This is of courseasimplifiedmodel since the size of the object will affect the possibility of detecting its presence.However, or he purpose of aiding in selecting photon energy, his is no severerestriction. The signal-to-noise ratio for the detection of the attenuation difference canbe defined as

where thesubscripts1and2denote he mainmaterial and hecontrastingdetailrespectively. If the filter function has the appropriate gain, the expectation values ofthe reconstructed linear attenuation coefficients, E ( p i 0, 0)) , can be replaced with thetrue attenuation coefficients p i . The signal-to-noise ratio can then be written as

For monoenergetic radiation the expectation values for the transmitted fractionof thephotons are exponentials of the linear attenuation oefficients of the materials compris-ing heobject. Ifwe allow for a shell surrounding heobject, we canwrite thesignal-to-noise ratio as

where p , , p 2and p 3 are the mass attenuation coefficients of the main material, thedetail and he shell respectively, d is the diameter of the object, AR the samplingdistance, t the slice thickness, No the number of incident photons per sampling pointover the field area ARt, and S the thickness of the shell. If we neglect the effects ofscattered radiation the absorbed dose in the centre of the phantom can be written as

where +is the primary photon energy fluence through the centre of the phantom andp e n / p s the mass energy absorption coefficient.

When h v is the photon energy, equations (3) , (6) and ( 7 ) combined give

This equation differs from that derived by Carlsson (1981) for conventional projectionradiography by a factor, cfilter,hich takes into account the effect of the reconstructionfilter and also that high contrast details are allowed. This model also makes t possibleto include a shell surrounding the object, e.g. to simulate the imaging of the skull ofa small laboratory animal. cfi l ters energy independent and its introduction thereforehas n o effect on the optimum energy.Instead of using an equation for D, it is possible to derive an expression for thenumber of photons necessary for each ray path through the object. When absorbeddoses are of less concern, e.g. in materials testing, minimising the number of photonsmay be a better optimisation criterion than minimising the absorbed dose. It can be


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X-ray energy optimisation i n CMT 683

where c l and c2 are constants and f is a function of the linear attenua tion coefficients.Since the energy depe nden ces of equations (9 ) and (10) are different, minimising theabso rbed do se doe s not result in the same opti mu m energy as minimising the numberof photons. This explains why Grodzins (1983) reports a much higher optimum energyfo ra 300 mm water phan tom han Brooks and Di Chiro 1976). From the aboveanalysis is it also possible to conc lude that the optim um pho tonnergy will be differentdepending on where in the phantom the absorbed dose is minimised. Furthermore, ifthe imaging time is to be m inimised, it is necessary to consider also the ph oton fluencerate characteristics of the radiation source at different energies.3. CalculationsCalculations of the absorbed dose and the numb er of photons required to achieve asignal-to-noise ratio of 5 in the detection of a volume element with a linear attenuationcoefficient different from the m ain object were performe d using mass attenuation andmass energy absorption coefficients from Storm and Israel (1970) and H ubbell (1977).A sampling distance of 0.5% of the object diameter was used, resulting in a 200 x 200image matrix. The sp atial resolution is therefore a constant fraction of the object sizein all calculations. The reaso n for keeping this fraction constant, which m eans thatthe object diameter and the spatial resolution are bothvaried at th e sa me time, is thatthere are practical constraints on the size of the image m atrix because of imaging timelimitationsanddatahandlingcapabili t ies.The pixel sizes corresponding o 10, 1,0.1 mm objects are therefore 50 , 5 an d 0.5 p m respectively. A signal-to-noise ratio of5 was cho sen since this is in the suggested range for the threshold where a detail inan image becomes visible to the hum an eye (cf. Motz and Da nos (1978) for a comm enton this thresh old). All calculations were ma de for alice thickness equal to the sam plingdistance. A ram p fil ter was assumed to be used for the reconstruction. The numberof sampling angles were assum ed to be op timum (He rma n 1980). O ne very interestingpotentialapplication of computedmicrotomo graphy is non-destructive maging oftissue samples, e.g. needle biopsies. One set of calculations was theref ore ma de forwater objects with diame ters in the rang e 0.1-10 mm an d with different contrastingmaterials (figures 1 and 2). The atomic comp osition for fat was obtained from IC RUReport 23 (ICRU 1975).I n order to com pare this analysis with Grodzins', calculations of the num ber ofph oto ns resulting in a signal-to-noise ratio of 5 were performed . Figure 3 shows theresults for water ph anto ms with contrasting details of a fat. Calculations were alsoma de for water pha nto ms with contrasting details of water with a 1% higher density.Shown in figure 4 are heoptim um energies esulting rom hesecalculationsasfunctions of the object d iameter. The two curves inigure 4 are the esults of optimisingthe nu mb er of incident photons and the absorbed dose at the centre of the phantom.The curve for the absorbed dose has been broken above 50 keV to empha sise that theneglection of scattered radiation is a poor approxima tion n his energy region, Infigure 5 the absorbed doses n the centresof water objects of 20 ,30 and40 mm diameter


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684 P Spanne

7 l 1 /-

l .^

l l l l I0 20 40Photon e n e r g y (keVi

Figure 1. The absorbed dose at the centres of c ircular water phantoms result ing from he detection of acontrastingdetail of fatwithasignal-to-noise atioof 5 . Thephantomdiametersare: x, 0.1; 0 , 1 ; +,10mm . The diameter of the contrast ing detai l is 1/200 of the phantom diameter.

are shown as functions of the photon energy. The contrasting detail is water with a1% lower density than the main object.4. Results and discussionFrom equation (8 ) it can be seen that the absorbed dose for a specified signal-to-noiseratio is strongly dependent on thesamplingdistance,and herebyalso hespatialresolution. Figure 1 shows that the absorbed dose varies rapidly with photon energyand increases rapidly with decreasing object size. Note that a decrease in object sizein figure 1 also means a higher spatial resolution and that all points on the curves arefor hesame imagequality.The curves in figure 1 alsoshow hat here exists anoptimum energy for each object and that this energy decreasesith diminishing objectsize. The minima in the curves mean that a broad bremsstrahlung spectrum degradesthe image as compared to monoenergetic radiation.Also note that for the small objectsassumed here, the optimum energies are below those which can generally be obtainedwith rotating-anode x-ray tubes for clinical use.

In figure 2 the dependence of the absorbed dose on the atomic number of thecontrastingdetail is shownforsomematerials.For hepixel-sizedsmalldetails nthese calculations, the optimum energy is almost constant for all the contrast materialsstudied, but if the detail is allowed to extend over a larger fraction of the object, the


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X- ra y energy optimisation in CMT 685

1 Fat>

I I I I I0 20 40

Photon energy ( k e V )Figure 2. The absorbed dose a t the cent res of wa te r phantomsf 1mm diameter required to detect contrast ingdetails of fat (O ) , i r ( x ) a nd C a ( +) with a signal-to-no ise ratio of 5 . The diamete r of the contrast ingdetail is 1/200 of the phantom diameter.

opt imum energy depends on the atomic num ber of the contrasting detail. Figure 2also shows the very strong depende nce of the absorbed doseon the difference betweenthe inearattenuation coefficients, which app ears qua red in thedenominator ofequation ( 8 ) .Figure 4 shows that the optim um energy for CT of a w ater object of 30 cm diameteris around 70 keV when the expression used here for the absorbed dose at the centreof the phan tom , neglecting scattered radiation, is minimised. This is less than one -tenthof that predicted by Grodzins (1983) . The difference is due to theact tha t he minimisedthe num be r of ph oto ns necessary to achieve a certain precision in the determinationof the linear atten uation coefficient. The difference between the op timu m energies forthe two optimisation criteria is most prono unce d for large objects an d diminishes w henthe objectdiameter is decreased.The argedifferenceclearlycalls oracarefulconsideration of whatoptimisationcriterion ochoose npractical maging.Over-estimating the optimum energy by a factor of 10 creates problems both in the choiceof radiation sou rce a nd the d etection of the rad iatio n. Even though scattered rad iationhas been neglected and the expression used here for the absorbe d dose at the centreof an object is a po or approxim ation at large object sizes, the large difference resultingfrom the two optimisation criteria will remain whe n scattered radiation is included inthe calculations. Including scattered radiation will increa se the calcu lated values for


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686 P SpanneIO"

LWnv)0c0LncC0C

IO 6

0.1 mm

I I 1 1 . 1 . 1 1 I I , 1 ,,.J10 102 io3

Photon energy (keV)Figure 3. The numbe rof incide nt pho tons per ay path resulting in a signal-to-noise ratio f 5 in the detectionof contrasting details of fat in c ircular water phantom s w ith diam eters in the rang e.1-100 mm. The diamete rof the contrast ing detail is 1/200 of the phantom diameter.

10.1 1 10 100 1000Object diameter (mm1

Figure 4. The optimal photon energy for the detect ion of a% density difference in circular water phantomsas a function of the diameter of the phantom. The diamete r f the contrast ing detai l s l j 2 0 0 of the phantomdiameter. Optimisation criteria: x , minimum absorbed dose a t cent re of phantom; 0 ,minimum numberof incident photons.


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X- ra y energy optimisation in CMT 687MO

300

100

0 I50 100Photon energy IkeV)

Figure 5. The absorbed dose a t the cent res of circular water phantom s resulting from the detection of acontrasting detail with a signal-to-noise ratio of 5 . The contrast ing detai l is water with a 1% higher densi tythan the rest of the object . The phantom diameters are: X , 20 mm; 0 ,30mm; 0 , 0 mm. The diamete r ofthe contrast ing detai l is 1/200 of the phantom diameter.

the absorbed dose a t the centref the ob ject, and the increasewill be larger the higherthe pho ton energy since incoherent scattering then becom es more important relativeto photoelectric absorption. The optimum energy will therefore be shifted owardslower energies when scattered radiation is taken into consideration.An other difference n theoptimisation esults is due o he fact hatGrodz insdefined the signal-to-noise ratio as the inverse value of the relative standard deviationof he econstructed inearattenu ation coefficient an d therefo redidnot explicitlyinclude any contrasting detail. In the analysis presented here the signal is defined asa difference between a tten ua tion coefficients. From figure 3, which shows the numbersof photo ns necessary to obtain a signal-to-noise ratio of 5, it can be seen hat herequired number of photons a t the optimum energys constant a t low optimum energiesbut starts to increase at higher optimu m energies. This is so becau se the differencebetween the attenuation coefficients decreases when the photon energy increases andphotoelectric absorption becom es less imp ortant. If the contrasting detail has d ifferentdensity but the same atomic compo sition as the rest of the object, no increase w ouldbe observed for higher optimum energies. For m aterials with high atom ic numbers,the difference between optimising the num ber of photo ns and the absorb ed dose willappear at smaller object diameters.A very consp icuou s feature of figure 4 is how the op tim um energy, when minimisingthe absorbed dose, changes rapidly with the object diameter between 30 and 40 mm .Th is shou ld not be mis interp reted as if the choi ce of energy is especially critical inthis photon energy region. It is, on the con trary, n ot p articula rly ensitive in this regionas can be seen in figure 5, which shows the absorbed dose as a function of photonenergy for water phantoms of 20 to 40 mm diameter with contrasting water detailswith 1% higher density than the main object. The curve for the 30 mm object shows


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688 P Spannetwo minima, and the rapid changen optimum energy appears w hen the object diameteris increased so that the absolute minimum is changed from one of these local minimato the other . In practice the absorbed doses at the two min ima differ so little that an yenergy between 20 an d 50 keV will result n almost the same image quality for thesame absorbed dose. The two minima can be connected o hose par ts of equatio n(8 ) which contain the numb er of incident photons and the mass energy absorptioncoefficient respectively. This is illustrated in igure 4, which sh ow s hat forobjectssmalle r than 30 mm optimising the absorbed dose gives the sam e result as optimisingthe number of incident photons, while at larger diameters the mass energy absorptioncoefficient makes the optimum energy for the ab sorbed dose d eviate from that for thenumber of incident photons.A com parison of the calculations for the nu mber o f photons and the ab sorbed do seshows that the optimum energy depends o n the quanti ty optimised. The optimisat ionmethod emphasised here rel ies on minimising the absorbed dose in the centre of thephantom. This is a reasonab le approach for com puted microtomograph y f very smallobjects where radiation damage is expected to limit the imag ing possib ilities and forimaging of sma ll aborato ry anim als whe re ong-term biolog ical effects can be dis-regarded. For arger objects , such as the hum an runk, he meth od presente d hereshould b e applied with caution, s ince scattered radiat ion has been neglected. Further-mo re, in most cases of imaging human s the stochastic effects of radiation are the mainconcern. In this case the effective dose equivalent in the patient s hou ld be m inimis ed.A future ref inement of the optimisat ion could be to use imparted f ract ions calculatedby MonteCarlo echniquesdescr ibed by Persl iden 1983)andPersl idenand AlmCarlsson (1984) .

The present analysis is valid for the central pixel in a circular object. In practice,imagedobjectsareoften rregular o r asymmetric, which maycause severe beamhardening artefacts. With the possibility of using mon oene rgetic radiation from syn-chrotron radiat ion sources these ar tefacts can be e l iminated.t is, however, not possibleto derive a simple formula similar to equation8) for an asymmetr ic object , and insteadnumerical methods have to be used to f ind the optimum energies. I t should a lso benoted that correla t ions have been neglected in this analysis .

5. ConclusionsA formula which can be used n est imating the optimum photon energy for computedx-ray transmission microtomog raphy has been der ived. I t is foun d that the optim umenergy varies depending o n whether the numb er of incident photons or the absorbeddose in the centre of the object is minimised. F or large o bjects these tw o o ptimisat ionmeth ods yield optimum energies that m ay differ by as m uch as an orderof magnitude.For water objects with d iameters smaller than 100 mm both optimisa tion criteria canbe used wi tho ut any significant difference in imag e quality .

AcknowledgmentsThis esearch was supported npar t by TheSwedishMedica lResearchCounc i l(MFR -7557, MFR -8633) , The Swedish Society of Medicine,TheSweden-AmericaFoundation, N I H (USPHS Fel lowship05- TW03735) ,O EDE- AC 02-76CH00016) .


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X-ray energy optimisation in CMTResumeOptim isation de ICnergie des rayons x en microtomographie assistee par ordinateur.

689

Les auteursproposentdesexpress ions dCcrivant ladoseabsorbteet la nomb redephotons ncidentsntcessaires pour la dttectionparscanographie dedttailsducontrastedansu n f a n t h e c ir cu la ir e; c esexpressions eposentsur les coefficients iniiquesdatttnuationdesmattr iauxcon stitu ant Iobjet et lesdtteils.Une nveloppe onsti tute unmatiriau diffCrent peut btre introduite fin e imuler estomographies de cr lnes de peti ts animaux de laboratoire. Ces tquations sont uti l isies pour estimer I tnergieoptimale des photons danse cas de la microtomographie e transmission par rayons x assist te par ordinateur.Linergie optimale dCpend soit du nombre de photons incidents, soit de la dose absorbCe dan s Iobjet auniveau du point ob elle est minimiste. Pour un objet Cquivalent-eau de 300 mm d e dia mt tre les deux crittresdoptimisation conduisent, pour les photons, A deux energies optimales different entre elles dun o rdr e degrandeur.

ZusammenfassungOptimierung der Rontgenenergie in der Mikrotomographie.Gleichungen zur Beschreibung der Energiedosis und der Anzahl der einfallenden Photonen, die fur denNachweis kontrastbildender Details in derRontgen-Transmission-CT-Abbildungines kreisformigen Phan-toms notig sind w urden abgeleitet als Funktion der l inearenSchwachungskoeffizienten der Materialien, ausdenen das Objekt und die Details bestehen. Ein anderes Material kann kann also Schale hinzugenomm enwerden, um so die Simulation von CT -Abbildungen des Schadels kleiner Labortiere zu ermoglichen. DieGleichungenwerdenverwendetzu rBestimmungderoptimalenPhotonenenergie nde rRontgentrans-missions-Mikro-Computertomographie.ie optimale Energie hangt davon ab, ob die Anzahl der einfallendenPhotonen oder die Energiedosis in einem Punktm Objekt minimiert wird. Fur ein Wasserobjektmit 300 mmDurchmess er fiihren beide Optimierungskriterien zu optimalen Photonenenergien, die sich um eine ganzeGroBenordnung unterscheiden.

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