Fluorescent heterogeneities in turbid media: limits for detection, characterization, and comparison with absorption

Transcription

1 Fluoresent heterogeneities in turbid media: limits for detetion, haraterization, and omparison with absorption Xingde Li, Britton Chane, and Arjun G. Yodh The fundamental limits for detetion and haraterization of fluoresent phosphoresent inhomogeneities embedded in tissuelike highly sattering turbid media are investigated. The absorption and fluoresene ontrast introdued by exogenous fluorophores are also ompared. Both analyses are based on pratial signal-to-noise ratio onsiderations. For an objet with fivefold fluorophore onentration and lifetime ontrast with respet to the bakground tissue, we find the smallest detetable fluoresent objet at 3-m depth in tissuelike turbid media to be 0.25 m in radius, whereas the smallest haraterizable objet size is 0.75 m in radius, given a model with 1% amplitude and 0.5 phase noise. We also find that, for fluoresene extintion oeffiients m 1 M 1, the fluoresene measurement mode is superior to the absorption mode for deteting an inhomogeneity. The optimal hoie of modulation frequeny for the frequeny-domain fluoresene measurements is also disussed Optial Soiety of Ameria OCIS odes: , , , , , Introdution Near-infrared NIR diffusive light is expeted to be a useful diagnosti and imaging probe for highly sattering media suh as biologial tissue. 1 One interesting and potentially important biomedial appliation is to detet and loalize heterogeneities suh as hematomas and tumors in thik tissue. 2 Most studies thus far have foused on endogenous tissue optial properties suh as loal variations in optial absorption and sattering for ontrast between a tissue abnormality and its surroundings. Fundamental limits for detetion and haraterization of endogenous absorptive and sattering optial inhomogeneities have been investigated. 3 In 6-m-thik slabs, for example, it has been shown that the smallest detetable When this study was performed, all the authors were with the University of Pennsylvania, Philadelphia, Pennsylvania X. D. Li was with the Department of Physis and Astronomy and the Department of Biohemistry and Biophysis. B. Chane was with the Department of Biohemistry and Biophysis, and A. G. Yodh was with the Department of Physis and Astronomy. X. D. Li is now with the Department of Eletrial Engineering and Computer Sienes, Massahusetts Institute of Tehnology, Cambridge, Massahusetts Reeived 4 Marh 1998; revised manusript reeived 3 June $ Optial Soiety of Ameria objet size is 0.30 m in diameter given an approximately three-fold absorption or approximately twofold sattering ontrast, whereas the smallest haraterizable objet size is 1.0 m in diameter given a pratial signal-to-noise ratio 0.3% amplitude noise and 0.08 phase noise. Although biologial tissues ontain several intrinsi fluorophores, 4 none is available in the NIR region, and the high absorption makes these fluorophores disadvantageous for thik-tissue diagnosis. Thus exogenous fluoresent and phosphoresent ontrast agents in the NIR region have been onsidered as means to enhane the sensitivity and speifiity for tumor detetion, imaging, and treatment in deep tissues The underlying ideas in this ase bear substantial similarity to positron emission tomography, fluoresene mirosopy, and gadoliniumenhaned magneti resonane imaging 15 MRI. The approah relies on the fat that some fluorophores e.g., hematoporphrin and fluorophore-tagged antibodies preferentially aumulate within the tumor tissue, 16 have environmentally sensitive lifetimes e.g., owing to fluoresene quenher onentration variation, or both. Clearly it is desirable to estimate the extent to whih fluoresent light an be used for tumor detetion and haraterization. In this paper we aim to quantify these detetion and haraterization limits. In addition to emitting at the fluoresene wave- 1 Otober 1998 Vol. 37, No. 28 APPLIED OPTICS 6833

2 length em, fluorophores absorb light at an exitation wavelength ex. Therefore all exogenous fluorophores also provide absorption ontrast at ex. A seond aim in this paper is to larify the irumstanes under whih absorption or emission detetion modes should be used with exogenous ontrast agents. The answer will depend on suh fators as fluoresene quantum yield and extintion oeffiient. Finally, diffusive wave modulation frequeny an elevate or suppress the fluoresene ontrast, and we disuss the optimal hoie of modulation frequeny for fluoresene measurements in the frequeny domain. We onsider a simple breast-tumor model system onsisting of a single spherial objet embedded in a highly sattering turbid medium, and we onsider the frequeny-domain experimental sheme throughout this paper. We use analytial solutions of fluoresent diffuse photon density waves 10 FDPDW s to alulate the exat fluoresene amplitude and phase at the sample boundaries. We then obtain simulated experimental data by adding random noise to the exat fluoresene signal amplitude and phase. First we address the fundamental limits for using FDPDW s to detet and haraterize tumors by onsidering a pratial fluorophore onentration lifetime ontrast and a realisti signal-to-noise ratio. We then ompare the fluoresene and absorption ontrast introdued by exogenous fluorophores. Finally we suggest a simple riterion for hoosing a modulation frequeny to elevate the fluoresene ontrast. The results of this analysis may serve as a guideline for design and seletion of exogenous ontrast agents. 2. Summary of Analytial Solutions In a thik highly sattering medium suh as biologial tissue, NIR light transport is often well approximated as a diffusive proess. Mirosopially, when the medium s absorption length is muh longer than its sattering length as is the ase for NIR light in many biologial tissues, eah photon experienes multiple sattering events before its absorption or its transmission through the sample boundaries. When a sinusoidal intensity-modulated light soure is oupled into the medium as in all frequenydomain experiments, the diffusive photons add inoherently to produe a salar photon density wave propagating outward from the soure. This salar wave is referred to as a diffuse photon density wave DPDW. 1 Next, suppose that the turbid medium ontains a distribution of fluorophores. The fluorophores may be exited by the inident DPDW at optial wavelength ex. When the exited fluorophores radiatively relax, fluoresent photons are generated that, again, propagate diffusively through the turbid medium. Colletively a FDPDW is reated by addition of the emissions from all fluorophores in the medium. We adopt the following notation in the present disussion: optial properties related to the homogeneous bakground medium i.e., outside the objet Table 1. a1 are indiated by a subsript 1, e.g., a1, s1 ; optial properties related to the objet are indiated by a subsript 2, e.g., a2, s2 ; optial properties related to the fluoresent or emission DPDW s are indiated with an extra subsript f. We use a supersript to indiate hromophore-related parameters see Table 1. The analytial solution for the FDPDW in a pieewise homogeneous infinite turbid medium onsisting of a spherial objet in an otherwise homogeneous bakground an be written as the superposition of the homogeneous FDPDW homo and a sattered FDPDW s Ref. 10; also see Appendix A ; i.e., hetero Chromophore Optial Properties at ex and em and Soure Modulation Frequeny a Bakground Inhomogeneity ex em ex em s1 a1f s1f r s, r d,, a homo r s, r d, s r s, r d,, a q 1 1 N 1 1 i 1 F 1 r s, r d, q 2 2 N 2 F 2 r s, r d,, a. (1) 1 i 2 Here r s and r d are the soure and the detetor positions, respetively, a is the radius of the inhomogeneity, and is the angular soure modulation frequeny; e.g., 2 f, where f is what we generally all the modulation frequeny. N 1 and 1 are the fluorophore onentration and lifetime, respetively, in the homogeneous bakground medium e.g., outside the spherial inhomogeneity ; N 2 and 2 are the onentration and lifetime inside the inhomogeneity. is the fluorophore extintion oeffiient at the exitation wavelength ex. 1 2 is the fluoresene quantum yield outside inside the objet. For simpliity, we set % in this disussion. q 1 q 2 is the quenhing fator defined below for the fluorophores outside inside the objet. The analytial solution Eq. 1 has been verified with finitedifferene numerial alulations. 10 Suppose that the natural radiative deay rate of the fluorophore is 0. In the presene of quenhers, fluorophores in exited states an also deay to their ground states through nonradiative hannels. Therefore the total deay rate inreases; e.g., 0 K D Q. Here Q is the quenher onentration and K D is the Stern Volmer quenhing onstant. 20 The quenhing fator q is the ratio of the radiative deay rate 0 to the total deay rate ; e.g., q 0 0, where 0 and are, respetively, the radiative lifetime and the total lifetime of the fluorophore in the presene of the quenher. In biology, paramag- a2 s2 a2f s2f a The optial properties given in this table are in units of inverse entimeters. The soure modulation frequeny is 200 MHz unless stated expliitly elsewhere APPLIED OPTICS Vol. 37, No Otober 1998

3 neti oxygen 3 O 2 is a ommon quenher for almost all fluorophores. The oxygen onentration Q 2 in the tumor an be two to four times lower that in the surrounding normal tissues Q 1 beause of the high oxygen onsumption rate within the tumor. 21 In this ase fluorophores within the tumor are quenhed less than the fluorophores in the surrounding normal tissues, and the lifetime within the tumor, 2, is longer than that of the surrounding normal tissues, 1. The quenhing fators outside and inside the tumor are q and q We find that the quenhing fator is proportional to the lifetime, and it is higher within the tumor than that in the normal tissue q 2 q 1 if the oxygen onentration is lower in the tumor. It has been suggested that this quenhing effet q 1,2 ombined with the demodulation fator 1 1 1,2 provides us the physiologial and biohemial basis for using fluorophore lifetime ontrast for tumor detetion. 14,17,19 The first term on the right-hand side of Eq. 1 is the analytial solution of the FDPDW in a homogeneous fluoresent turbid medium; e.g., homo q 1 1 N 1 1 i 1 F 1, where the funtion F 1 has a simple form for details see Appendix A. It depends on the optial properties of homogeneous bakground medium at both exitation and emission wavelengths. In the presene of exogenous fluorophores the total bakground absorption oeffiient is the sum of the endogenous hromophore absorption and the exogenous fluorophore absorption; e.g., the total absorption at ex of the homogeneous fluorophoreontaining bakground medium is a1 a1 N 1, where a1 is the endogenous absorption that is due to the tissue hromophores and N 1 is the onentration of exogenous bakground fluorophore. Most fluorophores have rather broad absorption spetra. In addition, we also aount for the self-absorption effet of fluorophores by approximating the fluorophore extintion oeffiient at fluoresene wavelength em as one tenth of that at ex, e.g., f 0.1. The total absorption at em is then a1f a1f f N 1. We an easily obtain the absorption for the medium inside the objet by replaing N 1 with the fluorophore onentration inside the objet, N 2. The hange of the sattering oeffiients an also be inorporated into Eq. 1. In general the moleular size of the fluorophore is small. So the hange of sattering introdued by the fluorophore is small. Therefore we simply neglet the hange of the sattering oeffiient that is due to the presene of the exogenous fluorophores. The seond term on the right-hand side of Eq. 1 denotes the sattered wave, e.g., s q 2 2 N 2 1 i 2 F 2. F 2 has a rather ompliated funtional dependene on many parameters, suh as the optial properties at ex and em, the fluorophore onentration inside outside the objet, the size and the position of the objet, the soure detetor positions, and the soure modulation frequeny see Appendix A and Ref. 10. It an be written as the sum of multipole moments of all orders spherial Bessel funtions and spherial Hankel funtions of the first kind, e.g., Fig. 1. Infinite slab with one side at z 0 m and the other side at z 5 m. a Single soure detetor geometry. The soure and the detetor are fixed as shown. A single objet is entered between the soure and the detetor at 0, 2.5 m. The FDPDW is alulated under this single soure detetor onfiguration for the detetion limits analysis. b Soure position fixed as shown. The detetor sans along one side of the slab x axis in steps of 0.2 m. The fluoresene amplitudes and phases at 21 detetor positions are alulated for haraterization limits analysis. s monopole diopole quadrapole higher order. The dominant term is the monopole moment. Different moments have different funtional dependenes on the radius of the objet a and on the fluorophore onentration inside the objet N 2. If we Taylor expand different moments with respet to the radius a and the onentration N 2, we find that the monopole moment is related to the onentration and radius through N 2 a 3, whih is the total number of fluorophores inside the objet; the dipole moment is related to N 2 a 5. Inasmuh as objet size a and fluorophore onentration N 2 appear in the sattered wave as a produt, it is essential that at least two multipole moments e.g., the monopole and the dipole moments exeed the noise level to haraterize a and N 2 simultaneously. We disuss this point in detail in Subsetion 5.C. On the other hand, the lifetime dependene of the moments is simple. Different moments have the same funtional dependene of the moments is simple. Different moments have the same funtional dependene on the lifetime through q 2 1 i i 2. This fat an be seen from Eq. 1, in whih the lifetime is an overall fator assoiated with the sattered wave. Thus we see that the simultaneous haraterization of lifetime 2 and onentration N 2 is relatively easier than the simultaneous haraterization of onentration N 2 and radius a. 3. Models and Criteria The model that we onsider in this paper is an infinitely long slab with a finite width of 5 m Fig. 1. An exitation light soure modulated at 200 MHz is plaed upon the surfae of one side of the slab, and a detetor is plaed upon the opposite side. A single spherial inhomogeneity is embedded at the enter of the slab, and it is aligned with the soure. Extrapolated zero boundary onditions are applied, leading to an infinite but onvergent series of image soure objet pairs Fig. 2. We then an use the analytial solution for an infinite medium to alulate the forward FDPDW s that orrespond to the real soure objet pair and eah image soure objet pair, and superpose those FDPDW s to obtain the total 1 Otober 1998 Vol. 37, No. 28 APPLIED OPTICS 6835

4 Table 2. Quantum Yield, Bakground Fluorophore Conentration a N 1, Lifetime 1, and Extintion Coeffiients and f N 1 1 at ex f at ex 10% 0.1 M 1.0 ns m 1 M m 1 M 1 Fig. 2. Extrapolated zero boundary onditions are inorporated into the forward FDPDW alulations by introdution of a series of image soure objet pairs. Planes B 1 and B 2 are the two physial surfaes of the slab turbid medium. Planes P 1 and P 2 are the extrapolated boundaries a distane z b s from the orresponding physial surfaes B 1 and B 2. The fluoresene photon fluene is approximated to be zero on the extrapolated boundaries P 1 and P 2. The thikness of the slab is denoted w. The soure S is at z 0, the objet O is at z d, and the detetor D an be anywhere on the physial surfaes or within the slab. S 1 O 1 is the image soure objet pair of the real soure objet pair S O with respet to plane P 1 ; S 1 O 1 and S O are mirror symmetri about plane P 1. S 2 O 2 is the image soure objet pair of the real soure objet pair S O with respet to plane P 2 ; S 2 O 2 and S O are symmetri about plane P 2. S 3 O 3 is the image soure objet pair of the image soure objet pair S 2 O 2 with respet to plane P 1 ; S 3 O 3 and S 2 O 2 are symmetri about plane P 1. Series of image soure objet pairs an go on following a simple observation. The signs of the image soures are also indiated here. The total FDPDW is the superposition of the FDPDW s generated by the real soure objet pair S O and all the image soure objet pairs S i O i with appropriate signs for the image soures. The series onverges fast beause the fluoresene photon fluene deays exponentially with respet to the image soure detetor separation. FDPDW for the slab geometry. For the detetion limit study, we alulate the fluoresene wave for a single detetor geometry in whih the detetor is aligned with the soure suh that the objet is entered between the soure and the detetor Fig. 1 a. For the haraterization study, we keep the soure position fixed and san the detetor along one side of the slab x axis over a 4-m 2, 2 region with 0.2-m step size Fig. 1 b. The fluoresene diffusive wave amplitude and phase is then alulated for 21 different detetor positions. Throughout the disussion, the endogenous hromophore optial properties of the objet and the surrounding bakground medium are kept fixed while we vary the fluorophore onentration and lifetime as well as the objet size. The riteria that we use to determine the detetion and haraterization limits are based on a signal-tonoise ratio analysis. We assume a perfet optial filter to rejet the bakground exitation light ompletely. We ome bak to disuss the pratial requirements for the optial filter in Setion 6. In general our system is not limited by shot noise. Consider a 3-mW exitation light soure. This orresponds to an exitation photon fluene of photons s. Given typial fluorophore onentration, lifetime, and optial properties see Tables 1 and 2, the deteted fluoresene fluene is photons s a The quantum yield is fixed at 10% for bakground fluorophores and fluorophores inside the objet. The fluorophore extintion oeffiient at the fluoresene wavelength em is one tenth of that at the exitation wavelength ex, e.g., f 0.1. Throughout the paper the extintion oeffiients are fixed to these values unless stated otherwise. for a 5-m soure detetor separation and an ative detetion area of 0.3-m diameter. Even when we take into aount a loss by another fator of 10 3 that is due to the optoeletrial onversion quantum effiieny of the detetor and to other optial ouplings, we still have a fluoresene fluene of 10 7 photons s, whih orresponds to a shot noise of in 1 s. In pratie, other noise from eletronis or positional unertainties will exeed this shot noise. For simpliity, we then onsider 1% 10 2 amplitude and 0.5 phase random noise in our noise model. For studies of detetion limits we onsider the relative amplitude and phase of the FDPDW, e.g., hetero homo 1 and Arg hetero Arg homo, respetively. They represent the frational amplitude and the relative phase perturbation to the FDPDW aused by the inhomogeneity. When either the frational amplitude or the relative phase perturbation is greater than the system noise level, e.g., hetero homo 1 1% or hetero homo 1% in amplitude or Arg hetero Arg homo 0.5 in phase, we say that the inhomogeneity is detetable. For studies of haraterization limits we first alulate the amplitude and the phase of the fluoresene diffusive wave, using the analytial solutions. 10 We simulate the experimental data by adding 1% noise randomly to the amplitude and 0.5 noise randomly to the phase. We then employ a 2 fitting proedure to determine the fluorophore onentration or lifetime or the size of the inhomogeneity, while assuming that all other parameters are known e.g., are determined by other imaging modalities suh as x-ray, ultrasound, or MRI independently or in onjuntion with the optial method. Clearly the situation that we onsider here is a best-ase one. When the fitted parameters have less than 20% frational unertainties relative to their true values, we say that the inhomogeneity is haraterizable. 4. Detetion Limits Consider a turbid medium of the slab geometry shown in Fig. 1 a. The amplitude and the phase of the FDPDW will depend on the onentration and lifetime as well as on other parameters. We onsider fluoresent dyes, e.g., ICG, whih is approved by the U.S. Food and Drug Administration for diagnosti purposes, that have lifetimes of roughly 1 ns and onentrations of roughly 0.1 M. 25 We first onsider a fixed bakground-fluorophore onentration 6836 APPLIED OPTICS Vol. 37, No Otober 1998

5 Fig. 3. Contour plots of frational amplitude and relative phase hange versus fluorophore onentration N 2 N 1 and lifetime variation 2 1 for different-sized objets. The radii are indiated in units inverse entimeters. a Curves, 1% fration amplitude perturbation ontours. Hathed areas, frational amplitude hange greater than 1% or less 1%. b Curves, 0.5 relative phase hange ontours. Hathed areas, relative phase hange greater than 0.5 or less than 0.5. N M and a lifetime ns Table 2. Thereafter the perturbation to the FDPDW introdued by the inhomogeneity will depend only on the relative values of fluorophore onentration ontrast N 2 N 1 and lifetime ontrast 2 1. Later we disuss how the absolute bakground-fluorophore onentration and lifetime affet the relative amplitude and phase of FDPDW, hetero homo 1 and Arg hetero Arg homo, for a fixed fluorophore onentration ontrast N 2 N 1 and lifetime ontrast 2 1. The administration of exogenous fluoresent ontrast agents is more desirable when the optial inhomogeneity is intrinsially weak. In this paper we assume that the differene of endogenous optial properties between the inhomogeneity and the surrounding bakground tissue is small Table 1. Thus the detetion ontrast is mainly from the exogenous ontrast agents. A. Relative Conentration and Lifetime Variation Contrast For different sizes of spherial inhomogeneities the amplitude and the phase of the FDPDW are alulated over a broad range of fluorophore onentration and lifetime ontrast. The frational amplitude perturbation and relative phase hange ontours are plotted in Fig. 3. The urves represent the 1% frational amplitude ontours and 0.5 relative phase ontours for different objet sizes. The hathed areas indiate the frational amplitude relative phase shift greater than 1% 0.5. Before making a detailed analysis, we find qualitatively that higher fluorophore onentration ontrast N 2 N 1 and higher lifetime ontrast 2 1 result in greater amplitude and phase perturbations and therefore permit detetion of smaller objets. Now let us study why the ontours follow the trends indiated by Fig. 3. First onsider the dependene of the FDPDW amplitude on the fluorophore onentration inside the objet, N 2. The fluorophore onentration has two opposing effets on the amplitude of the FDPDW. On the one hand, given a higher fluorophore onentration, more fluorophores per unit volume will be exited and therefore more fluoresent photons will be produed; on the other hand, higher fluorophore onentration inreases the total absorption oeffiients for both the exitation and the emission light. This result will then redue the exitation and emission photon fluene. The latter effet will not be important when the absolute onentration of the ontrast agent is low. From the pratial and linial point of view, the fluorophore onentration has to be low enough e.g., 1 M to avoid serious phototoxiity. For fluorophore extintion oeffiients near m 1 M 1 the additional absorption introdued by the exogenous fluorophore at 1- M onentration is roughly in the range of the endogenous tissue hromophore absorption, e.g., N m 1. Under this ondition, the perturbation to homogeneous fluoresent photon fluene by the objet will be roughly proportional to the fluorophore onentration. As shown in Fig. 3 a, for a given fluorophore lifetime 2, as the fluorophore onentration N 2 inreases, the objet size required to produe the same amount of amplitude 1% and phase 0.5 perturbation dereases, and therefore a smaller objet an be deteted. The dependene of fluoresene amplitude on the fluorophore lifetime inside the objet is apparent. Equation 1 indiates that the fluene is proportional to 2 1 i , where is the angular intensity modulation frequeny of the soure and 2 is the fluorophore lifetime in the objet e.g., tumor. Consider the hange of lifetime 2 that is due to quenhing q 2. An inrease in lifetime 2 introdues two ompeting effets on the fluoresene fluene. On the one hand, longer lifetimes indiate less quenhing for the fluorophores beause q 2 2 ; therefore the fluoresene fluene will inrease. On the other hand, longer lifetimes ause a greater demodulation 1 1 i 2 ; therefore the fluoresene fluene will derease. Combining these two effets, we find that overall the fluoresene fluene will first inrease as the lifetime 2 inreases at a fixed fluorophore onentration N 2. When the lifetime reahes a value that is greater than the soure modulation period, e.g., 2 2, these two effets anel and the fluoresene fluene then reahes its saturation state Fig. 3 a. The dependene of relative phase shift on fluorophore onentration N 2 is less obvious. Loosely speaking, the phase is related to the average photon path length of the deteted photons. A longer path length orresponds to a larger phase shift. Higher fluorophore onentration inreases the absorption oeffiient. Consequently, the survival probability of photons with longer path length dereases. The resultant average path length is then redued. Therefore the relative phase shift dereases as the fluorophore onentration inreases. The relative phase shift is related to the lifetime in the tumor by tan 1 2 see Eq. 1. A longer lifetime orresponds to a greater phase shift. This effet on the relative phase shift is opposite that whih results from the inrease of the fluorophore onentration. 1 Otober 1998 Vol. 37, No. 28 APPLIED OPTICS 6837

6 Therefore in the lower-left orner of the phase ontour plot in Fig. 3 b, where the phase shift is dominated by the onentration ontrast, we observe negative relative phase hanges, whereas in the upper-right orner, where the phase shift is dominated by the lifetime ontrast, we observe positive relative phase hanges. The detetion limits depend on the fluorophore ontrast N 2 N 1 and 2 1 as well as on the detetion system s noise level. Suppose that in pratie we have fivefold fluorophore onentration and lifetime variation N 2 N 1 5 and If our detetion system is limited by 1% noise in amplitude and 0.5 in phase, we find from Fig. 3 that the smallest detetable objet size is 0.25 m in radius. Given a higher fluorophore onentration ontrast e.g., N 2 N 1 10, a larger lifetime variation e.g., , and a lower detetion system noise level, a smaller objet e.g., 0.15 m in radius an then be deteted. B. Effet of Absolute Conentration and Lifetime Variation The perturbation of the FDPDW also depends on the absolute bakground-fluorophore onentration N 1 and lifetime 1. The total bakground absorption inreases as the bakground-fluorophore onentration inreases. If we keep the onentration ontrast N 2 N 1 fixed, the perturbation dereases slightly in a nonlinear fashion as the bakground onentration N 1 inreases. This is so simply beause less exitation light will reah the objet from the soure and less fluoresent light will reah the detetor from the fluoresent objet. Following the same analysis as that in the early part of this setion and assuming that we have the same ontrast N 2 N 1 5 and and the same detetion sensitivity 1% in amplitude and 0.5 in phase, when the bakground-fluorophore onentration inreases from 0.1 to 0.5 M we find that the smallest detetable objet is 0.30 m in radius ompared with 0.25 m in radius for lower bakground onentration N M. On the other hand, the perturbation of FDPDW has a simpler dependene on fluorophore lifetimes 1 and 2, e.g., the amplitude perturbation and phase perturbation tan 1 2 tan 1 1 tan tan 1 1. If we keep the relative lifetime ontrast 2 1 fixed, both the amplitude and the phase perturbation derease as the bakground-fluorophore lifetime 1 inreases. For example, given the same relative onentration and lifetime e.g., N 2 N 1 5 and 2 1 5, when the absolute bakground-fluorophore lifetime inreases from 1.0 to 5.0 ns we find that the smallest detetable objet size is 0.32 m in radius ompared with 0.25 m in radius for a shorter bakground lifetime ns. 5. Charaterization Limits Detetion of an objet does not imply that we an haraterize the objet. Charaterization is more diffiult. Complete haraterization inludes determination of all parameters of a heterogeneous medium. For a simple ase in whih there is one spherial objet embedded in an otherwise homogeneous-bakground turbid medium, these parameters are the endogenous absorption and sattering oeffiients of the objet and the bakground at the exitation and the emission wavelengths eight parameters, the fluorophore onentration and lifetime inside and outside the inhomogeneity four parameters, and the size and the position of the inhomogeneity four parameters. So we have a 16-parameter set. To explore the limits to whih the FDPDW an be fully used to haraterize a fluoresent objet, we simplify the problem by assuming that only a few parameters are unknown and all others are known e.g., determined by other modalities suh as ultrasound or MRI independently, or in onjuntion with the optial method. There are two situations that we onsider. First, we simultaneously haraterize the fluorophore onentration N 2 and lifetime 2 inside the inhomogeneity with an assumption that all other parameters are known. Seond, we simultaneously haraterize the interior fluorophore onentration N 2 and the size of the inhomogeneity a, assuming again that all the other parameters are known. The two situations that we onsider here are therefore best-ase ones. A. Charaterization of an Objet with a Known Size For this ase we alulate the FDPDW by using the exat analytial solutions for whih the objet size is kept at 0.5 m in radius. The soure position is fixed, and the detetor sans over 2.0, 2.0 m along the x axis with steps of 0.2 m Fig. 1 b. We then have 21 amplitude and 21 phase exat data points in total. The experimental data measurements are simulated by addition of 1% amplitude and 0.5 phase random noise to the exat data. We then apply the 2 fitting proedure to haraterize simultaneously the fluorophore onentration and lifetime of the fluoresent objet, assuming that all other parameters are known. The 2 to be minimized is the weighted differene between the measurements and the theoretial estimate, i.e., 21 2 N 2, 2 i 1 A i, M N 2, 2 A i, Th N 2, 2 2 A 2, (2) i, M N 2, 2 i, Th N 2, where A i,m and i,m are the amplitude and the phase of the ith measurements, respetively one of 21 measurements along the x axis, and A i,th and i,th are the amplitude and the phase of the orresponding theoretial preditions. A and are the amplitude and phase errors in orresponding measurements. The minimization is done by use of the Simplex Down-hill subroutine in Numerial Reipes in C. 26 For any given fluorophore onentration and lifetime, we take five sans by using different seeds for the random-noise generator. We then fit these five data 6838 APPLIED OPTICS Vol. 37, No Otober 1998

7 Fig. 4. Charaterization limits for an objet of a known size radius a 0.5 m. a haraterization unertainties in fluorophore onentration N 2. For a given lifetime the unertainty is smaller for a higher onentration; for a given fluorophore onentration the unertainty is smaller for a longer lifetime. b Charaterization unertainties in lifetime 2. For a given lifetime the unertainty is smaller for a higher onentration; for a given onentration the unertainty is smaller for a greater lifetime variation. See Subsetion 5.A for explanations. Fig. 5. Example of 2 distribution versus fitting parameters in haraterization of an objet with a known size. Fluorophore onentration N 2 and lifetime 2 are the two fitting parameters, whih we haraterize simultaneously. a 2 versus onentration for a long and a short lifetime 2. A longer lifetime orresponds to a narrower valley. b 2 versus lifetime for a high and a low onentration N 2. A higher onentration N 2 N 1 6 orresponds to a narrower valley. sets separately. The fitting results an be different for these five data sets. The standard errors N 2 and 2 of the fitted parameters are alulated for five sans, and the frational unertainties in the onentration and lifetime haraterization N 2 N 2 and 2 2 an then be obtained. We repeat the proedure for different fluorophore onentration and lifetime ontrast N 2 N 1 and 2 1. The ontours of the frational unertainties in the fluorophore onentration and lifetime haraterization are illustrated in Fig. 4 for a broad range of fluorophore onentration ontrast x axis and lifetime variation y axis. In Setion 4 we saw that higher onentrations N 2 and longer lifetime 2 introdue greater perturbation to the FDPDW, and therefore smaller haraterization errors are expeted as these parameters are inreased. This result is in agreement with the overall trends in Fig. 4. It shows that the haraterization error is smaller for higher onentration and longer lifetime. We also notie that the frational haraterization error in lifetime is greater when the lifetime gets longer, whih appears surprising in light of the fat that longer lifetimes give rise to greater perturbations. However, note that we onsider a 0.5 absolute random phase noise and that the absolute phase error propagates into the lifetime through the phase tan 1 2. Beause of the saturation harateristis of the tan 1 x funtion this 0.5 random phase noise will orrespond a larger lifetime unertainty for a larger 2. As shown in Fig. 4, if the objet size is known by some other means, the fluoresent objet an the be haraterized very aurately over a broad range of fluorophore onentration ontrast 2 N 2 N 1 10 and lifetime variation , e.g., within 6% frational unertainties. For an objet of a smaller size e.g., 0.3 m in radius and over the same range of onentration ontrast and lifetime variation, the haraterization unertainties will be muh larger, e.g., 20%, whereas for an objet of a greater size e.g., 0.8 m in radius the haraterization unertainties will be muh less, e.g., 3%. Loal 2 minima are problemati for any fitting proedure. One way to evaluate the fitting quality is to hek the 2 distribution versus fitting parameters. A good fit will orrespond to a minimum 2 with a narrow valley. Figure 5 a illustrates an example of 2 versus fluorophore onentration N 2 for two different fluorophore lifetimes and We hose these two lifetimes as representatives of a short and a long lifetime. A lear single valley exists for both short and long lifetimes. As we expet, a longer lifetime orresponds to a narrower valley. For omparison, 2 versus the fluorophore lifetime 2 for two different onentrations N 2 N 1 2 and N 2 N 1 6 is shown in Fig. 5 b. In this ase the 2 urve has a narrower valley for higher onentration. For other onentrations or lifetimes the 2 urves follow the general trends, as illustrated in Fig. 5. B. Charaterization of an Objet with an Unknown Size We next onsider the haraterization limits for an objet with unknown size. For this ase we apply the 2 minimization proedure and fit for size and onentration simultaneously with all other parameters known. For simpliity, we assume that there is no lifetime ontrast e.g., 2 1. We follow a proedure similar to that used in Subsetion 5.A. The simulated experimental data again have 1% random noise in amplitude and 0.5 random noise in phase. The only differene is that the fitting parameters are the onentration N 2 and the radius a in this ase; we minimize 2 by varying these two parameters. The ontours of the frational unertainties in the fluorophore onentration and objet radius are presented in Fig. 6 over a broad range of fluorophore onentration x axis and objet size y axis. For a given size, the higher the onentration, the smaller the haraterization unertainties; for a given fluorophore onentration, the bigger the objet size, the smaller the haraterization unertainties. 1 Otober 1998 Vol. 37, No. 28 APPLIED OPTICS 6839

8 Fig. 6. Charaterization limits for an objet of an unknown size. a Charaterization unertainties in fluorophore onentration N 2. b Charaterization unertainties in radius a. In both a and b, for a given objet size the unertainties are smaller for a higher onentration; for a given fluorophore onentration the unertainties are smaller for a bigger size. The smallest haraterizable objet size is 0.75 m in radius, onsidering a fivefold onentration ontrast and a 20% haraterization unertainty. Qualitatively speaking, it is easier to haraterize a bigger objet with higher onentration ontrast. From these ontour plots Fig. 6 we see that, when the objet exeeds 0.75 m in radius, the haraterization unertainties are 20% or less for a fivefold onentration ontrast or higher. If we onsider a fivefold onentration ontrast as a pratial obtainable ontrast and onsider 20% haraterization unertainty as aurate in pratie, we find that the smallest haraterizable objet size is 0.75 m in radius. For a higher onentration ontrast e.g., N 2 N 1 20 we an haraterize an objet with an even smaller size e.g., 0.6 m in radius. The 2 urves are also plotted in Fig. 7 to ensure that our fitting results are not trapped in loal minima. C. Multiple Moments Analysis Given a fivefold onentration ontrast, although the smallest detetable objet size is 0.25 m in radius, the objet is not haraterizable unless the radius exeeds 0.75 m beause of the funtional dependene of the sattered FDPDW on the objet size and the onentration. As we disussed in Setion 2, the sattered FDPDW is the superposition of different multipole moments. The monopole is related to N 2 a 3, whereas the dipole is related to N 2 a 5 at the limit of low onentration and small objet size e.g., Fig. 7. Example of 2 distribution versus fitting parameters in haraterization of an objet with an unknown size: a 2 versus onentration and b 2 versus radius. Fig. 8. a Frational amplitude and b relative phase hanges for different moments versus objet size. The onentration ontrast is assumed to be fivefold. Horizontal dashed lines indiate 1% amplitude and 0.5 phase noise level. The geometry is given in Fig. 1 a. when we Taylor expand the moments with respet to N 2 and a. For detetion, only the monopole need exeed the noise level; e.g.,,l 0 s homo 1% or Arg ;l 0 s homo 0.5. On the other hand, whenever two parameters appear as a produt in a funtion, to fit these two parameters simultaneously it is required mathematially that at least two different funtional dependenies of the funtion on these two parameters exist. Therefore for simultaneous haraterization of the onentration and size it is neessary for both the monopole and the dipole moments to exeed the noise level. This multipole moments analysis provides us with another way to investigate the detetion and haraterization limits. Given a fivefold onentration ontrast and no lifetime ontrast, we alulate the perturbation of the FDPDW for different multipole moments that are due to objets of different sizes. The amplitude,l 0,1,2 s homo and phase Arg,l 0,1,2 s homo perturbation versus objet size for the monopole, the dipole, and the quadrapole are plotted in Fig. 8 for different-sized objets, where we also indiate the 1% amplitude and 0.5 phase noise level by horizontal dashed lines. We find that when the objet radius exeeds 0.25 m the amplitude and the phase of the monopole moment exeed the noise level 1% in amplitude and 0.5, and this objet is thus detetable. Only when the objet radius exeeds 0.6 m do both the monopole and the dipole moments exeed the noise level, and this objet therefore is haraterizable. The results of this moment analysis are in agreement with the results from 2 fitting and atually provide a more rapid method to assess detetion and haraterization feasibility. 6. Comparison of Fluoresene and Absorption Contrast Exogenous fluorophores not only provide fluoresene ontrast for tumor detetion but also enhane absorption ontrast. To ompare the fluoresene and absorption ontrast indued by exogenous fluorophores we arry out an analysis similar to that in Setion 4 for both FDPDW and exitation DPDW; e.g., we are interested only in the amplitude and phase perturbation of the exitation DPDW and FDPDW. Inas APPLIED OPTICS Vol. 37, No Otober 1998

9 Fig. 9. Contours of frational amplitude and relative phase hanges versus onentration and radius. The lifetime ontrast is set to 1 for the fluoresene detetion. a and b show the absorption ontrast; and d show the fluoresene ontrast introdued by exogenous fluorophores. muh as lifetime does not play a role in the absorption measurement, to ompare the absorption and fluoresene under the same onditions we simply assume that the lifetimes inside and outside the objet are equal 2 1 1ns and therefore that there is no lifetime ontrast for the fluoresene detetion. Using the geometry in Fig. 1 a and analytial solutions for exitation DPDW and FDPDW, 10,27 we alulate the amplitude and phase perturbation for both the exitation DPDW s and the FDPDW s that are due to different-sized objets over a broad range of fluorophore onentration ontrast N 2 N 1 ; e.g., ex ex hetero homo forthe exitation DPDW and hetero homo for the FDPDW. The optial properties and other parameters are given in Tables 1 and 2. 1% frational amplitude and 0.5 phase perturbation ontours of the exitation DPDW are shown in Figs. 9 a and 9 b for different-sized objets as a funtion of fluorophore onentration; Figs. 9 and 9 d illustrate the 1% frational amplitude and 0.5 phase perturbation ontours of the FDPDW. Assuming a detetion system with 1% amplitude and 0.5 phase noise and utilizing the fluorophore-indued ontrast for both absorption and fluoresene we see that, for fivefold onentration ontrast, the smallest detetable objet is 0.5 m in radius when we probe the turbid medium using exitation DPDW, whereas the smallest detetable fluoresent objet under the same onditions is 0.25 m in radius. Comparing Figs. 9 a and 9 b with Figs. 9 and 9 d, we find that, in general, for a given fluorophore onentration and objet size, both the amplitude and the phase perturbation to the FDPDW are greater than those to the exitation DPDW. Given a perfet optial filter to rejet the exitation light in the fluoresene measurement, apparently the detetability is better in the fluoresene mode than in the absorption mode. In fluoresene measurements, however, given a 3-mW exitation light soure that emits photons s, the bakground exitation photon fluene is photons s for a 5-m soure detetor separation and an ative detetion area of 0.3 m in diameter. Under the same onditions the fluoresent photon fluene is photons s see Tables 1 and 2 for quantum yield, extintion oeffiient, and lifetime. Thus the bakground exitation signal an be 3 orders of magnitude larger than the fluoresene signal. A small amount of exitation light leakage e.g., 1% an overwhelm the weak fluoresene signal. Therefore, to measure the fluoresene signal and utilize fully the higher fluoresene ontrast introdued by fluorophores for deteting hidden objets, we need an optial filter with a rejetion optial density (OD) of 4.0 to suppress the strong bakground exitation light. As we know see Eq. 1, the fluoresene signal is proportional to the fluorophore quantum yield. A filter with a higher rejetion OD is orrespondingly needed for fluorophores with lower quantum yield. Mathematially the required rejetion OD of a filter is saled by the logarithm of the inverse of fluorophore quantum yield. In the above analysis we assumed a 10% quantum yield. In pratie, for fluorophores with muh lower quantum yields e.g., 0.1% the fluoresene measurement an be diffiult. The required rejetion OD of the optial filter an be as great as 6. Different fluoresent ontrast agents generally have different extintion oeffiients. The relative fluoresene absorption ontrast varies with the extintion oeffiient. Consider the geometry in Fig. 1 a with a spherial objet of 0.5-m radius. Figures 10 a and 10 b show the ontours of frational amplitude and phase for the absorption signal at exitation wavelength em, and Figs. 10 and 10 d show the ontours of frational amplitude and phase for the fluoresene signal at emission wavelength ex. We find that, for a fivefold onentration ontrast, the perturbation to the FDPDW an exeed 5% in amplitude and 1 in phase for fluorophores with extintion oeffiients lower than m 1 M 1, whereas the perturbation to the exitation DPDW exeeds 5% in amplitude only when the extintion oeffiient is greater than m 1 M 1. Note that the phase perturbation to the exitation DPDW is very small e.g., 0.5 over a broad range of the extintion oeffiients and onentration ontrast. Therefore it is diffiult to detet the inhomogeneity by use of fluoresent ontrast agents from only the phase measurements of the exitation DPDW. Roughly speaking, if we have an optial filter with an OD of 4 to rejet bakground exitation light, the fluoresene signal provides better ontrast for fluoresent ontrast agents with lower extintion oeffiients; only for higher extintion oeffiients 1 Otober 1998 Vol. 37, No. 28 APPLIED OPTICS 6841

10 Fig. 11. a Frational amplitude perturbation versus modulation frequeny for different lifetime ontrasts; b relative phase perturbation versus modulation frequeny for different lifetime ontrasts. Given a lifetime ontrast 2 1, we find that the amplitude and the phase perturbation that are due to the lifetime ontrast will be optimally elevated when the soure modulation frequeny is appropriately hosen, e.g., 2 1. Fig. 10. Contours of frational amplitude and relative phase hanges for a, b absorption and, d fluoresene ontrast versus onentration and extintion oeffiient. The objet size is 0.5 m in radius, and the soure detetor geometry is shown in Fig. 1 a. For fluorophores with an extintion oeffiient near m 1 M 1, we see that the fluoresene ontrast is in general greater than the absorption ontrast. e.g m 1 M 1 is the absorption ontrast superior to the fluoresene ontrast. The bakground-fluorophore onentration is taken to be N M in the above disussion. At a fixed relative onentration ontrast N 2 N 1 the bakground onentration N 1 determines the absolute onentration ontrast, and therefore it will also affet the perturbation to both the FDPDW and the exitation DPDW. As we disussed at the end of Setion 4, for a fixed relative onentration N 2 N 1 the inrease in the bakground onentration N 1 redues the detetability of an objet probed by the FDPDW. Similar analysis shows that the inrease in bakground onentration N 1, on the other hand, enhanes the detetability through the absorption measurement. We have found that, for a given bakground fluorophore onentration N 1, the fluoresene ontrast is better than the absorption ontrast at deteting a hidden objet when the absorption introdued by the exogenous bakground fluorophore is low, e.g., N 1 a1, where a1 is the endogenous bakground absorption. 7. Disussion and Conlusion It is essential to have multiple data points to haraterize an inhomogeneity by use of a 2 fitting proedure. So far we have onsidered multiple measurements by sanning the detetor. We hose our san region from 2 to 2 m in the above disussion. This san region was hosen so that soure and the detetor would be lose enough to ensure that the amplitude and the phase perturbations are greater than the system noise level. Measurements made at large soure detetor separation will not help to improve the detetion or haraterization if the perturbation is less the noise level. An alternative approah to obtaining multiple measurements is to employ multiple modulation frequenies for a single soure detetor position. 28 We have found that high modulation frequenies are not desirable. Higher modulation frequenies produe larger wave attenuation fators and therefore lower exitation and fluoresene fluenes. As we disussed above e.g., see Setion 4, the lifetime is oupled to the modulation frequeny through the demodulation fator 1 i. The demodulation introdues an amplitude perturbation and a phase shift tan 1 2 tan 1 1. The amplitude perturbation is roughly proportional to the lifetime ontrast 2 1 at low modulation frequenies e.g., 1, At high modulation frequenies e.g., 1,2 3, the amplitude perturbation diminishes and beomes independent of the lifetime ontrast. The phase shift follows the trend of the funtion tan 1 x. It an be shown that the peak of the phase shift, tan 1 2 tan 1 1 tan tan 1 1, whih is due to the lifetime ontrast 2 1, ours when Considering all other fators that ontribute to the phase shift suh as the onentration ontrast and the endogenous absorption ontrast, we find that 2 1 gives a simple estimate for the modulation frequeny at whih we will approximately obtain a maximum phase perturbation. These features of the amplitude and phase perturbations versus the modulation frequeny are shown in Fig. 11 for several lifetime ontrasts 2 1, where we assume that we have a 0.5-m-radius objet and a fixed onentration ontrast N 2 N 1 5. Other parameters are given in Tables 1 and 2. Clearly the hoie of an appropriate modulation frequeny an help in optimal use of the ontrast provided by the lifetime ontrast, and an inappropriate hoie of the 6842 APPLIED OPTICS Vol. 37, No Otober 1998

11 modulation frequeny an redue or demolish the ontrast. We an obtain a maximum amplitude perturbation when we employ a d soure 0, but we will lose the phase information. The rule of thumb for hoosing the modulation frequeny for fluoresene measurements is that 2 1, whih will ensure that we have enough sensitivity to the phase perturbation that is due to the lifetime ontrast without degrading the amplitude perturbation too muh. This disussion an also apply to phosphoresene with milliseond lifetime ranges for whih the optimal modulation frequeny would be of the order of kilohertz. We draw the following onlusions: 1 If our detetion system is limited by 1% noise in amplitude and by 0.5 in phase, the smallest detetable fluoresent objet size is 0.25 m in radius, given a fivefold fluorophore onentration and lifetime ontrast. 2 An inhomogeneity an be well haraterized within 6% haraterization unertainties if a priori information about its size is known. 3 When no a priori information about the objet size is available, the haraterization is then more diffiult. The smallest haraterizable objet size in this ase is 0.75 m in radius. For systems with lower noise levels or higher fluorophore onentration ontrast, we an then detet and haraterize a smaller objet. We an also statistially redue the haraterization errors by inreasing the number of measurements e.g., the haraterization error 1 N, where N is the total number of measurements. Finally, in our omparison of the absorption and fluoresene ontrast introdued by exogenous fluorophores we find that, given a suffiiently large fluroesene filter OD e.g., 4 and fluoresene quantum yield e.g., 10%, the fluoresene mode of measurements is in general superior to the absorption mode for fluorophores with moderate extintion oeffiients. Some fluorophores with higher affinity for tumor tissue have been under investigation. 29,30 These high-ontrast fluorophores make the fluoresent probe more promising and attrative for use in tumor detetion and imaging. Appendix A As was shown in Ref. 10, for a fluoresent spherial objet embedded in an infinite fluoresent turbid medium, the FDPDW is the superposition of the FDPDW in a homogeneous fluoresent medium home and the sattered FDPDW s ; i.e., hetero r s, r d,, a homo r s, r d, s r s, r d,, a. (A1) Here r s and r d denote the soure and the detetor positions, respetively; is the soure angular modulation frequeny; and a is the objet radius. The homogeneous FDPDW home is q 1 1 N 1 homo r d, r s, A 1 1 i 1 exp ik 1 r d r s 4 r d r s exp ik 1f r d r s 4 r d r s. (A2) Here A 1 is a onstant that depends on the optial properties at both exitation and emission wavelengths ex and em as well as on the exitation soure strength; k 1 and k 1f are the wave numbers of the diffusive photon density wave that orrespond to the optial properties at the exitation and emission wavelengths, respetively; is the fluorophore extintion oeffiient; and N 1, 1, 1, and q 1 are the bakground fluorophore onentration, lifetime, quantum yield, and quenhing fator, respetively. The sattered FDPDW s an be written as q 2 2 N 2 s r d, r s,, a A 2 1 i 2 lm Y lm. B lm h l 1 k 1f r C lm h l 1 k 1 r (A3) Here A 2 is a onstant that depends on the optial properties at both exitation and emission wavelengths ex and em as well as on the objet size and the exitation soure strength; N 2, 2, 2, and q 2 are the fluorophore onentration, lifetime, quantum yield, and quenhing fator inside the objet, respetively; and B lm and C lm are ompliated onstants that depend on the optial properties inside and outside the objet at the exitation and emission wavelengths ex and em, the objet size and the exitation soure position. h 1 l is the spherial Hankel funtion of the first kind, and Y lm is the spherial harmonis., k 1 and k 1f are the same as in Eq. A2. We are grateful to T. Durduran for helping with some of the relevant data analysis and to D. A. Boas and W. Ralston for some of the relevant disussions. We thank Mary Leonard for exellent drafting. A. G. Yodh aknowledges support from the National Siene Foundation under grant DMR B. Chane aknowledges support in part from the National Institutes of Health under grants CA and CA Referenes and Note 1. A. Yodh and B. Chane, Spetrosopy and imaging with diffusing light, Phys. Today 48 3, , and referenes therein. 2. See related studies in B. Chane, R. R. Alfano, and A. Katzir, eds., Optial Tomography and Spetrosopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, Pro. SPIE D. A. Boas, M. A. O Leary, B. Chane, and A. G. Yodh, Dete- 1 Otober 1998 Vol. 37, No. 28 APPLIED OPTICS 6843