{"id":"4a1cb1d6-5ffc-4ebe-97b1-580ca511d439","arxiv_id":"2411.15698","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"For nonzero fluorescence lifetime, the peak time of the time-domain FDOT response is approximated by an explicit nonlinear equation, enabling bisection and boundary-scan localization of point targets.","lead":"The authors find a simple formula for the moment a fluorescent signal peaks in scattering tissue, for markers that stay lit for a while, and use that peak time to work out where the markers are. They test the formula in numerical experiments and design fast algorithms for finding one or several glowing points.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The multi-target boundary-scan algorithm rests on an unproved equivalence between local minima of the measured peak-time landscape and true target locations; this is the weakest load-bearing assumption in the paper's central claim.","rationale":"I read the paper in good faith and find the single-target asymptotic derivation and bisection algorithm credible: the explicit peak-time equation (3.3) is derived from a Laplace-type asymptotic, the uniqueness theorem 3.1 is sound under its stated condition, and the numerical studies in Section 3 and Example 5.1 give plausible relative errors. The strongest single concern is not in the single-target analysis but in the multi-target extension. Corollary 2.4 proves an asymptotic dominance statement only for S-D pairs satisfying condition (2.18), yet Definition 4.4 builds the notion of 'well-separated' directly from the desired output of Algorithm 3, namely J local minima of the measured peak-time landscape. This is definitionally circular as a justification of the algorithm. The paper itself concedes in Section 4.3 that the reasoning is speculative ('we reasonably speculate'), and the numerical success in Example 5.3 covers only a single well-separated configuration with two targets and modest noise. The missing piece is a theorem or systematic numerical study showing that for targets that are 'well-separated' in a geometric sense (e.g., separation comparable to depth, or separation large relative to the scan-grid spacing), the measured peak-time landscape must have exactly J local minima, each within one grid cell of the corresponding target's horizontal projection. Without this, the boundary-scan algorithm can fail in ways not captured by Example 5.3. My recommended verdict is therefore unchanged: CONDITIONAL, with the multi-target precondition needing explicit verification.","tokens_in":23514,"tokens_out":17028,"duration_ms":150916,"concrete_test":"Run Algorithm 3 with noise-free data for two targets x1=(10,10,20) and x2=(10+delta,10,20) for delta = 4, 6, 8, 10, 12, 14 mm, using the scan grid (4.18) with M=N=40 and L=2, computing t_peak by numerical integration of (2.1). For each delta, record the number of local minima of the scanned peak-time landscape and the reconstructed positions; also check whether for each target there is a grid S-D pair satisfying both (2.18) and (4.17). If for any delta below a threshold either the number of local minima is not 2, a reconstructed target deviates by more than one grid cell from its true horizontal position, or no grid S-D pair satisfies (2.18)+(4.17), then the well-separated assumption of Definition 4.4 is violated and the boundary-scan algorithm's central premise is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Corollary 2.4 justifies the multi-target approximate peak-time equation (3.8) only for S-D pairs satisfying the dominance condition (2.18), i.e. one target strictly minimizes |x_d-x_c^(j)|^2 + |x_s-x_c^(j)|^2. Definition 4.4 then defines targets as 'well-separated' precisely by requiring the measured peak-time landscape to have J local minima, which is the conclusion Algorithm 3 needs rather than a proven premise. No theorem establishes that the number of local minima of t_peak equals J, that each such minimum is located at a grid S-D pair satisfying (4.17) for a true target, or that the discrete scan grid Xi contains such a pair for every target. The paper's only multi-target test, Example 5.3, uses two targets separated by about 14 mm at 16-18 mm depth on a 21x21 grid, and even then the reconstructed positions (3.00,5.00,15.67) and (17.00,17.00,17.72) are visibly shifted by the grid spacing; at noise level 1% the second target shifts to (18.00,17.00,17.84). The algorithm also requires an ad hoc 3x3 moving-average smoothing step to remove spurious local minima, which itself signals that the measured peak-time landscape does not robustly encode the target count. For closer targets, where (2.18) fails on some S-D pairs, the sum-of-contributions peak can have fewer, extra, or shifted local minima, and Algorithm 3 could report spurious or missing targets. This is the central unsupported step for the multi-target reconstruction claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies an inverse problem for time-domain fluorescence diffuse optical tomography (FDOT) in a half-space with point fluorescent targets having nonzero fluorescence lifetime. The authors start from the coupled diffusion model (1.1)-(1.2), replace the emission solution by an asymptotic expansion in the fluorescence lifetime (2.2)-(2.3), and then use a Laplace-type asymptotic argument to derive the approximate peak time equation (3.3), whose root defines the approximate peak time. Uniqueness of this root is proved in Theorem 3.1. The equation is extended to multiple targets under a nearest-target dominance condition (2.18), and two reconstruction algorithms are proposed: a bisection algorithm for a single target and a boundary-scan algorithm for multiple well-separated targets. Numerical experiments in Section 5 test the approximation and the algorithms under noiseless and noisy data.","tokens_in":23892,"tokens_out":12303,"duration_ms":111835,"significance":"If the results are valid, equation (3.3) is a genuinely useful reduction: peak-time localization becomes one-dimensional root-finding once the horizontal target coordinates are known, and the derivation is parameter-free with no constants fitted to data. The paper's strengths include the explicit derivation from (2.2) through Theorem 2.3, the uniqueness theorem for the approximate peak time, the reproducible numerical verification over a range of optical parameters, and the demonstrated robustness of the single-target bisection algorithm. However, the multi-target reconstruction claim depends on an unproved equivalence between local minima of the peak-time landscape and true target positions, and the dominance condition underlying the multi-target asymptotic formula is insufficient at the peak-time scale. The single-target contribution is sound and publishable, but the multi-target claim currently needs substantial additional support.","major_comments":[{"comment":"The dominance claim in Corollary 2.4 is not justified at the time scale used in the rest of the paper. Condition (2.18) only asserts that one target has the smallest value of |x_d-x_c^(j)|^2+|x_s-x_c^(j)|^2. But Theorem 2.3 evaluates the time integral at the saddle s = λ^(l) k^{-1/2}, so the relevant comparison time in equation (3.3) is t ≈ λ^(l)/√k. The exponential ratio controlling the contribution of a competing target j is then exp(-((λ^(j))^2-(λ^(l))^2)/t) = exp(-√k ((λ^(j))^2-(λ^(l))^2)/λ^(l)). This quantity is not necessarily o(1): if the target separation in depth is fixed while λ^(l) grows, the exponent tends to 2√k Δλ, a constant, so the competing target contributes at O(1) rather than o(1). Therefore (2.19), (2.20), and the multi-target approximate peak time equation (3.8) require a stronger separation condition, such as (λ^(j))^2-(λ^(l))^2 ≫ λ^(l), which is neither stated nor verified in Example 5.3.","section":"Corollary 2.4, Eqs. (2.19)-(2.20) and (3.8)"},{"comment":"Definition 4.4 defines well-separated targets by assuming that the peak-time landscape P has J local minima. This is exactly the conclusion that Algorithm 3 needs in order to reconstruct J targets, not a proven premise. No theorem establishes that the number of local minima of the measured t_peak equals J, that each local minimum is attained at an S-D pair satisfying (4.17), or that the discrete scan grid Ξ contains such a pair for every target. In Example 5.3, the reconstructed two-dimensional positions are (3.00,5.00) versus the true (3.3,5.2) and (17.00,17.00) versus (17.4,16.7), showing grid quantization; at 1% noise the second target shifts to (18.00,17.00). The ad hoc 3×3 moving-average smoothing used in Example 5.3 to remove spurious local minima indicates that the peak-time landscape does not robustly encode the target count. This is the central unsupported step for the multi-target reconstruction claim.","section":"Definition 4.4 and Algorithm 3"},{"comment":"The statement that the approximate peak time has excellent accuracy under practical parameters is not supported for all parameter values shown. In Figure 3.1(d), with default parameters (3.7), the relative error exceeds 10% at fluorescence lifetime ℓ = 500 ps, and the error decreases only gradually as ℓ increases, consistent with the ℓ ≫ 1 assumption in expansion (2.2). The paper does not quantify the joint validity region in (ℓ, λ, x_c3) and gives no error bound for equation (3.3). Since Stage 2 of the single-target reconstruction solves (3.4) with the measured t_peak as input, a quantitative statement about the approximation error is needed to assess the bias that the algorithm can introduce.","section":"Section 3.1 and Figure 3.1(d)"}],"minor_comments":[{"comment":"The monotonicity sentence after Eq. (4.13) has the signs reversed: for fixed λ^(1), the left-hand side t ln(λ1/λ2) decreases as λ2 increases, while the right-hand side increases. The uniqueness conclusion is correct and can be proved from this one-signed behavior, but the proof as written should be corrected.","section":"Proof of Theorem 4.2, text near Eq. (4.13)"},{"comment":"Eq. (4.10) contains an apparent typo: the threefold equality ta,(2)_peak = ta,(2)_peak = ta,(2)_peak should presumably define the two quantities in (4.11), which are the supremum-based and infimum-based characterizations.","section":"Eq. (4.10)"},{"comment":"The notation 'ROI := (xb, x_t)' and 'ROI := (x_b, x_t)' is inconsistent and ambiguous; the one-dimensional interval passed to Algorithm 2 should be written consistently, e.g. (x_b, x_t) or (x_l, x_r).","section":"Algorithm 1, Step 3"},{"comment":"The perturbation in Eq. (5.1) is described as time jitter, but it is a relative multiplicative perturbation of t_peak by a uniform random variable. It is not a jitter in the measured temporal response function. Please clarify the noise model.","section":"Eq. (5.1)"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the single-target part of the paper is technically sound and could be published after a modest revision. The multi-target part, however, is the advertised novelty and currently rests on an unproved identity between local minima of the peak-time landscape and true target locations, and on a dominance condition that is insufficient at the peak-time scale. I would not accept the paper in its present form until that gap is either proven under stated conditions or removed from the claims."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the single-target result is the real content here, and it holds up. The multi-target boundary-scan is the soft spot, and the stress-test note is fair on that point. The paper is not hiding anything, but the gap is real.\n\nWhat is new: eq. (3.3) gives an explicit asymptotic equation for approximate peak time in FDOT when fluorescence lifetime is nonzero, which the prior work [4,5] only covered for zero lifetime. The derivation from the integral representation (2.2) through Theorem 2.3 is explicit and checkable. Numerical tests in Figure 3.1 cover representative optical parameters and show small relative errors in the intended regime—large lifetime and large lambda. The bisection algorithm for a single point target follows from Theorem 4.2 and the examples, including noisy data, look reasonable. No constants are fitted, so the circularity burden is genuinely low.\n\nSoft spots: the approximation needs ell and lambda large. Figure 3.1(d) shows relative errors above 10% at shorter lifetimes; the text mentions this but could state it as an applicability limitation. The multi-target algorithm is the main weakness. Corollary 2.4 justifies the asymptotic peak time only for source-detector pairs where one target dominates, and Definition 4.4 defines 'well-separated' operationally by requiring J local minima in the measured peak-time landscape. That is not a derivation: no theorem proves geometric separation implies distinct local minima, nor that the scan grid contains a pair satisfying (4.17) for each target. Example 5.3 is the only multi-target test, and the positions shift by the grid spacing; at 1% noise the algorithm needs a 3x3 moving-average to suppress spurious local minima, which itself indicates the landscape can encode extra minima. Also, no code or data are released, so the numerics are not independently reproducible.\n\nWho this is for: people working on time-domain FDOT or inverse problems with point targets. The single-target equation is a useful, citable tool. The multi-target part is promising but needs stronger justification, either a proof under explicit separation conditions or a systematic numerical study of separation limits.\n\nRecommendation: accept for peer review. Ask the authors to sharpen the multi-target claims, quantify the lifetime/depth regime more carefully, and release code or detailed data. But the core derivation deserves a proper referee.","headline":"Single-target peak-time equation for nonzero fluorescence lifetime is a solid new result; the multi-target boundary-scan is heuristic and needs more support, but the paper deserves serious review.","tokens_in":24402,"tokens_out":4079,"would_cite":true,"duration_ms":35862,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","35K20"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single explicit equation approximates the peak time of time-domain fluorescence diffuse optical tomography, reducing target localization to a one-dimensional root-finding problem.","keywords":["FDOT","peak time","fluorescence lifetime","asymptotic analysis","diffuse optical tomography","point targets","boundary-scan reconstruction","bisection algorithm"],"falsifier":"Compute the true peak time by numerical integration of (2.1) for two targets placed so that, for the central source-detector pair, $|x_d-x_c^{(1)}|^2+|x_s-x_c^{(1)}|^2 = |x_d-x_c^{(2)}|^2+|x_s-x_c^{(2)}|^2$ (violating (2.18)), and compare the true peak time with the roots of (3.8) for each target; if neither root approximates the true peak time, or if Algorithm 3 returns a spurious local minimum between the two targets, the approximate equation's dominance assumption is the failing link.","tokens_in":23289,"feed_emoji":"⏱️","tokens_out":14560,"duration_ms":109255,"temperature":0.7,"pith_summary":"This paper tries to establish that, for a nonzero fluorescence lifetime $\\ell>0$ in the half-space diffusion model of fluorescence diffuse optical tomography, the peak time of the measured fluorescence response is well approximated by the unique positive root of the explicit equation $\\lambda e^{-(\\sqrt{kt}-\\lambda)^2/t} = \\pi^{1/2}\\ell^{-1}t^{3/2}$, where $\\lambda$ encodes the source-target and detector-target distances. The derivation runs through an asymptotic analysis of the time integral of the zero-lifetime solution, and the approximation is verified numerically for practical tissue parameters. From this equation the authors build two reconstruction algorithms: a bisection scheme for one point target and a boundary-scan scheme for well-separated multiple point targets. The practical value is that peak time is the most noise-resistant feature of the temporal response, so if the equation holds, the inverse problem collapses to a one-dimensional root-finding problem once horizontal coordinates are determined.","feed_headline":"Peak time of fluorescent light locates a target by one equation","feed_subtitle":"A new formula ties the peak light time to target distances, making localization a one-dimensional root search.","key_machinery":"The load-bearing object is the approximate peak time equation (3.3), $P(t;\\lambda)=\\lambda e^{-(\\sqrt{kt}-\\lambda)^2/t}-\\pi^{1/2}\\ell^{-1}t^{3/2}=0$, whose unique positive root is the approximate peak time. Here $\\lambda:=\\big((|x_d-x_c|^2+|x_s-x_c|^2)/(2vD)\\big)^{1/2}$ summarizes the target's location relative to each source-detector pair, and all uniqueness, order, and symmetry results are statements about how the root depends on this single number. The equation is obtained from the asymptotic expansion of Theorem 2.3, $\\int_0^t u^a_m(s)\\,ds \\sim k^{-3/4}(\\pi\\lambda)^{1/2}u^a_m(\\lambda k^{-1/2})$ for $\\lambda\\gg 1$, which converts the convolution integral in the fluorescence response into a point evaluation at the saddle time $s=\\lambda k^{-1/2}$, with $u^a_m$ the large-depth profile of the zero-lifetime solution. The same machinery carries to multiple targets by selecting, for each source-detector pair, the target that dominates the squared-distance sum.","core_discovery":"The paper's central claim is that for a single fluorescent point target in a highly scattering half-space with Robin boundary condition and nonzero fluorescence lifetime, the peak time $t_{\\mathrm{peak}}$ solves, to leading asymptotic order, $P(t;\\lambda)=0$ with $P(t;\\lambda):=\\lambda e^{-(\\sqrt{kt}-\\lambda)^2/t}-\\pi^{1/2}\\ell^{-1}t^{3/2}$, where $k=\\mu_av$ and $\\lambda^2=(|x_d-x_c|^2+|x_s-x_c|^2)/(2vD)$. For multiple well-separated targets, the same equation holds with $\\lambda$ computed from whichever target minimizes $|x_d-x_c^{(j)}|^2+|x_s-x_c^{(j)}|^2$ for that source-detector pair. The authors prove uniqueness of the root under a lower bound on $\\ell$, prove that the approximate peak time is monotonically ordered by $\\lambda$, and confirm numerically that the root tracks the true peak time to within a few percent across practical values of absorption, diffusion, lifetime, and depth. On the strength of these properties they assert that peak-time localization reduces to locating the minimal peak time over a boundary scan (horizontal coordinates) and then solving the one-dimensional equation for depth.","pith_inferences":["A consequence the authors leave implicit is that any monotone function of $\\lambda$, not just the peak time, could drive the same two-stage reconstruction; the bisection algorithm is essentially minimizing $\\lambda$ over the boundary, so alternative statistics of the temporal response that preserve that ordering would work.","The separation condition behind the multiple-target result is checkable from data only indirectly; a practical safeguard would be to verify near each detected local minimum that the measured peak time satisfies the single-target equation (3.8) at the reconstructed depth, flagging pairs whose residual exceeds the noise level as unresolved.","The saddle-point asymptotic of Theorem 2.3 is generic, so the same approximate equation should extend to other boundary conditions ($\\beta=0$, $\\beta=\\infty$) and, as the authors note, to curved measurement surfaces via parabolic-scaling Green functions; testing the formula on those geometries is a natural next step."],"forward_implications":["The explicit equation turns depth reconstruction into a one-dimensional root-finding problem once horizontal coordinates are known, eliminating iterative forward solves and regularization.","For a single target, the bisection algorithm converges to the true location, with numerical reconstructions accurate to a few percent relative error even when the measured peak time is perturbed by up to 5% noise.","For well-separated multiple targets, the boundary-scan algorithm recovers both horizontal coordinates from the local minima of the peak-time landscape and then solves each depth independently.","The paper's numerics show the approximation error decreases with increasing fluorescence lifetime and with smaller diffusion constants, so the formula is most trustworthy for nanosecond-lifetime fluorophores in tissues with smaller diffusion coefficient."],"supporting_citations":[{"why":"Supplies the large-depth asymptotic profile of the zero-lifetime solution used in Lemma 2.1 and the bisection reconstruction algorithm that the paper extends to nonzero fluorescence lifetime.","marker":"[4]"},{"why":"Derives explicit approximate peak time equations for the zero-lifetime case, the baseline whose nonzero-lifetime extension this paper provides.","marker":"[5]"},{"why":"Gives the coupled diffusion equation model with fluorescence decay convolution, the equations (1.1)-(1.2) whose solution the analysis starts from.","marker":"[16]"}],"fun_headline_variants":["Peak time equation locates fluorescent target","One equation gives target depth from peak time","Simplify FDOT localization to a single peak time equation","Fluorescence peak time root yields depth directly","Peak time equation turns FDOT into one root search"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"For multiple targets, the method assumes that from every source-detector pair one target is strictly closer in the squared-distance sense $|x_d-x_c^{(j)}|^2+|x_s-x_c^{(j)}|^2$ than all others, and that the measured peak-time landscape has exactly one local minimum per target; if two targets are close enough to violate this, the approximate equation and the scanning search can misidentify or miss targets.","fun_headline_variants_meta":{"raw":{"variants":["Peak time equation locates fluorescent target","One equation gives target depth from peak time","Simplify FDOT localization to a single peak time equation","Fluorescence peak time root yields depth directly","Peak time equation turns FDOT into one root search"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001946,"raw_usage":{"total_tokens":7630,"prompt_tokens":986,"completion_tokens":6644,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":602,"completion_tokens_details":{"reasoning_tokens":6573}},"tokens_in":602,"tokens_out":6644,"duration_ms":39850,"temperature":1.0,"reasoning_tokens":6573,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:00:33.034673+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the true peak time by numerical integration of (2.1) for two targets placed so that, for the central source-detector pair, $|x_d-x_c^{(1)}|^2+|x_s-x_c^{(1)}|^2 = |x_d-x_c^{(2)}|^2+|x_s-x_c^{(2)}|^2$ (violating (2.18)), and compare the true peak time with the roots of (3.8) for each target; if neither root approximates the true peak time, or if Algorithm 3 returns a spurious local minimum between the two targets, the approximate equation's dominance assumption is the failing link.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the coupled diffusion equation model with fluorescence decay convolution, the equations (1.1)-(1.2) whose solution the analysis starts from."}],"review_version":1}