Extreme transport of light in spheroids of tumor cells

Rogue waves (RWs) are statistically rare, extreme perturbations observed across complex systems, including oceans, plasmas, and optical media. Their formation involves linear and nonlinear processes such as chaos, turbulence, scattering, and topology, and is highly sensitive to microscopic system properties. While RWs can concentrate and transport energy in disordered systems, their occurrence in biological structures had not been reported. Biological tissues are thick, absorbing, and highly scattering, making deep visible-light concentration challenging. The study aims to determine whether optical extreme waves can form in dense, living tumor-cell spheroids and, if so, to characterize their statistics, physical origin (linear vs nonlinear), spatial structure, controllability via input conditions, and potential for biomedical light delivery.

The paper situates its work within extensive research on rogue waves across physics, noting mechanisms ranging from turbulence to nonlinearity and topology. In optics, RWs have been observed in fibers, lasers, Kerr media, and integrable turbulence, with ongoing debates on unifying mechanisms. Prior optical studies highlight both linear (e.g., random focusing) and nonlinear origins. In biophotonics, deep light delivery has focused on wavefront-shaping and guidestar approaches, but optical extreme events in biological aggregates were not documented. The authors distinguish their findings from long-tail statistics in OCT and diffuse reflectance, where anomalous PDFs are attributed to complex scatterer shapes and tissue multiscale structure, not nonlinear extreme events. They also relate to theories of eigenchannels in disordered media and to potential applications such as neuromorphic computing and phototherapy.

• Samples: Millimeter-sized three-dimensional tumor models (3DTMs) bioprinted from human pancreatic PANC-1 cells mixed with 5% w/v alginate, crosslinked with CaCl₂ and cultured. Average spheroid diameter: 3 ± 0.2 mm. Growth and stability monitored by optical/fluorescence microscopy; experiments performed two weeks after growth.
• Optical setup: CW laser at λ = 532 nm (up to 2 W), linearly polarized. Wavefront shaped by a reflective phase-only SLM (Hamamatsu X13138), with N = 64 input modes (120×120 pixels each; 2¹⁰ phase levels in [0, 2π]). Beam is filtered and focused (f = 100 mm, NA = 0.4) into 3DTM immersed in fresh culture medium inside a 10 mm quartz cuvette. Transmitted intensity imaged onto a CMOS camera (Basler a2A1920-160umPR, 12-bit) via objective (f = 75 mm, NA = 0.5). Setup in an incubator at 35 °C and 5% CO₂. Input power P measured before the sample.
• Coupling conditions: Weak vs strong coupling controlled by the distance d between the 3DTM and the input focal plane (e.g., d = 0 for weak, d = 1 mm for strong). This alters the transverse size of modes impinging on the 3DTM.
• Optical properties at 532 nm: Measured diffuse reflectance ρ = 0.3; absorption coefficient μ_a = 0.25 cm⁻¹. Low-power transport is diffusive; RWT predicts Rayleigh speckle statistics.
• Data acquisition: For fixed conditions, L = 640 transmitted intensity patterns are recorded at 10 Hz for different random input phase masks. A 200 × 200 µm² ROI is selected using the average intensity center of mass as origin to ensure statistical homogeneity.
• Statistical analysis: Intensity values normalized to mean. Local maxima extracted to build event series. Rogue wave threshold defined by oceanographic criterion I_RW > 2 I_s, with I_s the mean intensity of the highest third of events. PDFs computed and compared to Rayleigh and fitted to a Weibull distribution f(I) = γ λ^{-γ} I^{γ-1} exp(-(I/λ)^γ). Power dependence of shape γ and scale λ parameters measured. Euler Characteristics (EC) of the excursion set of amplitude η = √I at level h computed and compared with expectations for linear (Gaussian) and nonlinear (Tayfun) random wave surfaces.
• Spatial/coupling analysis: Positions and sizes of intense events mapped under weak and strong coupling to assess localization and size relative to speckle grain.
• Transmission matrix (TM): Reconstructed at P = 60 mW using L = 10×N = 640 random inputs over N = 64 input modes and M = 900 output modes (binned to speckle size ≈ 7 ± 1 µm). Complex TM estimated from intensity-only data via phase retrieval. Transmission eigenvalues τ_n (from t†t) analyzed; eigenchannels (singular vectors) and their input spectra examined to identify modes linked to RWs.
• Photothermal modeling: Evaluated RW-induced local temperature rise ΔT in water containing 20 nm Au nanoparticles using a thermal transfer model that neglects convection and assumes uniform background conductivity. RW areas defined as regions where I > 2 I_RW occurs across inputs; total transmitted intensity in that area used as I_0. Approximately 1 mW total power within ROI assumed; NP parameters from prior work. Input masks producing extreme events compared against homogeneous/random inputs for ΔT.

• Heavy-tailed transmission statistics: The PDF of transmitted intensity through 3DTMs exhibits a pronounced heavy tail relative to Rayleigh statistics, indicating frequent extreme events (rogue waves). The distribution is well modeled by a Weibull function.
• Weibull parameters and power dependence: Deviations from RWT (γ = λ = 1) emerge with increasing input power. Strong RWs occur within an optimal power range P ≈ 20–70 mW, then diminish; for P > 100 mW, the output becomes non-stationary. In the RW regime, the Weibull parameters converge to power-independent values λ_RW = 0.16 ± 0.07 and γ_RW = 0.45 ± 0.04, indicating a characteristic transported intensity fraction.
• Nonlinear origin: Capturing a RW at high power and gradually decreasing P reveals non-monotonic evolution of the peak intensity, with enhancement at intermediate power (≈ 20 mW), evidencing self-interaction and nonlinearity. EC analysis of amplitude excursion sets matches a nonlinear (Tayfun) random wave model rather than a linear Gaussian one, corroborating nonlinear generation.
• Spatial localization and size: Under strong coupling (e.g., d = 1 mm), RWs form localized optical filaments confined to specific regions, with transverse extent 10 ± 1 µm, larger than mean speckle size 7 ± 1 µm. Under weak coupling (d = 0), PDFs approach RWT and intense events are sparse and quasi-homogeneously distributed.
• RW identification and counts: Using the oceanographic criterion (I_RW > 2 I_s), the intensity maxima PDF shows abundant peaks exceeding I_RW (example value I_RW = 5.7 for a dataset of 25,600,000 points within a single spheroid), with approximately 2700 RWs detected.
• Transmission channels: In strong coupling, most transmission eigenchannels have reduced transmittance, but a few exhibit super-transmittance. RWs associate primarily with the largest eigenvalues (notably the 1st), whose input spectra display sharp peaks at specific wavenumbers k_RW that efficiently excite RWs. Similar localized spectral peaks are seen for the 2nd and 3rd eigenchannels.
• Photothermal potential: Leveraging RW localization, modeled local temperature rises within RW channels (I > 2 I_RW) exceed 10 °C for inputs that generate extreme waves, significantly surpassing homogeneous or random inputs under comparable conditions.
• Additional optical characteristics: Measured diffuse reflectance ρ = 0.3 and absorption coefficient μ_a = 0.25 cm⁻¹ at 532 nm indicate diffusive low-power transport. RWs weaken or are suppressed at very high power, consistent with nonlinear instability and randomization.
• Robustness to chemotherapy: Parallel experiments on 3DTMs treated with Gemcitabine (100 µM) show no systematic differences in intensity statistics or RW properties compared to untreated samples.

The results demonstrate that living, dense tumor-cell spheroids can support nonlinear optical rogue waves that act as localized, high-transmission channels for light transport across millimeter-scale, strongly scattering biological media. The power-dependent dynamics, EC analysis consistent with nonlinear wave surfaces, and sensitivity to coupling geometry support a nonlinear origin, likely involving optically induced thermal effects and structural rearrangements within the cell network. The identification of a small number of high-transmission eigenchannels, with distinct input spectral peaks (k_RW), offers a route to address and control RW formation through input wavefront engineering. Spatial localization on micrometer scales and enhanced transmittance enable substantial local energy delivery, making RWs promising for deep-tissue phototherapies (e.g., photothermal approaches), where limited penetration depth is a major bottleneck. The robustness of RW statistics to chemotherapy in these models suggests the phenomenon may persist under certain pharmacological treatments, though dependence on cell type and structural heterogeneity remains an open question. Overall, the findings address the central question of whether extreme optical events can form and be harnessed in biological aggregates, revealing a practical mechanism for concentrating light in complex tissues.

Macroscopic tumor spheroids bioprinted from human pancreatic cells exhibit optical extreme waves with Weibull-distributed transmission statistics, indicating frequent, localized intense events. These rogue waves are nonlinear, form localized optical filaments with enhanced transmission, and can be selectively excited via specific input modes identified through transmission eigenchannel analysis. By shaping the input wavefront, RWs can deliver concentrated optical power to micrometric regions deep within scattering tissue, enabling substantial local temperature increases relevant to phototherapy. Future work should explore RW formation across different cell types and tissue architectures, assess differences between healthy and malignant spheroids, refine control via eigenchannels under varying nonlinear regimes, and translate these mechanisms to in vivo-like conditions and therapeutic implementations.

• Non-stationarity at high power (P > 100 mW) limits stable operation and indicates regime-dependent behavior.
• RW occurrence depends on coupling geometry and local 3DTM structural heterogeneity; spatial localization varies across regions.
• Temperature-rise estimates use a simplified thermal model that neglects convection and assumes uniform background conductivity; results depend on nanoparticle parameters and assumed power within the ROI.
• Experiments were conducted on bioprinted 3DTMs (PANC-1); generalization to other cell types, healthy spheroids, and in vivo tissues remains to be established.
• Statistical analyses focus on a 200 × 200 µm² ROI per acquisition; global sample-scale variations may not be fully captured.