Measurement of the neutral axis in avian eggshells reveals which species conform to the golden ratio.

Avian eggs represent a striking evolutionary adaptation for which shell thickness is crucial. An understudied eggshell property includes the neutral axis, a line that is drawn through any bent structure and whose precise location is characterized by the k-factor. Previous studies have established that, for chicken eggs, mean k corresponds to the golden ratio (Φ = 1.618, or 0.618 in its reciprocal form). We hypothesized whether such an arrangement of the neutral axis conforms to the eggshell of any bird or only to eggshells with a certain set of geometric parameters. Implementing a suite of innovative methodological approaches, we investigated variations in k of 435 avian species, exploring which correspond to Φ. We found that mean k is highly variable among birds and does not always conform to Φ, being much lower in spherical and ellipsoid eggs and higher in pyriform eggs. While 21 species had k values within 0.618 ± 0.02 (including four falcon species) and the Falconinae subfamily (six species) revealed a mean of 0.618, it is predominantly domesticated species (chicken, ducks, and geese) that lay eggs whose neutral axis corresponds to the golden ratio. Thus, the study of the mathematical secrets of the eggshell related to the golden ratio of its neutral axis suggests its species-specific signatures in birds.

Can Pre-treatment Quantitative Multi-parametric MRI Predict the Outcome of Radiotherapy in Patients with Prostate Cancer?

RATIONALE AND OBJECTIVES: To investigate whether pre-treatment quantitative multiparametric MRI can predict biochemical outcome of prostate cancer (PCa) patients treated with primary radiotherapy (RT). MATERIALS AND METHODS: Fifty-one patients with biopsy confirmed PCa underwent prostate multiparametric MRI on 3T MR scanner prior to RT. Thirty-seven men (73%) were treated with external beam RT alone, 12 men (24%) were treated with brachytherapy monotherapy, and two men (4%) were treated with external beam RT with brachytherapy boost. The index lesion was outlined by a radiologist and quantitative apparent diffusion coefficient (ADC), T2 and DCE parameters were measured. Biochemical failure was defined using the Phoenix criteria. RESULTS: After a median follow-up of 65 months, seven patients had biochemical failure. ADC had an area under the receiver operating characteristic curve of 0.71 for predicting RT outcome with significantly lower ADC (0.78 ± 0.17 vs 0.96 ± 0.26 µm2/ms, p = 0.04) of the index lesion in men with biochemical failure. Ideal ADC cutoff point (Youdens index) was 0.96 µm2/ms which had a sensitivity of 100% and specificity of 48% for predicting biochemical failure. Kaplan-Meier analysis showed that lower ADC values were associated with significantly lower freedom from biochemical failure (FFBF, p = 0.03, no failures out of 20 men if ADC ≥ 0.96 µm2/ms; seven of 31 with failures if ADC < 0.96 µm2/ms). On multivariable analysis, ADC was associated with FFBF (HR 0.96 per increase in ADC of 0.01 um2/ms [95% CI, 0.92-1.00]; p = 0.042) after accounting for National Comprehensive Cancer Network risk category (p = 0.064) and receipt of androgen deprivation therapy (p = 0.141). Quantitative T2 and DCE parameters were not associated with biochemical outcome. CONCLUSION: Our results suggest that quantitative ADC values of the index lesion may predict biochemical failure following primary radiotherapy in patients with PCa. Lower ADC values were associated with inferior biochemical control.