Development and verification of the signal to noise ratio for a layer of turbulence in a multi-layer atmosphere.

Wide-field image correction in systems that look through the atmosphere generally requires a tomographic reconstruction of the turbulence volume to compensate for anisoplanatism. The reconstruction is conditioned by estimating the turbulence volume as a profile of thin homogeneous layers. We present the signal to noise ratio (SNR) of a layer, which quantifies how difficult a single layer of homogeneous turbulence is to detect with wavefront slope measurements. The signal is the sum of wavefront tip and tilt variances at the signal layer, and the noise is the sum of wavefront tip and tilt auto-correlations given the aperture shape and projected aperture separations at all non-signal layers. An analytic expression for layer SNR is found for Kolmogorov and von Kármán turbulence models, then verified with a Monte Carlo simulation. We show that the Kolmogorov layer SNR is a function of only layer Fried length, the spatio-angular sampling of the system, and normalized aperture separation at the layer. In addition to these parameters, the von Kármán layer SNR also depends on aperture size, and layer inner and outer scales. Due to the infinite outer scale, layers of Kolmogorov turbulence tend to have lower SNR than von Kármán layers. We conclude that the layer SNR is a statistically valid performance metric to be used when designing, simulating, operating, and quantifying the performance of any system that measures properties of layers of turbulence in the atmosphere from slope data.

Germline single nucleotide polymorphisms associated with response of urothelial carcinoma to platinum-based therapy: the role of the host.

BACKGROUND: Variations in urothelial carcinoma (UC) response to platinum chemotherapy are common and frequently attributed to genetic and epigenetic variations of somatic DNA. We hypothesized that variations in germline DNA may contribute to UC chemosensitivity. PATIENTS AND METHODS: DNA from 210 UC patients treated with platinum-based chemotherapy was genotyped for 80 single nucleotide polymorphisms (SNPs). Logistic regression was used to examine the association between SNPs and response, and a multivariable predictive model was created. Significant SNPs were combined to form a SNP score predicting response. Eleven UC cell lines were genotyped as validation. RESULTS: Six SNPs were significantly associated with 101 complete or partial responses (48%). Four SNPs retained independence association and were incorporated into a response prediction model. Each additional risk allele was associated with a nearly 50% decrease in odds of response [odds ratio (OR) = 0.51, 95% confidence interval 0.39-0.65, P = 1.05 × 10(-7)). The bootstrap-adjusted area under the curves of this model was greater than clinical prognostic factors alone (0.78 versus 0.64). The SNP score showed a positive trend with chemosensitivity in cell lines (P = 0.115). CONCLUSIONS: Genetic variants associated with response of UC to platinum-based therapy were identified in germline DNA. A model using these genetic variants may predict response to chemotherapy better than clinical factors alone.

Reproductive biology of squaretail coralgrouper Plectropomus areolatus using age-based techniques.

The squaretail coralgrouper Plectropomus areolatus was identified as a fast-growing, early maturing and relatively short-lived aggregation-spawning epinephelid. Examinations of sectioned otoliths found females and males first maturing at 2 and 3 years, respectively, suggesting protogynous hermaphroditism; however, no transitionals were observed in samples. Age distribution for the two sexes was similar and both were represented in the oldest age class; however, significant sex-specific differences in size-at-age were identified. Both sexes fully recruit into the fishery at age 4 years and reach 90% of asymptotic length by age 3 years. Underwater visual assessments, combined with the gonado-somatic indices, revealed a 5 month reproductive season, with interannual variability observed in the month of highest density within the spawning aggregation. Catch restrictions on adults during spawning times and at reproductive sites, combined with gear-based management and enhanced enforcement, are recommended to maintain spawning stocks. Based on the available evidence, the sexual pattern for this species is unresolved.

Prostate cancer risk in men with prostate and breast cancer family history: results from the REDUCE study (R1).

BACKGROUND: To what degree the associations between PCa risk and family history of prostate cancer (PCa) and/or breast cancer (BCa) are attributable to screening biases is unclear. We examined these questions within the REDUCE study, where biopsies were largely independent of prostate specific antigen (PSA) minimizing screening biases. METHODS: Data were from REDUCE, which tested dutasteride 0.5 mg daily for PCa risk reduction in men with PSA 2.5-10.0 ng mL(-1) and a negative prestudy biopsy. Among men undergoing at least one on-study biopsy with complete data (n = 6415; 78.1%), the association between family history and PCa risk was tested using multivariate logistic regression adjusting for clinicodemographic characteristics. RESULTS: A family history of PCa alone was associated with increased PCa diagnosis (OR: 1.47, 95%CI: 1.22-1.77). In North America, PCa family history was not related to PCa diagnosis (OR: 1.02, 95%CI: 0.73-1.44), whereas outside North America, PCa family history was significantly related to diagnosis (OR: 1.72, 95%CI: 1.38-2.15) (P-interaction = 0.01). A family history of both PCa and BCa (OR: 2.54, 95%CI: 1.72-3.75) but not BCa alone (OR: 1.04, 95%CI: 0.84-1.29) was associated with increased PCa risk versus no family history and irrespective of geographical region. CONCLUSIONS: In REDUCE, PCa family history was significantly related to PCa diagnosis, although only for men outside North America. The presence of both PCa and BCa family history significantly increased risk versus PCa family history alone, irrespective of geographical region. Ultimately, our observations may support the need for changes in how we address family history in terms of both risk of PCa diagnosis and general risk stratification.

Robotic retroperitoneal lymph node dissection for primary and post-chemotherapy testis cancer.

The role of retroperitoneal lymph node dissection (RPLND) in testicular cancer is well established in both the primary and post-chemotherapy setting. The aim of this study was to report our 2 years oncological outcomes of robotic RPLND. A retrospective review was performed of all patients undergoing robotic RPLND by a single surgeon at Princess Margaret Cancer Centre. Demographic, perioperative, and oncologic data were analyzed using descriptive statistics. Between September 2014 and June 2020, 141 patients underwent an RPLND [33 (23.4%) were primary, 108 (76.6%) were post-chemotherapy]. 27 (19.1%) patients underwent a robotic bilateral template nerve-sparing RPLND. RPLND indication was primary (i.e. pre-chemotherapy) in 18 (66.7%), and post-chemotherapy in 9 (33.3%) patients. Stage at RPLND was 2A (n = 15, 55.6%), 2B (n = 9, 33.3%), 2C (n = 1, 3.7%) and 3 (n = 2, 7.4%). Median OR time (incision to closure) was 525 min and blood loss was 200 ml. Nerve sparing was performed in all but one case. Six (22.2%) adjuvant procedures were performed including two (7.4%) vascular repairs. Median length of stay was 2 days. Viable tumor was detected in 17 (63%) and teratoma in 9 (33.3%). Median follow-up was 31.3 months. No adjuvant chemotherapy was given. Three patients (11.1%) relapsed: 2 out-of-field and 1 with both in-field and out-of-field disease. Robotic RPLND can be performed safely. Long-term follow-up of series such as ours, enriched with patients with viable disease and/or teratoma, and not treated with adjuvant chemotherapy is required to ensure oncological outcomes are comparable to the open approach.

Confidence limit variation for a single IMRT system following the TG119 protocol.

PURPOSE: To evaluate the robustness of TG119-based quality assurance metrics for an IMRT system. METHODS: Four planners constructed treatment plans for the five IMRT test cases described in TG119. All plans were delivered to a 30 cm x 30 cm x 15 cm solid water phantom in one treatment session in order to minimize session-dependent variation from phantom setup, film quality, machine performance, etc. Composite measurements utilized film and an ionization chamber. Per-field measurements were collected using a diode array device at an effective depth of 5 cm. All data collected were analyzed using the TG119 specifications to determine the confidence limit values for each planner separately and then compared. RESULTS: The mean variance of ion chamber measurements for each planner was within 1.7% of the planned dose. The resulting confidence limits were 3.13%, 1.98%, 3.65%, and 4.39%. Confidence limit values determined by composite film analysis were 8.06%, 13.4%, 9.30%, and 16.5%. Confidence limits from per-field measurements were 1.55%, 0.00%, 0.00%, and 2.89%. CONCLUSIONS: For a single IMRT system, the accuracy assessment provided by TG119-based quality assurance metrics showed significant variations in the confidence limits between planners across all composite and per-field evaluations. This observed variation is likely due to the different levels of modulation between each planner's set of plans. Performing the TG119 evaluation using plans produced by a single planner may not provide an adequate estimation of IMRT system accuracy.

Long-term use of 5-alpha-reductase inhibitors is safe and effective in men on active surveillance for prostate cancer.

BACKGROUND: Although 5-alpha-reductase inhibitors (5ARIs) have been shown to benefit men with prostate cancer (PCa) on active surveillance (AS), their long-term safety remains controversial. Our objective is to describe the long-term association of 5ARI use with PCa progression in men on AS. MATERIALS/SUBJECTS AND METHODS: The cohort of men with low-risk PCa was derived from a prospectively maintained AS database at the Princess Margaret (1995-2016). Pathologic, grade, and volume progression were the primary end points. Kaplan-Meier time-to-event analysis was performed and Cox proportional hazards regression was used to determine predictors of progression where 5ARI exposure was analyzed as a time-dependent variable. Patients who came off AS prior to any progression events were censored at that time. RESULTS: The cohort included 288 men with median follow-up of 82 months (interquartile range: 37-120 months). Among non-5ARI users (n = 203); 114 men (56.2%) experienced pathologic progression compared with 24 men (28.2%) in the 5ARI group (n = 85), (p < 0.001). Grade and volume progression were higher in the non-5ARI group compared with the 5ARI group (n = 82; 40.4% vs. n = 19; 22.4% respectively, p = 0.003 for grade progression; n = 87; 43.1% and n = 15; 17.7%, respectively for volume progression p < 0.001). Lack of 5ARI use was independently positively associated with pathologic progression (HR: 2.65; CI: 1.65-4.24), grade progression (HR: 2.75; CI: 1.49-5.06), and volume progression (HR: 3.15; CI: 1.78-5.56). The frequency of progression to high-grade (Grade Group 4-5) tumors was not significantly different between the groups. CONCLUSIONS: Use of 5ARIs diminished both grade and volume progression without an increased risk of developing Grade Groups 4-5 disease.

Declining grouper spawning aggregations in Western Province, Solomon Islands, signal the need for a modified management approach.

Globally, groupers (Epinephelidae) that form fish spawning aggregations (FSAs) are highly vulnerable to overfishing and often require site-specific approaches to management. Over 5-years (2009-2013), we conducted underwater visual censuses (UVC) at a well-known spawning site at Njari Island, Gizo, Western Province, Solomon Islands, that supports aggregations of squaretail coralgrouper (Plectropomus areolatus), camouflage grouper (Epinephelus polyphekadion) and brown-marbled grouper (E. fuscoguttatus). Findings show that while there were species-specific variations in the duration and timing of the spawning season, aggregation densities peaked from March to June, representing the main spawning season for all three species. For P. areolatus, gonad analysis from samples taken from 2008 to 2011 confirmed reproductive activity in support of density trends observed through UVC. Over the 5-year UVC monitoring period, FSA densities declined for P. areolatus and E. polyphekadion. Conversely, following the first year of monitoring, E. fuscoguttatus densities increased. These inter-specific differences may reflect variable responses to fishing as shown elsewhere, or for example, differences in recruitment success. In response to known declines in FSAs of these species, in 2018 the Solomon Islands government placed a nationwide ban on these species' harvest and sale between October and January. As this study shows, this ban does not encompass the peak aggregation period at Njari and will offer limited protection to other FSAs of these species that are known to vary in reproductive seasonality across the Solomon Islands. A more biologically meaningful and practical management strategy would be to implement a nationwide ban on the harvest and sale of these groupers each month between full and new moons when these FSAs form consistently throughout the country. Since effective management of FSAs typically requires a combined approach, spatial management that protects both spawning sites and reproductive migratory corridors is warranted.

Partial orchiectomy: The Princess Margaret cancer centre experience.

INTRODUCTION: Radical orchiectomy (RO) is the standard treatment for a testis cancer. Organ sparing surgery can be considered in the setting of a solitary functioning testis or bilateral tumors. It has also been suggested as an alternative to RO for small lesions. In this study we report our partial orchiectomy (PO) experience. METHODS: We performed a retrospective review using our prospectively maintained database analyzing PO. RESULTS: Between 1983 and 2018, 77 patients underwent PO. Mean age was 31.3 years (range 17-56). A lesion was palpable in 70 (90.9%) and median lesion size 14.1 mm (range 3-35 mm). Reasons for PO included ``small lesion" in 39 (50.6%); solitary functioning testis in 30 (39%); bilateral lesions in 6 (7.8%); or assumed benign lesion in 1 (1.3%). Median follow-up was 43.5 months (range 1-258). Lesion histology was benign in 25 (32.5%). A positive surgical margin was noted in 6 (7.8%) with none developing local or distant recurrence. Sixteen (20.8%) patients underwent salvage ipsilateral RO at a median of 3 months (range 0-46). Reasons for salvage RO included a radiologically detected lesion in 7, palpable lesion in 4, positive surgical margin in 3 and adverse pathology in 2 patients. Malignant histology was present in 12 (75%) of the salvage RO specimens. There were no reported Clavien-Dindo Grade 3 to 5 complications. CONCLUSION: Organ sparing surgery is a safe and feasible approach to small testis lesions. For the third with benign disease, and even those with malignant histology, a RO can be avoided in carefully selected patients.

Leaf epicuticular waxes.

The external surface of the higher plants comprises a cuticular layer covered by a waxy deposit. This deposit is believed to play a major part in such phenomena as the water balance of plants and the behavior of agricultural sprays. The wax contains a wide range of organic compounds. These complex mixtures are amenable to modern microchromatographic and microspectrometric analytical procedures. The few surveys which have been made of the species distribution of certain classes of constituents indicate that such distribution may be of limited taxonomic value; however, the wax composition of a species may differ for different parts of the same plant and may vary with season, locale, and the age of the plant. This fascinating subject, in which the disciplines of botany, biochemistry, chemistry, and physics overlap and interact, is still in a very active state. Much remains to be learned about the composition and fine structure of the wax deposits, and, for this, experimental study of wax crystallization and permeation through artificial membranes will be required. Enzymic studies, radiolabeling, and electron microscopy will be needed to reveal the mode of biogenesis of the wax constituents and their site of formation and subsequent pathway through the cuticle to the leaf surface.

Automated characterization of multiple alpha peaks in multi-site electroencephalograms.

The identification of alpha rhythm in the human electroencephalogram (EEG) is generally a laborious task involving visual inspection of the spectrum. Moreover the occurrence of multiple alpha rhythms is often overlooked. This paper seeks to automate the process of identifying alpha peaks and quantifying their frequency, amplitude and width as a function of position on the scalp. Experimental EEG was fitted with parameterized spectra spanning the alpha range, with results categorized by multi-site criteria into three distinct classes: no distinguishable alpha peak, a single alpha peak, and two alpha peaks. The technique avoids visual bias, integrates spatial information, and is automated. We show that multiple alpha peaks are a common feature of many spectra.

Performance of a video-image-subtraction-based patient positioning system.

PURPOSE: We have developed and tested an interactive video system that utilizes image subtraction techniques to enable high precision patient repositioning using surface features. We report quantitative measurements of system performance characteristics. METHODS AND MATERIALS: Video images can provide a high precision, low cost measure of patient position. Image subtraction techniques enable one to incorporate detailed information contained in the image of a carefully verified reference position into real-time images. We have developed a system using video cameras providing orthogonal images of the treatment setup. The images are acquired, processed and viewed using an inexpensive frame grabber and a PC. The subtraction images provide the interactive guidance needed to quickly and accurately place a patient in the same position for each treatment session. We describe the design and implementation of our system, and its quantitative performance, using images both to measure changes in position, and to achieve accurate setup reproducibility. RESULTS: Under clinical conditions (60 cm field of view, 3.6 m object distance), the position of static, high contrast objects could be measured with a resolution of 0.04 mm (rms) in each of two dimensions. The two-dimensional position could be reproduced using the real-time image display with a resolution of 0.15 mm (rms). Two-dimensional measurement resolution of the head of a patient undergoing treatment for head and neck cancer was 0.1 mm (rms), using a lateral view, measuring the variation in position of the nose and the ear over the course of a single radiation treatment. Three-dimensional repositioning accuracy of the head of a healthy volunteer using orthogonal camera views was less than 0.7 mm (systematic error) with an rms variation of 1.2 mm. Setup adjustments based on the video images were typically performed within a few minutes. The higher precision achieved using the system to measure objects than to reposition them suggests that the variability in repositioning is dominated by the ability of the therapist to make small, controlled changes in the position of the patient. CONCLUSION: Using affordable, off-the-shelf technology, we have developed a patient positioning system that achieves repositioning accuracy normally associated with fractionated stereotactic systems. The technique provides real-time guidance and can be used to easily and quickly correct patient setup before every treatment, thus significantly reducing overall random positioning error. This improved positioning capability provides the precision required to realize the potential gains of conformal radiotherapy.

The potential for normal tissue dose reduction with neoadjuvant hormonal therapy in conformal treatment planning for stage C prostate cancer.

PURPOSE: Preirradiation hormonal cytoreduction of prostate cancer has been proven to reduce exposure of normal structures by decreasing the size of the target volume. Dose-volume histogram (DVH) analysis, however, does not always appear to demonstrate a strong positive benefit with the use of neoadjuvant hormone therapy. This study analyzes various other factors influencing dose to normal organs, which may determine the success or failure of neoadjuvant hormonal therapy in achieving its goals. METHODS AND MATERIALS: Patients with bulky clinical Stage C adenocarcinoma of the prostate were given 3 months of hormone treatment consisting of oral Flutamide and monthly Zoladex injections prior to irradiation. Computerized tomography (CT) scans of the pelvis were obtained both prior to and following hormonal treatment. Treatment plans were generated by three-dimensional (3D) conformal treatment planning. The change in the volume of the prostate was assessed along with the percentage of prescribed dose delivered to the rectum and bladder. Various factors such as prostate size, bladder/rectum size, and organ shape were studied. Both dose-volume histograms (DVH) and dose-surface area histograms (DSH) were used for analysis. RESULTS: Six of seven patients had reduction in the size of their prostates. The mean volumes of the prostate before and after hormonal manipulation were 129.1 +/- 32.9 standard deviation (SD) cm3 and 73.0 +/- 29.5 SD cm3, respectively (p = 0.0059). The volume of rectum receiving 80% of the prescribed dose was reduced in five of seven patients from a mean of 83.2 to 59.9 cm3 (p = 0.045). The volume of bladder receiving 80% of the prescribed dose was also reduced in five out of seven patients from a mean of 74.5 to 40.2 cm3 (p = 0.098). Correlation between the size of the prostate and volume of rectum and bladder treated was not always consistent: greater reduction in prostate size did not necessarily result in large decreases in dose to bladder or rectum. The total size of the bladder and rectum were found to be important factors in normal tissue radiation exposure; the benefits of hormone therapy may be lost if the bladder and rectum are allowed to decrease in size. Also, the bladder may be prone to sagging into the pelvis of some patients following hormone therapy, resulting in a less optimal therapeutic ratio. CONCLUSION: Reduction in prostate size by neoadjuvant hormonal manipulation does decrease the amount of normal tissue irradiated in most patients. However, the correlation between the reduction in prostate size and amount of rectum or bladder treated is not linear if other variables are not controlled. Factors such as the shape of the organs, as well as the distensible nature of the bladder and rectum, play major roles in dose to normal tissues. These facts may mask the benefits of cytoreduction and could be obstacles in realizing consistent benefits from preirradiation hormonal treatment in the clinical setting if they are ignored.

Comparison of static conformal field with multiple noncoplanar arc techniques for stereotactic radiosurgery or stereotactic radiotherapy.

PURPOSE: Compare the use of static conformal fields with the use of multiple noncoplanar arcs for stereotactic radiosurgery or stereotactic radiotherapy treatment of intracranial lesions. Evaluate the efficacy of these treatment techniques to deliver dose distributions comparable to those considered acceptable in current radiotherapy practice. METHODS AND MATERIALS: A previously treated radiosurgery case of a patient presenting with an irregularly shaped intracranial lesion was selected. Using a three-dimensional (3D) treatment-planning system, treatment plans using a single isocenter multiple noncoplanar arc technique and multiple noncoplanar conformal static fields were generated. Isodose distributions and dose volume histograms (DVHs) were computed for each treatment plan. We required that the 80% (of maximum dose) isodose surface enclose the target volume for all treatment plans. The prescription isodose was set equal to the minimum target isodose. The DVHs were analyzed to evaluate and compare the different treatment plans. RESULTS: The dose distribution in the target volume becomes more uniform as the number of conformal fields increases. The volume of normal tissue receiving low doses (> 10% of prescription isodose) increases as the number of static fields increases. The single isocenter multiple arc plan treats the greatest volume of normal tissue to low doses, approximately 1.6 times more volume than that treated by four static fields. The volume of normal tissue receiving high (> 90% of prescription isodose) and intermediate (> 50% of prescription isodose) doses decreases by 29 and 22%, respectively, as the number of static fields is increased from four to eight. Increasing the number of static fields to 12 only further reduces the high and intermediate dose volumes by 10 and 6%, respectively. The volume receiving the prescription dose is more than 3.5 times larger than the target volume for all treatment plans. CONCLUSIONS: Use of a multiple noncoplanar conformal static field treatment technique can significantly reduce the volume of normal tissue receiving high and intermediate doses compared with a single isocenter multiple arc treatment technique, while providing a more uniform dose in the target volume. Close conformation of the prescription isodose to the target volume is not possible using static uniform conformal fields for target shapes lacking an axis of rotational symmetry or plane of mirror symmetry.

Functional imaging in treatment planning of brain lesions.

PURPOSE: Explore the use of functional imaging data in radiation treatment planning of brain lesions. METHODS AND MATERIALS: Compare the treatment-planning process with and without the use of functional brain imaging for clinical cases where functional studies using either single photon emission computed tomography or magnetic resonance imaging are available. RESULTS: A method to register functional image data with planning image studies is needed for functional treatment planning. Functional volumes are not simply connected regions. One activation study may produce many isolated functional areas. After finding the functional volumes and registering the functional information with the planning imaging data, the tools used for conventional three-dimensional treatment planning are sufficient for functional treatment planning. However, the planning system must provide dose-volume histograms for volumes of interest that consist of isolated pieces. Treatment plans that spare functional brain while providing identical target coverage can be constructed for lesions situated near the functional volume. However, the dose to other areas of the brain may be increased. CONCLUSIONS: Functional imaging will make determination of dose response of eloquent areas of the brain possible when combined with volumetric dose information and neuropsychological evaluation prior to and after radiation therapy. Realizing the full potential of functional imaging studies will require improved delineation of activated volumes and determination of the uncertainties in functional volume delineation. Optimization of treatment plans by minimizing dose to volumes activated during functional imaging studies should be used cautiously, because the dose to "silent," but possibly eloquent, brain may be increased.

Using vascular structure for CT-SPECT registration in the pelvis.

UNLABELLED: The authors outline a method for three-dimensional registration of pelvic CT and 111In-labeled monoclonal antibody capromab pendetide (111In MoAb 7E11.C5) images using 99mTc-labeled red blood cell SPECT data. METHODS: This method of CT-SPECT registration relies on the identification of major blood vessels in the CT and 99mTc SPECT images. The vessels are segmented from the image datasets by outlining them on transverse planar slices using a mouse-based drawing tool. Stacking the transverse outlines provides a three-dimensional representation of the vascular structures. Registration is performed by matching the surfaces of the segmented volumes. Dual isotope acquisition of 111In and 99mTc activities provides precise SPECT-SPECT registration so that registration in three dimensions of the 111In MoAb and CT images is achieved by applying the same transformation obtained from the 99mTc SPECT-CT registration. RESULTS: This method provided accurate registration of pelvic structures and significantly improved interpretation of 111In MoAb 7E11.C5 exams. Furthermore, sites of involvement by prostate cancer suggested by the 111In MoAb examination could be interpreted with the bony and soft tissue (nodal) anatomy seen on CT. CONCLUSION: This method is a general clinical tool for the registration of pelvic CT and SPECT imaging data. There are immediate applications in conformal radiation therapy treatment planning for certain prostate cancer patients.

Ventilator dependence in acute severe asthma due to a variant presentation of Guillain-Barré syndrome.

A 23-year-old white woman presented with severe status asthmaticus requiring prolonged ventilator support and subsequently developed a flaccid areflexic quadraparesis which resolved after plasmapheresis. This case represents a variant presentation of Guillain-Barré syndrome (GBS) that is difficult to distinguish from critical illness polyneuropathy (CIP).

A model for optimizing normal tissue complication probability in the spinal cord using a generalized incomplete repair scheme.

The purpose of this study was to determine the treatment protocol, in terms of dose fractions and interfraction intervals, which minimizes normal tissue complication probability in the spinal cord for a given total treatment dose and treatment time. We generalize the concept of incomplete repair in the linear-quadratic model, allowing for arbitrary dose fractions and interfraction intervals. This is incorporated into a previously presented model of normal tissue complication probability for the spinal cord. Equations are derived for both mono-exponential and bi-exponential repair schemes, regarding each dose fraction and interfraction interval as an independent parameter, subject to the constraints of fixed total treatment dose and treatment time. When the interfraction intervals are fixed and equal, an exact analytical solution is found. The general problem is nonlinear and is solved numerically using simulated annealing. For constant interfraction intervals and varying dose fractions, we find that optimal normal tissue complication probability is obtained by two large and equal doses at the start and conclusion of the treatment, with the rest of the doses equal to one another and smaller than the two dose spikes. A similar result is obtained for bi-exponential repair. For the general case where the interfraction intervals are discrete and also vary, the pattern of two large dose spikes is maintained, while the interfraction intervals oscillate between the smallest two values. As the minimum interfraction interval is reduced, the normal tissue complication probability decreases, indicating that the global minimum is achieved in the continuum limit, where the dose delivered by the "middle" fractions is given continuously at a low dose rate. Furthermore, for bi-exponential repair, it is seen that as the slow component of repair becomes increasingly dominant as the magnitude of the dose spikes decreases. Continuous low-dose-rate irradiation with dose spikes at the start and end of treatment yields the lowest normal tissue complication probability in the spinal cord, given a fixed total dose and total treatment time, for both mono-exponential and bi-exponential repair. The magnitudes of the dose spikes can be calculated analytically, and are in close agreement with the numerical results.