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At least 325 records · Page 18

Towards Automatic Mapping of Vulnerabilities to Attack Patterns using Large Language Models

With the advent of new devices and applications, cyber attack surface is continuously evolving due to the emergence of new attack techniques and vulnerabilities. Hence, security management tool must assess the cyber risk of an enterprise at regular interval basis through comprehensively identifying associations among attack techniques, weakness, and vulnerabilities. However, existing repositories providing such associations are incomplete (i.e., missing associations), inducing the likelihood of undermining the risk of particular set of attack techniques. Moreover, such associations still rely on manual interpretation, which is slow compared to attack speed and ineffective for the increasing list of vulnerabilities and attack actions. Therefore, there is an urge to develop methodologies for automatically associating vulnerabilities to all relevant attack techniques. In this paper, we present a framework, named VWC-MAP, that can automatically identify all relevant attack techniques of a vulnerability via weakness based on their text descriptions, applying natural language process (NLP) techniques. To achieve that, we present a novel two-tiered classification approach, where the first tier classifies vulnerabilities to weakness, and the second tier classifies weakness to attack techniques. This research has improved the scalability of the current state-of-the-art tool to make vulnerability to weakness mapping significantly faster. Moreover, this paper presents two novel approaches for weakness to attack technique mapping applying Text-to-Text and link prediction techniques. Our experiment results cross-validated through cyber-security experts show that VWC-MAP can associate vulnerabilities to weakness types with 87% accuracy and to new attack patterns with 80% accuracy.

Das, Siddhartha Shankar↗

City-Wide Distributed Roof-Top Photovoltaic System Adoption Forecast, Grid Impact Simulation, & Neighborhood Microgrid Contribution Assessment

The adoption of distributed photovoltaic (PV) systems grew significantly in recent years. Market projections anticipate future growth for both residential and commercial installations. To understand grid impacts associated with distributed PV, useful hosting capacity studies require accurate representations of the spatial distribution of PV adoptions. Prediction of PV locations and numbers depends on median income data, building use zoning maps, and permit records to understand existing trends and predict future adoption rates and locations throughout an entire city. Using the PV adoption data, advanced and realistic simulations were performed to capture the distributed PV impacts on the grid. Also, using graph theory community detection hundreds of neighborhood microgrids can be discovered for the entire city by identifying densely connected loads that are sparsely connected to other communities. Then, based on the PV adoption predictions, this work identified the contribution of PV within each of the newly discovered graph theory defined microgrid communities.

24 POWER TRANSMISSION AND DISTRIBUTION↗

Bayesian Approach to Estimation of Water Table Elevations Using Historical Rasters as Prior Information 2019 - 20430

In cases of complex but only partially known geology and a lack of spatial control in observation well locations, water table elevation estimation is very challenging. In some cases, auxiliary information, such as observations of the movement of tracers, operation of injection and extraction wells, and calibration of groundwater models against the historical elevation data, can be combined with expert judgement to estimate flow directions in areas of sparse data and to aid in the production of more reliable contour maps (and associated rasters) than could be produced by relying on sparse well elevation data alone. Given a historical sequence of these raster maps, the question arises how to automate, to the extent possible, the process of producing new raster maps to reflect data from previous times, the current data and the operation of expert judgement. One solution is to adopt a Bayesian point of view and to regard the historical well elevation data, auxiliary information and historical raster maps as prior information. The well elevations for water table wells, as well as those for injection/extraction wells and the data associated with other relevant variables, can be viewed as predictors for the raster surface. From this prior information, we can, conditional on the values of these predictors for a new time period, compute an expected value map and a standard deviation map for the new raster. These then can be taken to specify a prior predictive distribution for the pixels in the new raster map. Then we condition the pixels, corresponding to water level observation wells within the raster, on the observed values in those wells (which in general will differ from the regression estimate) for the new time period. Given the smoothness of the water table surface, we then smooth the surface of deviations from the mean surface, based on the variograms of the historical rasters, and add this smoothed surface to the regression mean surface. The error structure of the produced raster map is defined by the regression error structure and the error due to smoothing based on the estimated variograms. This methodology has been developed and is being further refined for groundwater monitoring and remediation at LANL. It is a very flexible method that can also be applied with a variety of other predictors applied to model the water level wells in the area of interest over the historical record. The smoothness of the spatial process and its possible evolution over time can then be estimated from the residuals from this regression. This can be augmented by expert hydrogeological opinion based on site topography and hydrogeology. (authors)

54 ENVIRONMENTAL SCIENCES↗

Mapping the Universe in hydrogen deuteride

Hydrogen deuteride (HD) is prevalent in a wide variety of astrophysical environments, and measuring its large-scale distribution at different epochs can, in principle, provide information about the properties of these environments. In this paper we explore the prospects for accessing this distribution using line-intensity mapping of emission from the lowest rotational transition in HD, focusing on observations of the epoch of reionization (z~6–10) and earlier. We find the signal from the epoch of reionization to be most promising, through cross correlations within existing [CII] intensity mapping surveys. While the signal we predict is out of reach for current-generation projects, planned future improvements should be able to detect reionization-era HD without any additional observations, and would help to constrain the properties of the star-forming galaxies thought to play a key role in reionization. We also investigate several avenues for measuring HD during “cosmic dawn” (z~10–30), a period in which HD could provide one of the only complementary observables to 21 cm intensity maps. We conclude that existing and planned facilities are poorly matched to the specifications desirable for a significant detection, though such a measurement may be achievable with sustained future effort. Finally, we explain why HD intensity mapping of the intergalactic medium during the cosmic dark ages (z≳30) appears to be out of reach of any conceivable experiment.

79 ASTRONOMY AND ASTROPHYSICS↗

A Method for Projecting Cloud Shadows Onto a Central Receiver Field to Predict Receiver Damage

This work demonstrates methods of mapping high-spatial-resolution direct normal irradiance (DNI) data from satellites, Total Sky Imagers (TSIs), and analogous data sources onto a heliostat field for characterizing the spatial and temporal variation of the incident flux on a central receiver tower during cloud transient events. The mapping methods are incorporated into an optical software module that interfaces with CoPylot–SolarPILOT’s python API– to provide computationally efficient optical simulation of the heliostat field and the solar power tower. Eventually, this optical model will be incorporated into optimization models whereby a plant operator can understand the effects of cloud transient events on overall power production and receiver lifetime due to creep-fatigue damage and therefore make better informed decisions about receiver shutdown events. By more accurately modelling the effects of cloud events on receiver flux maps, this work may determine the magnitude and frequency of thermal cycling on receiver tubes and panels using actual or realistic cloud shapes instead of averaged DNI values–which may undercount the total cycle number. This work may also prevent unnecessary plant shutdowns due to overly precautionary control strategies and characterize the relative impact of various cloud types on receiver life. We plan to eventually integrate this methodology into the System Advisor Model (SAM) to improve performance model accuracy during periods of cloudiness. In this paper, we demonstrate generating DNI maps and mapping them to a solar field in CoPylot using 10 m resolution data from publicly available Sentinel-2 satellite data over the Crescent Dunes plant.

Mullin, Matthew↗

Development of a shock noise prediction code for high-speed helicopters - The subsonically moving shock

A previously defined airfoil subsonic shock-noise prediction formula whose result depends on a mapping of the time-dependent shock surface to a time-independent computational domain is presently coded and incorporated in the NASA-Langley rotor-noise prediction code, WOPWOP. The structure and algorithms used in the shock-noise prediction code are presented; special care has been taken to reduce computation time while maintaining accuracy. Numerical examples of shock-noise prediction are presented for hover and forward flight. It is confirmed that shock noise is an important component of the quadrupole source.

Tadghighi, H.↗

Geographic variation in the relationships of temperature, salinity or sigma sub t versus plant nutrient concentrations in the world ocean

A NODC data set representing all regions of the world ocean was analyzed for temperature and sigma-t relationships with nitrate, phosphate or silicic acid. Six cubic regressions were for each ten degree square of latitude and longitude containing adequate data. World maps display the locations that allow the prediction of plant nutrient concentrations from temperature or sigma-t. Geographic coverage improves along the sequence: nitrate, phosphate, and silicic acid and is better for sigma-t than for temperature. Contour maps of the approximate temperature of sigma-t at which these nitrients are no longer measurable in a parcel of water are generated, based on a percentile analysis of the temperature or sigma-t at which less than a selected amount of plant nutrient occurs. Results are stored on magnetic tape in tabular form. The global potential to predict plant nutrient concentrations from remotely sensed temperature of sigma-t and to emphasize the latitudinally and longitudinally changing phytoplankton growth environment in present and past oceans is demonstrated.

Kamykowski, D.↗

Potential evapotranspiration and the likelihood of future drought

The possibility that the greenhouse warming predicted by the GISS general-circulation model and other GCMs could lead to severe droughts is investigated by means of numerical simulations, with a focus on the role of potential evapotranspiration E(P). The relationships between precipitation (P), E(P), soil moisture, and vegetation changes in GCMs are discussed; the empirically derived Palmer drought-intensity index and a new supply-demand index (SDDI) based on changes in P - E(P) are described; and simulation results for the period 1960-2060 are presented in extensive tables, graphs, and computer-generated color maps. Simulations with both drought indices predict increasing drought frequency for the U.S., with effects already apparent in the 1990s and a 50-percent frequency of severe droughts by the 2050s. Analyses of arid periods during the Mesozoic and Cenozoic are shown to support the use of the SDDI in GCM drought prediction.

Rind, D.↗

An approach for fast and accurate simulation of phase change material based thermal energy storage in buildings

Latent heat thermal energy storage (LHTES) has significant potential for mitigating peak electricity demand and enabling load shifting in buildings. Phase Change Material embedded heat exchangers (PCM-HX) can significantly improve energy demand management due to high storage capacity. However, PCM-HX evaluation typically depends on computationally expensive fully transient simulations, posing significant challenges for scalable system- and building-level energy assessments across different climates and system architectures. This paper presents a generalized, accurate, and computationally efficient methodology for simulating building energy systems integrated with LHTES. The PCM-HX transient performance is represented by performance maps generated using a Generalized Resistance-Capacitance Model (GRCM) that enables accurate predictions of arbitrary PCM-HXs at low computational cost. The feasibility of the proposed approach was verified using a case study considering a dual-mode heat pump-thermal energy storage (HP-TES) system simulated in Modelica with Spawn of EnergyPlus™ for a DOE prototype small office building in two locations: Tampa, FL, and International Falls, MN. The PCM-HX performance maps provided accurate predictions of PCM-HX transient behavior, with mean absolute percentage deviations within 2–4% compared to GRCM while also achieving at least 1800× reduction in computational time. Moreover, the HP-TES system achieved energy savings of up to 17.4% in Tampa, FL, and 62.2% in International Falls, MN, demonstrating the broader applicability of the proposed methodology across different climate zones. This work highlights the importance of robust PCM-HX models in enabling accurate and computationally efficient building-level simulations and enabling future research opportunities for investigating optimized HP-TES designs and advanced control strategies for grid-interactive buildings.

Modelica↗

Determination and prediction of Magellan's orbit

The Magellan spacecraft has been systematically mapping the surface of Venus since September 15, 1990, using a synthetic aperture radar. The spacecraft orbit about Venus is nearly polar, with an orbital period of 3.26 hours and periapsis altitude of 295 km. The radiometric measurements and the data reduction method used to determine and predict the spacecraft state are described. Orbit determination and prediction results are given for the first 146 days of mapping (through February 8, 1991, 60 percent of the first rotation of Venus). Orbit accuracy requirements of 150 meters in the radial position, and 1 km in the along-track and cross-track positions are shown to be met, but with exceptions. All error requirements were exceeded during a combined period of limited in-plane orbit observability due to earth-orbit relative geometry, and increased measurement noise due to superior conjunction.

Engelhardt, D. B.↗

Optimizing Cell-based Antimicrobials through Pooled Genomic Libraries

DNA synthesis and assembly technologies ushered in through synthetic biology have great promise for biomanufacturing, bioremediation, and the development of living therapeutics. Unfortunately, predicting sequence to function relationships, including for biosynthetic pathways expressed in a new host organism, is difficult and often requires many iterative cycles of design, construction, and testing. We are working to develop data-driven approaches to identify the genetic determinants of growth defects and productivity for the expression of a cell-based antimicrobial. We assayed the growth, pigment production, and antimicrobial activity of a collection of over 10,000 genetic mutants of the violacein biosynthetic pathway and sequenced the genetic variation of these mutants. Through this project, we have developed an innovative codebase to automate the determination of pigmentation and antimicrobial clearing diameter for tens of thousands of genetic mutants cultivated on agar dishes. Further, we have written DNA sequence analysis code to demultiplex & provide consensus sequences from high-throughput PacBio long-read circular consensus sequencing (CCS) datasets. From this foundation, we plan to map DNA sequence to function to predict an optimal genetic design to maximize antimicrobial activity while minimizing deleterious growth effects. The workflows and algorithms developed through this project can be broadly applied to other engineered functions in microbes, uncovering sequence to function relationships for complex phenotypes where function impacts fitness.

59 BASIC BIOLOGICAL SCIENCES↗

Understanding the fusion yield dependencies in OMEGA DT-layered implosion experiments using a physics-based statistical mapping model

Improving the performance of inertial confinement fusion implosions requires physics models that can accurately predict the response to changes in the experimental inputs. Good predictive capability has been demonstrated for the fusion yield using a statistical mapping of simulated outcomes to experimental data [Gopalaswamy et al., Nature 565(771), 581–586 (2019)]. In this paper, a physics-based statistical mapping approach is used to extract and quantify all the major sources of degradation of fusion yield for direct-drive implosions on the OMEGA laser. Here, the yield is found to be dependent on the age of the deuterium tritium fill, the ℓ = 1 asymmetry in the implosion core, the laser beam-to-target size ratio, and parameters related to the hydrodynamic stability. A controlled set of experiments were carried out where only the target fill age was varied while keeping all other parameters constant. The measurements were found to be in excellent agreement with the fill age dependency inferred using the mapping model. In addition, a new implosion design was created, guided by the statistical mapping model by optimizing the trade-offs between increased laser energy coupling at larger target size and the degradations caused by the laser beam-to-target size ratio and hydrodynamic instabilities. When experimentally performed, an increased fusion yield was demonstrated in targets with larger diameters.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

A New Objective Technique for Verifying Mesoscale Numerical Weather Prediction Models

This report presents a new objective technique to verify predictions of the sea-breeze phenomenon over east-central Florida by the Regional Atmospheric Modeling System (RAMS) mesoscale numerical weather prediction (NWP) model. The Contour Error Map (CEM) technique identifies sea-breeze transition times in objectively-analyzed grids of observed and forecast wind, verifies the forecast sea-breeze transition times against the observed times, and computes the mean post-sea breeze wind direction and speed to compare the observed and forecast winds behind the sea-breeze front. The CEM technique is superior to traditional objective verification techniques and previously-used subjective verification methodologies because: It is automated, requiring little manual intervention, It accounts for both spatial and temporal scales and variations, It accurately identifies and verifies the sea-breeze transition times, and It provides verification contour maps and simple statistical parameters for easy interpretation. The CEM uses a parallel lowpass boxcar filter and a high-order bandpass filter to identify the sea-breeze transition times in the observed and model grid points. Once the transition times are identified, CEM fits a Gaussian histogram function to the actual histogram of transition time differences between the model and observations. The fitted parameters of the Gaussian function subsequently explain the timing bias and variance of the timing differences across the valid comparison domain. Once the transition times are all identified at each grid point, the CEM computes the mean wind direction and speed during the remainder of the day for all times and grid points after the sea-breeze transition time. The CEM technique performed quite well when compared to independent meteorological assessments of the sea-breeze transition times and results from a previously published subjective evaluation. The algorithm correctly identified a forecast or observed sea-breeze occurrence or absence 93% of the time during the two- month evaluation period from July and August 2000. Nearly all failures in CEM were the result of complex precipitation features (observed or forecast) that contaminated the wind field, resulting in a false identification of a sea-breeze transition. A qualitative comparison between the CEM timing errors and the subjectively determined observed and forecast transition times indicate that the algorithm performed very well overall. Most discrepancies between the CEM results and the subjective analysis were again caused by observed or forecast areas of precipitation that led to complex wind patterns. The CEM also failed on a day when the observed sea- breeze transition affected only a very small portion of the verification domain. Based on the results of CEM, the RAMS tended to predict the onset and movement of the sea-breeze transition too early and/or quickly. The domain-wide timing biases provided by CEM indicated an early bias on 30 out of 37 days when both an observed and forecast sea breeze occurred over the same portions of the analysis domain. These results are consistent with previous subjective verifications of the RAMS sea breeze predictions. A comparison of the mean post-sea breeze winds indicate that RAMS has a positive wind-speed bias for .all days, which is also consistent with the early bias in the sea-breeze transition time since the higher wind speeds resulted in a faster inland penetration of the sea breeze compared to reality.

Case, Jonathan L.↗

Getting around Antarctica: New High-Resolution Mappings of the Grounded and Freely-Floating Boundaries of the Antarctic Ice Sheet Created for the International Polar Year

Two ice-dynamic transitions of the Antarctic ice sheet - the boundary of grounded ice features and the freely-floating boundary - are mapped at 15-m resolution by participants of the International Polar Year project ASAID using customized software combining Landsat-7 imagery and ICESat/GLAS laser altimetry. The grounded ice boundary is 53 610 km long; 74% abuts to floating ice shelves or outlet glaciers, 19% is adjacent to open or sea-ice covered ocean, and 7% of the boundary ice terminates on land. The freely-floating boundary, called here the hydrostatic line, is the most landward position on ice shelves that expresses the full amplitude of oscillating ocean tides. It extends 27 521 km and is discontinuous. Positional (one-sigma) accuracies of the grounded ice boundary vary an order of magnitude ranging from +/- 52m for the land and open-ocean terminating segments to +/- 502m for the outlet glaciers. The hydrostatic line is less well positioned with errors over 2 km. Elevations along each line are selected from 6 candidate digital elevation models based on their agreement with ICESat elevation values and surface shape inferred from the Landsat imagery. Elevations along the hydrostatic line are converted to ice thicknesses by applying a firn-correction factor and a flotation criterion. BEDMAP-compiled data and other airborne data are compared to the ASAID elevations and ice thicknesses to arrive at quantitative (one-sigma) uncertainties of surface elevations of +/-3.6, +/-9.6, +/-11.4, +/-30 and +/-100m for five ASAID-assigned confidence levels. Over one-half of the surface elevations along the grounded ice boundary and over one-third of the hydrostatic line elevations are ranked in the highest two confidence categories. A comparison between ASAID-calculated ice shelf thicknesses and BEDMAP-compiled data indicate a thin-ice bias of 41.2+/-71.3m for the ASAID ice thicknesses. The relationship between the seaward offset of the hydrostatic line from the grounded ice boundary only weakly matches a prediction based on beam theory. The mapped products along with the customized software to generate them and a variety of intermediate products are available from the National Snow and Ice Data Center.

Bindschadler, R.↗

Analysis and Prediction of VFR Vs IFR Traffic Behavior to Support Uncrewed Aircraft Flight Operations at Regional Airports

Uncrewed Aircraft flight operations at a regional airport will be affected by the uncertainty in the Visual Flight Rules traffic around. This paper analyzes Visual Flight Rules traffic behavior, compares it with Instrument Flight Rules traffic and develops traffic prediction methods to support uncrewed aircraft flight operations. The spatio-temporal distribution of traffic operating under Visual Flight Rules and Instrument Flight Rules was analyzed from one month of historical track data around Fort Worth Alliance airport as a representative regional airport. The traffic behavior was visualized as occupancy maps generated at different altitudes. The Instrument Flight Rules traffic was concentrated in fewer regions of the airspace along structured routes where the risk of interacting with one or more flights reached close to fifty percent in some areas. Visual Flight Rules traffic was spread over more regions, mostly segregated from the Instrument Flight Rules regions, and the risk of interaction was lower reaching up to twenty-five percent on average over the month in some regions. The interaction risk was predicted using predictive occupancy maps over multiple time horizons and conditioned on time and partial observation of traffic in the vicinity. The month-to-month predictability of Visual Flight Rules risk was lower than that of the Instrument Flight Rules traffic, consistently over all the conditions analyzed. However, the prediction and its accuracy were demonstrated to be sensitive to the conditions used. The predictive models generated can be used to support both strategic planning and in-flight decision-making by uncrewed aircraft during flight operations while maintaining an acceptable risk of interacting with other traffic.

VFR↗

Analysis and Prediction of VFR Vs IFR Traffic Behavior to Support Uncrewed Aircraft Flight Operations at Regional Airports

Uncrewed Aircraft flight operations at a regional airport will be affected by the uncertainty in the Visual Flight Rules traffic around. This paper analyzes Visual Flight Rules traffic behavior, compares it with Instrument Flight Rules traffic and develops traffic prediction methods to support uncrewed aircraft flight operations. The spatio-temporal distribution of traffic operating under Visual Flight Rules and Instrument Flight Rules was analyzed from one month of historical track data around Fort Worth Alliance airport as a representative regional airport. The traffic behavior was visualized as occupancy maps generated at different altitudes. The Instrument Flight Rules traffic was concentrated in fewer regions of the airspace along structured routes where the risk of interacting with one or more flights reached close to fifty percent in some areas. Visual Flight Rules traffic was spread over more regions, mostly segregated from the Instrument Flight Rules regions, and the risk of interaction was lower reaching up to twenty-five percent on average over the month in some regions. The interaction risk was predicted using predictive occupancy maps over multiple time horizons and conditioned on time and partial observation of traffic in the vicinity. The month-to-month predictability of Visual Flight Rules risk was lower than that of the Instrument Flight Rules traffic, consistently over all the conditions analyzed. However, the prediction and its accuracy were demonstrated to be sensitive to the conditions used. The predictive models generated can be used to support both strategic planning and in-flight decision-making by uncrewed aircraft during flight operations while maintaining an acceptable risk of interacting with other traffic.

VFR↗

Grassmannian diffusion maps based surrogate modeling via geometric harmonics

Abstract A novel surrogate model based on the Grassmannian diffusion maps (GDMaps) and utilizing geometric harmonics (GH) is developed for predicting the response of complex physical phenomena. The method utilizes GDMaps to obtain a low‐dimensional representation of the underlying behavior of physical/mathematical systems with respect to uncertain input parameters. Using this representation, GH, an out‐of‐sample extension technique, is employed to create a global map from the input parameter space to a Grassmannian diffusion manifold. GH is further employed to locally map points on the diffusion manifold onto the tangent space of a Grassmann manifold. The exponential map is then used to project the points in the tangent space onto the Grassmann manifold, where reconstruction of the full solution is performed. The performance of the proposed surrogate model is verified with three examples. The first problem is a toy example used to illustrate the technique. In the second example, errors associated with the various mappings are assessed by studying response predictions of the electric potential of a dielectric cylinder in a homogeneous electric field. The last example applies the method for uncertainty prediction in the strain field evolution in a model amorphous material using the shear transformation zone theory of plasticity.

42 ENGINEERING↗

MAPS of Cancer

Our goal was to produce an interactive visualization from a mathematical model that successfully predicts metastases from head and neck cancer. We met this goal early in the project. The visualization is available for the public to view. Our work appears to fill a need for more information about this deadly disease. The idea of this project was to make an easily interpretable visualization based on what we call "functional maps" of disease. A functional map is a graphic summary of medical data, where distances between parts of the body are determined by the probability of disease, not by anatomical distances. Functional maps often beat little resemblance to anatomical maps, but they can be used to predict the spread of disease. The idea of modeling the spread of disease in an abstract multidimensional space is difficult for many people. Our goal was to make the important predictions easy to see. NASA must face this problem frequently: how to help laypersons and professionals see important trends in abstract, complex data. We took advantage of concepts perfected in NASA's graphics libraries. As an analogy, consider a functional map of early America. Suppose we choose travel times, rather than miles, as our measures of inter-city distances. For Abraham Lincoln, travel times would have been the more meaningful measure of separation between cities. In such a map New Orleans would be close to Memphis because of the Mississippi River. St. Louis would be close to Portland because of the Oregon Trail. Oklahoma City would be far from Little Rock because of the Cheyenne. Such a map would look puzzling to those of us who have always seen physical maps, but the functional map would be more useful in predicting the probabilities of inter-site transit. Continuing the analogy, we could predict the spread of social diseases such as gambling along the rivers and cattle rustling along the trails. We could simply print the functional map of America, but it would be more interesting to show meaningful patterns of dispersal. We had previously published the functional map of the head and neck, but it was difficult to explain to either patients or surgeons because that view of our body did not resemble anatomy. This discrepancy between functional and physical maps is just a mathematical restatement of the well-known fact that some diseases, such as head and neck cancer, spread in complex patterns, not always to the next nearest site. We had discovered that a computer could re-arrange anatomy so that this particular disease spreads to the next nearest site. The functional map explains over 95% of the metastases in 1400 patients. In a sense, we had graphed what our body "looks like" to a tumor. The tumor readily travels between adjacent areas in the functional map. The functional map is a succinct visual display of trends that are not easily appreciated in tables of probabilities.

Gray, Lincoln↗