The formula for Basic COCOMO Embedded is:

where SM is Staff Months, the politicaly correct term formerly known as the Mythical Man Month, and KSLOC is thousand (kilo) source lines of code.

Basic COCOMO is based on a database of sixty-three software projects at TRW, Boehm’s then employer, during the 1970s. The Embedded Mode model is based on twenty-eight (28) of these projects that Boehm classified as Embedded projects. The projects were written in FORTRAN (24), COBOL (5), Jovial (5), PL/I (4), Pascal (2), Assembly (20), and miscellaneous other languages (3). None of these is commonly used today. Nonetheless, in the author’s experience, Basic COCOMO Embedded gives a rough order of magnitude (ROM) estimate of the effort for mathematical software projects such as implementing a video codec in C/C++ today (2012).

The Measurement Free Zone

Remarkably, despite the growing cost and importance of software, it is difficult to find publicly available information on the cost, schedule, and effort of software projects. There are a number of consulting firms and proprietary cost and schedule estimation tools but these do not disclose their databases of historical data. Indeed, many organizations, including many commercial businesses, do not seem to use historical data on the cost and schedule of software development to plan projects!

Donald Reifer’s 2004 Software Productivity Data

In 2004, software engineering expert Donald J. Reifer of Reifer Consultants, a colleague of Barry Boehm, published an article in The DoD SoftwareTech News, now The Journal of Software Technology, “Industry Software Cost, Quality and Productivity Benchmarks” giving the software productivity numbers, broken down by categories such as “Scientific” or “Web Business” for the most recent 600 of 1800 projects in his database of projects. These were projects from the last seven years prior to 2004 (about 1997 to 2004).

Table One below is a subset of Reifer’s data from Table 1 in his paper. These are the categories — Command and Control, Military – Airborne, Military – Ground, Military – Missile, Military – Space, and Scientific — that are similar (Command and Control, Military) or the same (Scientific) as mathematical software. The category “Web Business” is included as a point of reference.

Reifer uses equivalent source lines of code (ESLOC). For new code, ESLOC is equivalent to a line of code. For “legacy” code that is modified or reused, ESLOC applies a weighting factor to the line of code such as 0.4. This way data on maintenance or modifications of existing software can be combined with writing new software. Reifer uses equivalent source lines of code as defined by the Software Engineering Insitute.

Note that productivity in KESLOC (One Thousand Equivalent Source Lines of Code) is significantly higher for the Web Business category. This actually understates the difference because the “Web Business” projects, as indicated elsewhere in Reifer’s article, are usually written in so-called Fourth Generation Languages (4GLs), scripting languages such as Python, Perl, PHP, and so forth, whereas the other software categories are typically written in lower level languages such as C/C++. A single line of a 4GL language such as Python often corresponds to several lines of a language such as C/C++.

Scientific software has an average productivity of 195 ESLOC per Staff Month (SM). Note that there is a wide range of variation: 130 to 360 ESLOC per Staff Month (SM). This is for fairly large projects ranging from 28,000 lines of code to 790,000 lines of code.

Basic COCOMO Embedded predicts a productivity of 142 lines of code per Staff Month for a project with 28,000 lines of code. It predicts a productivity of 73 lines of code per Staff Month for a project with 790,000 lines of code. It predicts a productivity of about 280 lines of code per Staff Month for a project with 1,000 lines of code.

Basic COCOMO Embedded is quite similar to the numbers for Military Airborne, Missile, and Space.

Software productivity numbers are close to meaningless without an associated measure of the quality of the software. Reifer uses the number of bugs/errors/defects per thousand equivalent source lines of code (KESLOC). The error rates upon delivery to the customer show the difference between Web Business and the other categories. When the quality must be high, ideally no errors for mission critical life/death software such as airplane control software (avionics), then the number of lines of code per Staff Month is correspondingly lower.

Application Domain	Number Projects	Error Range (Errors/KESLOC)	Normative Error Rate (Errors/KESLOC)	Notes
Command & Control	45	0.5 to 5	1	Command centers
Military — All	125	0.2 to 3	< 1.0	See subcategories
— Airborne	40	0.2 to 1.3	0.5	Embedded sensors
— Ground	52	0.5 to 4	0.8	Combat center
— Missile	15	0.3 to 1.5	0.5	GNC system
— Space	18	0.2 to 0.8	0.4	Attitude control system
Scientific	35	0.9 to 5	2	Seismic processing
Web Business	65	4 to 18	11	Client/server sites

Table 8 (Abridged): Error Rates upon Delivery by Application Domain

Quality Requirements for Mathematical Software

The required quality for many types of mathematical software is often very high, meaning less than one error per thousand lines of code. For example, a video codec such as used by YouTube or Skype, generates the output, the video, seen and used by the customers. Almost any bug in a video codec will result in visible artifacts at best and often completely destroys the video. Many readers have probably noticed occasional blurriness or other anomalies in YouTube or other Web video; these are problems that remain after extensive debugging of the video software.

Many video, image, and audio processing applications have similar quality requirements to video codecs. Similarly, encryption and decryptions such as the Advanced Encryption Standard (AES) usually requires extremely high quality since even a single bit error will result in gibberish. Many other types of mathematical software require similarly high levels of quality. Many seem to have quality requirements in practice similar to avionics and other demanding applications modeled by Basic COCOMO Embedded.

Where Are All The Super Programmers?

It is not uncommon in verbal conversations or comments on Web blogs to encounter programmers who claim to routinely write five to ten-thousand lines of code per month. Nonetheless, Reifer’s data shows little evidence of this performance level. With some exceptions, studies of software productivity usually show much smaller numbers.

There is tremendous variation in software projects. The author once implemented the Advanced Encryption Standard (AES) in about one week. This is about 1500 lines of code. This would translate to 6000 lines of code per month if naively extrapolated. However, this was clearly unusual and stands out in the author’s memory precisely because the project went so quickly and smoothly.

It is probably possible to write many lines of working usable code for certain kinds of simple straight-forward business and user interface software. For example, the top productivity for the Web Business category in Reifer’s published data is 985 lines of code/month.

It is clear though that the average performance for the vast majority of software engineers, including most exceptional software engineers, is much less than 5000 lines of code per month for most categories of software projects, with the possible exception of some types of business and user interface software, if one requires reasonable quality.

Conclusion

In the author’s experience, it is common to encounter extremely optimistic ideas about the size, scope, and difficulty level of mathematical software projects. Many people appear to be genuinely unaware of how complex, how many lines of code, many commonly used examples of mathematical software such as video codecs actually are. Similarly, many people seem to be unaware of the quality level needed to produce an acceptable end-user/customer experience such as an enjoyable streaming video. Many people, even technical people who should know better, often seem to consciously or unconsciously use software productivity numbers like 5-10,000 lines of code per Staff Month even though these are not supported by most historical experience.

How should one use models like Basic COCOMO Embedded that are based on historical data or historical software productivity numbers like Donald Reifer’s data? These are good for rough order of magnitude (ROM) estimates including basic sanity checks. If one only has resources for a two week project and Basic COCOMO says the project is a six month project, one should probably reevaluate one’s plans. On the other hand if one has the resources for a six month project and Basic COCOMO says seven months, the difference is probably not meaningful given the large variation between actual effort and estimated effort. The same applies to blindly plugging in numbers like Reifer’s average 195 lines of code per Staff Month for Scientific software.

These models and data are not good for precise scheduling. There is substantial variation between actual and estimated effort. Software seems to inherently involve large variations in effort that are difficult or impossible to predict in advance.

Suggested Reading/References

Barry Boehm, Software Engineering Economics, Prentice-Hall, Englewood Cliffs, NJ, 1981

© 2012 John F. McGowan

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

Possibly related articles:

1. Estimating the Cost and Schedule of Mathematical Software
2. The Scope of Mathematical Programming Projects
3. LAME: A Case Study in Mathematical Programming

Mathematical software may offer solutions to many pressing problems such as curing cancer by helping develop systems of smart drugs that perform mathematical or logical calculations to identify and selectively kill cancer cells (see the previous post Animations of a Possible Cure for Cancer).

Mathematical software can also solve many small problems such as the need to relax with entertaining new audio/video effects for computer games and movies (see the previous post, Creating Cartoon Voices with Math).

The successful solution of problems using mathematics and mathematical software usually requires estimating the cost and schedule of mathematical software projects based on historical experience. A good coach and quarterback plan the strategy and plays for a successful football team based on what the players and team can actually do.

Estimating the Cost and Schedule of Mathematical Software

Mathematical software development is an uncommon area, unlike mainstream software development such as business web sites and user interface software development. User interface software, for example, is often extremely easy today. With modern scripting languages like Python and GUI (Graphical User Interface) Builders, it is possible to create working user interfaces rapidly with little risk if one sticks with standard GUI components such as buttons, sliders, and data entry fields. Mathematical software development is usually much harder than modern user interface software development — taking longer per line of code, involving much more debugging — and much less predictable.

Individuals and groups often have extremely optimistic ideas about the size and scope of mathematical software projects. Many people appear to be unaware of the size and complexity of various commonly used examples of mathematical software. For example, the widely used open-source x264 h.264 video encoder is over 62,000 lines of code developed between 2004 and 2011 with at least 18 contributors. The H.264 video compression standard is the video compression used in many YouTube videos, BluRay discs, and other high performance video systems.

The popular open source Independent JPEG Group JPEG image encoder/decoder used by many commercial and open-source image editors and viewers is over 52,000 lines of code developed between 2000 and 2011 with at least 13 contributors.

The LAME MP3 audio encoder, best known as the MP3 encoder plugin for the Audacity audio editor, is over 87,000 lines of code (about 40,000 lines of algorithmic C/C++ code and about 47,000 lines of Bourne shell installer code) developed between 1998 and 2011 with at least nine primary developers and at least 21 total contributors.

A line of code is comparable to at least one moving part in a physical machine. As a point of reference, the Space Shuttle Main Engine, one of the most sophisticated engines in the world, has about 50,000 moving parts. These mathematical programs are comparable in complexity to the most sophisticated physical machines in the world and often fail catastrophically due to tiny errors just as a rocket engine will.

Software engineering expert Barry Boehm’s original software cost estimating model COCOMO (Embedded) which stands for the Constructive Cost Model for Embedded Software Development — appears to give a rough estimate of the time it takes to develop these low level mathematical programs, although there is substantial variation between estimates and actual effort.

The model is:

where Boehm’s Man Month is 152 man-hours (19 man-days) and KDSI is 1000 (Kilo) Delivered Source Instructions (lines of code). Blank lines and comments are not counted.

A few quick numbers from COCOMO Embedded:

1000 lines of code 3.6 man months
2000 lines of code 8.3 man months
5000 lines of code 24.8 man months

Basic COCOMO (Embedded)

If the project is using consultants at an hourly rate, one should multiply the number of man months times 152 hours times the hourly rate of the consultants. If the project is using direct employees, one should use the cost of the employees per month. Boehm’s model omits one man-day per month for the average paid time off of the employee.

Cost = (Estimated Man-Months)*(152 hours)*(hourly rate)

or

Cost = (Estimated Man-Month)*(Monthly Salary and Overhead)

There is substantial variation between the actual and estimated effort from this simple model. In his book Software Engineering Economics (p. 84), Boehm notes:

From a practical standpoint, it is important to note that Basic COCOMO estimates are within a factor of 1.3 of actuals only 29% of the time, and within a factor of 2 only 60% of the time.

Barry Boehm advises against using his model for projects smaller than 2000 lines of code.

In the author’s experience, shorter projects, such as 1000 lines of code, are still in the same ballpark, on average, as predicted by Basic COCOMO Embedded but there is even more variation between the estimates and actual effort. The author once implemented the Advanced Encryption Standard (AES), about 1500 lines of code, in one week which was much faster than the model would predict or the author’s experience with other projects. The effort varies even more on small projects depending on the details of the algorithm and other factors that are hard to know in advance.

The free open-source CLOC (Count Lines of Code) utility is available for the major programming platforms: Windows, Mac OS X, Linux, and other common forms of Unix. There are now many free open-source programs that implement known mathematics and algorithms such as x264 and the other examples cited above. It is thus often possible to get a rough estimate of the size and scope of mathematical software projects that involve known mathematics and algorithms.

Limitations of Lines of Code

It is important to keep in mind that a line of code (LOC) is a rough estimate of the size and complexity of a computer program. In the C or C++ programming languages, these are both a single line of code:

a = 1;

if ( (a > b && a < c) || d < e) { a = sin(b+c) } else { a = tanh(a + b + c)/(d+e) };

The second example line of code would usually require more actual effort than the first example line. This is one of the reasons cost and schedule estimates based on counts of the lines of code vary a lot compared to the actual effort.

Because of the many problems with using lines of code for cost and schedule estimation, other methods such as function points have been developed. Function points are currently popular in books and articles on software cost and schedule estimation. However, function points were developed for business and user interface software. Function point estimation generally involves counting the number of inputs and outputs to the program such as data entry fields in a business program. This often predicts the actual effort well because many business and user interface programs have simple internal logic or mathematics and the effort is proportional to the number of inputs and outputs of the program. Business software usually uses only basic arithmetic, adding columns of numbers and similar simple operations.

Mathematical programs are generally extremely complex internally but often appear as only a few inputs and outputs. For example, a video compression program takes one input, the uncompressed raw video, and returns one output, the compressed video. Thus, methods such as function points tend to grossly underestimate the size and scope of mathematical software projects.

This weakness of function points has been recognized for many years and there are more advanced versions of the function point method that attempt to better estimate the size and complexity of complex algorithms hidden from the end user. However, it is still better to rely on lines of code for estimating the size and scope of mathematical software, despite the obvious limitations of using lines of code for cost and schedule estimation.

Scripting Languages (Matlab) Versus Low-Level Compiled Languages (C/C++)

One well known way to speed up the development of mathematical software is to use mathematical scripting languages such as Matlab, Mathematica, Octave (a free open-source program that is mostly compatible with Matlab), and many others. These are scripting languages similar to Python or PHP that have large well-integrated libraries of mathematical function combined with a list (e.g. Mathematica) or numerical array/matrix data type (e.g. Matlab).

In the author’s experience, the speed of development of mathematical software using Octave, MATLAB, or similar tools is generally 2-3 times faster on average than C/C++. This is mostly because the number of lines of code is reduced by a factor of 2-3. The Basic COCOMO Embedded model still gives a useful rough estimate of the actual effort required, but the number of lines of code input to the cost model is reduced!

There is a lot of variation in the increased speed of development from using Octave, Matlab, or similar tools, depending on the details of the algorithm and just plain luck. Some algorithms are well adapted to implementation in Octave/MATLAB and the speed of development gain can be 10-20 times the speed to develop in C/C++. Mathworks, which markets MATLAB, plays up cases like this. There are also some algorithms where there is no gain; the Octave/MATLAB code is pretty much the same as the C/C++ code.

Unfortunately, the speed of execution of the programs in Matlab, Octave, and similar tools is often significantly less than compiled code written in C, C++, or similar programming languages. With languages like Matlab that use numerical arrays, the penalty is not as great as it was a decade ago. Some operations such as the Fast Fourier Transform (FFT) often seem to be just as fast in Matlab or similar tools as compiled versions. However, one should generally plan for a penalty of 2-3 times in speed of execution. It is still often not practical due to speed of execution and memory usage problems to develop computationally intensive mathematical software such as video compression programs using tools such as Octave, Matlab, or Mathematica.

Conclusion

Mathematics and mathematical software can deliver large improvements in speed and efficiency as well as new useful features. Success is much more likely with estimates of the size and scope of mathematical software development based on historical experience.

Suggested Reading/References

Barry Boehm, Software Engineering Economics, Prentice-Hall, Englewood Cliffs, NJ, 1981

© 2012 John F. McGowan

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

Possibly related articles:

1. Octave: An Alternative to the High Cost of MATLAB
2. The Scope of Mathematical Programming Projects
3. LAME: A Case Study in Mathematical Programming

As some of you know, I run a service called Any New Books?, which emails you a list of new books that are related to the categories of your choice each week. For the most part I pulled this list from the weekly staff picks there throughout this past year, just in time for your Christmas shopping.

The books are ordered by their current sale rank on Amazon (from the most popular to the least popular at the moment, with hardcovers first). I hope this page will help you discover a few titles you may have not have noticed yet.

[Programming book list]

	The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy Stores: USA | UK | Canada | Italy | Kindle | UK Kindle By Sharon Bertsch McGrayne ISBN: 0300169698 Publisher: Yale University Press Publication date: May 17, 2011 Binding: Hardcover Estimated price: $17.60 Share this book on Twitter |Facebook
	The Man of Numbers: Fibonacci’s Arithmetic Revolution Stores: USA | Canada | Italy | Kindle | UK Kindle By Keith Devlin ISBN: 0802778127 Publisher: Walker & Company Publication date: July 5, 2011 Binding: Hardcover Estimated price: $14.50 Share this book on Twitter |Facebook
	The Mathematics of Life Stores: USA | Canada | Italy | Kindle By Ian Stewart ISBN: 0465022383 Publisher: Basic Books Publication date: June 7, 2011 Binding: Hardcover Estimated price: $12.13 Share this book on Twitter |Facebook
	The Art of Computer Programming, Volume 4A: Combinatorial Algorithms, Part 1 Stores: USA | UK | Canada | Italy By Donald E. Knuth ISBN: 0201038048 Publisher: Addison-Wesley Professional Publication date: January 22, 2011 Binding: Hardcover Estimated price: $53.15 Share this book on Twitter |Facebook
	SuperCooperators: Altruism, Evolution, and Why We Need Each Other to Succeed Stores: USA | Canada | Italy | Kindle By Martin Nowak, Roger Highfield ISBN: 1439100187 Publisher: Free Press Publication date: March 22, 2011 Binding: Hardcover Estimated price: $8.33 Share this book on Twitter |Facebook
	Undocumented Secrets of MATLAB-Java Programming Stores: USA | UK | Canada By Yair M. Altman ISBN: 1439869030 Publisher: Chapman and Hall/CRC Publication date: December 12, 2011 Binding: Hardcover Estimated price: $60.57 Share this book on Twitter |Facebook
	Number-Crunching: Taming Unruly Computational Problems from Mathematical Physics to Science Fiction Stores: USA | UK | Canada | Italy | Kindle | UK Kindle By Paul J. Nahin ISBN: 0691144257 Publisher: Princeton University Press Publication date: August 28, 2011 Binding: Hardcover Estimated price: $18.74 Share this book on Twitter |Facebook
	Stats: Data and Models (3rd Edition) Stores: USA | UK | Canada | Italy | Kindle By Richard D. De Veaux, Paul F. Velleman, David E. Bock ISBN: 0321692551 Publisher: Addison Wesley Publication date: January 8, 2011 Binding: Hardcover Estimated price: $98.99 Share this book on Twitter |Facebook
	Functional Analysis: Introduction to Further Topics in Analysis Stores: USA | UK | Canada | Italy | Kindle | UK Kindle By Elias M. Stein, Rami Shakarchi ISBN: 0691113874 Publisher: Princeton University Press Publication date: September 11, 2011 Binding: Hardcover Estimated price: $60.00 Share this book on Twitter |Facebook
	Cluster Analysis Stores: USA | UK | Canada | Italy | Kindle | UK Kindle By Brian S. Everitt, Dr Sabine Landau, Dr Morven Leese, Dr Daniel Stahl ISBN: 0470749911 Publisher: Wiley Publication date: March 8, 2011 Binding: Hardcover Estimated price: $59.99 Share this book on Twitter |Facebook
	Viewpoints: Mathematical Perspective and Fractal Geometry in Art Stores: USA | UK | Canada | Italy | Kindle | UK Kindle By Marc Frantz, Annalisa Crannell ISBN: 0691125929 Publisher: Princeton University Press Publication date: July 25, 2011 Binding: Hardcover Estimated price: $35.31 Share this book on Twitter |Facebook
	The Glorious Golden Ratio Stores: USA By Alfred S. Posamentier, Ingmar Lehmann ISBN: 1616144238 Publisher: Prometheus Books Publication date: November 22, 2011 Binding: Hardcover Estimated price: $14.46 Share this book on Twitter |Facebook
	One, Two, Three: Absolutely Elementary Mathematics Stores: USA | UK | Canada | Italy | Kindle By David Berlinski ISBN: 0375423338 Publisher: Pantheon Publication date: May 10, 2011 Binding: Hardcover Estimated price: $9.89 Share this book on Twitter |Facebook
	Game Theory and the Humanities: Bridging Two Worlds Stores: USA | UK | Canada | Italy By Steven J. Brams ISBN: 0262015226 Publisher: The MIT Press Publication date: March 4, 2011 Binding: Hardcover Estimated price: $28.81 Share this book on Twitter |Facebook
	Majority Judgment: Measuring, Ranking, and Electing Stores: USA | UK | Canada | Italy By Michel Balinski, Rida Laraki ISBN: 0262015137 Publisher: The MIT Press Publication date: January 28, 2011 Binding: Hardcover Estimated price: $31.89 Share this book on Twitter |Facebook
	The Oxford Handbook of Random Matrix Theory Stores: USA | UK | Canada | Italy By Gernot Akemann, Jinho Baik, Philippe Di Francesco ISBN: 0199574006 Publisher: Oxford University Press, USA Publication date: September 25, 2011 Binding: Hardcover Estimated price: $173.05 Share this book on Twitter |Facebook
	An Elementary Introduction to Statistical Learning Theory Stores: USA | UK | Canada | Italy By Sanjeev Kulkarni, Gilbert Harman ISBN: 0470641835 Publisher: Wiley Publication date: August 2, 2011 Binding: Hardcover Estimated price: $73.99 Share this book on Twitter |Facebook
	The Ultimate Challenge: The 3x+1 Problem Stores: USA | UK By Jeffrey C. Lagarias ISBN: 0821849409 Publisher: American Mathematical Society Publication date: January 14, 2011 Binding: Hardcover Estimated price: $59.00 Share this book on Twitter |Facebook
	Connections in Combinatorial Optimization Stores: USA | UK | Canada | Italy By Andras Frank ISBN: 0199205272 Publisher: Oxford University Press, USA Publication date: June 1, 2011 Binding: Hardcover Estimated price: $94.04 Share this book on Twitter |Facebook
	The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein Stores: USA | UK | Canada | Italy | Kindle | UK Kindle By Burt C. Hopkins ISBN: 0253356717 Publisher: Indiana University Press Publication date: September 7, 2011 Binding: Hardcover Estimated price: $30.00 Share this book on Twitter |Facebook
	The Evolution of Principia Mathematica: Bertrand Russell’s Manuscripts and Notes for the Second Edition Stores: USA | UK | Canada | Italy By Bernard Linsky ISBN: 110700327X Publisher: Cambridge University Press Publication date: July 11, 2011 Binding: Hardcover Estimated price: $138.09 Share this book on Twitter |Facebook
	Comparing Groups: Randomization and Bootstrap Methods Using R Stores: USA | UK | Canada | Italy By Andrew S. Zieffler, Jeffrey R. Harring, Jeffrey D. Long ISBN: 0470621699 Publisher: Wiley Publication date: June 15, 2011 Binding: Hardcover Estimated price: $64.99 Share this book on Twitter |Facebook
	Lectures in Game Theory for Computer Scientists Stores: USA | UK | Canada | Italy By Editors at Cambridge University Press ISBN: 0521198666 Publisher: Cambridge University Press Publication date: February 14, 2011 Binding: Hardcover Estimated price: $50.00 Share this book on Twitter |Facebook
	Probability Concepts and Theory for Engineers Stores: USA | UK | Canada | Kindle | UK Kindle By Harry Schwarzlander ISBN: 0470748559 Publisher: Wiley Publication date: March 1, 2011 Binding: Hardcover Estimated price: $67.99 Share this book on Twitter |Facebook
	Advanced Topics in Linear Algebra: Weaving Matrix Problems through the Weyr Form Stores: USA | UK | Canada | Italy By Kevin O’Meara, John Clark, Charles Vinsonhaler ISBN: 0199793735 Publisher: Oxford University Press, USA Publication date: September 16, 2011 Binding: Hardcover Estimated price: $71.55 Share this book on Twitter |Facebook
	The Art of R Programming: A Tour of Statistical Software Design Stores: USA | UK | Canada | Kindle | UK Kindle By Norman Matloff ISBN: 1593273843 Publisher: No Starch Press Publication date: October 12, 2011 Binding: Paperback Estimated price: $21.99 Share this book on Twitter |Facebook
	Here’s Looking at Euclid: From Counting Ants to Games of Chance – An Awe-Inspiring Journey Through the World of Numbers Stores: USA | Canada | Italy | Kindle By Alex Bellos ISBN: 1416588280 Publisher: Free Press Publication date: April 19, 2011 Binding: Paperback Estimated price: $4.50 Share this book on Twitter |Facebook
	Matlab, Second Edition: A Practical Introduction to Programming and Problem Solving Stores: USA | UK | Canada | Kindle | UK Kindle By Stormy Attaway ISBN: 0123850819 Publisher: Butterworth-Heinemann Publication date: August 11, 2011 Binding: Paperback Estimated price: $35.44 Share this book on Twitter |Facebook
	The Best Writing on Mathematics 2011 Stores: USA | UK | Canada | Kindle | UK Kindle By Editors at Princeton University Press ISBN: 0691153159 Publisher: Princeton University Press Publication date: November 27, 2011 Binding: Paperback Estimated price: $12.14 Share this book on Twitter |Facebook
	Algorithmic Puzzles Stores: USA | UK | Canada | Kindle By Anany Levitin, Maria Levitin ISBN: 0199740445 Publisher: Oxford University Press, USA Publication date: October 14, 2011 Binding: Paperback Estimated price: $21.13 Share this book on Twitter |Facebook
	Number Sense Routines: Building Numerical Literacy Every Day in Grades K-3 Stores: USA | Canada By Jessica F. Shumway ISBN: 1571107908 Publisher: Stenhouse Pub Publication date: April 10, 2011 Binding: Paperback Estimated price: $17.88 Share this book on Twitter |Facebook
	Agent-Based and Individual-Based Modeling: A Practical Introduction Stores: USA | UK | Canada | Kindle | UK Kindle By Steven F. Railsback, Volker Grimm ISBN: 0691136742 Publisher: Princeton University Press Publication date: November 6, 2011 Binding: Paperback Estimated price: $41.39 Share this book on Twitter |Facebook
	Foundations of Geometry (2nd Edition) Stores: USA | UK | Canada By Gerard Venema ISBN: 0136020585 Publisher: Addison Wesley Publication date: July 16, 2011 Binding: Paperback Estimated price: $60.35 Share this book on Twitter |Facebook
	Taming the Infinite: The Story of Mathematics from the First Numbers to Chaos Theory Stores: USA | UK | Canada | Italy By Ian Stewart ISBN: 1847247687 Publisher: Quercus Publication date: November 1, 2011 Binding: Paperback Estimated price: $6.03 Share this book on Twitter |Facebook
	The Number Mysteries: A Mathematical Odyssey through Everyday Life Stores: USA | Canada | Italy | Kindle By Marcus du Sautoy ISBN: 0230113842 Publisher: Palgrave Macmillan Publication date: May 24, 2011 Binding: Paperback Estimated price: $10.66 Share this book on Twitter |Facebook
	The Number Sense: How the Mind Creates Mathematics, Revised and Updated Edition Stores: USA | UK | Canada | Italy | Kindle By Stanislas Dehaene ISBN: 0199753873 Publisher: Oxford University Press, USA Publication date: April 29, 2011 Binding: Paperback Estimated price: $20.04 Share this book on Twitter |Facebook
	Introduction to Proof in Abstract Mathematics Stores: USA | UK | Canada | Italy By Andrew Wohlgemuth, Mathematics ISBN: 0486478548 Publisher: Dover Publications Publication date: February 17, 2011 Binding: Paperback Estimated price: $12.89 Share this book on Twitter |Facebook
	Wizardry: Baseball’s All-Time Greatest Fielders Revealed Stores: USA | UK | Canada | Italy | Kindle | UK Kindle By Michael Humphreys ISBN: 0195397762 Publisher: Oxford University Press, USA Publication date: March 30, 2011 Binding: Paperback Estimated price: $11.00 Share this book on Twitter |Facebook
	Differential Geometry: Bundles, Connections, Metrics and Curvature Stores: USA | UK | Canada | Kindle By Clifford Henry Taubes ISBN: 0199605874 Publisher: Oxford University Press, USA Publication date: December 1, 2011 Binding: Paperback Estimated price: $36.20 Share this book on Twitter |Facebook
	Fatal Numbers: Why Count on Chance Stores: USA | UK | Kindle | UK Kindle By Hans Magnus Enzensberger ISBN: 1935830015 Publisher: Upper West Side Philosophers, Inc. Publication date: February 2, 2011 Binding: Paperback Estimated price: $10.05 Share this book on Twitter |Facebook
	Matroid Theory Stores: USA | UK | Canada | Italy By James Oxley ISBN: 0199603391 Publisher: Oxford University Press, USA Publication date: April 22, 2011 Binding: Paperback Estimated price: $53.16 Share this book on Twitter |Facebook
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Cancer is the second leading cause of death in the United States. Over five-hundred thousand people died from cancer in 2007. If current trends continue, about one in three of readers will die from cancer.

Since 1971 the United States has spent about $200 billion on research into cancer. The National Cancer Institute has an annual budget of over $5 billion. This is comparable to the Manhattan Project that invented the atomic bomb and the first nuclear reactors continued for forty years. The results have clearly been quite disappointing. Is there a way to get better results from the many years of hard work, billions of dollars, and mountains of knowledge collected? Are there ways to apply today’s powerful computers and mathematics to defeat this disease?

Cancer is now thought to be caused by mutations of genes, cancer genes or oncogenes and tumor suppressor genes, that control complex networks of proteins that regulate the division, growth, and differentiation of cells in the body. Differentiation refers to the process by which cells turn into specialized kinds of cells such as skin, blood, and nerves. As we age, we accumulate mutations of these genes in some cells. It requires several mutations of several different genes to produce most forms of cancer. Many different sets of mutated genes cause cancer.

While a medical doctor or pathologist may identify a cancer as breast cancer or skin cancer, at a molecular and genetic level, skin cancer is thought to be many different cancers caused by many different sets of mutated genes. In total, cancer is now thought to be thousands of different diseases. This makes finding a single chemical similar to penicillin, for example, that can kill all cancers either impossible or very difficult, at least by starting from the individual cancer genes and the proteins they produce.

Even worse, cancer cells are generally thought to become genetically unstable and mutate much more rapidly than normal cells. Hence, the cancer cells begin to evolve in the body and can develop immunity to anti-cancer drugs such as chemotherapy agents.

While cancer varies enormously at the level of genes and proteins, the part level, cancer cells may have common system-level features. For example, pathologists can identify cancer cells or tissues from biopsies under an optical microscope as cancer. Another common characteristic is that many, perhaps all, cancer cells have an abnormal number of chromosomes, often too many. This article considers targeting the abnormal number of chromosomes.

The Bathtub Mechanism, developed by the author several years ago, is an algorithm, which can be implemented by a relatively simple set of molecules, that may be able to selectively destroy cell with an abnormal number of chromosomes. This system of drugs is like a bathtub with several running faucets, one for each chromosome, and a single drain. If there are too many faucets, chromosomes, the water level, the concentration of the cell killer, will rise and overflow the bathtub. If there are the right number, forty-six, or too few, less than forty-six, faucets, the drain can remove the water being added and the water level never rises. The water level remains almost zero; the concentration of the cell killer is far too low to harm the cell.

One can kill cells with too few chromosomes (less than forty-six) by swapping the roles of the drain and the source. The drain is now a feature of the chromosomes. The source is the constant numerical feature of the cells. Thus, if there are too few chromosomes, there are not enough drains to remove the cell killer produced by the source. The bathtub has one big faucet and many small drains, one for each chromosome. The water level, the concentration of the toxin, rises if there are too few drains/chromosomes.

It may be possible to create proteins that react directly with the source and drain features in the cell. On the other hand, it may be necessary to use a source and a drain catalyst that bind to the source and drain features and become active catalysts only when binding to the source or drain features. In this article the first case is considered. The source and drain catalysts are discussed in more detail in the previous two articles.

Molecular Building Blocks of the Bathtub Mechanism

(A (BC)) harmless Precursor
(BC) Cell Killer
B harmless fragment
C harmless fragment

IN Inhibitor Precursor
I Bacteriophage Inhibitor
N harmless fragment

D Drain
IS Inhibitor Source
S Source (on or associated with chromosome, may be a DNA sequence)

The bathtub mechanism requires two features in the cell: a numerical or quantitative feature that is proportional to the number of chromosomes and a feature that is constant in all cells, both normal and cancerous. Some obvious features that probably vary with the number of chromosomes are the telomeres at the end of the chromosomes and the centromeres at the center of the chromosomes.

There are many molecular structures in the chromosomes and associated with the chromosomes. It seems probable, although not certain, that one can find a numerical or quantitative feature that varies with the number of chromosomes that could be used. A more serious problem with the bathtub mechanism is the constant feature that is the same in both healthy cells and cancer cells, especially since cancer cells are thought to be constantly mutating and changing. This may be a show-stopper.

Since the cancer cells may be mutating, it may be impossible to find a constant feature in the cancer cells. The feature could disappear entirely or change in size or number. There is at least one possible way to add such a feature artificially to the cells, both healthy and malignant.

A bacteriophage is a kind of virus that attaches to the exterior membrane of a cell and injects its genetic material into the cell. The bacteriophage’s genetic material then takes over the machinery of the cell and directs it to make more bacteriophages. The bacteriophage consists of a protein sheath that looks something like a science fiction bug with several arms (see animations below) that grab the surface of the cell and a spherical or polyhedral chamber that carries the genetic material.

In principle, one could modify the genetic material of the bacteriophage to create cells (the commonly used E. Coli bacteria, for example) that make not the virus, but the protein sheath with a payload of other proteins or non-coding DNA sequences, in particular DNA sequences that regulatory proteins bind to. These pseudo-bacteriophages would inject their protein or non-coding DNA payloads into cells instead of the genetic material of the naturally occurring bacteriophage. They would not be infectious like a normal virus.

If, and this is a big if, one could modify the protein sheath so it would only inject the protein or non-coding DNA payload into a cell without an inhibitor protein I that is generatd by inhibitor sources (IS) in the payload, one could inject a payload that contained an artificial constant drain feature D and the inhibitor sources IS into the cell. The inhibitor protein I might work, for example, by blocking the arms of the bacteriophage from attaching to the exterior membrane of the cell, which presumably triggers the injection of the payload.

Once the new drain feature was added to the cell, the pseudo-bacteriophages would stop injecting payloads into the cell because it now also contained the inhibitors. Thus, a constant number of features could be added to each cell, both healthy and cancerous.

The Pseudo-Bacteriophage Payload is either a string of protein units or non-coding DNA with repeated sequences of regulatory protein binding sites, drains D and inhibitor sources IS

Series of reactions:

ABC (Precursor) ==> Source (S) (telomere or other chromosome feature) ==> A + BC (Cell Killer)

BC (Cell Killer) ==> Drain (D) ==> B (Harmless Fragment) + C (Harmless Fragment)

IN (Inhibitor Precursor) ==> IS (Inhibitor Source) ==> I (Bacteriophage Inhibitor) + N (Harmless Fragment)

The pseudo-bacteriophage payload is:

DDDDDDDDD(IS)(IS)(IS)(IS)(IS)(IS)(IS)

In the animations below:

The inhibitor I and the inhibitor source IS are represented by the blue spheres in the payload string

The drain D is the orange spheres in bacteriophage payload

The bacteriophage payload is shown as a string of blue and orange spheres in the first four animations below, mostly clearly in the fourth closeup animation. The inhibitors are shown in the second animation as blue spheres on the surface of the cell that prevent the bacteriophage from injecting a second payload string (drain) into a cell.

The payload is a single strand of protein sub-units or non-coding DNA. When the cell divides, the payload should end up in only one daughter cell. The other daughter cell will lack the payload and the inhibitor sources. The pseudo-bacteriophages will then add another payload string with the drain to the drainless daughter cell.

Alternatively, if the payload is a non-coding DNA string, not proteins, it may be possible to integrate the DNA string into the cell’s DNA, the chromosomes, as a single inherited drain. In this case, the drain will be inherited by both daughter cells when the cell divides.

Animations

The following animations illustrate the Bathtub Mechanism, a basic concept. The animations were created by the author using the free POV-Ray (Persistence of Vision Ray Tracing Program) for Windows 3.62 on a PC running Windows XP Service Pack 2. The POV-Ray scene description files contain a very simple mathematical model of the bathtub mechanism. The rendered frames were combined into MPEG-4 video files using the free, open-source ffmpeg video encoding utility. These animations illustrate a basic concept. They are not a quantitative mathematical model or simulation of cells, even at low fidelity.

This animation shows a pseudo-bacteriophage injecting a drain payload into a cell:

This animation shows a pseudo-bacteriophage prevented from injecting a second drain payload into a cell that already has a drain. The blue spheres are the inhibitors that prevent the pseudo-bacteriophage legs from attaching to the cell membrane.

This animation shows a wide angle view of the harmless precursor (red cone with green sphere cap) converted to the cell killer (red cone) by the telomere (yellow end of cylinder) of a single chromosome and then neutralized by the drain payload (shown as a string of orange drain spheres and blue inhibitor source spheres):

This animation shows a closeup view of the harmless precursor (red cone with green sphere cap) converted to the cell killer (red cone) by the telomere (yellow end of cylinder) of a single chromosome and then neutralized by the drain payload (shown as a string of orange drain spheres and blue inhibitor source spheres):

This animation shows a normal cell with forty-six chromosomes (represented by a simple blue sphere for clarity). The drain is represented by a simple green and gray sphere for clarity. The drain is green when it can process a cell killer, converting it to a harmless fragment (represented by a white sphere for clarity) which is excreted by the cell. The drain is black when it is processing a cell killer and cannot convert another. The drain has a maximum throughput. In a normal cell, the drain can remove as many cell killers as are added by the sources, the chromosomes. The concentration of the cell killer, the number in the lower right corner of the animation, remains low, never reaching the lethal level of two-hundred.

This animation shows the cell killer concentration rising and killing a cancer cell with too many chromosomes (represented by two blue spheres for two sets of chromosomes). The cell killer concentration is the number displayed in the lower right corner. The drain cannot remove the cell killers as rapidly as they are added. The concentration rises to the lethal level of two-hundred and the cell disintegrates. The membrane is shown decaying by making it more and more transparent as the cell killer concentration rises.

Future Steps

Many technical details and difficulties have been omitted to present the idea. While it might be possible to research and develop the bathtub mechanism entirely empirically at a laboratory bench through extensive trial and error, it should be possible to substantially accelerate the development process by simulating the molecular mechanisms using today’s powerful computers. In practice, it would probably require careful tuning of the chemical reaction rates in the cell to produce the desired selective destruction of cells with abnormal numbers of chromosomes or other features associated with cancer.

The next logical step is to construct a mathematical model and simulation of the bathtub mechanism in real cells, iteratively increasing the level of fidelity. This would enable evaluation of the feasibility of the concept and of specific variants of the concept, as many variations are possible and more will become evident with detailed simulation and working through of the concept. Perhaps more importantly a detailed simulation would make it easier for specialists in various fields of biology and organic chemistry — chromosomes, bacteriophages, proteins, many others — to see where their expertise could fit into the concept or resolve otherwise intractable problems.

Naturally occurring networks of proteins and other molecules in cells seem to be able to perform many complex mathematical and logical calculations, such as the feedback control networks that seem to malfunction in cancer. While one cannot be certain, it is not unlikely that a relatively simple network of proteins and other molecules can implement the bathtub mechanism or something similar.

Even engineering a single molecule such as genetically engineered insulin for diabetics is a daunting task at present. So a system of even a few molecules would be a substantial and difficult undertaking. Nonetheless it is probably doable now or in the near future.

However, the underlying biology is unknown. Even though there are over one-million research papers on cancer, it is difficult to get a clear picture of the role of aneuploidy in cancer. Most modern cancer research is conducted within the framework of the oncogene theory and an implicit assumption that the way to cure or treat cancer is to target either a protein generated by a cancer gene or the gene directly.

Chromosomal anomalies, both abnormal numbers of chromosomes and the rearrangements of chromosomes that are common in many cancers, are usually discussed as an aside to the putative cancer genes. This translocation of chromosome X mutated the key cancer gene ABC, or the duplication of chromosome X resulted in two copies of the key cancer gene ABC.

It could be that killing cancer cells with the wrong number of chromosomes would have no effect on the disease. It would simply result in a cancer with the correct number of chromosomes in the surviving cancer cells. It could slow the disease if the abnormal number of chromosomes is related to the malignancy of the cancer cells. In the best case, it might cure the disease, if the abnormal number of chromosomes is either the cause of cancer or essential in some way to the malignant characteristics of the cancer cells.

Conclusion

Everyone faces about a one in three chance of dying from cancer. Cancer researchers would like more impressive results to show policy makers and the general public, especially when seeking continued or increased funding. Pharmaceutical and biotechnology companies should desire improved anti-cancer drugs and treatments to maintain and increase their profits. Defeating cancer would free up resources and researchers to tackle other diseases of old age and even the aging process itself.

It may be possible to cure or effectively treat cancer with a system of smart drugs that perform a simple mathematical or logical calculation to selectively destroy cancer cells or probable cancer cells while sparing most normal healthy cells. These systems of smart drugs may be able to identify system level features of cancer cells independent of the confusing plethora of cancer genes and tumor suppressor genes.

The bathtub mechanism discussed in this article is one possible example of such a system of smart drugs. Mathematics and computers can enable or greatly accelerate the development of such systems of smart drugs.

Given the multitude of cancer genes and tumor suppressor genes that have been discovered in the last forty years, we should look at other aspects of cancer such as possible system level features for a cure or effective treatment. Today’s powerful computers, mathematics, and physics combined with the vast biological knowledge acquired in the last forty years may make it possible to attack cancer successfully in ways that were not practical even a few years ago.

© 2011 John F. McGowan

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

Possibly related articles:

1. Can Mathematics Cure Cancer?
2. Tackling Cancer with Math

Steve Jobs inspiring commencement address to Stanford University is also a poignant reminder of the ephemeral nature of words like “cured” and “curable” in cancer research and treatment. Steve Jobs may well have believed his rare form of pancreatic cancer was “cured” or “curable” as he claimed his doctors told him.

Steve Jobs death also highlights the limited benefits of today’s extremely powerful computers and electronics in fields outside of computers and electronics. Despite the frequently hyped promise of multi-Gigahertz and multi-core CPUs, these impressive chips have rarely translated into substantial progress in medicine, power, propulsion, and other essential areas.

One need only consider the many tragic deaths from cancer, the current rising energy prices, and the seeming wars over dwindling supplies of inexpensive oil and natural gas that plague the world today. Steve Jobs and his team at Apple have created many impressive gadgets such as the iPhone and iPad, but they were unable to exploit their computer expertise to defeat cancer. Is there a better way? Can we harness the unused power of today’s computers to solve these pressing problems?

The enormous power of today’s computers is useless without concepts, mathematics, and algorithms that use this power to solve real problems.

There has been impressive progress in some areas including video compression such as the H.264 and related standards used by YouTube, Skype and many other tools, audio compression such as MP3, image compression such as the widely used JPEG standard, computer generated images for movies, television, and computer games, the Global Positioning System or GPS that tells people where they are, and even speech recognition which is slowly finding some practical use despite many difficulties.

There is currently a fad to develop and implement recommendation engines such as Netflix’s Cinematch system to recommend purchases to customers using advanced statistical methods.

At best recommendation engines can increase sales by only a tiny amount, a few percent, and can never solve critical, trillion dollar market size, problems such as cancer, the diseases of old age, and energy shortages. Improved video compression in the form of video conferencing tools such as Skype may well help solve the current energy crisis. Video conferencing, however, cannot substitute for most energy needs. Other advances are needed. As Steve Jobs death shows, many major problems have not been solved at all.

This article discusses some ways that math and computers might be used to develop a cure for cancer. It is a follow-on article to the previous article Can Mathematics Cure Cancer?

This article discusses ways that mathematics might be used to identify and selectively destroy cancer cells. It discusses a specific approach and algorithm, “The Bathtub Mechanism,” that may be able to selectively kill cells with an abnormal number of chromosomes, a common feature of many cancers, and presents a sketch of some ways this algorithm might be implemented using cellular and molecular building blocks that may be known to present day biology, avoiding the need to construct nanorobots, something still far in the future.

All About Cancer

The current prevailing theory of cancer is the oncogene or “cancer gene” theory. This is viewed as a proven fact by many molecular biologists. Cancer is now said to be hundreds, even thousands of different diseases. While a medical doctor or pathologist may identify something as “breast cancer” or “skin cancer” or a similar general category, at a molecular and genetic level, “breast cancer” is actually many different diseases.

It is thought that cancer is caused by the accumulation of many mutations of many different oncogenes and tumor suppressor genes that control complex networks of proteins that direct the growth, functioning, and differentiation of cells. In biology, differentiation refers to the process by which cells “differentiate” during growth into various specialized types of cells such as neurons in the brain, blood cells, and skin cells with different specific properties and functions.

One type of breast cancer may have genes A,B,C, and D mutated while another has genes W, X, Y, and Z mutated. Not only this, but the cancers are thought to be continually mutating and evolving in the body, developing immunity to chemotherapy drugs for example. Thus, there does not seem to be a common molecular target that an anti-cancer drug can target in the way that penicillin or other antibiotics can kill a wide range of different bacteria, for example.

There may be system level features of cancer cells that identify them. Traditional chemotherapy drugs were designed to kill dividing cells on the theory that cancer cells divide rapidly. However, healthy cells divide also and traditional chemotherapy has very limited benefits for most cancers. Only surgical removal of a tumor before it spreads — becomes metastatic in cancer jargon — appears to be able to cure cancer using the common sense definition of “cure”. While targeting cell division largely does not work, targeting other system level characteristics of cancer may work.

It may be possible, with great difficulty, to produce a small system of interacting drugs that perform a mathematical or logical calculation in the cell and selectively kill cancer cells or probable cancer cells while sparing normal cells. It is here that mathematics may be of use. To achieve success in the near future, the simpler the mathematics the better. Even engineering a single molecule such as genetically engineered insulin for diabetics is a daunting task at present. So a system of even a few molecules would be a substantial and difficult undertaking.

The Selective Destruction of Cells with Abnormal Numbers of Chromosomes

(NOTE: This section largely repeats the section with the same title in the previous article Can Mathematics Cure Cancer? If you are familiar with the concept, you may skip this section and jump to the following section which discusses how to implement the bathtub mechanism.)

One common characteristic of many cancers is an abnormal number of chromosomes, known as aneuploidy. This is often an excess number of chromosomes. A normal healthy human cell has forty-six (46) chromosomes. Cancer cells often have more than forty-six chromosomes. This was discovered long before the modern genetic era.

One historical theory, now out of favor, is that the abnormal number of chromosomes causes cancer. This theory is usually credited to the German biologist Theodor Boveri. The most prominent modern advocate of the role of aneuploidy and chromosomes in cancer is the extremely controversial researcher Peter Duesberg who has published some articles on his theories in cancer research journals and a popular article in Scientific American in 2007 (“Chromosomal Chaos and Cancer”, Scientific American, May, 2007).

A number of other researchers such as Angelika Amon at MIT have been investigating the role of chromosomes and aneuploidy in cancer in recent years; references are given in the previous article Can Mathematics Cure Cancer?.

The abnormal number of chromosomes or the other chromosomal anomalies often seen in a wide range of cancers may be a system-level characteristic of cancer that could be targeted despite the extreme variation in gene-level mutations (part-level characteristics of cancer).

Even though there are over one-million research papers on cancer, it is difficult to get a clear picture of the role of aneuploidy in cancer. Most modern cancer research is conducted within the framework of the oncogene theory and an implicit assumption that the way to cure or treat cancer is to target either a protein generated by a cancer gene or the gene directly.

Chromosomal anomalies, both abnormal numbers of chromosomes and the rearrangements of chromosomes that are common in many cancers, are usually discussed as an aside to the putative cancer genes. This translocation of chromosome X mutated the key cancer gene ABC, or the duplication of chromosome X resulted in two copies of the key cancer gene ABC.

It could be that killing cancer cells with the wrong number of chromosomes would have no effect on the disease. It would simply result in a cancer with the correct number of chromosomes in the surviving cancer cells. It could slow the disease if the abnormal number of chromosomes is related to the malignancy of the cancer cells. In the best case, it might cure the disease, if the abnormal number of chromosomes is either the cause of cancer, essential to the malignant nature of the cancer cells, or simply always associated with malginancy for some other reason.

It may be possible to kill cells with an abnormal number of chromosomes using a system of five molecules: a harmless precursor A, a source catalyst S, a cell killer B, a drain catalyst D, and a neutralized cell killer C that the cell can safely digest or excrete.

The source catalyst S is inactive until it bonds to a numerical or quantitative feature on the chromosomes such as the telomeres at the ends of the chromosomes or the centromeres at the center. It becomes an active catalyst S* when it bonds to the chromosomes. Then the activated catalyst S* catalyzes the conversion of a harmless precursor A into a cell killer B. The activated catalyst S* has a maximum throughput. If the concentration of the precusor A is high enough in the cells, the catalyst S* will add the cell killer to the cell at a rate proportional to the number of chromosomes in the cell.

The cell killer B is relatively harmless in low concentrations. It needs to build up to a high level to kill the cell. So far, this will happen in all cells. However, if there is a drain catalyst D that bonds to a numerical feature in the cell that is the same in both normal cells and abnormal cells (cancer cells) and becomes an active drain catalyst D* that removes the cell killer B by converting it to the neutralized cell killer C, then the concentration of B can be engineered to rise to lethal levels only in cells with too many chromosomes.
A ==>S*==> B
B ==>D*==> C

This system of drugs is like a bathtub with several running faucets, one for each chromosome, and a single drain. If there are too many faucets, chromosomes, the water level, the concentration of the cell killer B, will rise and overflow the bathtub. If there are the right number, forty-six, or too few, less than forty-six, faucets, the drain can remove the water being added and the water level never rises. The water level remains almost zero; the concentration of the cell killer B is way too low to harm the cell.

One can kill cells with too few chromosomes (less than forty-six) by swapping the roles of the drain and the source. The drain catalyst bonds to the chromosomes. The source catalyst bonds to the constant numerical feature of the cells. Thus, if there are too few chromosomes, there are not enough activated drains to remove the cell killer B produced by the source catalyst. The bathtub has one big faucet and many small drains, one for each chromosome.

In principle, one could eliminate all cells with either too many or too few chromosomes by first treating the patient with a system of drugs that kills cells with too many chromosomes and then a system of drugs that kills cells with too few chromosomes. Cancer cells are frequently reported to have too many chromosomes, but sometimes too few is also reported.

A computational system of this type would now (2011) be easy to implement using mechanical components like the gears and springs used in traditional mechanical clocks, vacuum tubes and other traditional analog electronics components, or an integrated circuit. The problem is that as simple as such a computational system is, it is extremely challenging to implement using our current ability to engineer proteins and molecular biological systems in the cell.

How to Implement the Bathtub Mechanism

The bathtub mechanism requires two features in the cell: a numerical or quantitative feature that is proportional to the number of chromosomes and a feature that is constant in all cells, both normal and cancerous. It is sometimes reported that cancer cells have abnormal numbers of antigens on the membranes of the cells. Hence, the bathtub mechanism may not require a feature that varies with the number of chromosomes, but this article is about targeting abnormal numbers of chromosomes rather than antigens.

Some obvious features that probably vary with the number of chromosomes are the telomeres at the end of the chromosomes and the centromeres at the center of the chromosomes. These are both involved in cell division. There should be concern that the source or drain catalyst binding to the telomere or centromere may interfere with cell division. The bathtub mechanism must kill all the cancer cells and spare most or all of the healthy cells. It may be possible to use the telomeres or centromeres, but it could be impossible.

A more promising feature may be some of the non-coding sequences in the chromosome DNA, the so-called “junk DNA.” It is currently thought that the vast majority of DNA in the chromosome has no function. On theoretical grounds, the author finds this implausible as do many. However, the genes that appear to code for the proteins in the body seem to comprise only a few percent of the DNA in the chromosomes. The rest seems to do nothing. Sequences of non-coding DNA are used in DNA profiling, for example. Depending on the actual function of the junk DNA, if any, it may be possible to safely bind a source or drain catalyst to non-coding sequences that vary in quantity with the number of chromosomes.

There are many molecular structures in the chromosomes and associated with the chromosomes. It seems probable, although not certain, that one can find a numerical or quantitative feature that varies with the number of chromosomes that could be used. A more serious problem with the bathtub mechanism is the constant feature that is the same in both healthy cells and cancer cells, especially since cancer cells are thought to be constantly mutating and changing. This may be a show-stopper.

Since the cancer cells may be mutating, it may be impossible to find a constant feature in the cancer cells. The feature could disappear entirely or change in size or number. There is at least one possible way to add such a feature artificially to the cells, both healthy and malignant.

Bacteriophage P2 using Transmission Electron Microscope

A bacteriophage is a kind of virus that attaches to the exterior membrane of a cell and injects its genetic material into the cell. The bacteriophage’s genetic material then takes over the machinery of the cell and directs it to make more bacteriophages. The bacteriophage consists of a protein sheath that looks something like a science fiction bug (see pictures) with several arms that grab the surface of the cell and a polygonal chamber that carries the genetic material.

3D Model of T4 Bacteriophage

In principle, one could modify the genetic material of the bacteriophage to create cells (the commonly used E. Coli bacteria, for example) that make not the virus, but the protein sheath with a payload of other proteins. These pseudo-bacteriophages would inject their protein payloads into cells instead of the genetic material of the naturally occurring bacteriophage. They would not be infectious like a normal virus.

If, and this is a big if, one could modify the protein sheath so it would only inject the protein payload into a cell without an inhibitor protein I that is part of the payload, one could inject a payload that contained an artificial constant feature F and the inhibitors I into the cell. Once the new feature that the drain or source catalysts would bind to was added to the cell, the pseudo-bacteriophages would stop injecting payloads into the cell because it now also contained the inhibitors. Thus, a constant number of features could be added to each cell, both healthy and cancerous.

Math and Computers

This is a simplified sketch of the bathtub mechanism, a basic concept. Many technical details and difficulties have been omitted to present the idea. While it might be possible to research and develop the bathtub mechanism entirely empirically at a laboratory bench through massive trial and error, it should be possible to substantially accelerate the development process by simulating the molecular mechanisms using today’s powerful computers. In practice, it would probably require careful tuning of the chemical reaction rates in the cell to produce the desired selective destruction of cells with abnormal numbers of chromosomes or other features associated with cancer.

One should not expect the computer simulations to be perfect. They would probably be far from perfect at first. Rather, the use of mathematical models and computers should be part of an iterative process in which the models and simulations are continuously compared to laboratory bench experiments and improved. The basic concept may also need to be modified iteratively as new data is collected. This has been the usual process in most genuine breakthroughs.

Conclusion

It may be possible to cure or effectively treat cancer with a system of smart drugs that perform a simple mathematical or logical calculation to selectively destroy cancer cells or probable cancer cells while sparing normal healthy cells. These systems of smart drugs may be able to identify system level features of cancer cells independent of the confusing plethora of cancer genes and tumor suppressor genes. The bathtub mechanism discussed in this article is one possible example of such a system of smart drugs. Mathematics and computers can enable or greatly accelerate the development of such systems of smart drugs.

The author suggests that cancer researchers, business leaders, and policy makers should direct a significant amount of time and resources to the investigation of such systems of smart drugs. This should be a diversified effort not focusing on any one particular approach such as the bathtub mechanism. While there should be some redundancy, there is probably no point in having dozens of competing research groups all trying the same basic approach as seems to be the case with the current attempts to apply differential equations to modeling the growth and spread of cancer, the major current example of applying mathematics to cancer research and treatment. A more diverse effort that is willing and able to question more assumptions is more likely to succeed based on the history of scientific research and technological development.

The successful application of mathematics and computers to cancer and biology requires a professional working relationship based on mutual respect between experts in several fields: computers, mathematics, physics, and traditional biology. The recent appearance of extremely powerful computers presages a sea change in biology and many other fields where computers and mathematics play a much more important role than in the past. Computer experts, mathematicians, and physicists need to respect the hard earned experience of traditional biologists. There is no way the bathtub mechanism could be implemented successfully, if possible, without the expertise of molecular biologists, cell biologists, organic chemists, and others familiar with the detailed structure and function of the chromosomes and cells in the human body. The same can be said of other possible systems of smart drugs and algorithms that may be able to selectively kill cancer cells.

So too, biologists need to respect the expertise of computer experts, mathematicians, and physicists. Successful mathematical modeling is usually a tedious, time consuming process taking months or years, typically longer than many quick few week biology experiments. Even a Nobel Prize in molecular biology or other impressive credentials does not make one an expert in mathematical modeling or other techniques that will be needed to apply mathematics and computers successfully to cancer and other problems. Management level issues such as technical feasibility, scope, difficulty, and complex technical issues will arise in a collaboration between biologists and mathematicians. These will need to be discussed freely in an adult manner to succeed.

There are many pressing problems in the world today like cancer. As current headlines attest, we are doing a poor job solving many of these problems. For the most part, the enormous power of today’s computers has not been applied successfully to these problems. In some cases, there has been no attempt. In other cases, the favored approaches have failed despite decades of effort and genuinely new or simply unpopular ideas should be tried. The War on Cancer is probably an example of the latter case.

Steve Jobs will be remembered for entertaining gadgets like the iPad, the iPhone, and the Macintosh. What an accomplishment it would be if these gadgets went on to successfully solve major problems like the cancer that felled their creator.

© 2011 John F. McGowan
About the Author
John F. McGowan, Ph.D. solves problems by developing complex algorithms that embody advanced mathematical and logical concepts, including video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.
 
 

Credits

http://commons.wikimedia.org/wiki/File:Bacteriophage_P2.jpg

English: Bacteriophage P2 using Transmission Electron Microscope
Author: Mostafa Fatehi
This file is licensed under the Creative Commons Attribution 3.0 Unported license.

http://commons.wikimedia.org/wiki/File:T4_rendered.jpg
An artist’s rendering of a T4 bacteriophage.
Source Self-modeled in Blender.
Author: Mysid
This file is in the public domain

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The Mayfield case is probably the most famous case of an incorrect fingerprint identification. The Mayfield case is an example of a “cold hit” in which a huge biometric database was searched for a possible match to an unknown fingerprint taken from a crime scene. Unlike suspects with plausible links to the crime, there was nothing specific to connect Mayfield to the crime other than the database search match.

There are subtle and serious mathematical and statistical problems with cold hits, which occur with both DNA profiling and fingerprint identification. This article explores in detail the mathematics and statistics of the cold hit problem.

The cold hit problem is closely related to a well-known problem in probability and statistics known as the birthday problem. Imagine a room full of people: Bob, Frank, Mary, Estelle, and others. Each person has a birthday: May 1, December 13, March 11, July 17, and so on.

Not knowing the birthdays of the people in the room, what is the probability that at least two people in the room have the same birthday? How many people need to be in the room for there to be an even (50/50) chance that at least two people in the room have the same birthday?

A naive and incorrect answer would be to reason as follows. There are three-hundred and sixty-five (365) days in the year. The probability that two people have the same birthday is 1/365. Therefore, the probability that at least one pair of people in a room with N people have the same birthday is about N/365. Thus the room needs about 183 people for an even chance of a match. The actual answer is twenty-three (23) people, much smaller than 183!

Let us consider the problem in detail. First, what is the probability that Bob and Frank have the same birthday? There is a 1/365 chance that Bob was born on January 1. There is a 1/365 chance that Frank was born on January 1. Thus, there is a 1/(365*365) chance that both Bob and Frank were born on January 1. There are, however, three hundred and sixty-five days in the year, so the probability that Bob and Frank were born on the same day is 365/(365*365) or 1/365.

We need to find the probability that at least one pair of people in the room (Bob and Frank, Bob and Mary, Bob and Estelle, Frank and Mary, Frank and Estelle, Mary and Estelle, and all other possible distinct pairs) have the same birthday. If there are N people in the room, there will be distinct possible pairs of people. Each pair will have a probability of 1/365 of having the same birthday.

The probability that at least one pair of people have the same birthday is:

P = 1.0 - (Probability that the Pair Does Not Have the Same Birthday)^(Number of Distinct Pairs of People)

which is

P = 1.0 - (Number of Distinct Pairs of People)(Probility that the Pair Does Not Have the Same Birthday)

or 

P = 1.0 - (1.0 - 1/365)^(N(N-1)/2)

It turns out that P is 0.50048, almost exactly even, for N = 23. The number of distinct pairs of people in the room is proportional to the square of the number of people in the room , not the number of people in the room (N). Hence, it takes far fewer people in the room than one would naively expect for there to be an even chance that at least two people in the room have the same birthday.

Probability At Least Two People in Room Have Same Birthday

The plot of the probability of at least two people in a room having the same birthday was generated using the two Octave scripts below: birthday.m and plot_bday.m.

Octave is a free open-source numerical programming environment that is mostly compatible with MATLAB.

birthday.m

function [p] = birthday(n, m, bTrace)
% p = birthday(n [, m, bTrace])
% probability that at least one pair of members of set of N have same birthday (M days in year)
% n  number of people
% m  number of "days" in year (default value = 365)
% bTrace flag to trace operation of function (default value = false)
%
% (C) 2011 John F. McGowan
% E-Mail: jmcgowan11@earthlink.net
% 

if nargin < 2
	m = 365;
	bTrace = false;
end

if nargin < 3
	bTrace = false;
end

p = 0.0;

p_no_pair = 1.0; % probability no pair of people in the sample have the same birthday

% loop over pairs of people in the sample (room full of people)
% brute force
% for i = 1:n
	% for j = i+1:n
	% p_pair = m*(1/m)*(1/m); % probability i and j have same birthday
	% p_no_pair = p_no_pair*(1.0 - p_pair);
	% end
% end

% fast
number_pairs = n * (n-1)/2;
p_pair = m*(1/m)*(1/m);
p_no = 1.0 - p_pair;
if bTrace
	printf("number_pairs: %d  p_pair: %f p_no: %f\n", number_pairs, p_pair, p_no);
	fflush(stdout);
end % if

p_no_pair = p_no_pair*power( p_no, number_pairs);

p = 1.0 - p_no_pair;

end % function

plot_bday.m

% plot probability of at least two people having the same birthday
% in a room full of N people
%
% (C) 2011 John F. McGowan, Ph.D.
% E-Mail: jmcgowan11@earthlink.net
%

p = zeros(1,100);

for i=1:100
	if mod(i, 10) == 0
		printf("processing %d people in the room\n", i);
	end
	p(i) = birthday(i);
end

printf("displaying graph");
fflush(stdout);

figure(1);
plot(p);
title('Probability At Least Two People Have Same Birthday');
ylabel('P');
xlabel('Number of People in Room');

printf("writing plot to file prob_bday.jpg");
fflush(stdout);

print('prob_bday.jpg');

What does the birthday problem have to do with fingerprint identification, DNA profiling, or other forms of biometric identification? Replace the people in the room with fingerprints or other biometric identifiers (DNA profiles, iris images, faces,...) in a database.

Replace the three-hundred and sixty-five distinct birthdays with thousands, millions or more distinct biometric identification codes derived from the fingerprint, DNA profile, iris, or other form of identification. The pairs of people with the same birthday become pairs of people with the same fingerprint or other biometric identifier: the actual criminal who commits a crime and at least one other innocent person.

What happens if a fingerprint database has 100 million people and the chance of two people having the same fingerprint (we are referring to the same partial prints such as a thumb print lifted from a crime scene) is only one in a trillion ().

Astonishingly, the probability of at least two people in the database having the same fingerprint is almost one (1.0). This is because there are (100,000,000)(99,999,999)/2 possible pairs of people in the database — about five quadrillion (1,000 trillion) possible pairs. Even though the probability of any two people having the same fingerprint is extremely low, at least one misidentification occurring somewhere in the system is almost certain (probability 1.0).

The FBI fingerprint database contains about 200 million people, accumulated since the 1920s, and the probability of two people having identical or indistinguishable partial fingerprints (or even all ten fingerprints) is unknown.

DNA profiles are currently claimed to have a probability of two people having the same profile of about one in ten trillion. With cold hits, with a search of a large database of DNA profiles such as are currently being collected, it is actually likely that there will be incorrect matches somewhere in the system.

Brandon Mayfield probably fell victim, in part, to the counter-intuitive statistics of the birthday problem. As the size of biometric databases collected by governments, law enforcement agencies, intelligence agencies, and private companies grows, the cold hit problem will grow — as the square of the number of entries in the databases.

If everyone, all of the nearly seven billion people on Earth, was in the databases, one could produce a list of all possible suspects based on fingerprint or other biometric identification alone. This could easily be hundreds or thousands or even more people.

How does one handle possible suspects who lack an adequate alibi and could have flown to a crime? How many of those possible suspects will have some tenuous seven degrees of separation connection to the crime? Brandon Mayfield was a Muslim American who had represented an alleged Islamic terrorist in a child custody case: a tenuous but possible connection to the terrorists responsible for the Madrid train bombings. This is the crux of the cold hit problem.

© 2011 John F. McGowan

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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If you’ve always wanted to start your own technical blog, perhaps about a mathematical or scientific topic, but never got around to do it or failed to attract a following, read on.

My book about technical blogging is now available in Beta from The Pragmatic Bookshelf:
http://pragprog.com/book/actb/technical-blogging

It’s the kind of book that will teach you everything you need to realistically know to succeed at blogging. In it I provide a complete road map that can be applied whether you blog about math, science, technology, programming, or just about any other professional area.

Check it out and let me know what you think. If you’re just hearing about this book for the first time now, be sure to read the free introduction and excerpts to get an idea of how practical and useful this book is.

Happy technical and scientific blogging!

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The Spanish police recovered a bag containing detonating devices which had a latent fingerprint that the Spanish shared with the United States Federal Bureau of Investigation (FBI).

The FBI apparently ran a check on the fingerprint using the FBI’s Automatic Fingerprint Identification System (AFIS). AFIS uses a pattern recognition algorithm to generate an ordered list of possibly matching fingerprints.

One of the fingerprints in this list matched Brandon Mayfield, a Muslim American attorney from the Portland, Oregon region. Expert latent fingerprint examiners from the FBI proceeded to positively identify the fingerprint from the Madrid train bombing as belonging to Brandon Mayfield; at least, this is what the FBI claimed at the time.

Mayfield was arrested as a material witness in the bombing and a great deal of information about him seems to have been leaked to the press. Meanwhile, the Spanish police matched the fingerprint to an Algerian man whom they arrested. The Spanish police directly challenged the FBI identification of Mayfield, leading to his eventual release.

Mayfield later successfully sued the FBI for his treatment.

A Single Fingerprint

Fingerprint identification has been in widespread use in the United States since the 1920′s where popular culture has, until recently, held that fingerprints are unique. According to some reports, people have even been executed based solely on a fingerprint identification.

This is striking since most human and automatic pattern recognition abilities and algorithms have significant false positive and false negative error rates. The author had the experience in 2002, prior to the Mayfield case, of trying to locate scientific studies confirming the accuracy of latent fingerprint identification without success. In fact, there have been a number of cases prior to the Brandon Mayfield case in which fingerprint identification was shown to have been wrong.

The suspect had an airtight alibi. DNA tests contradicted the fingerprint identification and cleared the suspect. These cases of incorrect fingerprint identification have always been blamed on fraud or error by the human latent fingerprint examiners rather than a case of identical fingerprints.

Some facts about fingerprints. Identical twins usually, perhaps always, do not have the same fingerprints. This means a fingerprint test can discriminate between the otherwise identical twins. Dramatic demonstrations of this remarkable fact helped convince juries in the 1920′s to accept fingerprint identification. However, these demonstrations do not constitute rigorous scientific statistical studies of the accuracy of latent fingerprint identification.

A small minority of people do not have fingerprints. Some people have fingers with very shallow ridges which, in practice, makes fingerprint identification more difficult. Contrary to some claims, fingerprints can be altered by scarring and wear and tear. Automatic fingerprint recognition algorithms have had substantial problems with people who work with their hands.

Discussions of the accuracy of fingerprint identification often confuse the accuracy for comparisons of all ten prints, all ten fingers, and the rates for a single or a few prints. For example, automatic fingerprint recognition algorithms were very accurate with all ten fingerprints in 2002 but much less accurate for a single print such as the forefinger or thumb. It is difficult to get all ten prints in the real world.

Latent fingerprint identification is performed by human examiners. There are automatic fingerprint recognition programs such as AFIS but these are probably not as accurate as human beings. This is not unusual. In general, human pattern recognition abilities are significantly better than automatic methods based on mathematical, statistical, or scientific methods: artificial intelligence, pattern recognition, machine learning, and other synonyms.

This is something to keep in mind when scientists, attorneys, or others denigrate eyewitness testimony. Nonetheless, human pattern recognition abilities are imperfect. There are false positive and false negative rates. Eyewitnesses do misidentify people and objects. Human fingerprint examiners almost certainly have non-zero false positive and false negative rates.

In the wake of the Mayfield case, an FBI Laboratory review committee evaluated the scientific basis of friction ridge examination (fingerprint identification) and recommended scientific research including a study of the accuracy of latent fingerprint examiners (!).

The National Research Council (NRC) also identified the need for evaluations of fingerprint examination decisions in a study in 2009. The FBI recently published a report on such a study in the Proceeding of the National Academy of Sciences (Accuracy and reliability of forensic latent fingerprint decisions, PNAS, April 25, 2011). This study found a 0.1% false positive rate and a 7.5% false negative rate.

It is worth considering this for a moment. Fingerprint identification is in widespread use in the United States. People are routinely convicted or cleared of crimes based solely or in part on fingerprint identification. Fingerprints have long been portrayed and perceived as unique.

Fingerprint identification is usually perceived as a highly scientific form of identification. Yet basic scientific studies of the accuracy of the technique appear to have been lacking until recently. This lack has only become apparent recently as unfavorable comparisons to the seemingly rigorous statistical basis of DNA profiling (formerly known as DNA fingerprinting) have been made as well as the extensive publicity received by the Mayfield case, much higher than previous misidentifications which lacked the post 9/11 terrorism angle.

The uniqueness of fingerprints seems to be one of those things that “everyone knows” that has a remarkably weak basis in fact. Indeed, this seems to happen from time to time in supposedly fact-based scientific and engineering fields. A significant number of scientific and technological breakthroughs have occurred when someone went back and questioned the underlying evidence or data behind something “everyone knew.”

In most crimes, only one or a few partial fingerprints are recovered, such as the thumb and forefinger used to hold an object, e.g. the bag in the Madrid train bombing. There are over six billion people on Earth. Suppose that one in a million people have the same or essentially the same partial prints; an examiner cannot tell the difference. This means that, in fact, there would be about six thousand (6,000) possible matches including the guilty party.

With automobiles, trains, and especially air travel, it is probable that a substantial proportion of these six thousand suspects live within traveling time of the crime and lack an alibi. There was a small possibility that Brandon Mayfield traveled secretly from Portland, Oregon in the United States to Madrid, Spain to participate in the bombings. Unlikely, but certainly possible. Even a very small false positive rate raises a reasonable doubt.

Especially since the Mayfield case, there has been more questioning of the scientific basis of fingerprint identification both by authorities such as the FBI and the National Research Council as well as in popular culture. The TV show Numb3rs, discussed in the previous post The Magical Mathematics of Numb3rs, features an episode, probably inspired by the Mayfield case, in which a man is wrongly convicted due to an error in fingerprint identification.

In their book The Numbers Behind NUMB3RS, mathematicians Gary Lorden and Keith Devlin have a chapter questioning some of the mathematical basis of fingerprint identification. DNA profiling is seemingly based on detailed rigorous scientific studies of the frequency of the various genetic markers used in the DNA tests.

Comparable studies seem to be lacking where fingerprints are concerned, hence the studies of fingerprint identification that the FBI is now performing and publishing. The comparison between DNA profiling and fingerprinting has led to questions about the accuracy of fingerprint identification.

It may also be the case that questions about the scientific basis of fingerprints may be a way of marketing DNA profiling as a more “scientific” and reliable replacement for now “old fashioned” (“legacy” in the parlance of the software industry — usually meaning it works and the market is saturated so we need to sell a new replacement technology) fingerprint identification.

The uncritical acceptance of fingerprint identification for over eighty years, without apparently performing adequate rigorous studies of the accuracy, illustrates the enormous hypnotic power of mathematics and science in our culture. The popular image of mathematics and science is that they give exact, black and white answers.

“Scientific” tests give reliable yes/no answers. The Madrid bomber was Brandon Mayfield. The bomber was not Mayfield. There are no false positive or false negative error rates. Two plus two is four, not 3.999 plus or minus 0.012. Yet, this is very rarely the case in the real world. In fact, one should almost always demand to know the error rates of numbers and be suspicious of numbers quoted without error rates or other qualifications.

Suggested Reading/References

Simon Cole, Suspect Identities: A History of Fingerprinting and Criminal Identification, Harvard University Press, Cambridge, Massachusetts, 2001

Accuracy and reliability of forensic latent fingerprint decisions

Bradford T. Ulery(a), R. Austin Hicklin (a), JoAnn Buscaglia(b),1, and Maria Antonia Roberts(c)

(a) Noblis, 3150 Fairview Park Drive, Falls Church, VA 22042;
(b) Counterterrorism and Forensic Science Research Unit, Federal Bureau of Investigation
Laboratory Division, 2501 Investigation Parkway, Quantico, VA 22135;
(c) Latent Print Support Unit, Federal Bureau of Investigation Laboratory
Division, 2501 Investigation Parkway, Quantico, VA 22135

Edited by Stephen E. Fienberg, Carnegie Mellon University, Pittsburgh, PA, and approved March 31, 2011 (received for review December 16, 2010)

Proceedings of the National Academy of Sciences (PNAS)
April 25, 2011

The Numbers Behind Numb3rs: Solving Crime with Mathematics
Keith Devlin, Ph.D. and Gary Lorden, Ph.D.
Penguin Books, New York, 2007

Credits

The fingerprint image is from Wikimedia Commons and is in the public domain. Fingerprint Image at Wikimedia Commons

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

No related posts.

Actor David Krumholtz (Charlie Eppes in Numb3rs)

Real life mathematicians including Caltech statistics professor Gary Lorden consulted for the show. Gary Lorden is listed in the credits for each episode as “Math Consultant”. In later seasons, Stephen Wolfram’s Wolfram Research also provided consulting advice to the series. There are scenes with references to Mathematica, Wolfram’s flagship product, and close-up shots of Wolfram’s magnum opus A New Kind of Science on Charlie Eppes desk. Wolfram also has a Caltech connection; he received his Ph.D. in Physics from Caltech in 1979. Gary Lorden and fellow mathematician Keith Devlin published a popular book The Numbers Behind Numb3rs: Solving Crime with Mathematics in 2007: “A companion to the hit CBS crime series Numb3rs presents the fascinating ways mathematics is used to fight real-life crime.” The shadowy National Security Agency (NSA), probably the largest patron of mathematics and mathematicians in the United States and the world, makes several appearances in Numb3rs.

Millikan Library at Caltech (appears often in Numb3rs)

CBS, Texas Instruments (a leading maker of digital signal processor or DSP chips), and the National Council of Teachers of Mathematics (NCTM) developed the “We All Use Math Every Day” initiative, sometimes abbreviated WAUMED, to inspire students to achieve more in math by showing how the subject is relevant to their lives:

Using the hit CBS television show, Numb3rs, the “We All Use Math Every Day” initiative provides free classroom activities online at cbs.com/Numb3rs that help students understand how the math they are learning in the classroom applies to the real world. The activities explore the math derived from the concepts used to solve cases in the FBI crime-solving show.

The show used the “We All Use Math Every Day” tagline in the opening introduction to the show in the first and second seasons.

How Realistic is Numb3rs?

Although explicitly fiction, in many respects Numb3rs paints a picture of mathematics and science that is similar to ostensibly factual popular science such as Scientific American articles, PBS/Nova video programs, Congressional testimony by leading scientists, and informal discussions at fundraising cocktail parties — unless the scientists or mathematicians are in the rare and unusual position of having to explain an obvious failure to a lay audience:

“Well, Senator, as everyone knows, science is a risky enterprise. Eighty to ninety percent of our research projects fail. Surely your staff briefed you on that; the proposal committee mentioned this clearly in italics in footnote 83 in Appendix C of the Proposal for the New Manhattan Project that Will Produce Miraculous Results by the Next Election.”

In several important respects, Numb3rs is very unrealistic. It is also the case that many people ranging from Silicon Valley executives trying to use mathematical methods for their businesses — for example, the current fad trying to use machine learning for recommendation engines in social networking and search businesses — to practicing scientists and engineers who one might think would know better often have expectations similar to what is portrayed in Numb3rs. These misconceptions almost certainly contributed in a major way to the multi-trillion dollar global housing bubble and crash through the widespread use of invalid mathematical models for the valuation of mortgage-backed securities. With business and political leaders seemingly floundering in the current economic difficulties, these misconceptions may wreak even greater havoc.

Before launching into a critique of Numb3rs, it is important to realize that there have been many successes in applied mathematics and mathematical software including impressive advances in video compression such as used by YouTube and Skype, audio compression such as the widely used MP3 standard, still image compression such as JPEG images, computer generated imagery in movies and video games, the Global Positioning System (GPS) that tells people where they are, and even speech recognition which is finally finding some practical use. Modern computers are extremely powerful, comparable to the supercomputers of previous decades; this power is mostly unused because we do not have the mathematics to put this power to practical use. Today’s powerful computers and new mathematics probably can solve or help solve many pressing problems, even trillion dollar problems such as energy shortages or major diseases such as cancer. Success in solving problems with mathematics requires realistic expectations, realistic planning, and adequate time and resources.

In The Numbers Behind Numb3rs (page 208), the mathematicians Keith Devlin and Gary Lorden, a full professor at Caltech, write:

One thing that is completely unrealistic is the time frame. In a fast-paced 41-minute episode, Charlie has to help his brother solve the case in one or two “television days.” In real life, the use of mathematics in crime detection is a long and slow process. (A similar observation is equally true for the use of laboratory-based criminal forensics as depicted in television series such as the hugely popular CSI franchise.)

Also unrealistic is that one mathematician would be familiar with so wide a range of mathematical and scientific techniques as Charlie. He is, of course, a television superhero — but that’s what makes him watchable. Observing a real mathematician in action would be no more exciting than watching a real FBI agent at work! (All that sitting in cars waiting for someone to exit a building, all those hours sifting through records or staring at computer screens… boring.)

It’s also true that Charlie seems able to gather masses of data in a remarkably short time. In real-life applications of mathematics, getting hold of the required data, and putting it into the right form for the computer to digest, can involve weeks or months of labor-intensive effort. And often the data one would need are simple not available.

(Emphasis Added)

In their discussion of the episode “Manhunt” (Airdate: May 13, 2005,The Numbers Behind Numb3rs, page. 78), in which Charlie Eppes uses Bayesian statistics to predict the actions and location of an escaped killer, Devlin and Lorden also write:

As is often the case with dramatic portrayals of mathematics or science at work, the length of time available to Charlie to produce his ranking of the reported sightings [of the escaped killer] is significantly shortened, but the idea of using the mathematically based technique of Bayesian analysis is sound.

(Emphasis Added)

Real-life mathematics and mathematical software development involves much more time, much more trial and error, much more debugging, and much more risk than depicted in Numb3rs. Scientists often claim an eighty to ninety percent failure rate in their research projects, frequently when explaining an obvious failure to disappointed graduate students, donors, policy makers, and others who expected more. Charlie Eppes almost never fails! There is historical evidence that the failure rate in genuine “breakthroughs” is higher, quite possibly ninety-nine percent or worse. Some of the mathematics that Charlie whips up in a few “television days” in the show would actually qualify as breakthroughs in real-life, notably some mathematics and algorithms for artificial intelligence and pattern recognition (see below). Historically, genuine breakthroughs have usually involved at least five years of effort when successful. To give a recent example, Grigoriy Perelman’s proof of the Poincare Conjecture took him at least seven years. There appear to have been about one hundred failed published attempts to prove the conjecture by mathematicians prior to Perelman’s success.

The reality is, in fact, worse than Devlin and Lorden concede in their book. Numb3rs has several episodes that portray artificial intelligence (AI), pattern recognition, machine learning, and similar technologies far superior to reality at the time the show aired (2005-2010) or even today (2011). In one episode, Charlie whips up an image/object recognition algorithm in a few hours to enable the NSA to track a yellow truck carrying a contraband missile guidance system through their satellite images of LA to a terrorist (“Finders Keepers,” Original Air Date: January 12, 2007). Similarly, remarkably effective face recognition algorithms play a role in several episodes. Many of Charlie’s AI and pattern recognition algorithms and the other pattern recognition technology shown in Numb3rs works much better than the real algorithms and math.

The Specter of 9/11

Numb3rs is a fast-paced entertaining show with sexy, idealistic, highly effective heroes and heroines. Although it is sometimes critical of security agencies like the CIA and powerful institutions like pharmaceutical companies, in many respects it is Hollywood product placement for the post 9/11 world of massive, expensive high-tech surveillance and security measures both overseas and at home — in which mathematics plays an important and growing role. It reminds one of President Eisenhower’s speeches during the 1950′s:

The worst to be feared and the best to be expected can be simply stated.

The worst is atomic war.

The best would be this: a life of perpetual fear and tension; a burden of arms draining the wealth and the labor of all peoples; a wasting of strength that defies the American system or the Soviet system or any system to achieve true abundance and happiness for the peoples of this earth.

Every gun that is made, every warship launched, every rocket fired signifies, in the final sense, a theft from those who hunger and are not fed, those who are cold and are not clothed. This world in arms is not spending money alone.

It is spending the sweat of its laborers, the genius of its scientists, the hopes of its children.

The cost of one modern heavy bomber is this: a modern brick school in more than 30 cities.

It is two electric power plants, each serving a town of 60,000 population.

It is two fine, fully equipped hospitals. It is some 50 miles of concrete highway.

We pay for a single fighter plane with a half million bushels of wheat.

We pay for a single destroyer with new homes that could have housed more than 8,000 people.

This, I repeat, is the best way of life to be found on the road the world has been taking.

This is not a way of life at all, in any true sense. Under the cloud of threatening war, it is humanity hanging from a cross of iron.

Chance for Peace (April 16, 1953)
President Dwight David Eisenhower (shortly after the death of Joseph Stalin)

President Dwight D. Eisenhower

Eisenhower and his advisers were no shrinking violets. They were well aware the world can be a nasty, dangerous place. They presided over a massive military buildup and controversial covert operations in Guatemala, Iran, Vietnam, and other countries. By the end of his Presidency Eisenhower and his advisers found that it was never enough. Even thousands of nuclear weapons, ships, tanks, spies, and what we now know was a massive lead over the Soviet Union was not enough to satisfy what he famously labeled the “military industrial complex” in his Farewell Address. Eisenhower found himself attacked by Republicans and Democrats alike for not spending even more money on guns and preparations for war!

Osama Bin Laden: The Trillion Dollar Man

Following the reported death of Osama Bin Laden, Andrea Millen Rich, writing in the libertarian Reason magazine, estimated the direct cost of getting Bin Laden at $1.1 trillion. Tim Fernholz and Jim Tankersley, writing in The Atlantic estimated the total cost at $3 trillion over fifteen years. Sam Stein of the Huffington Post, citing a Congressional Research Service report of March 29, 2011, put the cost at at least 1.283 trillion.

According to the United States Centers for Disease Control, the leading causes of death in the United States in the calendar year 2007 were:

Number of deaths for leading causes of death

* Heart disease: 616,067
* Cancer: 562,875
* Stroke (cerebrovascular diseases): 135,952
* Chronic lower respiratory diseases: 127,924
* Accidents (unintentional injuries): 123,706
* Alzheimer’s disease: 74,632
* Diabetes: 71,382
* Influenza and Pneumonia: 52,717
* Nephritis, nephrotic syndrome, and nephrosis: 46,448
* Septicemia: 34,828

All homicides, of which terrorist attacks are a small fraction even in 2001, do not make the top ten. In 2007, the Centers for Disease Control listed all homicides as the 15th leading cause of death:

All homicides

* Number of deaths: 18,361
* Deaths per 100,000 population: 6.1
* Cause of death rank: 15

It is worth noting that the US invasion of Iraq in 2003 resulted in a dramatic drop in Iraqi oil production, undoubtedly contributing substantially to the large increases in oil and energy prices in the last decade. So too the US invasion of Afghanistan in 2001 seems to have scuttled any chance of constructing a pipeline for natural gas from Turkmenistan to the Indian Ocean, also undoubtedly contributing to high energy prices.

It is difficult to improve on President Eisenhower’s words today. Bayesian statistical analyses that predict terrorist attacks, even if they work, don’t make up for dwindling supplies of inexpensive oil and natural gas. They don’t feed people. They don’t cure diseases like cancer or prevent heart attacks. How much more could have been and could still be accomplished if today’s powerful computers and new mathematics were applied to substantive problems such as energy, food, and health instead of the will-o’-the-wisp of perfect security or the pseudo-scientific financial engineering that helped cause the current Great Recession? Mathematicians, scientists, business leaders, and policy makers can do better than we have done.

Conclusion

Numb3rs is a fun, entertaining show. If you are a mathematician, it will probably make you feel great about your profession unless you are in the unfortunate position of dealing with an employer, client, investor, or funding agency that expects you to do what Charlie Eppes does in every episode of Numb3rs. Some of the math and science in Numb3rs is completely realistic. Some of the math is somewhat exaggerated. Some of the math is pure science fiction even though it generally seems very real and believable. As Devlin and Lorden admit in their book, the time frame is, in most cases, completely unrealistic.

The world is presently confronted with serious and worsening problems, possibly due to a dwindling supply of inexpensive oil and natural gas. The political and economic leadership of the world appears paralyzed and unable to deal with the problems, bickering over debt ceilings and other silliness. We do have vast unused resources in the computational power of hundreds of millions of computers and other devices. With the proper mathematics and creative thinking, we may be able to harness this power to resolve many of the current problems, without waiting for paralyzed governments or blundering Too Big To Fail banks to act wisely.

Most mathematics and mathematical software has been developed by individuals and small teams working over periods of several months to several years with total costs of tens of thousands to a few million dollars per project. Success requires realistic expectations about the size, scope, difficulty level, and risks of developing and implementing mathematics and mathematical software. In these difficult times, mathematicians and scientists must gain support for realistic projects that can find real solutions to our pressing problems, and honestly reject the fantasy elements of Numb3rs.

Suggested Reading/References
The Numbers Behind Numb3rs: Solving Crime with Mathematics
Keith Devlin, Ph.D. and Gary Lorden, Ph.D.
Penguin Books, New York, 2007

The Shadow Factory: The Ultra-Secret NSA from 9/11 to the Eavesdropping on America
James Bamford
Doubleday, New York, 2008

The Cost of Iraq, Afghanistan, and Other Global War on Terror Operations Since 9/11
Amy Belasco, Congressional Research Service, Washington, D.C, March 29, 2011

Credits
The picture of actor David Krumholtz at the Serenity Premiere is from Wikimedia Commons, licensed under the Creative Commons Attribution 2.0 Generic license.

This image was originally posted to Flickr by RavenU at http://flickr.com/photos/36330825119@N01/45967991. It was reviewed on 10:00, 30 April 2007 (UTC) by the FlickreviewR robot and confirmed to be licensed under the terms of the cc-by-2.0.

Millikan Library at Caltech Image from Wikimedia Commons.

Official Portrait of President Dwight D. Eisenhower, May 29, 1959

(from Wikipedia) This image is a work of an employee of the Executive Office of the President of the United States, taken or made during the course of the person’s official duties. As a work of the U.S. federal government, the image is in the public domain.

The image of Usama Bin Laden (Osama Bin Laden) is from the FBI Ten Most Wanted Poster.

© 2011 John F. McGowan

About the Author

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing video compression and speech recognition technologies. He has extensive experience developing software in C, C++, Visual Basic, Mathematica, MATLAB, and many other programming languages. He is probably best known for his AVI Overview, an Internet FAQ (Frequently Asked Questions) on the Microsoft AVI (Audio Video Interleave) file format. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech). He can be reached at jmcgowan11@earthlink.net.

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