Looking for a small, light, quality camera? – II

Jun 30, 2011 2 comments

Just over a year ago I wrote a blogpost Looking for a small, light, quality camera?, which collected together the size and weights of lightweight quality cameras. The focus was on mirrorless interchangeable lens cameras.

Things have moved on since then: manufacturers have released new cameras and lenses. I’ve repeated the exercise with today’s crop of cameras. If camera size and weight is an important consideration for you, then these tables may be of some help. Note I make no attempt to compare the quality or ergonomics of any of these cameras, there are plenty of camera review sites that do that.

Mirrorless cameras

The following table gives the sizes of various mirrorless cameras, and their weights with various lenses. The pancake lens is the manufacturer’s wide angle prime lens. The zoom lens is the manufacture’s nearest equivalent to a 28-84mm (full frame) lens. The superzoom is the manufacture’s nearest equivalent to a 28-300mm (full frame) lens. All weights include batteries.

Model Weight Weight
(pancake)
Weight
(zoom)
Weight
(superzoom)
Dimensions LCD Dots
Olympus E-P3 369g 440g 481g 659g 122 x 70 x 34 mm 614,000
Olympus E-PL3 313g 384g 425g 603g 110 x 64 x 37 mm 460,000
Olympus E-PM1 254g 325g 366g 544g 110 x 64 x 34 mm 460,000
Panasonic GF2 310g 410g 475g 770g 113 x 68 x 33 mm 460,000
Panasonic GF3 264g 364g 429g 724g 108 x 67 x 32 mm 460,000
Sony NEX-C3 225g 299g 439g 749g 110 × 60 × 33 mm 921,600
Samsung NX100 282g 367g 480g N/A 120 × 71× 35 mm 614,000

The overall lightest camera/lens combination, at 299g is the Sony NEX-C3 with 16mm lens. This is thanks to the low weight of the Sony camera. With zoom lenses, the inherent size/weight advantage of micro4/3 comes into play: the Olympus EPM-3 provides the lightest camera with standard zoom combination and superzoom combinations (at 366g and 544g respectively). (Of course slightly lighter combinations could be obtained by mixing Panasonic and Olympus lenses and bodies.)

Compact cameras

For comparison, here are the weights and dimensions of some of the higher end compact cameras:

Model Weight Dimensions LCD Dots
Canon PowerShot G12 [28-140mm] 491g 112 x 76 x 48 mm 461,000
Canon PowerShot S95 [28-105mm] 195g 100 x 58 x 30 mm 461,000
Leica X1 [35mm] 306g 124 x 60 x 32 mm 230,000
Nikon Coolpix P7000 [28-200mm] 310g 114 x 77 x 45 mm 921,000
Panasonic Lumix DMC-LX5 [24-90mm] 271g 110 x 65 x 43 mm 460,000

The lightest mirrorless camerea with a standard zoom lens (Olympus E-PM1 with 28-84mm equivalent lens) weighs 425g and compares reasonably favorably, weightwise, with some of these compact cameras. The Sony NEX-C3 with 16mm lens (24mm equivalent) weighs 299g and also compares reasonably favorably weightwise.

DSLRs

For further comparison, here are the sizes and weight of some of the smaller and lighter DSLRs:

Model Weight Weight
(pancake)
Weight
(zoom)
Weight
(superzoom)
Dimensions LCD Dots
Canon EOS 1100D 495g N/A 695g 1090g 130 x 100 x 78 mm 230,000
Nikon D60 505g N/A 770g 1065g 126 x 94 x 64 mm 230,000

Lenses

For reference, here are the weights and sizes of some of the lenses available for mirrorless interchangeable lens cameras:

Model Weight Dimensions Filter diameter
Olympus 9-18mm[18-36mm] f4.0-4.6 155g 57 x 50 mm 52mm
Olympus 14-42mm[28-84mm] f3.5-5.6 150g 62 x 44 mm 40.5mm
Olympus 14-42mm[28-84mm] f3.5-5.6 II 112g 57 x 50 mm 37mm
Olympus 14-150mm[28-300mm] f4.0-5.6 290g 64 x 83 mm 58mm
Olympus 40-150mm[80-300mm] f4.0-5.6R 190g 64 x 83 mm 58mm
Olympus 75-300mm[150-600mm] f4.8-6.7 430g 70 x 116 mm 58mm
Olympus 12mm[24mm] f2.0 130g 56 x 43 mm 46 mm
Olympus 17mm[34mm] f2.8 71g 57 x 22 mm 37 mm
Olympus 45mm[24mm] f1.8 116g 56 x 46 mm 37 mm
Panasonic 7-14mm[14-28mm] f4.0 300g 70 x 83 mm -
Panasonic 14-42mm[28-84mm] f3.5-5.6 165g 61 x 64 mm 52 mm
Panasonic 14-45mm[28-84mm] f3.5-5.6 195g 60 x 60 mm 52 mm
Panasonic 14-140mm[28-280mm] f4.0-5.8 460g 70 x 84 mm 62 mm
Panasonic 45-200mm[90-400mm] f4.0-5.6 380g 70 x 100 mm 52 mm
Panasonic 100-300mm[200-600mm] f4.0-5.6 520g 74 x 126 mm 67 mm
Panasonic 8mm[16mm] f3.5 165g 61 x 52 mm N/A
Panasonic 20mm[40mm] f1.7 100g 63 x 26 mm 46 mm
Panasonic 25mm[50mm] f1.4 200g 63 x 55 mm 46 mm
Panasonic 45mm[90mm] f2.8 225g 63 x 63 mm 46 mm
Samsung 18-55mm[28-85mm] F3.5-5.6 198g 63 x 65 mm 58 mm
Samsung 20mm[40mm] f2.8 89g 62 x 25 mm 43 mm
Samsung 30mm[46mm] f2 85g 62 x 22 mm 43 mm
Samsung 50-200mm[77-308mm] F4-5.6 417g 70 x 101 mm 52 mm
Sony 16mm[24mm] f2.8 74g 62 x 23 mm 49 mm
Sony 18-55[27-83mm] f2.5-5.6 214g 62 x 60 mm 49 mm
Sony 18-200[27-300mm] f3.5-6.3 524g 76 x 99 mm 67 mm

Time for real-realpolitik

Feb 27, 2011 4 comments

Realpolitik is politics defined as

politics based on practical and material considerations, rather than based on ideological or ethical considerations.

Realpolitik has been practiced, to varying degrees, by the West since before the term was coined. It has been predominant since the end of the World War II. Realpolitik has resulted in various nefarious practices, including (but not limited to): seeking out and supporting “strong men” as bulwarks against Communism (and more recently as protectors of Western interests), using aid programmes to prop up or undermine regimes, and deposing democratically elected leaders.

As Brian Crozier says in his 1965 book “South-East Asia in Turmoil”:

In its search for local ‘strong men’, the United States used anti-Communism as a decisive credential… administrative capacity, and even honesty, were less important than unwavering anti-Communism plus willingness to become a partner of the United States. Nor did it matter if a strong man was also an oppressive despot, as Syngmnan Rhee of Korea and Ngo Dinh Diem of Vietnam turned out to be, so long as he kept repeating that he was anti communist.

Or put more succinctly in the apocryphal statement attributed to Franklin D. Roosevelt:

Somoza may be a son of a bitch, but he’s our son of a bitch.

(Some claim that this was actually a self-serving statement fabricated by Somoza himself.)

The trouble with realpolitik is that it just does not work. Sure, it can produce short term advantages, but those advantages don’t last, and the long term consequences can be dire. Even a cursory examination of history shows this.

The 1955 partition of Vietnam and the support of President Ngo Dinh Diem – that didn’t work out too well did it? The 1953 overthrow of democratically elected Dr. Mohammed Mossadegh of Iran and his replacement with Shah Mohammad Reza Pahlavi – that didn’t work out too well either. The subsequent US backing of Saddam Hussein in the 1980-1988 Iran-Iraq war, how did that go? The appeasement of Nazi Germany during the 1930s was arguably a form of realpolitik, since an ethical stance against Germany was replaced by a pragmatic one directly seeking peace. That could hardly be called a success. And recent events in the Middle East and North Africa show that the “stability” achieved by supporting despots and kleptocrats is short-lived.

It’s not actually very surprising that realpolitik doesn’t work. If we replace the phrase “practical and material considerations”, with “short term considerations” (since “practical considerations” are effectively things we believe are achievable in the short term) and if we replace “ideological or ethical considerations” with “long term considerations” (since our ideology and ethics is based on how we believe things should be, along with a recognition that those ideals may take some time to achieve), we then find the definition of realpolitik is changed from:

Realpolitik is politics based on practical and material considerations, rather than based on ideological or ethical considerations.

to

Realpolitik is politics base on short term considerations, rather than long term considerations.

When put like that, it’s not surprising realpolitik doesn’t work.

Advocates and practitioners of realpolitik often accuse those who advocate more ethically based policies of being naive. I argue that it is the practitioners of realpolitik who are simplistic in their belief that long term interests can be achieved by narrowly pursuing short term gains. Real-realists know that to achieve your long term goal you often need to make short term compromises.

It’s time for real-realpolitik. It’s time to recognise that we cannot achieve our long term goals by abandoning them as naive and unrealistic. It’s time to recognise that our foreign policy should have a much larger ethical component, not only because it is “the right thing to do”, but also because it is what works and it is what is in our own long-term self-interest.

NASA names best and worst sci-fi movies of all time – or not?

Jan 5, 2011 9 comments

A front page article in this week’s Sunday Times stated that NASA had named 2012 the most absurd science-fiction film of all time. In a follow-up article on page nine, the Sunday Times stated that NASA had recently held a conference at the JPL (Jet Propulsion Laboratory) to highlight the good and bad scientific practices of Hollywood. According to the Sunday Times, NASA and the Science and Entertainment Exchange produced the following lists:

Worst sci-fi movies:
1. 2012 (2009)
2. The Core (2003)
3. Armageddon (1998)
4. Volcano (1997)
5. Chain Reaction (1996)
6. The 6th Day (2000)
7. What the #$*! Do We (K)now!? (2004)

Most realistic films:
1. Gattaca (1997)
2. Contact (1997)
3. Metropolis (1927)
4. The Day The Earth Stood Still (1951)
5. Woman in the Moon (1929)
6. The Thing from Another World (1951)
7. Jurassic Park (1993)

I was curious about the reasons for some of the choices, so I thought I’d look this up on the NASA website. But I couldn’t find any mention of these films on NASA web site, the JPL web site, or the Science and Entertainment Exchange web site. There were plenty of articles reporting NASA’s choice of films, but none of them I looked at had links back to a primary source at NASA. Many had links to articles in other newspapers. And curiously, none of the articles were in American newspapers.

Then, today (Jan 4, 2011), The Science and Entertainment Exchange issued a statement on its blog:

The article in the London Sunday Times on January 2, 2011 “To Absurdity and Beyond: NASA damns flaws in sci-fi films” incorrectly attributed a top-ten worst sci-fi films list to the Science & Entertainment Exchange. We were not involved in creating the list.

This raises some interesting questions:

  • Did NASA, in fact, publish a list of best and worst sci-fi films?
  • Where did the lists published by the Sunday Times originate?
  • Did NASA have a conference about sci-fi films?
  • Why did so many newspapers publish a story about this without bothering to check the primary source?

The newspapers and journals that re-hashed this story include:

Update, 6th Jan 2011

Dave Kellam, at eightface.com, also wrote a blog post on this subject: NASA and bad science movies. Kellam emailed Donald Yeomans, the manager of NASA’s Near-Earth Object Program, who was quoted in John Harlow’s Sunday Times article. In his reply Yeomans stated:

There is no list and there was no meeting to put together such a list. NASA would never put together a list of “worst sci-fi films.” We are not movie critics.

According to Kellam, Yeomans stated that he was interviewed by a British journalist, but was subject to misquotes and manufactured quotes.

Browne Report and graduate tax

Oct 14, 2010 1 comment

The Browne report, Securing a Sustainable Future for Higher Education, has just been published.

The almost universal view among commentators is that Lord Browne has conclusively argued the case against a graduate tax.

In this blogpost I argue that the Browne Report has done no such thing. Rather it has set up a straw man version of a graduate tax, and shown this straw man to be unworkable. It is perfectly possible to set up a version of a graduate tax that avoids the Browne Report’s objections. (Note that in this post I am not arguing for or against a graduate tax, merely that the Browne Report has not made a convincing argument against a graduate tax.)

[A straw man argument is an informal fallacy based on misrepresentation of an opponent’s position. To “attack a straw man” is to create the illusion of having refuted a proposition by substituting a superficially similar yet unequivalent proposition (the “straw man”), and refuting it, without ever having actually refuted the original position.]

What some of the commentators say

The Times in its editorial of Wednesday 13 October 2010 says:

This series of proposals demolishes the case for centralised control of universities and they demolish the case for a graduate tax…

Vince Cable in his statement to the House of Commons says:

But Browne identifies serious problems with a ‘pure’ graduate tax. The proposal is unworkable; does not produce sufficient revenue to finance higher education until 30 years from now; weakens university independence; and is unfair to British graduates as opposed to graduates living overseas.

The Independent in its editorial of Monday the 11th October says:

Vince Cable has, mercifully, surrendered the notion of a graduate tax, paid retrospectively, which would have been a bureaucratic nightmare to administer.

The Guardian in its editorial of Wednesday 13 October 2010 says:

Browne’s plan, with its strong support for the less well-off, is also better than a full-on graduate tax.

Have any of these people actually read the Browne Report?

What the Browne Report actually says

The arguments against a graduate tax are in Chapter 7, Other Proposals we have Considered. I’ve repeated them below for reference:

Issues Graduate Tax Student Finance Plan
1 Cost of learning No upfront costs No upfront costs
2 Cost of living Will require support through loans – this means that graduates have to pay the additional tax as well as make loan payments Graduates make a single set of payments to cover the costs of learning and living provided upfront by Government
3 Payment terms Linked to income, no fixed mortgage style payments, payments continue indefinitely Linked to income, no fixed mortgage-style payments , payments stop when costs of learning and living are paid back – or 30 years – whichever is earlier
4 Protection for graduates on low incomes Graduates start paying when they cross the income tax threshold – £6,475 per year Graduates pay nothing until they earn £21,000 per year
5 Costs for graduates Uncapped, could be several multiples of the cost of the degree Maximum payment is equal to the charge for the degree. Majority of graduates will pay less
6 Funding to universities Tax revenues take time to build up – for first 25 years, model depends on Government filling that gap; after that, depends on Government enforcing a ring fence around graduate tax revenues Direct funding relationship between student and university
7 Burden on Government Additional £3bn a year until 2015-16 at least; additional spending continues until ca. 2041-42 No additional spending; continuing requirement to provide student finance
8 Relationship between students and universities Student experience does not matter to university for raising funding University depends on student willingness to pay for significant proportion of funding, so providing a high quality student experience is critical
9 Incentives for institution No variability in funding, so no incentives to focus on quality, access or student experience Sustaining income – or raising it – depends on improving quality, access and student experience

Virtually all of these objections assume a particular (and not very sensible) version of a graduate tax.

Lets look at each in turn:

1) Cost of learning – same for both options.
2) Cost of living – true, but the level of the graduate tax could be set to take this difference into account.
3) Payment terms – there is nothing inherent in a graduate tax that says it must continue indefinitely. It be formulated to stop when the tuition costs have been met and/or after 30 years. (Indeed, the Browne Report even mentioned it looked at the NUS proposal of a graduate tax with a capped maximum contribution.)
4) Protection for graduates on low incomes. Again there is nothing inherent in a graduate tax that says it must kick in as soon as the graduate starts to pay income tax – it could also kick in at an earnings threshold of £21,000.
6) Funding to universities – clearly a graduate tax cannot immediately replace the current system, it needs to be phased in over a number of years.
7) Burden on Government. As for (6) a transition period is clearly required. The Browne Report does not make it clear why the burden to government during that transition period would be any more than the current system.
8) Relationship between students and universities. The Brown Report’s statements are true if the graduate tax is paid to central government. However the report has not even considered the possibility of the graduate tax being paid directly to the university. Such a tie would mean the university would have a long term interest in the success of the student.
9) Incentives for institution. Similarly to (8), if the graduate tax is paid directly to the university, then the university has a long term interest in the student. This would provide incentives not only for good teaching, but also good careers advice and even for continuing education and refresher courses. It would also provide disincentives for admitting students just to make up the numbers, since if these students failed to graduate, the university would not gain any income from them.

Of all the arguments posed against a graduate tax, the only serious one is about how to manage and fund the transition from the current system to a graduate tax. The Browne Report doesn’t consider a transition period – it looks at an immediate changeover to a graduate tax with the funding gap filled by government: of course this is going to look unfavourable, it’s the most expensive (to government) way of funding the transition.

Conclusion

The Browne Report’s case against a graduate tax is based on a fallacious straw-man argument.

There may be legitimate arguments against a graduate tax, but the Browne Report has not made them.

Ancient stones from Olduvai

Oct 11, 2010 Leave a comment

I have been listening to the BBC’s A History of the World in 100 Objects narrated by the director of the British Museum, Neil MacGregor.

Inspired by the programme, I visited the British Museum. I was particularly interested in seeing two objects from the cradle of humanity, the Olduvai Gorge in northern Tanzania: the Olduvai stone chopping tool (made 1.8 million years ago) and the Olduvai handaxe (made 1.2 – 1.4 million years ago).


The two objects were, of course, behind glass. But the museum had made available alternative, equally old, stone tools that could be handled under the supervision of an expert. It was a great pleasure to be able to handle a stone chopping tool that was over 1.5 million years old. And the children who were playing with the stone tools before I got my turn seemed to be equally enjoying themselves.

MacGregor describes the significance of the finds at Olduvai:

Leakey’s discoveries in the warm earth of the Rift Valley did more than push humans back in time, they made it clear that all of us descend from those African ancestors, that every one of us is part of a huge African diaspora – we all have Africa in our DNA and all our culture began in the same place.

Listening to the news on the radio, it’s easy to imagine the world is divided into rival tribes and competing civilisations. So it’s good, it’s essential in fact, to be reminded that the idea of our common humanity is not just an enlightenment dream, but a genetic and a cultural reality. It’s something we’ll see again and again in this series.

Singer, songwriter and anthropologist Johnny Clegg puts it more poetically in his song “Scatterlings of Africa”:

Ancient bones from Olduvai
Echoes of the very first cry:
“Who made me, here and why?”
Beneath this copper sun
My very first beginnings
Beneath the copper sky
Lie deeply buried
In the dust of Olduvai

And we are scatterlings of Africa
Both you and I
We are on the road to Phelamanga
Beneath a copper sky
And we are scatterlings of Africa
On a journey to the stars
Far below we leave forever
Dreams of what we were
Hawu beke Mama-ye! Mama-ye!
In the beginning
Beneath the copper sky
Ancient bones
In the dust of Olduvai
Who made us, here, and why
Remember!

Contador – cheat or bad meat?

Oct 1, 2010 24 comments

Alberto Contador has tested positive for the banned substance clenbuterol. Contador claims that the positive test result was a result of food contamination and that he did not deliberately take the drug.

In this blogpost I do the pharmacokinetics calculation to see if Contador’s claim is feasible. My calculation is limited to this specific test result.

Here are some facts relevant to the case:

  • Contador tested positive for clenbuterol on the 21st July, with a urine concentration of 50 picograms per millilitre (see UCI press release)
  • A picogram is a very small amount, a millionth of a millionth of a gram.
  • The UCI has said further scientific investigation is required before any conclusion could be drawn.
  • Tests on July 19th and 20th showed no traces of the drug. After finding 50pg/ml on the 21st, Contador’s sample from the next day showed 20pg/ml and then only traces on the following two days.
  • In the European Union, it is illegal to use clenbuterol in animal feed.
  • Nevertheless some farmers use clenbuterol, since it increases the lean yield of livestock
  • There have been of cases of human clenbuterol poisoning from contaminated meat. These have occurred in both pork and beef and in various countries including Spain, France, Ireland, Mexico, and China. The European cases occurred in the 1990s. (see Tainted Meat: Clenbuterol use in the meat industry)
  • There have been no recent cases of human clenbuterol poisoning in the European Union.
  • In the European Union it is legal to use clenbuterol as a tocalytic (that is to surpress premature labour) in cattle.

So could Contador’s test result be due to contaminated food? We can do a calculation to find out. Given the concentration of clenbuterol in Contador’s urine we make an estimate of how clenbuterol much was ingested. We can then compare this estimate with the residue levels of clenbuterol in contaminated meat and the therapeutic dosages of clenbuterol and thus decide which is more likely: cheat or bad meat.

According to “Clenbuterol Residues in Bovine Feed and Meat” (see reference 1) clenbuterol levels in contaminated beef (in Mexico) have values in the range 0.1 to 2.3 micrograms of clenbuterol per kilogram of meat. So a 100g piece of steak could contain between 0.01 and 0.23 micrograms, that is between 10 and 230 nanograms, of clenbuterol. Contamination levels may of course differ in European meat.

The European Union Maximum Residue Limit (MRL) for clenbuterol is 0.1 microgram per kilogram for bovine muscle and 0.05 microgram per kilogram for bovine milk. So a 100g piece of steak could legally contain up to 0.01 micrograms, that is 10 nanograms, of clenbuterol.

Therapeutic dosages of clenbuterol are in the range of 20 to 80 micrograms per day (that is 20,000 to 80,000 nanograms).

According to my calculation (full workings at end of blogpost), Contador ingested approximately 540 nanograms of clenbuterol. This is slightly higher than expected if Contador ate contaminated meat, but much much less than if Contador had taken a therapeutic dose of clenbuterol. This dosage is much higher than could be obtained from meat that complied with EU regulations.

Conclusions

The amount of clenbuterol ingested by Contador (540 nanograms) is consistent with his assertion that he ate contaminated meat.

This amount is only 1/40th of the theraputic dose, and so seems unlikely to have resulted in any performance benefit.

On this occasion, I am inclined to believe Contador’s story.

Further comments

There have been suggestions in the press that Contador’s positive test is the result of an autologous blood transfusion (blood doping) – that is he re-infused so of his own blood that was collected earlier in the year – and that this blood contained clenbuterol. This is indeed possible, but autologous blood transfusions can be detected in a blood test, and, as winner of the Tour de France, Contador would undoubtably have been subjected to a blood test. My assumption is that if the UCI had detected blood doping, then they would have already released the test results. To clarify this matter, the UCI should make a formal statement and publish the results of any such blood tests.

Update – Sunday 3rd October 2010

There are currently three proposed explanations for Contador’s positive drug test:

  1. Contador deliberately took clenbuterol.
  2. Contador inadvertently took clenbuterol as a result of eating contaminated meat.
  3. Contador had a blood transfusion and the clenbuterol was in the tranfused blood.

The calculations in this blog post show that it is unlikely that Contador deliberately took clenbuterol, and that it is feasible that eating contaminated meat could have resulted in his test result (namely urine with a concentration of 50 picograms/millilitre).

There remains the question of the blood transfusion. Nowadays it is possible to detect if someone has had a blood transfusion, whether that transfusion is autologous (from one’s own blood) or homologous (from someone else’s blood). We know that cyclists are tested to see if they have had blood transfusions – Alexander Vinokourov was tested positive in 2007. I see four possibilities:

  1. Contador was tested for blood doping and the result was positive.
  2. Contador was tested for blood doping and the result was negative.
  3. Contador was tested for blood doping and the result was inconclusive.
  4. Contador was not tested for blood doping.

Contador himself has denied receiving blood transfusions

The UCI and the World Anti-Doping Agency have refused to comment on L’Équipe’s story that raises the possibility of blood doping.

In my view the UCI’s reticence on the blood doping issue is inexcusable. Both in fairness to Contador and for their own credibility they must release the results of any blood doping tests they have made on Contador.

References

  1. Clenbuterol Residues in Bovine Feed and Meat (Research Journal of Biological Sciences)
  2. Pharmacokinetics of plasma and urine clenbuterol in man, rat, and rabbit (PubMed)
  3. Tainted Meat: Clenbuterol use in the meat industry (Serendip)
  4. Press release – Adverse analytical finding for Alberto Contador (UCI press release)
  5. Contador maintains innocence (The Press Association)
  6. Maximum Residue Limits, Clenbuterol (European Agency for the Evaluation of Medicinal Products)
  7. Contador’s scientific expert De Boer details defense (Cycling News)

Calculation

From reference [2] we know that after ingesting clenbuterol reach a maximum level after 2.5 hours, remain at this level until 6 hours after ingestion. From this, and assuming that the rate of drug removal by the kidneys is proportional to its plasma concentration, we can model the concentration C of clenbuterol at time t (given in hours) in the urine as:

C(t) = \left\{ \begin{array}{l l l} mt  & \quad \mbox{0} \le {t} \le {2.5} \\ 2.5m  & \quad \mbox{2.5} \le {t} \le {6.0} \\ 2.5me^{-k(t-6)} & \quad \mbox{t} \ge {6.0} \end{array} \right.

The half life of clenbuterol in blood plasma is approximately 35 hours, so:

k = \frac{ln(2)}{35} = 0.01980

The cumulative urinary excretion is 20% of the dose, D after 72 hours, so:

\int_{0}^{72} C(t)\: \mathrm{d}t = 0.2D

that is

D = {5}\left(\int_{0}^{2.5} mt\: \mathrm{d}t + \int_{2.5}^{6} {2.5}m\: \mathrm{d}t +\int_{6}^{72} {2.5}me^{-k(t-6)}\: \mathrm{d}t\right)

\quad = {5}\left[{0.5}mt^{2}\right]_{0}^{2.5} + {5}\left[{2.5}mt\right]_{2.5}^{6} + {5}\left[\frac{-2.5m}{k}e^{-k(t-6)}\right]_{6}^{35}

so

D = 378.63m \quad  (equation 1)

Let’s assume Contador’s urine sample was taken 24 hours after ingestion of the clenbuterol and consists of the urine produced in the previous hour. Then, assuming a typical urine production rate of 50ml/hour, the quantity of clenbuterol excreted, per ml of sample, E is given by:

E = \frac{1}{50}\int_{23}^{24}C(t)\: \mathrm{d}t

\qquad = \frac{1}{50}\int_{23}^{24} {2.5}me^{-k(t-6)}\: \mathrm{d}t

\qquad = \left[\frac{-m}{20k}e^{-k(t-6)}\right]_{23}^{24}

so

E = 0.03536m \quad  (equation 2)

We know Contator’s test result – his urine sample contained 50 picograms per ml, this allows us to solve equation 2 for m and feed the result back into equation 1 to obtain the approximate dose of clenbuterol ingested by Contador. This gives an approximate dosage of 535,400 picograms, that is 540 nanograms (rounded).

Disclaimer: I’m a mathematician, not a medic or a pharmacologist. The above calculations have not been independently reviewed, so there may be errors in the model or the calculations. If you find any errors, please let me know and I will correct them.

Tony Blair pledges book proceeds to Royal British Legion

Aug 16, 2010 5 comments

Let me ask you one question,
Is your money that good?
Will it buy you forgiveness?
Do you think that it could?
I think you will find,
When your death takes its toll,
All the money you made
Will never buy back your soul.

Bob Dylan – Masters of War

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