Orac sent me a link to some more HIV denialist material, I assume under the assumption that since I’m already being peppered by insults from the denialist crowd, I might as well cover this now.<\/p>\n
\n To begin with, what were these people trying to do? They wanted to measure the As a mathematician, I was intrigued by the claim of John Maddox, editor of And they state in the same paragraph that virus producing cells can to ‘a good The antiretroviral agents used in this study, despite their differing Orac sent me a link to some more HIV denialist material, I assume under the assumption that since I’m already being peppered by insults from the denialist crowd, I might as well cover this now.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[27],"tags":[],"class_list":["post-155","post","type-post","status-publish","format-standard","hentry","category-hiv-denial"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-2v","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/155","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=155"}],"version-history":[{"count":1,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/155\/revisions"}],"predecessor-version":[{"id":3458,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/155\/revisions\/3458"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=155"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=155"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=155"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nWhat I’m looking at today is a paper by Mark Craddock called “HIV: Science by press conference”.<\/a> The paper is purportedly about how the AIDS research community, in cahoots with the media, are deceiving the public about the nature of results of AIDS research. In his words “One of the most disturbing aspects of what passes for AIDS research these days, is the separation between what researchers actually find, what they tell the press conference and what the media tells the public.”
\nAs an example of this, he discusses work by George Shaw, Xiping Wei, et al., which is one of the first papers to show viral dynamics of HIV in human lymph tissue after performing a human immunodeficiency virus (HIV) test<\/a>. Craddock’s criticism is allegedly based on the bad mathematics of Shaw & Wei. Below, I reproduce the section of Craddock’s paper where he explains the basis of his criticism. Since the paper is only online in bitmapped PDF form, I’ve transcribed it by hand; any transcription errors are entirely my fault, and I will correct them immediately if any are discovered. This comes from page 2 of Craddock’s paper; that’s page 13 of the PDF. I’ll interject a few brief comments that belong in-line, and hold the detailed discussion for after the quotation.<\/p>\n
\n rate at which both HIV and T cells are produced in infected people. The idea is
\n deceptively simple. You measure the viral load in a patient at a given time,
\n and then you pump them full of ‘antiviral’ drugs. The drugs reduce the amount
\n of virus present in the blood by some factor. (Claimed to be an implausible
\n 98.5% in these papers. Implausible because there is no possible way that viral
\n load can be measured as accurately as the figure of 98.5% suggests.)
\n (This is an interesting claim: that one cannot measure viral load to 2 significant figures. Because that’s all that’s mentioned here. The viral load after is 1.5% of the viral load before. One can determine a figure of 1.5% if load before and after are accurate to 2 significant digits. I find it astounding that Craddock is actually making such a ridiculous claim.)<\/em>
\n You then
\n wait till HIV magically mutates into ‘drug resistant’ strains, and wait till
\n the viral load returns to pre treatment levels.
\n (Another interesting tidbit; apparently, Professor Craddock doesn’t believe in mutation and evolution, at least where HIV is concerned.)<\/em>
\n This gives an estimate, through
\n some relatively simple mathematics, for the rate at which the virus replicates.
\n Both Ho and Shaw’s groups found that in the absence of viral clearance, the
\n total amount of virus in the body should double every two days. So suddenly,
\n the low levels of viral replication found over the past decade are thrown out
\n the window, and HIV is now the cause of a relentless battle, a battle that
\n takes place over a decade or more. The measurement of T cell production can be
\n made in much the same way.
\n (This is a rather obnoxious misrepresentation. The fundamental new discovery of the work being discussed is that it describes how HIV reproduces; in particular, they discovered that HIV replication occurs in the lymphatic system. Comparing the rates of viral replication found before this work in non-lymphatic tissue and the rates discussed in this work in lymphatic tissue is comparing apples and oranges.)<\/em>
\n So our question must be whether or not we should believe these results? After
\n all at the last big press conference, Ashley Haase’s group (Embretson et al.<\/em>,
\n Nature, March 25, 1993) found low levels of HIV RNA in the T cells of patients
\n studied (4 people, one of whom had no HIV proviral DNA at all) indicating ‘low
\n levels’ of viral replication. So what do we do when one press conference seems
\n to contradict the other? Clearly we have to examine both studies carefully.<\/p>\n
\n Nature, that the new results provide a new mathematical understanding of the
\n immune system. Unfortunately, my confidence in this claim was badly shaken when
\n it turned out that the very first page of the Shaw paper (Wei et al<\/em>., p 117,
\n Nature, Jan 12,, 1995) they make an appalling mathematical error. And in the
\n same paragraph, make an assumption which turns out, by their own admission to
\n have no basis in observation, and which they give no justification for.
\n (Take a look at that paragraph again. It’s going to be important later.)<\/em>
\n The
\n authors of Wei et al<\/em>. are attempting to give a mathematical formula for the
\n amount v<\/em> of free virus as time t<\/em>. They state that the virus is produced by
\n virus producing cells y<\/em>, at rate k<\/em>, and decays exponentially at rate u<\/em>.
\n These two statements are mutually contradictory but that is not a real problem.
\n If they change the word ‘decays’ to ‘is cleared’ then all is well.
\n (Because using “decays” instead of “clears” when you’re describing something that matches what’s called a “decay curve” clearly totally invalidates the mathematical statement.)<\/em>
\n This leads
\n to what is known as a differential equation for v<\/em> which may be solved easily.
\n (Craddock, letter to Nature unpublished), They state a formula for v<\/em> based on
\n their assumption, which unfortunately is completely wrong. Confidence that
\n anything good will come out of this paper plummets at this point. Their result
\n for v<\/em> is not only wrong, but it does not even look right.
\n (Because, you see, the solutions to differential equations are always intuitively obvious, and looking at a solution, one can easily tell if it looks right.)<\/em>
\n You do not have to
\n be a mathematician to realise that if the rate at which v<\/em> is produced depends
\n on ky(t)<\/em>, where y(t) is the number of virus producing cells at time t<\/em>, then
\n v<\/em> is going to depend upon ky(0)<\/em>, where y(0)<\/em> is the initial size of the
\n virus producing cell population. So one wonders how they manage to produce a
\n formula for v<\/em> which does not depend on ky(0) al all?<\/p>\n
\n approximation’ be assumed to decline exponentially. They then state a few lines
\n further down that they ‘have data only for the decline of free virus, and not
\n for virus producing cells’. If they have no data for virus producing cells,
\n then how can they possibly know that these cells decline exponentially? They
\n might do anything. That is the whole point of not having any data. You do not
\n know what is happening.
\n So, now we’ve seen the core of his criticism. Basically, he’s got two real problems with the math of the paper. First, he questions the validity of the way they fit the data to an exponential decline; and Second, he believes that their descriptive equation is invalid. (He has several reasons for claiming that it’s invalid, but they’re part of the same argument that the way the exponential curve was fitted to get an equation is incorrect.)
\n Before we look at what’s wrong with his criticism, it’s only fair to look at the passage from the Shaw paper that he’s criticizing. What follows is taken from
\n the Shaw paper, pages 2\/3 of the PDF, pages 117\/118 in the journal. Again, it is transcribed by hand, so any errors introduced are entirely my fault.
\n The overall kinetics of virus decline during the initial weeks of therapy with
\n all three agents corresponded to an exponential decay process (Figs 1 and 2a).<\/p>\n
\n mechanisms of action, have a similar overall biological effect in that they
\n block de novo<\/em> infection of cells. Thus the rate of of elimination of plasma
\n virus that we measured following the initiation of therapy is actually
\n determined by two factors: the clearance rate of plasma virus per se<\/em> and the
\n elimination (or suppression) rate of pre-existing, virus-producing cells. To a
\n good approximation, we can assume that virus-producing cells, decline
\n exponentially according to y(t) = y(0)e-at<\/sup><\/em>, where y(t)<\/em> denotes
\n the concentration of virus-producing cells at time t<\/em> after the initiation of
\n treatment and a<\/em> is the rate constant for the exponential decline. Similarly,
\n we assume that free virus v(t)<\/em> is generated by virus-producing cells at the
\n rate ky(t)<\/em> and declines exponentially with rate constant u<\/em>. Thus for the
\n overall decline of free virus, we obtain v(t) = v(0)[ue-at<\/sup> –
\n ae-ut<\/sup>]\/(u-a). The kinetics are largely determined by the slower of
\n the two decay processes. As we have data only for the decline of free virus,
\n and not for virus producing cells, we cannot determine which of the two decay
\n processes is rate-limiting. However, the half-life (t1\/2<\/sub><\/em>) of
\n neither process can exceed that of the two combined. With these considerations
\n in mind, we estimated the elimination rate of plasma virus and virus-producing
\n cells by three different methods: (1) first-order kinetic analysis of that
\n segment of the viral elimination curve corresponding to the most rapid decline
\n in plasma virus, generally somewhere between days 3 and 14; (2) fitting of a
\n simple exponential decay curve to all viral RNA determinations between day 0 and
\n the nadir or inflection point (Fig. 1); and (3) fitting of a compound decay
\n curve that takes into account the two separation processes of elimination of
\n free virus and virus-producing ceels, as described. Method (1) gives a
\n t1\/2<\/sub> of 1.9 +\/-0.9 days; method (2) gives a t1\/2<\/sub> of 3.0
\n +\/- 1.7 days; and method (3) gives a t1\/2<\/sub> of 2.0 +\/- 0.9 days for
\n the slower of the two decay processes and a very similar value, 1.5 +\/- 0.5
\n days for the faster one. These are averages (+\/- 1 s.d.) for all 22 patients.
\n Method (3) arguably provides the most complete assessment of the data, whereas
\n method (2) provides a simpler interpretation (but slightly slower estimate) for
\n virus decline because it fails to distinguish the initial delay in onset of
\n antiviral activity due to the drug accumulation phase, and the time required
\n for very recently infected cells to initiate virus expression, from the
\n subsequent phase of exponential virus decline. There were no significant
\n differences in the viral clearance rates in subjects treated with ABT-538,
\n L-735,524 or NVP, and there was also no correlation between the rate of virus
\n clearance from plasma and either baseline CD4+<\/sup> lymphocyte count or
\n baseline viral RNA level.
\n So, Craddock’s first criticism is of the use of exponential decay. He asserts that they “make an assumption which turns out, by their own admission to
\n have no basis in observation”. How does this stand up?
\n Very poorly. There are two good reasons why the exponential decay was used.
\n 1. Past observation has shown us that most infectious diseases respond to medications following an exponential decay; so we know that it’s a plausible pattern;
\n 2. Observing the data, it fits an exponential decay quite well.
\n So the data appears to follow an exponential pattern, and we have experience to show us that that is a likely outcome, and that the fit of the data to that kind of curve is not likely to be an artifact. This is what, in science, is known as “developing a hypothesis, and then testing it against the data”. Shaw et al. think that an exponential decay is a likely response; they take the data and analyze it, and the math shows a very clean and consistent fit. So we tentatively accept the assumption that the exponential curve is correct, either until more data contradicts it; or more data supports it to the point that we no longer consider it tentative.
\n What about Craddock’s second criticism? He claims that the equations that they fit to the data have several problems. The differential equation describing viral growth must include y(0), the initial<\/em> number of virus-producing cells.
\n Does that make sense? Well, no. We’re working in differential equations; that is, equations that measure rates of change<\/em>. In fact, it’s an elementary property of differential equations that they do not work in terms of raw values, in terms of rates of change<\/em>.
\n Think back to college calculus. What’s the integral of x2<\/sup>dx<\/em>? x3<\/sup>\/3 + c<\/strong><\/em>. Notice that “+c” there. What does it mean? Because the derivative – the differential equation – only measures the rate of change. The value of the rate of change for y = x3<\/sup>\/3<\/em> is not<\/em> dependent on the value of y<\/em> when x=0<\/em>.
\n (Note: the following paragraphs have been substantially rewritten for clarity. See the discussion in the first few comments to see how things were changed.)<\/em>
\n Remember, the original equation for describing the number of virus producing cells y(t) = y(0)e-at<\/sup><\/em>. And then using that equation, they generate the solution to<\/em> the differential<\/em> equation for decay rate v(t) = v(0)[ue-at<\/sup> – ae-ut<\/sup>]\/(u-a)<\/em>. This second equation is the important one, the one that is the focus of their attention. What does this second equation mean, and what is it used for?
\n The paper proposes a hypothesis: that the growth rate of the virus follows a pattern that can be modeled by an exponential function. It then goes on to test this hypothesis: they collected data about the decay rate in a number of infected patients, and see if the data matches the hypothesis. This equation is a template, whose behavior is determined by three variables. If the exponential curve fits the data correctly, two<\/em> of those variables should have essentially constant values: these are the variables that describe the growth rate of the virus, u<\/em> and a<\/em>. The third variable is v(0) – the initial population of the virus; this one varies by patient, because different patients have different initial viral loads; the load of an infected patient can vary by several orders of magnitude. (That’s not a statement about HIV; the viral population in different individuals infected by any<\/em> virus can vary by several orders of magnitude.)
\n So why does the equation include v(0)<\/em> (the initial population of virus), and not y(0)<\/em>, the original population of virus-producing cells? Because in the context of this equation, the way that it’s used, that leading value in the equation is basically a scaling factor: it’s the factor that describes the unique initial starting point of each individual patient. Since the experiment starts in an equilibrium state, it doesn’t matter<\/em> whether you use v(0) or y(0) – in the equilibrium state, the two are directly proportional. Choosing one of them will fix the specific curve that fits a particular patient in a different position – but the key properties of the curve – the rate of change – will be completely unchanged.
\n Craddock is supposed<\/em> to be a professor of mathematics, doing an unbiased analysis of the math of this paper. I don’t believe that anyone teaching college level math could have made a mistake like this by accident. This is deliberate: he’s practicing what I call obfuscatory mathematics<\/em>. That is, you want to slip something past people who aren’t particularly comfortable with math. So you talk fast, and use lots of mathematical words to obscure the fact that you’re saying something patently ridiculous; you count on the fact that the overwhelming majority of your readers will not notice the stupid lie hidden by your obfuscation.
\n His argument sounds good<\/em>: how can you calculate something that depends on the initial number of virus producing cells if you don’t include the initial number of virus producing cells? But when you look carefully: we’re creating a differential equation – an equation that describes the viral load in terms of its change over time. The number of virus producing cells is<\/em> important; but our differential equation is written in terms of the number of viruses. It includes the term v(0): the number of viruses at time 0. Since we’re measuring the change<\/em> in viral load, the important initial factor is the initial number of viruses<\/em>. The initial number of viruses is, of course, related to on the initial number of virus-producing cells: it’s directly proportional; but you don’t need y(0) to be an explicit part of the solution to the differential equation. It’s just a scaling factor which doesn’t change the result of the computation of viral replication speed<\/em>. Whether you use v(0), y(0), or some computed combination of the two doesn’t matter<\/em>. In the end, they’re effectively just constant scaling factors that have no effect<\/em> on the values of u<\/em> and a<\/em>, which are what will allow us to determine the rate of viral replication.
\n Craddock also gripes about the fact that there are actually two factors involved in the viral load, but they can only measure one. But just go back and read the text carefully: the explain how, mathematically, they can account for it. Once again, this is just how science works<\/em>: they’ve got a hypothesis; they’ve clearly stated their assumptions; and they’ve shown how the data fits their prediction. After the work is published, other people go back and try to reproduce it and refine it; and long term, either it stands up or it doesn’t. They’re very honest and open about the fact that they can only measure one of the two factors, and what assumptions they make for their model. In this paper, the very first paper to propose a model for HIV viral load in vivo<\/em>, their model produces quite respectable results; results which have continued to hold up well over time.
\n So, after a quick look, how do Craddock’s critiques hold up? Quite poorly. And as Craddock himself says, after his alleged discovery of an error on the first page of the Shaw paper: “Confidence that anything good will come out of this paper plummets at this point.” I couldn’t say it any better: after deliberate misrepresentations like what we see in the first two pages of Craddock, there’s really not much point in paying attention to the rest. Craddock is a just another true believer who’s willing to tell deliberate lies if they support his position.\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"