MIT LIBRARIES 3 9080 02245 1980 DUPL 28 14 M DEWEY 1 The Impact of novel Treatments on A6 Burden Disease: Insights from a Mathematical David L. Craft, in Model Lawrence M. Wein, Dennis Sloan Working Paper J. Number 4 56 January 31,2001 1 Alzheimer's Selkoe The Impact in of Novel Treatments on A/5 Burden Alzheimer's Disease: Insights from a Mathematical Model David L. Ciaft\ Lawrence M. Wein"^, Dennis J. Selkoe^* ^Operations Research Center, MIT. Cambridge, K4A 02139 '^ 3 Sloan School of Management, MIT, Cambridge, MA 02139 Center for Neurologic Diseases, Harvard Medical School, Brigham and Women's Hospital, Boston, 'Person to whom MA 02115 correspondence should be addressed: email: si'lkoe fend. bwh. harvard. eihi Abstract Motivated by recent therapeutic initiatives for Alzheimer's disease, mathematical model of the accumulation of amyloid /3-protein we developed a (A/i) in the brain. The model incorporates the production and clearance of kl5 monomers, and the elongation and fragmentation of polymers by monomer aggregation and Our break-off, respectively. analysis suggests that A/3 dynamics are dictated by a single tmitless measure re- ferred to as the polymerization ratio, which is the product of the production and elon- gation rates divided by the product of the clearance and fragmentation rates. Cerebral k(i burden attains a sustained growth if finite steady-state level this ratio is if this ratio greater than one. The is less than one, and undergoes highly nonlinear relationship be- tween the polymerization ratio and the steady-state A/3 burden implies that a modest reduction in the polymerization ratio (e.g., a a 7-secretase inhibitor) achieves a significant Our model 40% reduction (e.g., 18-fold) in the production rate by decrease in the A/3 burden. also predicts that after initiation or discontinuation of treatment, take months to reach a new steady-state suggest that the research A/3 burden. may Taken together, our findings comnumity should focus on developing agents that provide a modest reduction of the polymerization ratio while avoiding long-term nally, it toxicity. Fi- our model can be used to indirectly estimate several crucial parameters that are difficult to measure directly: the production rate, the fragmentation rate strength (i.e., percentage inhibition or enhancement) of treatment. and the Introduction The cerebral build-up of amyloid /^-protein (A/3) and its aggregation into oligomers and polymers are key components of the pathogenesis of Alzheimer's disease (AD) studies suggest that A/3 accumulation plays an important role in ^'^, AD '. Recent neurodegeneration and a variety of A(5 treatment strategies are being aggressively pursued, including the enhancement of A/? clearance via immunization with A/3 production by inhibiting 7-secretase, which precursor protein (APP) is ^'^^ and the reduction of A/3 one of the two enzymes that cuts the amyloid to release the A/3 fragment ^•^. The kinetics of A/3 production, aggregation (polymerization), disaggregation and clearance, and the effect of these novel treatments on the time-dependent variation of A/3 levels are not well understood. Although ^"^^ or plaque there exist several mathematical models that focus on either fibrillogenesis formation ^''^'^^ these studies neither consider the impact of treatment nor allow for a contiiniously renewable source of A/3 from APP molecules. Here we formulate and analyze a simple mathematical model that tracks the dynamics of A/3 production, polymer elongation, fragmentation and clearance during the course of treatment. Simple formulas and numerical results are presented that elucidate the impact of treatment on A/3 burden. Mathematical model In AD brain tissue, the level of extracellular (and perhaps intracellular) k[5 (particularly A/342, the tion. 42 amino-acid form of A/3) increases over time, which gives Some of A/342, rise to polymeriza- of these polymers cluster into light microscopically-visible particles consisting which can accumulate to create diffuse (amorphous) neuropathological lesion of AD. Further, platiues, the some polymers containing A/^42 apparent and/or initial A/340 t'^n eventually fold into long hlamentous assemblies called amyloid hbrils, and clumped masses of these are referred to as amyloid (senile) plaques. Our mathematical model idealizes this process in several iinjxjrtaiit ways. First, we only consider extracellular A/?, which be in and a dynamic equilibrium with intracellular A/i. A/?42, or do not also between soluble and insoluble AS, because there dynamic changes the current literature about the precise and preclinical We clinical is distinfi,uish the we restrict is we discuss how incorporate in the to understand number and size fibrillization or plaque formation model might be interpreted this in However, later in vivo. in terms of - and generalized to plaque formation. Our model an is infinite system of nonlinear differential equations that tracks the temporal evolution of the concentration of extracellular A/3 i-mers, for variant of the time in our attention to A/3 polymerization, and do not attempt to capture downstream processes of the paper between A/i4o in these species over the impact of treatment on total cerebral A/3 burden (as opposed to the of plaques), likely to information insufficient phases of AD. Finally, because our primary goal is Smoluchowski equation -'^^~\ which has been used 2 = for nearly a 1,2,.... It is a century to study aggregation processes such as polymerization, galaxy formation, crystallization, and cloud 2-2-23 formation vVfhiie this nrodel typically considers novelty of our model is to simultaneously incorporate fragmentation, a source (A/i production) and a sink (A/i For t = 1. 2, . denotes monomer, of change of r,(i), , . . t = aggregation in closed systems, the we monomer let c,(f) loss). denote the concentration of A/? z-mer at time 2 denotes dimer, etc. denoted by Cj(t), monomer The t\ differential equations dictate the i.e., i = 1 time rate and are given by oo ci(f) monomers = p /^c,(0 + s-l::^ production all ' fragmentations + ^c,(0)- -eci(t)(2c,(f) —— 1=2 ^ v all elongations ' lc,{t) loss (e.g., by degradation) ;i: + ec,(OQ(0 of and for c,(0 i > = monomers + er,(f)r,_i(0_ of elongation (? - l)-mers The model captures constant rate - /c,+i(0. ec,{t)r,{t fragmentation (i + l)-mers (3) 1. p. fragmentation of ^-mers elongation of /-mers of four processes - production, elongation, fragmentation, and loss monomers In equation (1), A/3 and are lost at time t cell mnemonic parameters are produced via cleavage of the at rate lci{t). time units on average before being cleared by That is, A/3 monomers are APP live for /"^ internalization (e.g., microglial ingestion), degradation by proteases or removal from the brain via the circulation. Parameter (2) fragmentation of dimers of dimers clearance) - and for ease of reference the corresponding displayed in Table at a elongation C2{t) 3, t-mers (i.e., fragmentation of 3-mers elongation diiners / ec,{t)c2{t) f<-3{t) factor 2 in equation (1) arises because the elongation to a The and the term //2 in equation (2) stems from the dimer requires two monomers, fact that a dimer can only fragment one location, while larger «-mers possess two potential fragmentation cancel each other out in the fco term in ecjuation alternate modeling approach is 2 and 1/2 (1). to allow direct clearance of monomers off of i-mers represent microglial ingestion of polymers), rather than requiring a two-step proce- (e.g., to dure of Fragmentation //2 and creates two monomers, and hence the factors of dimers occurs at rate An sites. in monomer fragmentation sult in the omission of the "all followed by fragmentations" term of / as an ingestion rate, but monomer and dimer a post-treatment steady-state. in clearance. This alternative would re- equation (1) and the re-interpretation would not change the qualitative nature of our would mainly change the precise conditions ues of steady-state monomer for a steady-state solution and the results: it relative val- concentrations, and slightly hasten the approach to Finally, while AjS treatment parameters are not explicitly incorporated into equations (l)-(3), treatment can be included in the model, as explained later. Results Steady-state solution Our analysis begins by seeking a steady-state (3). To describe our which is results in a concise Ci equilibrium) solution, Cj, (i.e., to equations (1)- manner, we define the polmerization ratio a unitless quantity that incorporates the four model parameters. side of equations (l)-(3) for (i.e., Setting the left the rates of change of i-mer concentration) to zero and solving reveals that there are two regimes: a steady-state (or subcritical) regime 6 where / < 1 . and a supercritical regime where > r In the former case, the A/? levels settle into a 1. steady-state where d = and y, c, = 2^r'-' for Hence, ?— mer concentrations decay geometrically for lent than dimers < r if and 0.5, less i > / 2, prevalent than dimers > 2. (5) and monomers are more prevaif r > In contrast, 0.5. r if > 1 then the A/j burden grows indefinitely. Because the A/? burden appears to remain relatively stable over time in symptomatic AD patients ^-^-'^s, we focus primarily on the steady-state regime, and defer until later a discussion of the supercritical regime. Equation number total which is implies that the steady-state value for the total A/3 concentration (5) of Af3 molecules in the system, whether they exist as denoted by r = ^j^i ic^ and referred to as the Ap burden, monomers is (i.e., or ?'-mers). given by Notice that the total Aft burden c approaches infinity as the polymerization ratio r appoaches 1. We now use equations (5)-(6) to estimate the values of the model parameters, and these parameter values, in turn, allow us to calibrate the key relationship (6) (i.e., A/i burden) to the clinical setting. Parameter estimation The value of the which is brains of A;:^ monomer / in Table 1 corresponds to a close to the crude experimental value of 38 APP transgenic mice primaril\- via A.J monomer a synthetic fibril analysis and loss rate protofibrils ^. Protofibrils addition, and ^'*, which is half-life of 41.6 minutes reported by one group ~', fibrils '' " and plaques we use an elongation ^^ 7 in the appear to grow rate e in Table 1 taken from within a factor of two of other estimates for ^' minutes, fibrils ^^ The fragmentation empirically. Fortunately, man AD / and the monomer production rate p are rate we can use equations brains, to estimate these 1.4% of A/? in the human brain is difficult to estimate (5)-{6), together witli A/i estimates from hu- two quantities as soluble, and found that primarily of monomers, dimers and trimers. A A follows. this soluble related study human erage (of A/i4o and A/?42) soluble fraction of A/i in study rec(Mit the fraction of A/? that consists of monomers, ci/c, to 0.013. soluble dimers and trimers in and setting AD brain. is only 0.05% We By By equations this expression equal to 0.013 yields the consists taking the and assuming '^'') monomers, we equate recognize that this analysis in the brain if there are large equate the A/i burden on the right side of equation (A/3.}o compartment measured the weighted av- '-^^ that the soluble dimers and trimers are exactly offset by any insoluble underestimate the percentage of soluble A/3 estimated that brains to be 1.2%. average of these two estimates (we ignore a third estimate that will * (5)-(6), Ci/c = - (2/(1 = polymerization ratio r amounts r)'' - 1)^\ We 0.84. of also the average of the total A/3 levels (6) to plus A/342) found in five different cortical regions of wet brain tissue of patients with a clinical of 2819 dementia rating (CDR) score of pmol/g is similar in magnitude to those in wet cortical tissue (which estimates for pmol/g / and 5.0 (severe dementia) in is about 87% water) r into the right side of is = equal to that of water. equation (6) times smaller than the value of the loss rate ^' ^ of - /, that directly ingests represented as a fragmentation enhancer. 1. estimate (this and equating 1. Substituting our this expression to = Finally, because r 0.84, together with our earlier estimates of yield the value for the fragmentation rate / in Table model an A/3 vaccine 1 by assuming that the density of '^), yields the value of the production rate p in Table the polymerization ratio estimate r Table Note that this value of / pe/{fl), e, is 2819 I and p, about 70 which suggests that within the context of our monomers off of oligomers can be accurately A/3 With burden we now return these parameter values in hand, A/3 burden c. to equation (6) for the steady-state total This nonlinear relationship between the polymerization ratio and the steady- state A/? burden is shown in Figure 1, and implies that a given treatment will generally achieve a greater (relative and absolute) reduction in steady-state A/? burden for patients To position with higher pretreatment A/3 burdens. context, ^. we convert the A/3 burden Figure in 1 this relationship within the clinical CDR scale to the using the data in Table This conversion allows us to explore the potential clinical impact (i.e., neurodegeneration associated with A/9 can be reversed) of a reduction ratio via treatment. is APP fall in transgenic mice at least partially reversible in mice. the right ordinates of Figures not of assuming that the in the polymerization Recent studies reveal that A/? peptide vaccination reduces not only A/3 burden but also cognitive impairment in neurodegeneration 1 1, 2 and 3 synchrony with A/3 burden in can be ignored humans; the if We ^~- ", implying that the CDR note that the the reader believes that latter ordinate is still scales on CDR will valid. 7-secretase inhibitors Because of their potentially adverse effect on Notch signalling 34-3(^ 7-secretase inhibitors might best be used to reduce only modestly the A/^ by 20-6U%. To assess the effect of such treatment, assuming that a symptomatic patient is in monomer production we numerically rate, solve equations (l)-(3) a pretreatment steady-state on days 0-250, and administered a 7-secretase inhibitor that decreases the A/j production rate p by 250-620 (i.e., for an 18-fold drop 1 year). in the A,3 As shown in Figure 2, is 40% on days the post-treatment steady-state represents burden and a theoretical decrease in the concommitant from 5.0 (severe dementia) to 0.0 (no dementia). The perturbation dynamics are rather sluggish: perhaps CDR score in Figure 2 the relaxation time, which we define to be the time from the start of 9 treatment until the A/? burden drops to state (i.e., 90% of the way towards the post-treatment steady- time until the A/i burden equals the steady-state post-treatment A/? burden tlie plus 0.1 times the difference of the steady-state pretreatment and post-treatment burdens) is 113 days. Interestingly, treatment for the A/3 it takes much longer ~ about 600 days after the discontinuation of burden to revert to 90% of a polymerization ratio r of 0.84 (Table Comparing Equation (6) **, different treatments allows us to provide a comparison across various treatment approaches. Witliin (e.g., four parameters: 7-secretase inhibitors all upstream mechanism by which A/3 or other agents that target the APP pretreatment steady-state (assuming 1)). the context of our model, treatments can affect ^' its is produced from promote the /j-secretase inhibitors), reduce the production rate p; agents that fragmentation of monomers from polymers and/or enhance the monomer increase the fragmentation rate / and loss rate and agents that deposition of A/3 production rate p monomers reduce is the elongation rate e is reduced by x%;. Similarly, / is y% By equations if has the same effect as a (4) c is x% higher than if the loss rate e is ^ inhibit the and (6), if the lower than the fragmentation rate / have somewhat more leverage than reducing the A/3 burden. Moreover, an respectively), e. reduced by x%, then the resulting A/^ burden Hence, the parameters p and rate, respectively) respectively; the elongation rate by y%, then the resulting A/3 burden in /, *• clearance rate is if increased increased by y%. and /, respectively, x% reduction in the production rate (elongation y% increase in the loss rate (fragmentation rate, where lOOx ^~ For example. Figure 2, 100 -x" which was computed using a 7-secretase inhibitor that reduces the production rate by 40%i, would also resvilt from a treatment that increased the 10 loss rate by 66.7%. Unfortunatt'ly, there by changes in p is no simple formula to compare the A/? burden reduction achieved / or by changes vs. in e vs. /. However, treatment steady-state Af3 burden (and corresponding in CDR Figure 3 we plot th(> post- score) as a function of Ijoth the percentage reduction in production rate p achieved by a 7-secretase inhibitor and the percentage increase in fragmentation rate / achieved by, 70 times larger than / in Table 10%) change in these rate still only a modest / is (e.g., Because A/3 monomers on dementia. this figure also off, tliat (recall that recjuired to achieve a several-fold reduction in A(3 is clinically significant effect need to be cleared after they break an A/i vaccine This graph strongly suggests 1). parameter values burden and potentially a e.g., shows that an infinite leads to a finite steady-state A/3 burden, given the limits of the fragmentation monomer loss rate /. Additional computational results (not shown here) reveal that a fragmentation rate enhancer has a smaller (i.e., faster) relaxation because the relaxation time depends time than a production rate inhibitor, probably in large part on the fragmentation rate. For example, an agent that increases the fragmentation rate by 118%! achieves the same post-treatment steady-state A/3 burden as the is 40% 7-secretase inhibitor in Figure but the relaxation time 2, only 40 days compared to 113 days for the 7-secretase inhibitor. Supercritical regime Thus far, we have assumed that /• < 1. so that a steady-state the A/3 burden grows indefinitely, eventually (perhaps after at rate p{r — l)/r. Moreover, for all r > 1, is many achieved. If r > 1 then years) increasing linearly the distribution of polymers no longer decays geometrically, but tends over time to a uniform distribution, where each ?-mer (starting with smaller values of t) successively achieves the concentration 11 ci = //e, c, = 2//e for i. > 2. Heme, in is there are three possibilities for the effect of treatment: (i) the A/3 burden a pretreatment steady-state and drops to a lower post-treatment steady-state, as in Figure 2; (ii) treatment causes the A/i burden to steady-state regime (Figure 4a); and the supercritical regime, although (iii) (ii) than in case from the supercritical regime to the treatment does not allow the A.i burden to exit does lower the growth rate (Figure 4b). it of Figure 4a to Figure 2 shows that steady-state in case shift may it nmch take Comparison longer to achieve a post-treatment (i). Sensitivity analysis Because we do not possess precise estimates sitivity analysis that varies the for our parameter values, we performed a sen- pretreatment polymerization ratio driver of the A/3 dynamics, according to our analysis. technique as before: monomer fix / and e to their values in ci/c, use the average A/? wide range of values we for r, which Table 1, in investigate the effect caused by a '^ which find r it the primary from the fraction of A/? to derive p, we now allow is same parameter estimation use the burden of 2819 pmol/g calculate /. Rather than eciuating Ci/c to 1.3%, a We /•, and use r and p to to vary in order to generate turn causes p and / to change. In Figures 5a and 5b, 40% reduction in the production rate (via a 7-secretase inhibitor) as a fvmction of the pretreatment polymerization ratio 7". Figure 5a plots the pre- treatment steady-state A/3 burden divided by the post-treatment steady-state A/3 burden (i.e., the fold-reduction in A/3 burden), and Figure 5b shows the relaxation time, which was defined earlier. Recalling our assumption that a three-fold reduction in AI3 burden leads to a clinically significant improvement in CDR score (Figure 1), Figure 5a shows that the 40% production inhibition appears capable of a significant clinical improvement for essentially practical values of r (i.e.. corresponding to percentages of monoiueric A/3, ci/c, 30%), and a l-to-2 log drop in A,3 burden as r increases 12 less all than from 0.76 to 0.93. Figure 5b shows = that our base case of r 0.84 polymerization ratio. Hence, is if on the steep portion of the curve of relaxation time versus the fraction of is actually 5% rather than 1% and the relaxation time decreases from 113 days of total A/i, then r drops from 0.84 to 0.7 to only monomeric A/3 two weeks. Discussion Despite the existence of some mathematical models for A/i fibrilli/ation and plaque growth, this paper appears to represent the attempt to use a mathematical model to assess first the effect of treatment on A/3 burden in Alzheimer's disease. Our model is monomer production purposefully rate (p), A/3 parsimonious, and contains only four parameters: A/3 monomer and polymer fragmentation rate loss rate (/), The most polymer elongation rate basic result of our analysis upon the value of the polymerization less than 1, ratio which equals pe/{fl). If this unitless quantity then the A/9 burden enters a steady-state regime and takes on a polymerization ratio is (/). that there are two possible regimes, depending is r, (e), finite value. If is the greater than one, then the A/3 burden grows indehnitely, eventually increasing linearly at rate p{r - l)/r. Researchers in other disciplines have found that the dynamics of their complex systems are dictated by a unitless measure that leads to a subcritical regime greater than in 1; epidemiology if the measure is less than 1 and a supercritical regime if the measure is two examples are the basic reproductive rate Rq of an infectious pathogen ^^ and the in these disciplines the traffic intensity p in queueing theory ^^. Moreover, as with AD, ultimate measures of performance (size of the epidemic and queue length, respectively) have a highly nonlinear relationship with the unitless measure. Applied researchers in these disciplines have adopted these unitless measures as a central part of \hc\r mental model of these complex systems, and routinely estimate them and monitor achieved reductions in them. We propose that 13 AD researchers follow suit by adopting the polymerization ratio pe/{fl) as the key "A/3 driver" in this disease. Several studies have with th(> ratio is shown that the A/3 burden does not appear duration or clinical severity of less than 1 for of plaque formation symptomatic '*' '^ ""^ - ^^ of plaque via a feedback may exist, patients. which suggests that the polymerization Although previous mathematical models requires that the disaggregation rate be modulated by the mechanism that minimizes the changes that occur While some feedback mechanism brain. 21-26^ have also proposed a dynamic equilibrium of aggregation and disaggregation, the model in amount AD AD to be tightly correlated - for in the example, a cytotoxic T-lymphocyte response our model shows that when A/3 production is A/3 burden can be achieved by a constant fragmentation rate incorporated, a steady-state (i.e., in the absence of any feedl)ack). Nonetheless, our estimate for r so is it conceivable that certain aggressive forms of the disease in amyloidogenic presenilin mutations) importantly, while studies measures is AP '^"^--^ may burden and or mice (strongly 1. More burden by counting plaques, a more recent study burden biochemically, and these more refined data suggest that A/3 burden (i.e., AD, which is not inconsistent with linear growth of the supercritical regime in our model). be required to further elucidate whether A/3 burden continual growth in symptomatic state humans achieve a polymerization ratio greater than assess Af3 correlated with clinical severity of A3 of the order of 10"' (rather than 10"^ or smaller), is AD is Although further research may in a steady-state or patients, our identification of is experiencing two regimes (steady- and supercritical) suggests that there are three types of possible treatment outcomes: a reduction in (Figure 1), A3 burden from a pretreatment steady-state to a post-treatment steady-state a change in regime from pretreatment growth to post-treatment steady-state (Figure 4a), and a reduction in growth rate from a pretreatment supercritical regime to a post-treatment supercritical regime (Figure 4b), These results suggest that the failure of a 14 drug to reduce the A/i burden drug inactivity, in a subset of mice or than is less 1, burden within several months after treatment is necessarily be due to but rather could signal a pretreatment supercritical regime. Also, even pretreatment polymerization ratio it humans may not a patient may if the revert to his pretreatment A/3 discontinued (see Figure 5b); consequently, is important to attempt to measure the A/3 burden throughout the course of treatment a cliniial trial. In contrast, if the pretreatment polymerization ratio after discontinuation of treatment a patient's A/3 burden will is greater than 1, always be smaller than in then if he had not received treatment. Equation (6) provides a simple formula for the steady-state A/3 burden in terms of the polymerization ratio pictured in Figure < 1) the A/3 burden is r 1, (provided the polymerization ratio, a modest reduction in r r and the production rate-to-loss rate ratio, pi I. As a highly nonlinear (increasing, convex) function of and approaches infinity as r approaches 1. Hence, the impact of appears to be sufficient to produce a many-fold reduction A/3 burden; for the parameter values in Table a 1, 40% reduction in the production rate via a 7-secretase inhibitor achieves an 18-fold reduction in A/3 burden. consistent with recently reported data in mice ^' * in the and appears to be Such a reduction sufficient to convert the disease process to a non-pathological state 2,3,29^ assuming neuronal dysfunction concommitantly decreased. Figure 1 also is is shows that, while the magnitude of the A/3 burden reduction increases with the strength of treatment, there are decreasing returns to scale as the treatment gets stronger. in A/3 Finally, our analysis predicts that the percentage reduction burden achieved by a given treatment then decreasing) of the polymerization ratio burden increases with would be achieved AD mutation ^', in r, is r. a unimodal function For r < 1, (i.e., first increasing, the percentage reduction in A/3 and hence one might expect that a larger percentage reduction forms of AD that are particularly aggressive the trisomy 21 nnitation that occurs in 15 (e.g., the Swedish familial Down's syndrome ''^). However, as explained in the previous paragraph, the A/3 treatment if the post-treatment polymerization ratio Our model affect burden continues to grow is flexible enough greater than is in the face of 1. to incorporate a variety of treatment approaches that any of the four model parameters. Equation (5) allows us to perform an apples-to- apples comparison of production inhibitors, elongation inhibitors, clearance enhancers, and For a given percentage change fragmentation enhancers. in parameters caused by a treat- ment, this analysis shows that the production inhibitors (clearance enhancers, respectively) are somewhat more effective in reducing the A/3 Equation tation enhancers, respectively). (7) burden than elongation inhibitors (fragmen- and Figure 3 provide an explicit comparison between production inhibitors and fragmentation enhancers, and computational results show that a fragmentation enhancer attains tion inhibitor. in its post-treatment steady state faster than a produc- These comparisons, coupled with future data on drug determining the optimal mix and level of toxicity, may be useful treatments that minimize the A/3 burden subject to toxicity constraints. Figure 2 shows that new post-treatment it takes several months for treatment to reduce the A.(5 burden to a steady-state, and the return to a pretreatment steady-state A/3 burden following the discontinuation of treatment that the rate of A/3 turnover is even more gradual. However, Figure 5b shows is highly sensitive to the pretreatment polymerization ratio suggesting that aggressive forms of AD (having high r) are likely to take much r, longer to get under control. Hence, our lack of a precise estimate for the pretreatment polymerization ratio prevents us some from accurately predicting the rapidity of response to treatment, but cases this response may be This general state of in two seminal papers the research HIV affairs is similar to ^^'•*. community considerably quicker than suggested in Figure The stability of into believing that HIV was 16 2. infection, as revealed several years plasma HIV levels over many in ago years lulled a relatively static disease process. but a perturbation of this steady-state by a powerful protease inhibitor revealed a highly dynamic equilibrium, where a high virus production rate was offset by an equally high virus chvirance rate. Hence, antiviral therapy led to a precipitous drop in plasma HIV discontinuation of treatment allowed for a rapid return of steady-state. To borrow A/9 level) in the bath APP) and AD (i) the analogy often used in the may change drain {A(5 loss) field, may be lessons learned by the HIV toxicities of various types of agents are non-overlapping, if the water level operating at deceptively high rates. It is the Achilles heel of is worthwhile clinical research dynamic equilibrium oi A/3 - which implies that and the interaction between Ap toxicity, which is (iii) (ii) (e.g., watch given the require chronic treatment - production and Notch signalling on developing drugs that avoid long-term community AD patients may and '*''; for community: then drug combinations HIV treatment the (i.e.. monomer from a 7-secretase inhibitor plus an A/i vaccine) are likely to outperform monotherapy; out for drug resistance, which and levels, pretreatment levels to the very slowly, but the tap (production of A/i mind the researchers to keep in HIV HW '^^-•^''^ efforts should focus beginning to plague the HIV ^^. Several of the model parameters, particularly the production rate and the fragmentation rate, are difficult to and (6) measure in vivo. benefit of our analysis is that equations (5) allow an indirect estimation of the production rate p and the fragmentation rate / given pul)lished data on the A/9 burden more One generally, ^ and the fraction of Afl that is monomer ^- ^^ (or, any quantity, such as mean polymer length, that can be derived from the steady-state distribution of i-mers). Equation (5) can also be used to measure the strength (i.e., percentage inhibition or enhancement) of a treatment treatment and post-treatment steady-state Al3 burdens. in vivo, given data on the pre- Moreover, a transient analysis of our model (not provided here) would allow for indirect estimation of a third model parameter from serial measurements of post-treatment A(i burden. 17 Our study has two important limitations. First, our entiate between the various forms of A/3 {A(3iQ vs. model does not currently A/342, soluble vs. differ- However, insoluble). while we have been careful to restrict the model to A/i polymerization per sc and ignore protofibril-to-fibril conversion and plaque growth, viewed as a representation of plaque growth plaques are primarily due to deposition onto plaques is monomer believed to be the more may be more complex than realistic = 2/3 or 1, model can, the be in fact, deposition onto plaques and fragmentation of main form monomer '^^' '*'', of aggregation in vitro Similarly, fragmentation of oligomers ^^. monomer release that is assumed here. it ''^ is and Hence, a model of plaque growth might use a variation of Smoluchowski's equation that allows i-mers and /-mers to coalesce A; clear that our addition and break-off, respectively. While quite likely that polymers coalesce in vivo plaques if it is (e.g., at a rate proportional to 'i''j^Ci{t)cj{t), where depending on the porosity of plaques) and allows fragmentation of monomers from the plaque surface (at rate fi^'^Ci{t)) or the entire plaque (at rate While such ftCi{t)). a model would drastically change the distribution of A/3 ?;-mers relative to the current model (skewing the distribution towards huge polymers that represent plaques), significantly alter the basic nature of our results pertaining to the A/3 burden. the extent that the model parameters including the effect of treatment A/342 or soluble vs. insoluble A/3, then these in - unlikely to is it However, to vary for A/34o vs. model generalizations would be worth pursuing the future. The second main weakness indicative of human AD brains: of our study that our parameter values is the loss rate / soluble. Our fraction of A/3 that is pH environment monomer with our parameter estimates are, a modest (several-fold) change^ in i'- '*. ^, and Also, the fraction of A/3 that sensitivity analysis (Figures 5a-5b) suggests that, regardless of 18 not necessarily was derived from the brains of mice the elongation rate e was measured from synthetic A/3 in a low we crudely equated the ar(> how is inaccurate a parameter value caused by lead to a clinically significant treatiiieiit will still (i.e., at last three-fold) reduction in the steady-state k(5 burden, but that the time to achieve this post-treatment steady-state relaxation time) is (i.e., quite sensitive to our parameter values. Despite the model's simplicity, our analysis elucidates the basic nat ure of the kinetics of the .\d burden in the face of etiologic treatments, and provides a framework metric, the polymerization ratio - ^' and a novel within which to interpret the laboratory and clinical results that are likely to be generated during the next few years. on the identity of the toxic moiety - ^^ , As more progress is made the elucidation of the inflammatory and neurotoxicity cascade, and the parameter values and mechanisms related to A/9 polymerization and plaque growth, our hope is that models such as this one will be refined and generalized to provide some guidance about the optimal way to employ the emerging anti-A/3 drug arsenal. Acknow^ledgment This research was partially supported by the Singapore-MIT Alliance Foundation for Neurologic Diseases (DJS). 19 (LMW) and by the References [1] Selkoe, D. Translating J. cell biology into therapeutic advances in Alzheimer's disease. Nature 399 (Supp). 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The relationship between steady-state A/9 burden score (right ordinate) score is based on Table 1 of'-, (left as explained in the text. 2.5 c 0) 1.5 < 0.5 % enhancement in fragmentation rate iC^ c •D 3 < 2 pretreatment a 3 n < .£ C o .E = o o E % of monomer A|5 MOV ^00' Date Due Lib-26-67 MIT LIBRARIES 3 9080 02245 1980