Extrapolation of Threshold-Limited Null Measurement Frequencies

Extrap olation of Threshold-Limited Null Measuremen t F requencies O. E. P ercus Coura n t Institut e/ NYU 251 Mercer Street New Y o rk, N Y 1 0012 J. K. P ercus Coura n t Instit ute/NYU 251 Mercer Street New Y ork, NY 10012 1 Abstract The total measurable lev el of a pathogen is du e to man y sources, whic h pro duce a v ariet y of pulses, o verlapping in time, that rise sud denly a nd then deca y . What is measured is the lev el of the total con tribution of the s o urces at a giv en time. But since we are only capable of measuring the total lev el ab o ve some t hresh o ld x 0 , w e wo uld like to predict the distrib ution b e lo w this leve l. Our p rincipal mo del assu mption is that of the asymptotic exp onent ial de- ca y of all pulses. W e sho w that this imp lie s a p o wer la w distribution for the frequencies of lo w amp lit ude observ ations. As a consequence, there is a simple extrap olat ion pro cedure for carrying the data to the r e gion b elo w x 0 . Keyw ords: exp o nen tial decay; p ow er-la w distribution; completion of data 1 In tro duction Acquiring sufficien t data of sufficien t a cc uracy is the standard problem in the use of applicable mathematics. R elianc e up on null measuremen ts—i.e., an a ns w er of y es or no—is often an in t e lligen t wa y o f attending to the latter desideratum, as in the familiar limiting dilution assay s [Lefk o witz and W aldman, 1979]. But the former frequen tly is con t r o lled by exp erimen tal inability , or p erhaps excessiv e exp ense, in dealing with some region o f da t a . If enough is surmised ab out the structure of the data, suc h regions can b e red uced by su itable extrap olation, but the implicit assumption [for an elegan t presen tation, see Berman, 2006] of some sort of analytic structure runs the 2 risk of b eing to o m uc h o f a mathematical band-aid unless it is justified b y a v ersatile underlying mo del. In this note, we address a situat io n of some generalit y . It is that in whic h a n organism, biolo g ical or mec hanical, is con tin ually sub jected to transient defects, e.g. pathogenic molecular sp ecies, internally or externally incited but so on eliminated. These inhibit its ability to effectiv ely deal with its env ironmen t. W e imagine that the net pathogen leve l A is measurable at o ccasional time interv als, but o nly if it exceeds some threshold x 0 (i.e. A ≥ x 0 ). A n ull measuremen t sequence would t he n giv e the relativ e frequency G ( x 0 ) of measuremen ts falling b elo w the threshold x 0 . W e would w ant e.g. to obtain from this the densit y function ρ ( A ) of amplitudes of t he pa t ho gen aggregate lev el, A , with par t icular attention to the unav ailable lo w amplitudes. The total pathogen load A at a giv en measuremen t w ould b e expected to b e the resultan t of the curren t amplitudes o f eac h of the sources; these sources may b e imagined as time-displaced v ersions of a discrete set of t yp es, and this is the mo del that we will study in detail. The mo del w as originally used in a somewhat differen t con text, tha t of the significance of “blips” in HIV viral lev el in pa tien ts undergoing mu lti-drug therap y . [see P ercus et al, 200 3] What we can adjust in this scenario is the threshold lev el ab ove x 0 , a nd then observ e the null frequency G ( x ) for x ≥ x 0 . The relatio ns hip b et w een t he in t rins ic ρ ( A ) and G ( x ) is ob vious G ( x ) = Z x 0 ρ ( A ) dA, (1) just the cummulativ e distribution of A . Our task is now to o btain the f orm of ρ ( A ) 3 from the mo del assumptions a nd use this e.g. to extrap olate the av ailable G ( x ) f or x ≥ x 0 to v alues 0 < x < x 0 . 2 The Un derlying Mo del ( F ) τ λ F F λ ( t ) 0 − b λ t ∆ Figure 1: Parameters o f T ypical Pulse Shap e W e imagine that the arriving pulses are all translatio ns in time of a basic set of shap es indexed by λ F λ ( t ) , a < t < b (2) These shap es are non-negativ e functions suc h that Z b a F λ ( t ) dt is finite, No w, place eac h of these f unc tions, indep end en tly on the in terv al ( − T , T )( − T < a < b < T ) ν λ times. T o do this, let ˆ τ b e a random v ariable uniformly distributed on the in t e rv al ( − T , T ) and ˆ ν λ a P ossion random v ariable with mean 2 T q λ , i.e. h λ ( ν λ ) ≡ P { ˆ ν λ = ν λ } = (2 T q λ ) ν λ ν λ ! e − 2 T q λ . (3) 4 The lo cation in time of F λ is determined, for example, so that its maxim um is at the origin (see Fig. 1). The time co ordinate of the maxim um p oint of the i th o ccurrence of F λ is then denoted b y ˆ τ λ i i = 1 , . . . , ˆ ν λ . The equation of the i th o ccurrence of the curv e F λ is then ˆ A λ i = F λ ( t − ˆ τ λ i ) . The total amplitude at an y sp ecified time, sa y t = 0 , is ˆ A = X λ ˆ ν λ X j =1 F λ ( − ˆ τ λj ) . (4) W e would lik e to find the probability densit y of the ra ndo m v ariable ˆ A . Let ρ ( A ) b e the probabilit y densit y function of the ra ndom v ariable ˆ A i.e. ρ ( A ) = ( ∂ /∂ A ) P r ( ˆ A ≤ A ) (5) W e will assume a steady state distribution of “pathog e ns” in the course of measure- men ts. This is a limitation of our approach: often the life-time of the organism may b e comparable to the “decay” of pathogen. Then the system is translation-inv arian t in time, which is wh y w e can choose, without loss of g en erality , the observ a t io n time t = 0, as in (4). Let us construct the generating function for ρ ( A ) w ( α ) ≡ E  e − α ˆ A  (6) Then w ( α ) = Z ∞ 0 e − α A ρ ( A ) dA. (7) 5 W e need E  e − α F λ ( − ˆ τ λ j )  = 1 2 T Z T − T e − α F λ ( τ ) dτ , (8) so that w ( α ) = E Y λ 1 2 T Z T − T e − α F λ ( τ ) dτ ! ν λ ! . (9) But from (3), E  Y ˆ ν λ  = e 2 T q λ ( Y − 1) , a nd w e see at once tha t w ( α ) = Y λ exp " q λ Z T − T  e − α F λ ( τ ) − 1  dτ # , (10) or letting T → ∞ , Z ∞ 0 e − α A ρ ( A ) dA = exp X λ  q λ Z ∞ −∞  e − α F λ ( τ ) − 1  dτ  , (11) whic h is our basic expression. Eq. (11) can b e expressed more concisely . Define ∆ τ λ ( F ) (see Fig. 1) a s the total time that the ordinate F λ ( τ ) ≥ F i.e. ∆ τ λ ( F ) ≡ Z θ ( F λ ( τ ) − F ) dτ where θ ( x ) = ( 0 if x < 0 1 if x ≥ 0 . Also note t hat ∆ τ ′ λ ( F ) ≡ d dF ∆ τ λ ( F ) = − δ ( F λ ( τ ) − F ) where δ ( x ) is the D ir a c δ function. Then for an y function f w e hav e Z f ( F ) ∆ τ ′ λ ( F ) dF = − Z f ( F ) Z δ ( F λ ( τ ) − F ) dτ dF = − Z f ( F λ ( τ )) d τ . It follow s that Z ∞ −∞  e − α F λ ( τ ) − 1  dτ = Z ∞ 0  1 − e − α F  ∆ τ ′ λ ( F ) dF , (12) so that if τ ( F ) ≡ X λ q λ ∆ τ λ ( F ) , (13) 6 w e ha v e the simple equality Z ∞ 0 e − α A ρ ( A ) dA = exp Z ∞ 0  1 − e − α F  τ ′ ( F ) dF . (14) 3 Rationale for Extrap olatio n Eq. (14) can of course b e solv ed for ρ ( A ) in nominal closed form by applying the in verse Lapla ce transform. But a less formal path is to use (1 4 ) to set up an equation that ρ ( A ) satisfies. F o r this purp ose, take t he logarithm of the equalit y (14) a nd apply the op eration − ∂ /∂ α to b oth sides, yielding Z ∞ 0 e − α A A ρ ( A ) dA = Z ∞ 0 e − α F ( − F τ ′ ( F )) dF Z ∞ 0 e − α A ρ ( A ) dA = Z ∞ 0 Z ∞ 0 Q ( F ) e − α ( F + A ) ρ ( A ) dA dF = Z ∞ 0 Z ∞ 0 Q ( F ) e − αA ρ ( A − F ) dF d A where Q ( F ) ≡ − F τ ′ ( F ) (15) and w e ha v e used the fact that ρ ( A ) = 0 for A < 0. No w the in v erse La pla ce transform (lo osely , tak e the co efficien t of e − α A on b oth sides) establishes that A ρ ( A ) = Z A 0 Q ( F ) ρ ( A − F ) d F. (16) Our inte rest is in the b eha vior of ρ ( A ), or G ( X ), for small v alues of A , o r X ; since F ≤ A in (3.2), this corresp onds to small v a lues of F . Now the anticipated nature of t he pulse profiles comes into play . A pulse form of type λ will b e initiated (see Fig. 1) at some time − b λ . If it is thereafter determined by any standar d c hemical kinetic sequence leading to its eve n tual disapp earance , it will asymptotically decay 7 as C λ e − a λ t for some a λ . Hence the low a mplitude leve l F duration will b e giv en by τ λ ( F ) = − b λ − 1 a λ ℓ n ( F /C λ ) . (17) Consequen tly , w e hav e f o r the total w eigh ted duration τ ( F ) = X λ q λ  − b λ + 1 a λ ℓn C λ  − X λ q λ a λ ! ℓn F , (18) from whic h Q ( F ) of (3.1) has the constan t v alue Q ( F ) = Q ≡ X λ q λ /a λ . (19) Eq. (16), with ρ ( F ) = 0 for F < 0 then b ecomes A ρ ( A ) = Q Z A 0 ρ ( F ) dF , (20) or in terms of the null measuremen t cum ulan t G ( x ) of (1.1), x G ′ ( x ) = Q G ( x ), with the solution G ( x ) = C x Q . (21) W e conclude tha t ℓn G ( x ) = ℓn C + Q ℓn x, (22) so tha t a standard linear extrap olation of ℓn G vs ℓ n x is v alid at sufficien tly small x Let us tak e a h yp othetical example. It is that of c hronic parasitic infection of an org anism , with contin ual birth of clusters of parasites, eac h of whic h is quenc hed b y the imm une system. There is a large fluctuation in parasite load A , sampled sequen tially in equiv alent test v olumes, measurable if ab o v e the threshold x 0 . If the data is acquired via n ull measuremen ts of the lo ad ab o v e virtual thresholds { x ≥ x 0 } , 8 w e wan t to extrap olate the ensuing G ( x ) to x < x 0 . Cho ose as typic al p opulation spik e, (with origin at τ = 0 rat he r than at max F λ —it mak es no difference) the form F λ ( − τ ) = C λ e − a λ τ  1 − e − d λ τ  , (23) and for definiteness, 1 ≤ c λ ≤ 5, 1 ≤ a λ ≤ 3, 1 ≤ d λ ≤ 5 o v er a p eriod 0 ≤ τ ≤ 10, with parameters distributed uniformly in their domains, and a ll q λ = 1. Ev aluating A of Eq. (6) for 1000 runs, the resulting ℓn G ( x ) is plotted a gainst ℓn x in Fig. 2. The feasible linear extrap olation region is indeed v ery lar g e. The conclusion (22) is not without assumptions that ha v e b een p oin t ed out, but it app ears to b e a r esult of some generalit y , exemplifying the assertion that extrap olation is a mo del-dep enden t procedure, and that recognitio n of this fact has important op erational significance. 9 0 -2 -4 -6 -8 -10 ln G ln A 1.5 2 2.5 3 1 Figure 2: Ty pical Dep endenc e of ℓn G on ℓn A 10 References Berman, S, 2006. Legendre P olynomial Kernel Estimation, Comm . Pur e and Appl. Math. 60 , 123 8. Lefk owitz, I and W aldman, H, 1979. Limiting Dilution Analysis of Cells in the Imm une System, Camb ridge Pr es s P ercus, P ercus, Mark ow itz, Ho, di Mascio, and P erelson, 2003. The distribution of viral blips observ ed in HIV-1, Bul l. Math. Bio. 65 , 26 3 –277. 11