A mathematical modelling tool for predicting survival = of individual patients following resection of glioblastoma: a proof of = principle

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Translational Therapeutics

British Journal of Cancer (2008) = 98,=20 113-119.
doi:10.1038/sj.bjc.6604125 = http://www.bjcancer.com/ = Published=20 online 4 December 2007

A=20 mathematical modelling tool for predicting survival of individual = patients=20 following resection of glioblastoma: a proof of = principle

K R Swanson^1,2, R C = Rostomily³=20 and E C Alvord Jr¹

¹Laboratory of Neuropathology, = Department of=20 Pathology, University of Washington, Seattle, WA, = USA

²Department of Applied Mathematics, = University=20 of Washington, Seattle, WA, USA

³Department of Neurological Surgery, = University=20 of Washington, Seattle, WA, USA

3D""

Correspondence to: Dr KR Swanson, Laboratory of=20 Neuropathology, Department of Pathology, University of Washington, = Box=20 359791, Harborview Medical Center, 325 9th Avenue, Seattle, WA = 98104-2499,=20 USA; E-mail: swanson@amath.washington.edu=

Revised=20 29 October 2007; accepted 6 November 2007; published online 4 = December=20 2007

3D""=20

The prediction of = the outcome of=20 individual patients with glioblastoma would be of great = significance for=20 monitoring responses to therapy. We hypothesise that, although a = large=20 number of genetic-metabolic abnormalities occur upstream, there = are two=20 'final common pathways' dominating glioblastoma growth - net rates of proliferation () and dispersal (D). These = rates can be=20 estimated from features of pretreatment MR images and can be = applied in a=20 mathematical model to predict tumour growth, impact of extent of = tumour=20 resection and patient survival. Only the pre-operative = gadolinium-enhanced=20 T1-weighted (T1-Gd) and T2-weighted (T2) volume data from 70 = patients with=20 previously untreated glioblastoma were used to derive a ratio=20 D/ for each patient. We developed a = 'virtual=20 control' for each patient with the same size tumour at the time of = diagnosis, the same ratio of net invasion to proliferation = (D/) and the same extent of resection. = The median=20 durations of survival and the shapes of the survival curves of = actual and=20 'virtual' patients subjected to biopsy or subtotal resection (STR) = superimpose exactly. For those actually receiving gross total = resection=20 (GTR), as shown by post-operative CT, the actual survival curve = lies=20 between the 'virtual' results predicted for 100 and 125% resection = of the=20 T1-Gd volume. The concordance between predicted (virtual) and = actual=20 survivals suggests that the mathematical model is realistic enough = to=20 allow precise definition of the effectiveness of individualised = treatments=20 and their site(s) of action on proliferation () and/or = dispersal=20 (D) of the tumour cells without knowledge of any other = clinical or=20 pathological information.

Keywords: glioblastoma; invasion; MRI; = mathematical=20 model; proliferation; resection

3D""=20

Gliomas are well known as extensively invasive = lesions with=20 a variety of genetic-metabolic abnormalities that contribute to = their=20 uncontrolled proliferation and invasion. A mathematical model has = been=20 developed based on the two final common paths of these mechanisms = - proliferation and invasion. Serial medical = imaging=20 can be used to track the spatio-temporal behaviour of the = detectable=20 portion of each lesion, but the undetectable and undefinable = fraction of=20 the neoplasm remains as a problem.

The=20 original mathematical model (Tracqui=20 et al, 1995), based on the analysis of a single patient = followed with serial CTs during his final year, provided values of = rates=20 of net proliferation () and net diffusion (D) that = were used as=20 averages for populations of hypothetical patients (Woodward=20 et al, 1996) to compare with populations of real = patients (Kreth=20 et al, 1993) subdivided only by extent of surgical = resection,=20 biopsy or 'gross total resection' (GTR). A difference in median = survival=20 of 7 weeks was predicted mathematically, and actually found, = although not=20 statistically significant, even with 115 patients (Kreth=20 et al, 1993). Other modelling efforts in the area of = glioma=20 have focused either on multicellular spheroids in vitro (Stein=20 et al, 2007) or on the averages of groups of patients = (Morris=20 et al, 2006). Our mathematical modelling approach has=20 increasingly focused on the individual patient, as in this=20 study.

The current study is = motivated=20 by several other recent applications of our modelling technique.=20 Specifically, our discovery that the model-predicted linear radial = growth=20 of the imaging-detectable portion of the lesion (at a velocity = dependent=20 on both and D) was accurate in a = population of=20 WHO grade II gliomas followed serially without intervening = treatment (Mandonnet=20 et al, 2003) which led to confirmation that the = velocity of=20 growth predicts survival time in a similar population (Pallud=20 et al, 2006). However, since higher grade gliomas=20 (glioblastomas) are typically treated immediately, a similar = population of=20 serially imaged untreated glioblastomas has been difficult to = obtain and=20 only a few cases have been reported (Swanson=20 and Alvord 2002; Harpold=20 et al, 2007). Here we demonstrate how our mathematical = model,=20 adapted to accommodate a level of resolution of MRIs, from which = could be=20 derived a ratio of D/ for any glioblastoma patient (Swanson,=20 1999; Swanson=20 and Alvord 2002; Swanson=20 et al, 2002; Alvord=20 et al, 2003; Swanson=20 et al, 2003a, 2003b),=20 can be used to help estimate the undetectable, diffusely invading=20 component of gliomas and to predict survival of individual=20 patients.

It should be = emphasised that=20 the model uses only currently available diagnostic pre-operative = MR=20 imaging characteristics of individual patients, the only necessary = requirement being the presence of contrast enhancement, and uses = no other=20 clinical or statistical interpretation, not even the histologic = diagnosis=20 of glioblastoma (although all of the present patients have such a=20 diagnosis). It should also be emphasised that model parameters = D=20 and are net, downstream from any = intracellular=20 metabolic activity controlling migration and proliferation, = irrespective=20 also of any microenvironmental factors that might further modify = them. In=20 contrast to the original comparison of virtual and real patients = (Woodward=20 et al, 1996), where individual sizes were not known and = where=20 averages from the scanty literature were available at the time of=20 diagnosis (Blankenberg=20 et al, 1995) and at death (Concannon=20 et al, 1960; Burger=20 et al, 1988) had to be used, the present series was = able to use=20 not only the individual sizes known for each patient at diagnosis, = but=20 also the confirmation of the completeness of resections by = post-operative=20 enhanced CT, not just the surgeon's opinion (Kreth=20 et al, 1993). The results suggest the potential utility = of such=20 modelling (avoiding all statistically defined qualifiers, = including age,=20 Karnofsky Performance Score (KPS), histological type and grade, = etc.) in=20 future clinical studies comparing specific biomarkers and = treatments of=20 gliomas in individual patients.

3D""=20

MATERIALS AND = METHODS

The=20 patients

The 70 patients = were=20 adults (22-79 years old, mean age 58 = years) with=20 supra-tentorial glioblastoma diagnosed, treated and followed at = the=20 University of Washington Medical Center in 1993-1995. Follow-up was complete to death in 66 = patients=20 and to 104, 208, 230 and 230 weeks in the 4 patients who were lost = to=20 follow-up. All of the patients received X-irradiation to the = tumour of=20 59.4 Gy or more, 58 received chemotherapy and 30 s operations. Our = Institutional Review Board approved this retrospective study of=20 patients.

Of the 70 patients, = 58 had=20 lobar tumours and 12 had tumours in deep or mixed location; 58 had = tumours=20 that anatomically made them eligible for 'gross total resection' = (GTR),=20 but 20 of these were reclassified as only 'subtotal resection' = (STR) when=20 post-operative enhanced CT revealed a nodule of residual tumour. = Of the 18=20 whose tumours were too extensive even to consider GTR, 11 had STR = and 7=20 only biopsy. Major clinical characteristics are summarised in Figure=20 1, which shows the durations of survival following the = specific=20 treatments correlated (P<0.01 Student's t-test) = with age=20 and KPS and demonstrates the broad ranges of clinical factors that = can be=20 accommodated by - but not incorporated = in - the mathematical model.

Figure 1
Scatter graphs=20 defining some of the major characteristics of the 70 = patients with=20 correlations of duration of survival for BX/STR patients (asterisks) and for the = GTR=20 patients (squares) with age, KPS and KPS-age. Note the broad = ranges,=20 illustrating the variety of the patients. Note also that = none of=20 these data enter into the mathematical=20 model.

The=20 diagnosis of glioblastoma was generally the result of a consensus = of four=20 neuropathologists at a time when one of the present authors (ECA) = was=20 chief of neuropathology and required necrosis (Nelson=20 et al, 1983) as one of the histologic criteria of=20 glioblastomas. No distinction was made between various subtypes, = including=20 primary or secondary, as later suggested by Kleihues=20 et al (2002).

We = used the=20 pre-operative tumour volumes as defined by gadolinium-enhanced = T1-weighted=20 (T1-Gd) and T2-weighted (T2) MR imaging. These volumes had been = digitised=20 and calculated using NIH Image software that added the areas on = each slice=20 and multiplied the sum by the thickness of each section to obtain = the=20 volume. Two independent observers had made the measurements and = the mean=20 was accepted. The percent differences between the two observers = was 7.6%=20 (=B17.3%) and 6.0% (=B14.7%)=20 for T1-Gd and T2 measurements, respectively. The total volumes = included=20 any central necrosis. From these volumes we calculated the radii = of=20 equivalent spheres (Figure=20 2), and these radii were the only data entered into the = mathematical=20 formulations.

Figure 2
The actual=20 data used for the present analysis: average radii of spheres = equivalent to the T1-Gd and T2-weighted MRI volumes for=20 glioblastomas of 70 actual patients subjected to biopsy or = subtotal=20 resection (BX/STR, asterisks, = N=3D38),=20 or to gross total resection (GTR, squares, N=3D32) as = defined=20 by the absence of tumour on post-operative enhanced=20 CT.

We used=20 other clinical data only to define appropriate subsets to be = compared as=20 to durations of survival, actual vs predicted (virtual). = These data=20 included statements as to whether the patient had received = steroids, what=20 type of surgery had been performed, and the duration of survival = from the=20 date of operation. The extents of resection included only = statements as to=20 biopsy only, subtotal and gross total resection, the latter two=20 arbitrarily defined by the presence or absence of residual tumour = on=20 post-operative contrast-enhanced CT, performed within 2 days = following=20 surgery.

Model=20 development

The = mathematical model=20 continues to be based purely on the classical definition of cancer = as=20 uncontrolled proliferation of cells with the potential for = invasion and=20 metastasis, simplified for gliomas, which practically do not = metastasise.=20 Thus, the model defines the behaviour of gliomas in words and = mathematics=20 as follows:

This is a = classical=20 conservation-diffusion equation (Murray,=20 2003), in which c(x, t) defines the = concentration=20 of malignant cells at location x and time t, = D=20 (mm² day^-1) is the random motility = (dispersal)=20 coefficient defining the net rate of migration of the tumour = cells,=20 (per day) represents the net = proliferation rate=20 of the tumour cells (including mitosis and cell loss), K is = the=20 limiting concentration of cells that a volume of tissue can hold = (i.e.,=20 the carrying capacity of the tissue) and ² represents the dispersal operator, the = Laplacian, expressed mathematically as the sum of three second = derivatives=20 in space (Strang,=20 1991). The model has been adapted to use the BrainWeb Atlas = (Collins=20 et al, 1998) to accommodate an irregularly shaped = tumour=20 located anywhere within 3-dimensionally continuous heterogeneous = tissue=20 with differences in grey and white matter, anatomically accurate = to 1=20 mm³ (Swanson,=20 1999; Swanson=20 et al, 2003a). The model can accommodate different = velocities=20 of glioma cell motility in grey and white matter (Swanson,=20 1999) but, since the original MRIs were not available, this = feature=20 was not used in the present analysis.

Equation=20 (1) implies mathematically that the 'edge' of the visible = tumour=20 advances asymptotically as a 'traveling wave,' (Swanson,=20 1999) which expands radially and linearly, according to = Fisher's=20 approximation or Skellam's model (Shigesada=20 and Kawasaki, 1997): 3D6604125e2.gif=20

.=20 Although there is no true edge to an infiltrating tumour, such as = a=20 glioma, any point on the 'gradient' between the T1-Gd and T2=20 circumferences (Figure=20 3) moves as part of the 'traveling wave.' In general, as shown = schematically in Figure=20 3, proliferation () tends to drive the wave up (but = not above the=20 carrying capacity K) and dispersal (D) tends to = drive the=20 gradient centrifugally.

Figure 3
Schematic=20 representations of the relationships between T1-Gd and T2 = images and=20 the concentrations of tumour cells and of the effects of = increasing=20 the net rates of diffusion (D) and proliferation = () on the 'traveling wave' of = the 'edge' of=20 an enlarging glioblastoma. The thresholds of detection for = T1-Gd and=20 T2 MR images (solid lines) are estimated at 80 and 16%,=20 respectively, of the maximum cell density K in Equation=20 (1).

The=20 'gradient' between the T1-Gd and T2 images can be expressed in a = different=20 way, involving the ratio D/. This 'gradient' has not been = quantitatively=20 defined but can be approximated from the observations of Kelly=20 et al (1987), Kelly=20 (1993) and Dalrymple=20 et al (1994), who reported that the T1-Gd circumference = approximates the edge of the 'solid tumour' and that the T2 = circumference=20 represents not only the extent of oedema but also a zone of a low=20 concentration of 'isolated tumour cells.' We hypothesised that = these=20 circumferences might represent concentrations of tumour cells = equal to 80=20 and 16%, respectively, of the maximum concentration (Figure=20 3). That tumour cells extend much farther then even the = imageable=20 abnormality is evidenced by malignant cells being cultured by Silbergeld=20 and Chicoine (1997) from as far away as 4 cm. A close study of = other=20 solutions of the model Eq.=20 (1) finds a highly non-linear relationship between the ratio=20 D/ and the average radii of spheres = equivalent to=20 the volumes defined by T1-Gd and T2, r_T1 and=20 r_T2, respectively (Harpold=20 et al, 2007). The equation by no means is a simple = ratio of any=20 part(s) of the MR images, but includes fractional exponents of = ratios of=20 differences that make it highly non-linear.

Our model consists of the following two parts: = (1) Eq.=20 1, the spatio-temporal bio-mathematical formulation of the=20 proliferation and dispersal of the tumour cells, both visible = (detectable=20 by scans) and invisible (diffusing into the surrounding = normal-appearing=20 tissue), and (2) the use of Fisher's approximation ( 3D6604125e3.gif=20

) = to=20 estimate the time required for the tumour to expand from its = detectable=20 actual size at diagnosis to its size at death (Woodward=20 et al, 1996).

Data=20 analysis

Statistical = analysis was=20 performed (SPLUS,=20 1998), as noted throughout the text. From the pre-operative = MRI=20 volumes, we calculated the average radii of equivalent spheres of = the=20 T1-Gd and T2 volumes for each of the 70 patients (Figure=20 2). These became the actual data from which all of the rest of = the=20 calculations derive (see Results section). The mean T1-Gd radius = was 2.01=20 cm, the median 2.13 cm and the range from 0.86 to 3.26 cm. The = mean=20 T2-weighted radius was 2.89 cm, the median 2.84 cm, and the range = from=20 1.05 to 4.75 cm.

We used = other clinical=20 data provided to classify each actual patient according to the = extent of=20 resection. We found that there is no statistically significant = difference=20 between the means of the T1-Gd radii of the 7 biopsy patients = (2.03 cm)=20 and the 31 STR patients (2.20 cm, P=3D0.43) or of the 7 = biopsy and 32=20 GTR patients (1.81 cm, P=3D0.44), in large part because of = the small=20 number of biopsy patients (N=3D7); but there is a = significant=20 difference between the means of the T1-Gd radii of the combined 38 = biopsy=20 and STR patients (mean=3D2.17 cm) and the 32 GTR patients = (P=3D0.01=20 using Student's t-test). Justification for this combination = of=20 patients is provided in the Results section.

From the diagnostic MRI tumour volumes, we have = also=20 calculated the actual ratio, D/, for each of the 70 patients and = found the=20 median to be 9.83 mm², the mean 10.52 mm² = and the=20 range from 0.24 to 35.92 mm². There is no significant=20 difference in the distributions of the ratio D/ between actual patients who had had = steroids=20 for a few days (N=3D22) and those who had not had steroid=20 (N=3D45) before their MRIs (Figure=20 4A, P=3D0.75 by Kolmogorov-Smirnov's test (K-S test)), = Watling=20 et al, (1994) having used steroids for a longer time to = see any=20 effect. There is also no difference in the distributions of the = ratio=20 D/ for the actual biopsy, STR and GTR = patients (Figure=20 4B, K-S test P=3D0.675).

Figure 4
Distributions=20 (%) of actual values of D/, in (A) for 22 = patients having had=20 steroid (asterisks) and for 45 patients not having had = steroid=20 (circles) before their MRIs, and in (B) for 38 = patients=20 having had biopsy or subtotal resection (circles) and 32 = patients=20 having had gross total resection (asterisks), as defined by = the=20 presence or absence of residual tumour in post-operative = enhanced=20 CT.

In the=20 absence of a second set of MRIs to define the actual velocity for = each=20 patient, we had to resort to estimates, applying the average = of 0.012 per day (Tracqui=20 et al, 1995; Woodward=20 et al, 1996) to the calculated actual ratios of = D/ to estimate D for each = actual patient.=20 We could then apply Fisher's approximation ( 3D6604125e4.gif=20

) = to=20 calculate an approximate radial velocity of expansion of the = 'edge' of=20 each tumour. We found the median velocity to be 0.075mm = day^-1,=20 the mean to be 0.0723 mm day^-1, and the range from = 0.0118 to=20 0.1438 mm day^-1. Since the ratio of D/ is known from the MRIs of each = patient, any=20 different value of would induce a corresponding change = not only in=20 this estimated value of D but also in the velocity = v. We=20 accepted the average size at death as equivalent to a sphere of = radius 3=20 cm (Woodward=20 et al, 1996) and allowed the model to project each = tumour at=20 its estimated velocity v and to determine the time required = to grow=20 from its actual diagnostic T1-Gd radius to the assumed fatal = radius of 3=20 cm. These times were the predicted virtual durations of survival, = which=20 could be calculated not only without treatment but also by = superimposing=20 virtual resections of any extent desired. Any or all of these = predicted=20 survival times could be compared with the appropriately matched = actual=20 durations of survival (see Results section).

Estimates of the predicted survival time for each = of the 70=20 patients were made following a variety of 'virtual' treatments: = (1) biopsy=20 only (no resection) and STR (treated as no resection, as in Woodward=20 et al, 1996), (2) resection of 100% of the T1-Gd volume = (the=20 smallest 'virtual GTR' that could have accomplished the CT-proven = result)=20 and (3) resection of 125% of the T1-Gd volume (an arbitrarily = chosen=20 estimate of a more extensive 'virtual GTR' that the surgeon could = have=20 easily undertaken). Groups 2 and 3 were predicted to develop a = range of=20 predictions for comparison with the actual GTR patients, in this = data set=20 defined by the absence of residual tumour on post-operative = enhanced CT.=20 It seems obvious that the exact extent of resection beyond = complete=20 removal of the CT-enhancing tumour was not determined in any = actual=20 patient and was not likely to have been the same in each patient. = Virtual=20 resection was simulated, as by Swanson=20 et al (2003a), by imposing a cell concentration of zero = in the=20 resected region and allowing the computer to proceed with the=20 model-predicted tumour growth defined by Eq.=20 (1).

3D""=20

RESULTS

We begin by presenting Figure=20 5, summarising the actual durations of survival for the 7 = patients=20 subjected to biopsy only, the 31 patients subjected to STR and the = 32=20 patients subjected to GTR as defined by post-operative enhanced = CT. Since=20 there were so few biopsy patients and since there was no = difference in=20 actual durations of survival compared with the STR patients=20 (P=3D0.671, ²-test), consistent with = previous=20 studies of patients receiving Bx or STR (Lacroix=20 et al, 2001) these two actual groups were combined in = further=20 comparisons. These actual results (Figure=20 5) are meaningless until matched with appropriate controls = (see Figure=20 6 below).

Figure 5
Actual=20 survival curves for 7 patients subjected to biopsy (BX, = pluses), for=20 31 subjected to subtotal resection (STR, triangles) and for = 32=20 subjected to gross total resection (GTR, asterisks) as = defined by=20 presence or absence of residual tumour on post-operative = enhanced=20 CT. Inset shows a close-up of the survival curves near the = median=20 survival times of 36.5 and 62 weeks,=20 respectively.

Figure 6
(A)=20 Survival curves for actual patients (asterisks) and virtual = patients=20 (squares) subjected to biopsy or subtotal resection (BX/STR, N=3D38). Inset shows a = close-up of the=20 survival curves near the median survival times of 32.4 and = 36.5=20 weeks. (B) Survival curves on a longer time scaling = following=20 gross total resection (GTR, N=3D32) in actual = patients=20 (asterisks) defined by the absence of residual tumour on=20 post-operative enhanced CT. The virtual patients (matched to = actual=20 pre-operative T1-Gd volume and D/ ratio derived from the T1-Gd = and T2=20 volumes) were subjected to no resection (BX/STR, squares), to resection of 100% of = the T1-Gd=20 volumes or radii, r_T1 (circles) and to = resection=20 of 125% of the T1-Gd volumes or radii, 1.25 = r_T1=20 (diamonds). Inset shows a close-up of the survival curves = near the=20 median survival times of 44.9, 55, 62 and 66.9=20 weeks.

In Figure=20 6, we have paired the virtual and real patients for = appropriate=20 comparisons. It should be noted that these comparisons are being = made=20 between real and virtual patients with exactly the same = pre-operative MRI=20 characteristics defined by the actual size (T1-Gd volume) and the = actual=20 ratio D/ (derived non-linearly from the = actual T1-Gd and=20 T2 volumes). The actual survival curves are represented as = asterisks and=20 the virtual survival curves as open boxes of various shapes = corresponding=20 to the various extents of resection. Although the degree of = overlap=20 represents the degree of accuracy of prediction, enlargements are = provided=20 in the insets to reveal the slight discrepancies present. In Figure=20 6A, the untreated virtual patients are compared with the = actual=20 patients subjected to biopsy and STR. The curves are practically = identical=20 (no significant difference P=3D0.693, ²-test), as predicted by = Woodward=20 et al (1996) and as found by Lacroix=20 et al (2001). In Figure=20 6B, the untreated and treated virtual (100 and 125% = resections,=20 respectively) and actual patients subjected to GTR are compared. = As=20 expected, the actual curve lies between the 100 and 125% 'virtual' = resections. Neither of the virtual curves was significantly = different from=20 the actual (P=3D0.695, 0.685, respectively, ²-test).

3D""=20

DISCUSSION

The real power of any model is its ability to = provide=20 meaningful predictions (Murray,=20 2003), as seen most graphically in Figure=20 6. The present mathematical model, although requiring a = computer for=20 solutions, is relatively simple and based biologically on the = classical=20 definition of cancer/gliomas. It = reduces the=20 significant variables to only two, the 'final common paths' of net = proliferation () and dispersal (D) rates. = All of the=20 genetic-metabolic abnormalities that are being revealed to = underlie the=20 mysteries of glioblastomas lie upstream but their downstream net = effects=20 can be quantified by the model. The model has allowed us to = provide almost=20 exact virtual controls, matched for pre-operative size (T1-Gd) and = D/, untreated and treated, for each = patient, and=20 to make accurate predictions of survival times following a broad = range of=20 surgical resections, as shown in Figure=20 6. As Woodward=20 et al (1996) predicted mathematically and as shown = actually in=20 Figure=20 5, there is no difference in duration of survival between real = patients receiving biopsy only and those receiving STR = (P=3D0.671,=20 ²-test). As for the = evaluation of the=20 effectiveness of GTR, a subject of almost limitless controversy = (Nazzaro=20 and Neuwelt, 1990; Kelly,=20 1993; Kreth=20 et al, 1993, 1999;=20 Kowalczuk=20 et al, 1997; Hess,=20 1999; Keles=20 et al, 1999; Lacroix=20 et al, 2001), we would emphasise Figure=20 6, where the virtual 'untreated' controls (asterisks) for each = subset=20 (biopsy/STR and GTR) differ in duration = of=20 survival (32.4 vs 44.9 weeks). There being no difference in = the=20 distributions of the actual ratios D/ (Figure=20 4B), the only remaining reason for these differences in = survival is=20 that the individually matched T1-Gd sizes differ for the two = subsets, as=20 can also be seen in Figure=20 2. Both Figure=20 2 and Figure=20 6 suggest that the 'better' patients were chosen for=20 GTR.

Thus, comparisons = between the real=20 (unmatched) subsets (Figure=20 5) are, in our opinion, not so appropriate as comparisons = between the=20 real and virtual pairs of matched subsets (Figure=20 6). We conclude that the 25.5 weeks advantage that actually = GTR=20 appears to provide over the actual (but unmatched) BX/STR controls (62 vs 36.5 weeks in Figure=20 5) can be shown by the present mathematical model to be = half-accounted=20 for by the inadvertent selection of many patients with smaller = tumours to=20 be subjected to GTR (c.f., Figure=20 2). If one compares the real GTR patients with their virtual = controls=20 matched by pre-operative imaging characteristics (Figure=20 6B), one can see that the advantage is only about 17.1 weeks, = this=20 being the difference between the median survival times of the = actual 32=20 patients (62 weeks) and their 32 model-matched (biopsy only) = virtual=20 controls (44.9 weeks). A comparison of the two sets of virtual = controls=20 shows that those for the GTR patients survived 12.5 weeks longer = than for=20 the BX/STR patients (median=3D44.9 = weeks in Figure=20 6B vs 32.4 weeks for Figure=20 6A). Clearly, the 'better' patients were chosen for=20 GTR.

We believe that the = present=20 mathematical model is a powerful tool for understanding and = interpreting=20 the images displayed by different MR imaging sequences of = glioblastomas=20 since it successfully combines two of the pre-operative imaging=20 characteristics of a glioblastoma, the 'size' and the 'gradient' = of=20 concentration of glioma cells between the 'edges' revealed by = T1-Gd and T2=20 weightings of the MRI (Kelly=20 et al, 1987; Dalrymple=20 et al, 1994). A significant part of the power of the = model is=20 the use of the actual sizes shown in the MRIs (Figure=20 2). The prognostic importance of the 'size' has also been=20 controversial, a few (Yamada=20 et al, 1993; Xue=20 and Albright, 1999; Jeremic=20 et al, 2004) reporting it to be significant but most = (Reeves=20 and Marks, 1979; Andreou=20 et al, 1983; Wood=20 et al, 1988; Hammoud=20 et al, 1996; Kowalczuk=20 et al, 1997; Keles=20 et al, 1999; Kreth=20 et al, 1999; Lacroix=20 et al, 2001) failing to find any relation. Xue=20 and Albright (1999) make the point that the volume should be=20 calculated as accurately as possible (i.e., planimetrically as the = sum of=20 the areas), as we have done, not modelled simply as spheres or = ellipsoids=20 with only the diameter(s) measured. The failures are easy to = understand=20 because of the biologic heterogeneity of all groups of patients so = far=20 studied, that is, with inherently different but unknowable rates = of=20 proliferation and invasion. These biologically fundamental rates = have been=20 subsumed within the histopathologic diagnosis (e.g., 'generally=20 circumscribed, slowly growing,' 'slow growth, and diffuse = infiltration,'=20 'rapid, infiltrative growth' etc.) (Kleihues=20 et al, 2002) and have previously not been more = specifically=20 considered, much less estimated. In fact, these turn out to be the = most=20 significant of all of the factors, even better than age and KPS. = It is not=20 surprising that such previously unidentified phenotypical = diversity would=20 be impossible to control for using standard statistical analysis = of groups=20 of patients without the advantage of a mathematical model to = characterise=20 virtual controls for each individual patient, untreated or = subjected to=20 various extents of resection.

Another=20 reason for previous failures (Reeves=20 and Marks, 1979; Andreou=20 et al, 1983; Wood=20 et al, 1988; Kreth=20 et al, 1993; Hammoud=20 et al, 1996; Keles=20 et al, 1999; Lacroix=20 et al, 2001) to find an effect of 'size' of the tumour = relates=20 to the inherent ambiguity of any ratio; that is, the ratio = D/ can remain the same if both = D and=20 change in the same direction. = Furthermore,=20 statistical techniques generally look for linear relationships, = whereas=20 the relationship we have found for D/, which we have associated with the = T1-Gd and T2=20 radii, is highly non-linear.

These=20 ambiguities and complexities relate to the invisible infiltration=20 (dispersal) of glioma cells. At one theoretical extreme, for = example, if=20 there had been no dispersal, both the volume and the radius of the = solid=20 tumour could have increased exponentially with time, as postulated = by Blankenberg=20 et al (1995) and Haney=20 et al (2001). With dispersal, however, the infiltrating = cells=20 become invisible and represent an unknowable fraction of the = total,=20 leaving the mass of those cells that remain visible (the = complementary but=20 still unknowable fraction of the total) to appear to grow quite=20 differently: the radius actually increases linearly with time = (according=20 to Fisher's approximation) and the volume as a cubic. Neither = radius nor=20 volume increases exponentially as the classical model with a = constant=20 volume-doubling time demands. This is mathematically true even = though the=20 total number of tumour cells, both visible and invisible, may = continue to=20 increase exponentially. The measurable difference between = exponential and=20 cubic growth is relatively slight over the time intervals = typically=20 available clinically, but the consequences for longer term = predictions are=20 quite marked. The concept may be subtle but the effect is real and = important. For another example, the 'volume-doubling time' that = can be=20 calculated at any moment in our model is not constant but = decreases=20 continuously, as actually shown in the report by Haney=20 et al (2001). The concept of a volume-doubling time is=20 meaningless for infiltrating tumours such as = gliomas.

Evidence that T2 reveals not just oedema is seen = in Figure=20 4, where the distribution of the values of D/ is the same whether steroid had = been given or=20 not. Of course, the short duration of administration of steroid in = these=20 patients may not have allowed any significant decrease in the T2 = image (Watling=20 et al, 1994). The evidence in Figure=20 4 suggests that the postulated difference between the = hypothesised=20 concentrations of tumour cells visualised by T1-Gd (80% of = maximum) and T2=20 (16% of maximum) images may be very close to true, even though the = reports=20 of Kelly=20 et al (1987), Kelly=20 (1993) and Dalrymple=20 et al (1994) were quantitatively far from=20 convincing.

3D""=20

CONCLUSIONS

Our mathematical model rests on only two = biological=20 factors, the net rates of proliferation () and diffusion (D), which = are the 'final=20 common paths' controlling cancerous growth. The comparison of the = virtual=20 results predicted by this mathematical model with the real results = of 70=20 patients has revealed the power of the model as a 'proof of = principle' not=20 only in predicting durations of survival following a wide range of = resections but also in accommodating the actual size of each = patient's=20 glioblastoma and the estimated 'gradient' of cells migrating from = the=20 'edge' of the tumour. A major deficiency of the present report is = the=20 absence of a second MRI before treatment, forcing us to use the = average=20 of 0.012 per day (Tracqui=20 et al, 1995) to calculate an approximate D and = an=20 approximate velocity of expansion for each patient. To paraphrase=20 Archimedes' request for a fulcrum and a strong lever, given a = second MRI=20 without intervening treatment, the model would have allowed = measurement of=20 the actual velocity (v) and calculation of actual rates of=20 dispersal (D) and proliferation () for each actual patient and the = direct=20 calculation of their matched virtual controls without all of the=20 approximations that we had to resort to. With that information we = should=20 also be able to determine not only the degree of effectiveness of = each=20 patient's individualised therapies but also of the specificity of = each=20 therapy for affecting some step in the intricate metabolic = pathways=20 controlling D and/or . Although we could not calculate = D and=20 explicitly for each of the = patients, the ratio=20 D/ fell within the range expected for=20 glioblastomas. We also need more data concerning the size at = death,=20 attainable by late follow-up MRI and autopsy, to define more = accurately=20 the end point, the postulated 'fatal tumour = burden.'

Acknowledgements

KRS=20 acknowledges the support of the Charles A Dana Foundation, the = University=20 of Washington's Academic Pathology Fund and the Shaw Professorship = in=20 Neuropathology.

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