Predicting drug pharmacokinetics and effect in vascularized tumors using computer simulation
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- Cite this article as:
- Sinek, J.P., Sanga, S., Zheng, X. et al. J. Math. Biol. (2009) 58: 485. doi:10.1007/s00285-008-0214-y
In this paper, we investigate the pharmacokinetics and effect of doxorubicin and cisplatin in vascularized tumors through two-dimensional simulations. We take into account especially vascular and morphological heterogeneity as well as cellular and lesion-level pharmacokinetic determinants like P-glycoprotein (Pgp) efflux and cell density. To do this we construct a multi-compartment PKPD model calibrated from published experimental data and simulate 2-h bolus administrations followed by 18-h drug washout. Our results show that lesion-scale drug and nutrient distribution may significantly impact therapeutic efficacy and should be considered as carefully as genetic determinants modulating, for example, the production of multidrug-resistance protein or topoisomerase II. We visualize and rigorously quantify distributions of nutrient, drug, and resulting cell inhibition. A main result is the existence of significant heterogeneity in all three, yielding poor inhibition in a large fraction of the lesion, and commensurately increased serum drug concentration necessary for an average 50% inhibition throughout the lesion (the IC50 concentration). For doxorubicin the effect of hypoxia and hypoglycemia (“nutrient effect”) is isolated and shown to further increase cell inhibition heterogeneity and double the IC50, both undesirable. We also show how the therapeutic effectiveness of doxorubicin penetration therapy depends upon other determinants affecting drug distribution, such as cellular efflux and density, offering some insight into the conditions under which otherwise promising therapies may fail and, more importantly, when they will succeed. Cisplatin is used as a contrast to doxorubicin since both published experimental data and our simulations indicate its lesion distribution is more uniform than that of doxorubicin. Because of this some of the complexity in predicting its therapeutic efficacy is mitigated. Using this advantage, we show results suggesting that in vitro monolayer assays using this drug may more accurately predict in vivo performance than for drugs like doxorubicin. The nonlinear interaction among various determinants representing cell and lesion phenotype as well as therapeutic strategies is a unifying theme of our results. Throughout it can be appreciated that macroscopic environmental conditions, notably drug and nutrient distributions, give rise to considerable variation in lesion response, hence clinical resistance. Moreover, the synergy or antagonism of combined therapeutic strategies depends heavily upon this environment.
Mathematics Subject Classification (2000)92C45
1 1 Introduction
While drug resistance to solid tumors is often a consequence of genetic factors, such as upregulation of anti-apoptotic proteins or overexpression of efflux mechanisms, factors at coarser physiological scales may also profoundly influence tumor therapeutic response . A tumor is a heterogeneous three-dimensional composite of fibrous and connective tissues, stromal components, vasculature, and multiple clones of cancer cells. Atop this intrinsic heterogeneity is layered the anatomical and functional irregularity of tumoral vasculature, characterized by erratic flow, collapsed vessels, diminished oxygen tension, and a large mean tissue-to-vessel distance [5,29,30,34, 54]. As a consequence, the tumor microenvironment is highly variable, marked by gradients of nutrient and oxygen, resulting in regions of hypoxia, acidity, and necrosis, and heterogeneous proliferation. In order for an anticancer agent to be effective, it must extravasate, diffuse through lesion tissue, and be transported into cells, where it must bind to its target and induce cell apoptosis or mitotic inhibition. The tumor environment is not conducive to these processes . The vessel bed’s blood flow and spatial distribution hinder uniform extravasation, calling into question the capability of drug molecules to adequately distribute throughout tissue. Recent experiments in vitro [38,73,81] and in vivo [40,57] demonstrate limited drug penetration through tumors, especially highly protein-bound molecules like doxorubicin and paclitaxel. Once a drug molecule has traversed lesion tissue from its point of extravasation and is presented to a cancer cell, the path from extracellular space to intracellular target is fraught with difficulties ranging from protonation due to the acidic environment, which renders anthracyclines incapable of traversing membrane, to intracellular removal by drug efflux pumps, to cellular processes that effect DNA repair and drug clearance [3,15,31,65,71,79]. In addition to pharmacokinetics, drug pharmacodynamics is equally impaired. Significant hypoxia and hypoglycemia throughout may induce cell quiescence, reducing the efficacy of cell-cycle chemotherapeutic agents like doxorubicin and cisplatin [18,19]. Hypoglycemia causes the glucose-regulated stress response detrimental to the action of topoisomerase II-directed drugs like doxorubicin [41,49,64,68].
The heterogeneity and three-dimensionality of the tumoral environment presents a challenge to drug assessment, both during development and in the clinic. Whereas a particular drug may show marked activity against a particular specimen in vitro, its potency may vanish or become far less reliable in vivo. This is evidenced by the differential between positive predictive accuracy of in vitro-assisted therapy selection (around 70%) and negative predictive accuracy (around 90%), a situation not remarkably changed over the years [26,27]. Supraoptimal delivery of drug to cultured cells eliminates the gauntlet of biobarriers in vivo described above, precluding the variability they induce. A drug that consistently works in vitro can therefore be expected to only sometimes work in vivo. Unraveling the myriad interactions of therapeutic determinants within the complex three-dimensional tumoral environment is evidently difficult, resulting in high costs of drug development and patient suffering.
Perhaps the crystal ball we are attempting to build is incomplete when made only of glass typically found in experimental labs; computer (in silico) simulations based on mathematical modeling and calibrated with experimental data might fulfill a key aspect of the lens. A significant capability of in silico experimentation (including simulated assays) is the complete control over and monitoring of the simulated in vivo tumor environment. Moreover, computer modeling can create hypothetical environments and conditions impossible to achieve otherwise, the study of which is nonetheless instrumental in unraveling disease and drug mechanisms. This expansive control, founded upon an adequately mechanistic mathematical basis, could facilitate the discovery of hypotheses as to why certain drugs or therapeutic strategies would or would not be effective, potentially on a patient-by-patient basis. The relative ease and cost-efficiency of performing simulations could furthermore enable a thorough investigation of strategies, revealing the optimal among them. The judicious combination of this burgeoning technology with the capabilities of the wet-lab is an attractive development in both drug discovery and the clinical management of cancer leading to the easing of patients’ burdens.
The past two decades have witnessed explosive growth in the mathematical and computational modeling of vascular and avascular tumors [1,4,7–9,13,36,43,44,56, 82]. As the basic elements of tumoral growth models have matured, specialized treatments of angiogenesis and flow [2,46,47,69], drug delivery and response [33,40,52,66,80], and effects of the tumoral microenvironment [23,45] have been developed. Some simulation and analysis has provided great insight using one-dimensional modeling (employing cylindrical or spherical symmetry) without discrete vasculature. More recently, powerful numerical methods have been developed so as to simulate multidimensional complex morphological progression and its relation to cell phenotype and the microenvironment involving, for example, nutrient and biomechanical tissue response [23,45,82].
In the present paper we examine therapeutic efficacy of two common drugs, cisplatin and doxorubicin, in relation to phenotypic and microenvironmental conditions. Model parameters govern extracellular drug/tissue diffusivity; cellular uptake, efflux, and metabolism; cellular density; and the effect of nutrient heterogeneity on drug action. Simulations are performed in a two-dimensional (non-symmetric) setting employing discrete vasculature, which enables the incorporation of morphological and topological influence on drug and nutrient distributions. The effect of these distributions on therapeutic efficacy is of special interest. Sinek et al.  had earlier performed a similar investigation; however, the pharmacokinetics and pharmacodynamics (PKPD) component was rudimentary, assuming one homogenous lesion compartment and not based upon experimentally acquired parameter values. In the present work we implement an extensive multi-compartment PKPD component whose parameter values are calibrated via published experimental data. This enables a comparison of the tissue and cell-level drug dynamics of the two drugs, and facilitates the generation of hypotheses to explain their in vivo characteristics. We ask that the reader consider that if doxorubicin and cisplatin were discovered only today, the simulations herein could be seen as providing insight into their anticipated in vivo performance, potentially streamlining and reducing costs of development. Indeed, the methodology presented herein could, with additional development, be applied to both established and nascent drugs to the end of refining clinical trials and assisting in clinical therapeutic strategy to improve patient comfort and survival.
2 2 Mathematical model and parameters
2.1 2.1 Model description
Both cisplatin and doxorubicin pass through cell membrane according to k12 (which includes possible pump and transporter activity, as do all other rates). From there, the drugs may efflux according to k21 or may bind to DNA according to k23. The kinetics differ from here for the two drugs. Cisplatin may be removed according to the rate k3, which destroys the functioning of the drug and repairs the DNA . Doxorubicin, however, has an off rate given by k32, and moreovermay be sequestered and released by lysosomes according to k24 and k42 [3,31,59]. Although lysosomal flow to membrane and exocytosis of sequestered drug plays a role in some drug resistant cell lines, we are not necessarily modeling drug resistance via this function, and so assume this process to be negligible in accordance with . On the other hand, we are concerned with the quantity of drug lysosomes can sequester, as this contributes to the cellular uptake of drug, and hence, its penetration characteristics.
2.2 2.2 Pharmacokinetics model parameters
A complete summary of baseline pharmacokinetics and pharmacodynamics parameters
Cell volume(fL cell−1)
Cell density (cells mL−1)
Nutrient/ECM diffusivity (µm2 min−1)
Drag/ECM diffusivity (µm2 min−1)
Nutrient metabolism (min−1)
Drug uptake (min−1)
Drug efflux (min−1)
Drug-DNA binding (min−1)
Drug-DNA release (min−1)
Drug-DNA repair (min−1)
Lysosomal sequestration (min−1)
Lysosomal release (min−1)
Drug-DNA capacity (fmole)
Phenomenological PD parameter
Phenomenological PD parameter
Nutrient effect parameter
Cisplatin parameters We begin with cisplatin, setting k24 and k42 to 0 since we assume only three compartments, and k32 to 0 since we assume the repair rate k3 is the dominant removal rate of DNA-bound drug. k3 is next obtained as follows. In experiments performed by Sadowitz et al. , adducts per million nucleotides on isolated peripheral blood mononuclear cell DNA fall from 75 to 5 and 185 to 40 in 2 h in two different experiments. Thus, assuming the exponential repair model ˙s3 = k3s3, we calculate the repair rate to be about 0.015 min−1. An initial estimate of k23 is then made as follows. Sadowitz shows that for 7 µM cisplatin, in 2 h peripheral blood mononuclear cells accumulate from about 25 (non-thiol-blocked cells) to 175 (thiolblocked cells) adducts per million nucleotides. Assuming that DNA consists of about 1.25E6 kbp, this converts to from 1.04E-4 to 7.27E-4 fmoles of Pt docked on the DNA (1 atom/adduct). Neglecting the cell membrane and supposing DNA to be exposed directly to the drug, we have the ODE ˙s3 = 7λ23 − k3s3, where λ23 is a clearance parameter (fL min−1). The solution is s3 = 7(λ}23/k3)(1 − exp(−k3t)). Substituting values of k3 = 0.015 min−1, t = 120 min, and 1.04E-4 ≤ s3 ≤ 7.27E0-4 fmole yields 0.27 ≤ λ23 ≤ 1.9 fL min−1. To convert this to a rate we use the relation k23 = λ23/Vc, arriving at 5.19E-4 ≤ k23 ⩽ 3.65E-3 min−1. The assumption that DNA was exposed directly to the cisplatin solution means that this rate is only a bootstrap approximation and must be refined. We note that the extremely low ratio of adducts per kbp implies that the saturation capacity of DNA with respect to cisplatin is never approached, and so set sm to ∞.
Doxorubicin parameters Proceeding to doxorubicin we first obtain an acceptable range for k12 and k21 from the literature. For a variety of anthracyclines, including doxorubicin, initial estimates of cell membrane permeability P are taken from experiments with SU-4 and SU-4R wildtype and resistant human lymphoma cells , from experiments with EHR2 and EHR2/DNR+ wildtype and resistant Ehrlich ascites tumor cells [15,16], and from experiments with MDA-468 breast cancer cells . The range reported is 2.4 ≤ P ≤ 1000 µ min−1. The relation k12 = PAc/Vc, where Ac represents the cell membrane area, can then be used to arrive at an initial range of 1.4 ≤ k12 ≤ 600 min−1, which will be refined later. In the case of passive diffusion, k21 = k12. We note that these values are far larger than those obtained for cisplatin previously. More generally, it has been remarked that cell membrane permeability for cisplatin is much lower than for doxorubicin, etoposide, and vinblastine, although all four drugs are thought to enter cells by passive diffusion .
We next turn our attention to DNA-binding affinity. Given the great DNA affinity of the anthracyclines, saturability of the DNA must be taken into account, requiring an estimate of sm. There is evidence a typical anthracycline molecule intercalation occludes from 3 to 10 binding sites in a manner that cannot be corrected exactly by a factor [48,59,74]; however, to a first approximation we assume that such a correction can be applied. Demant and Friche  report a DNA binding site concentration of about 5 mM within a cell volume of 1000 fL, yielding 5 fmoles of sites. A low value of 0.7 fmoles is obtained by using our assumed value of 1.25E6 kbp and the reported site exclusion parameter of about 3 from Rizzo et al. . Tarasiuk et al.  find that the DNA of human lymphocytes is comprised of about 6.0E6 kbp and that one intercalating molecule of doxorubicin requires 10 base pairs. Thus, Tarasiuk’s data implies a factor-corrected quantity of 1 fmole of binding sites, which we take as a representative value of sm.
DNA binding kinetics of the anthracyclines is nontrivial, perhaps requiring multiple steps and demonstrating sequence specificity [58,59]. Bearing this in mind, as an approximation it will suffice to assume non-specific, one-step binding and unbinding according to the chemical reaction A representative value for the binding coefficient in the above equation for doxorubicin is reported as kon = 4.2E8 M−1 min−1 and a value of the unbinding coefficient (identical with k32) as koff = 1800 min−1 . From kon we calculate a clearance parameter (as with cisplatin) given as λ23 = konsm = 4.2E8 fL min−1 (being cautious with the scales of our dimensions). k23 can then be calculated as λ23/Vc, given in Table 1.
We next turn our attention to the rates k24 and k42 governing lysosomal sequestration. Experiments by Hurwitz et al.  using U-937 myeloid leukemia cells and their dox-resistant variant U-A10 show that the ratio of DNA-bound to lysosomally sequestered drug is about 3 (Hurwitz uses daunorubicin, an anthracycline related to doxorubicin). In our modified model equations with all other parameters set as described above, the amount of sequestered drug at equilibrium is dependent only upon the ratio k24/k42. This ratio furthermore does not affect the equilibrium quantity of DNA-bound drug. Arbitrarily selecting k24 = 1 min−1, we find that the appropriate DNA-bound to lysosomally sequestered ratio is obtained by setting k42 to 0.007. Considering that lysosomalmembrane permeability is quite high , the lysosomally bound drug must achieve equilibrium quickly, which can be modified by changing k24 while keeping the ratio k24/k42 constant. We find that increasing k24 by a factor of 10 reduces the time required for the system (Eqs. 2) to achieve 95% of equilibrium value (max95) to about 300 min, below which further increases in k24 only reduce this time negligibly. Thus, we set k24 = 10 and k42 = 0.07.
To refine our initial range of k12 and k21, we use the modified version of Eqs. (2) to compare our simulated monolayer uptake profiles of total intracellular drug with those of DeGregorio et al.  using human Ewing’s sarcoma and rhabdomyosarcoma cells. At 5.40 min−1 both uptake profiles and equilibrium values compare favorably at three test concentrations.
Drug-tissue diffusivity The last pharmacokinetics parameter values needed are the diffusivities Ds of cisplatin and doxorubicin through tumor interstitium. Formolecules of their size (dox M.W. = 544, cis M.W. = 300), diffusivity should be about 30,000 µm2 min−1 [53,70]. However, doxorubicin faces particularly severe barriers due to binding to extracellular constituents such as hyaluronic acid [37,38], and its diffusivity in some tissues has been estimated to be as low as 1000 µm2 min−1 , which we take as our baseline value.
2.3 2.3 Pharmacodynamics model parameters
In order to calibrate the pharmacodynamics model (Eq. 4), we use in vitro data of Levasseur et al.  with A2780 ovarian cancer cells exposed in monolayer to both doxorubicin and cisplatin over a range of times and concentrations. We assume the previously discussed modified pharmacokinetics model along with the values derived, and simulate Levassuer’s exposures followed by approximately 24 h of drug washout in drug-free medium (s1 is set to 0). During this time, DNA-bound AUC is calculated. These data are then used in conjunction with Levasseur’s surviving fraction data to fit the parameters A and m in Eq. (4) using Microsoft Excel (Fig. 1b). During this process, nutrient is assumed plentiful (n = 1), thus bypassing the nutrient effect.
3 3 Computational simulations
3.1 3.1 Non-dimensionalization and numerical methods
3.2 3.2 In silico experiments
Three simulated tumors are grown using Zheng et al.’s model . Each lesion represents one replication of each experiment.
The pharmacokineticsmodel (Eqs. 5) is used to deliver drug to the lesions. In each case we hold the intravascular concentration of drug sv constant for 2 h, then set it to zero for eighteen more hours to allow washout.
DNA-bound AUC is computed by re-dimensionalizing s3 and time and integrating using Matlab. The result is then used in the pharmacodynamics model (Eq. 4) to compute cell inhibition. Nutrient (Eq. 1) is relevant when the nutrient effect is employed.
Each lesion is produced based upon the same set of growth and vasculature parameters (see Zheng et al. for a complete description), but randomness in the angiogenesis algorithm and slightly different initial shapes produce different vasculatures and morphologies. It is assumed that a tumor in vivo does not grow or regress appreciably during the 20-h course of the therapy we are attempting to simulate, hence we stop tumor and vascular growth during our in silico therapies. Intravascular concentrations are calibrated in each case to produce a total cellular growth inhibition of 50%. This concentration is referred to as the IC50. The sharp “square wave” delivery of drug is perhaps a caricature of clinical bolus administration, but it allows for consistent analysis and comparison of results.
Our first set of experiments compares DNA-bound drug AUC distributions of doxorubicin and cisplatin under the baseline conditions in Table 1. We furthermore show the homogenizing effect of doxorubicin retention on final DNA-bound AUC . We next investigate the impact of inhibition heterogeneity on dosing requirements, paying particular attention to the nutrient effect for doxorubicin under baseline conditions and improved penetration by, for example, removing hyaluronic acid [37,38]. In our third set of simulations we more deeply investigate the effect of doxorubicin penetration therapies under three circumstances: baseline tumor density, high tumor density, and baseline tumor density with Pgp efflux activity. These are chosen because they demonstrate a spectrum of possibilities due to their effect on cellular drug uptake. High tumor density increases uptake, while Pgp efflux decreases it. In order to simulate increased penetration, we increase Ds for doxorubicin from its baseline value to 5000 µm2 min−1 for a moderate increase, and 30,000 for the maximum increase, thus matching the performance of cisplatin. To simulate high tumor density we increase ρ by 50% to 1.5E9 cells ml−1. This has the effect of lowering the interstitial fraction F to 0.22, which in turn increases k′12 and k′21 while leaving all other rates unchanged. Pgp efflux is simulated by increasing k21 by a factor of 10, which has the effect of reducing all intracellular compartment concentrations by approximately the same factor. This is consistent with results of  that show Pgp activity can reduce intracellular concentrations of daunorubicin (an anthracycline related to doxorubicin) by up to a factor of 100. In our fourth and final set of experimentswe investigate permeabilization therapy with respect to cisplatin, whereby a detergent, such as digitonin, or electropermeabilization is used to increase the permeability of cell membrane [35,72]. We take an extreme case, increasing the rate constants k12 and k21 from baseline both by a factor of 100. Note that this does not increase the limiting intracellular or DNA-bound levels of drug attained in simulated monolayer, only the rate at which these come to equilibrium. Thus highly permeabilized, DNA-bound max95 is attained at 3.4 h of exposure; further permeabilization reduces this negligibly. For comparison, max95 is greater than 27 h for unpermeabilized cells. This therapy is simulated under both in vivo baseline and very high cell densities achieved by increasing the baseline density 75% to 1.75E9 cells ml−1. At this density, the interstitial fraction F drops to a mere 0.08. Both of these are further compared to monolayer results to probe the conditions under which in vitro assays can be used to predict clinical efficacy.
Although all treatments described are duplicated in each of the three in silico tumors, we display only representative plots with appropriate summaries of all data. The nutrient effect is only used where noted.
3.3 3.3 Results
First experiment We begin by examining DNA-bound AUC distributions at various times in the baseline simulated lesions (each lesion corresponding to a column, I, II, or III), shown in Fig. 2. From top to bottom, the times correspond to 2, 8, 14, and 20 h post-bolus initiation. Levels are normalized relative to the average AUC within viable lesion for comparison of heterogeneity. Although surrounding host tissue cells uptake and bind with drug differently than cancer cells, we make no distinction in these color plots; however, quantitative analytical results only consider DNA-bound drug within viable lesion. The two left column sequences (Lesions I and II) show doxorubicin AUC, while the rightmost column shows cisplatin. For both Lesions I and II, at 2 h doxorubicin AUC is seen to be about three times the average (dark red) in the vicinity of the vasculature, and almost 0 (blue) elsewhere. The distribution is only slightly more homogeneous by 8 h. By 14 h the heterogeneity has lessened, with the peaks close to the vasculature reaching only about 2.2. Finally, at the conclusion of washout 20 h after bolus initiation, the distribution has become much more homogeneous, with peaks only reaching about 1.7 times the average. In contrast, cisplatin distribution within Lesion III remains extremely homogeneous, right at the average, throughout the entire treatment.
The probability distributions at the bottom, corresponding to AUC at 20 h postbolus initiation, allow for a more quantitative comparison. The two corresponding to doxorubicin show much heterogeneity relative to cisplatin on the right. Using the leftmost distribution as an example, the average DNA-bound AUC is found to be 6.04 fmole min. 25% of tumor cells receive less than 1.66 fmole min each, while 25% of tumor cells receive more than 9.54 fmolemin. The remaining 50% of the tumor cells receive between these two values, a range of 7.88 fmole min. When normalized with respect to the average and expressed as a percent, this yields 131% (the interquartile range, or IQR), and gives a concise measure of distribution heterogeneity (the closer to 0, the more homogeneous). IQR’s are given at each of the other time points as well. All three tumors, despite varied lesion and vasculature morphologies, demonstrate similar results (not all shown). Doxorubicin AUC IQR’s typically lessen from about 250% at 2 h to 150% at 20 h; cisplatin AUC IQR’s drop from about 10 to 2%. Interestingly, in the run shown, the heterogeneity for cisplatin increases slightly in the last frame. This happens in one of the other two tumors as well.
Second experiment We next investigate the impact of drug and nutrient heterogeneity on cell inhibition distributions and IC50’s. Bolus administrations are simulated for cisplatin using baseline lesions exactly as in Fig. 2. The PD model (Eq. 4) is then used to calculate cell inhibition. For doxorubicin we use baseline lesions as well as lesions in which drug penetration therapy is applied. Experiments for dox are run both with and without the nutrient effect.
Mean ± SD of the IC50 and the logs of their ratios with respect to monolayer treatments for experiments to investigate the impact of drug and nutrient heterogeneity
0.482 ± 0.163
0.424 ± 0.138 (*p < 0.05)
1.34 ± 0.874
0.830 ± 0.261 (*p < 0.05)
Doxorubicin with penetration therapy
0.197 ± 0.0172
0.0511 ± 0.0371 (p < 0.05)
0.371 ± 0.0356
0.325 ± 0.0407 (*p < 0.05)
7.14 ± 0.0757
0.00529 ± 0.00462 (p > 0.05)
At the 5% significance level, one-tailed t tests show that the average log IC50 ratio is not greater than 0 for cisplatin, underscoring the homogeneity of its distribution. In contrast, out of the four experiments performed for doxorubicin from the combinations of nutrient effect and penetration therapy, three indicate that the average log ratios are greater than 0 at the 5% significance level. Within this group of four we can analyze the strength of the nutrient effect. For the baseline lesion, the nutrient effect increases the log IC50 ratio by 0.406 units (a factor of about 2.5). For the lesion with penetration therapy, the increase is 0.274 units (a factor of about 1.9). Paired t tests show that these differences are significant at the 5% level.
4 4 Discussion
In agreement with experimental observation, our simulations show that heterogeneities of drug, nutrient, and oxygen, caused in part by irregular vasculature and lesion morphology, exist and can significantly impact therapeutic results. Moreover, the sense and magnitude of their influence is not always intuitively obvious. A good example of this is that, despite its well-noted penetration difficulties, doxorubicin performs well clinically. Our simulations show that this may be somewhat explained by its retention in tissue removed from vasculature, causing homogeneity of exposure to increase long after the bolus has been terminated (Fig. 2). This phenomenon has been experimentally verified in with spheroids. Because of this, the resulting cell inhibition distribution is more homogenous than would otherwise be expected. On the other hand, cisplatin maintains an homogenous DNA-bound distribution at all times from bolus initiation to 20 h later, resulting in an extremely uniform cell inhibition distribution. This result, as well as the near equality of its IC50 and IC50,mono demonstrated in Table 2, has also been experimentally verified with spheroids [18,32,37].
While retention in tissue contributes to the performance of doxorubicin in vivo, Table 2 demonstrates that its heterogeneity of distribution contributes to increased serum drug concentrations to match the same cell inhibition in monolayer. In one case, the average amount of drug increases by nearly one log unit. It is reasonable to expect that heterogeneity of nutrient, resulting in hypoxia and hypoglycemia, should compound this problem for doxorubicin. Indeed, this is the case as can be seen by the approximate doubling of the IC50’s (0.482 µM vs. 1.34 and 0.197 vs. 0.371) when the nutrient effect is applied. By graphically and quantitatively showing corresponding cell inhibition distributions. Figure 4 offers further insight into these phenomena. It is easily seen that cell inhibition distributions are as heterogeneous as their corresponding DNA-bound AUC distributions, with areas of lesion removed from vasculature experiencing reduced cell inhibition. An examination of the probability plots in the upper block, corresponding to baseline lesions, shows that a full 24% of viable lesion undergoes no inhibition at all. It is clear from these graphs that penetration therapy greatly decreases heterogeneity of cell inhibition (and commensurately, IC50) as does removal of the nutrient effect. The latter may be clinically feasible through, for example, carbogen breathing or recombinant human Epo (rHuEPO) administration [63,75].
One puzzling behavior is that while both the color and probability plots demonstrate consistently increased heterogeneity brought about by the nutrient effect (as is evidenced by the broadening of the probability curves), the IQR actually decreases in the baseline case from 81 to 77%. This occurs with Lesions I and III as well. A solution to the mystery is obtained by noticing that, absent the nutrient effect, a large portion of the tumor either experiences no inhibition (about 24%, which is distant from the vasculature) or an already heterogeneous inhibition (about 31%). The remaining 45% receives a near homogeneous level of inhibition (the vertical portion of the curve), and this fraction corresponds to tissue close to the vasculature. Thus, the only significant heterogeneity that can be induced by the nutrient effect is within this fraction. Indeed, it is just this part of the curve that broadens in the second probability distribution, indicating greater heterogeneity, as expected. As the IQR is designed to measure heterogeneity somewhat more globally, it misses—in fact, misdiagnoses—the change occurring within this fraction.
In addition to dosing requirements, there is a second and subtler reason to consider heterogeneity of the inhibition distribution when treating clinical tumors. Heterogeneities in microenvironmental conditions have been linked to increased lesion fragmentation and invasiveness [12,25,39,45,55]. While the mechanisms underlying this phenomenon are complex, involving myriad protein signaling events and activities at the cellular level, they may at least partly rely on gross lesion effects, including those caused by drug administration.
That something along these lines is probably happening is demonstrated in Fig. 5b by the generally high IQR’s for the high density case versus the low IQR’s for the Pgp, and the large change in the high density triplet versus the Pgp triplet. Indeed, there is experimental evidence that Pgp and other transporters might facilitate drug transport further away from vasculature . This could be an important point when deciding upon appropriate therapies for tumors exhibiting different characteristics such as efflux mechanisms and relatively high or low densities. Any therapy involves risk. There may be less to gain under certain conditions, advising that the therapy not be performed or perhaps, that concomitant therapy be performed designed to optimize conditions.
The great homogeneity of both cisplatin AUC and cell inhibition demonstrated in Fig. 2 and Table 2 indicate that in vitro assays using this drug may have relatively high positive predictive accuracy. While our simulations do not yield enough resolution to claim that the results herein answer this question, the outcome shown in Fig. 6 is of interest. There is no statistical difference at the 0.01 significance level between improvement obtained by permeabilization therapy in the simulated tumors (even very dense ones) and monolayer. This compares favorably with the findings of  inwhich the improvement obtained via cisplatin permeabilization therapy in vitro is strongly reflected by the improvement in isolated lung perfusion in rats.
While it can and should be argued that the simulations herein fail to account for some (many!) critical aspects of tumor growth and drug response (such as clonal heterogeneity, cell phase sensitivity, and signaling pathways) and that parameter settings may in some cases be inexact, it should not be concluded that these shortcomings invalidate characteristics the simulations have revealed. Indeed, we have correctly post-dicted several results: that doxorubicin retention results in a more uniform AUC and cell inhibition than would otherwise be indicated by its penetration difficulties ; that Pgp and other transporters might facilitate drug penetration into lesion tissue ; that cisplatin achieves a highly uniform AUC, and its cellular in vitro monolayer inhibition can closely match that of in vitro spheroids [18,32,37]; and that improvement due to cisplatin permeabilization therapy in vitro has the potential to predict improvement in vivo . Through the model we have provided evidence that not only do macroscopic environmental conditions, namely, drug and nutrient distribution heterogeneity, potentially greatly impact therapeutic efficacy, but also that the outcome of therapeutic strategies can depend upon them in nonlinear and a priori unpredictable ways. The results of our third set of experiments (doxorubicin penetration therapy) provide an example. In light of this, it would be prudent to devote attention to factors residing at coarser and more global scales than solely the genetic.
One of our broader goals is to demonstrate how increasingly sophisticated in silico technology, driven by mathematical modeling and calibrated with experimental data, can and is being developed to provide an alternate investigative and clinical tool complementary to traditional methods [6,24,61,62,67]. It can well be imagined that were doxorubicin and cisplatin discovered today, the in vivo simulations herein presented could be used to anticipate their lesion- and cellular-scale pharmacokinetics, helping to refine clinical trial design and lower costs. In clinical application, the results could be used to guide therapeutic strategy. For example, any risks associated with doxorubicin penetration therapy could be minimized if it were known that the patient’s tumor were expressing Pgp or otherwise had lowered cellular uptake, according to the results given in Fig. 5. With further development, we anticipate that in silico models and methods similar to those in this paper will become increasingly accurate and useful. Towards that end we are working on incorporating a more veridical model of vasculature along the lines of McDougall and Stephanou, in which blood flow plays a key role in the formation of vasculature [46,69]. We are furthermore developing 3D models, which are proving successful at accurately simulating morphological evolution . One area of special interest is that of liposomal and nanovectored delivery. These have the potential of unprecedented accuracy and specificity of delivery . It would be interesting to adapt the PKPD model herein, paying particular attention to transport in and extravasation from blood vessels , to see what predictions can be made.
The power of in vitro experimentation lies in its ease of implementation while remaining in the biological realm. By its very nature, in vitro experimentation attempts to refine and isolate. Yet, much of what happens in vivo is the result of a nonlinear system whose behavior is more than the sum of its parts. The power of in silico simulation lies in its ability to integrate components into a virtual system capable of reproducing such behavior, implicitly taking into account circuits of information flow difficult to explicitly analyze. Accurately calibrated and rigorously validated, such an integrated model could provide a “dry-lab” to be used as a powerful complement to the traditional wet-lab in fundamental research, drug discovery, and the clinic. It could be used to probe scenarios and test hypotheses that are either difficult or impossible to instantiate in the body. Results could then suggest supportive in vitro and in vivo experimentation, the end result being new therapeutic targets or strategies. Simultaneously, weaknesses (or strengths!) of the in silico model could be uncovered and addressed. Computational models have the potential to facilitate an era of great discovery and progress in understanding and treating cancer, and providing new hope to its victims.
The authors would like to thank Steve Wise (U.C. Irvine) for assistance with numerical methods and programming, Ardith El-Kareh (U. of Arizona) for insights into the cellular pharmacokinetics and pharmacodynamics of doxorubicin and cisplatin, Hoa P. Nguyen (U.C. Irvine) for helpful discussions regarding experimental protocols and drug response, John Fruehauf (U.C. Irvine) for suggestions and corrections covering a broad spectrum of oncological knowledge, and the anonymous referees for their helpful comments.
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