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Spatial heterogeneity in drug concentrations can facilitate the emergence of resistance to cancer therapy

Feng Fu, Martin A. Nowak, Sebastian Bonhoeffer

arXiv:1411.6684v1q-bio.PEq-bio.TO

TL;DR

The paper asks how uneven drug penetration may contribute to acquired cancer resistance, beyond pre-existing genetic variation. Using stochastic branching models, it examines resistance across drug-heterogeneous compartments connected by cell migration. Resistance can emerge in low-drug sanctuary sites and spread to high-drug sites, while the effect of heterogeneity depends on migration rate.

  • Problem

    The study addresses how spatial variation in drug penetration within tumors and across metastases may contribute to acquired resistance during cancer treatment.

    Source: §I · p. 4

  • Method

    The authors use multitype stochastic branching models to study cancer-cell growth, resistance, and migration between compartments with different drug concentrations.

    Source: §I · p. 6, Appendix A · p. 43

  • Results

    Resistance is more likely to arise in low-drug sanctuary compartments before spreading to high-drug compartments, and heterogeneity accelerates resistance only below a critical migration rate.

    Source: §II · p. 11, §III · pp. 13–14

  • Takeaways & Limitations

    The results identify cell motility and sanctuary sites with little drug exposure as important to rapid resistance emergence in the modeled metastatic-cell populations.

    Source: §III · p. 14

Abstract

from arXiv · show

Acquired resistance is one of the major barriers to successful cancer therapy. The development of resistance is commonly attributed to genetic heterogeneity. However, heterogeneity of drug penetration of the tumor microenvironment both on the microscopic level within solid tumors as well as on the macroscopic level across metastases may also contribute to acquired drug resistance. Here we use mathematical models to investigate the effect of drug heterogeneity on the probability of escape from treatment and time to resistance. Specifically we address scenarios with sufficiently efficient therapies that suppress growth of all preexisting genetic variants in the compartment with highest drug concentration. To study the joint effect of drug heterogeneity, growth rate, and evolution of resistance we analyze a multitype stochastic branching process describing growth of cancer cells in two compartments with different drug concentration and limited migration between compartments. We show that resistance is more likely to arise first in the low drug compartment and from there populate the high drug compartment. Moreover, we show that only below a threshold rate of cell migration does spatial heterogeneity accelerate resistance evolution, otherwise deterring drug resistance with excessively high migration rates. Our results provide new insights into understanding why cancers tend to quickly become resistant, and that cell migration and the presence of sanctuary sites with little drug exposure are essential to this end.

I. INTRODUCTION

The paper asks how spatial differences in drug penetration and cell proliferation affect resistance during treatment, particularly across metastatic compartments. It uses a stochastic branching model to examine resistance pathways and the roles of migration and sanctuary sites.

I. INTRODUCTION

  • Cancer recurrence after treatment with single agents is almost certain due to acquired drug resistance; efforts to understand rapid resistance acquisition have particularly focused on pre-existing resistance arising prior to treatment.Source: §I · p. 4
  • The effect of spatial drug heterogeneity on cancer resistance remains incompletely addressed, despite evidence that drug gradients can accelerate antibiotic resistance.Drug-penetration differences occur both within solid tumors and across metastatic lesions.Source: §I · p. 4
  • The study asks whether cells acquire resistance in low-drug sites before migrating to high-drug sites, or migrate first and adapt there.These questions follow experimental findings that motility and chemotherapy gradients can be associated with resistant cells emerging in otherwise growth-inhibiting regions.Source: §I · p. 5
  • The model treats metastatic deposits as spatial compartments linked by cell dissemination and seeding, focusing on cancers with differing metastatic capacity.The authors contrast relatively distinct compartments in some solid tumors with the diminished compartment structure of highly fluid blood cancers.Source: §I · p. 5
  • Building on prior models of pre-existing primary-tumor resistance, the study incorporates metastatic compartments and quantifies how spatial drug-concentration heterogeneity shapes metastatic-cell resistance evolution during treatment.Focusing on resistance mutations that incur a fitness cost without drugs but confer an advantage under treatment, it finds resistant mutants arise through a mutation-migration pathway: they acquire costly resistance in sanctuary sites before migrating to high-drug compartments.Source: §I · p. 6

II. RESULTS

The results use compartmental models to examine how drug penetration, cell growth, and migration shape resistance. Resistance can emerge in sanctuary sites and spread to drug-rich compartments; the effects of spatial heterogeneity depend on migration rate and model assumptions.

II. RESULTS

  • At low mutation and migration rates, mutation in the sanctuary followed by migration to the drug compartment can produce resistance faster than migration followed by mutation when the fitness condition is met.The simplified condition is expected to hold in many cases of potent therapy, where δ > s.Source: §II · p. 8
  • With a fitness cost of resistance in the sanctuary, resistant cells that populate drug-rich compartments are more likely to originate in sanctuary sites than to evolve there in situ.The sanctuary's slower proliferation can delay resistance, but the mutation-migration pathway remains predominant under a range of conditions.Source: §II · pp. 9–10
  • Increasing the difference in drug concentration between metastatic compartments produces a sharp shift from likely eradication to treatment failure and shortens relapse time.In the simulated small-lesion case, the worst-case scenario yields relapse after approximately 103 days; larger lesions can relapse within weeks.Source: §II · p. 10
  • In the simulated metastatic lesions, lineages from the larger drug-rich lesion evolved resistance faster, but this ordering reversed when that lesion was smaller.The authors attribute the initial result to the larger lesion supplying more escaping cells and thus more opportunities for resistant mutations.Source: §II · p. 10
  • When one mutation cannot confer resistance to the highest drug concentration, multiple mutations may be required, and the time to sufficient resistance can rise abruptly beyond a concentration-difference threshold.In the very-high-difference case described, two-point mutants are most likely to arise in the sanctuary before migrating to the drug compartment.Source: §II · p. 11
  • Only below a critical migration rate does spatial drug heterogeneity speed resistance emergence; excessively high migration rates can slow it by increasing exposure to drug-rich compartments.The results also identify an optimum migration rate for fastest emergence.Source: §II · p. 11
  • In the extended model, strong resistance evolves faster with local than global migration, particularly for sensitive cells starting in low-drug sanctuary sites.Sequential local migration following newly acquired mutations contributes to selection for high-level resistance.Source: §II · pp. 11–12
  • In the modeled solid tumor, poor drug penetration creates outer sanctuary compartments where resistance is more likely to arise despite slower cell proliferation.Relapse occurs sooner as modeled spatial drug heterogeneity increases; under near-uniform drug penetration, the tumor can be eradicated.Source: §II · pp. 12–13

III. DISCUSSION

The model predicts that resistance can emerge in poorly exposed sanctuary sites and spread to high-drug compartments, with spatial heterogeneity speeding resistance only below a migration-rate threshold. The authors identify modeling boundaries and discuss implications for treatment of disseminated cancer.

III. DISCUSSION

  • Poor drug penetration can speed resistance: it is unlikely to arise in high-concentration compartments, instead emerging in sanctuary sites before spreading to them.The high-concentration compartments can completely inhibit growth of sensitive cells.Source: §III · p. 13
  • The framework addresses metastatic cancer more directly than prior microbial-evolution models, allowing realistic concentration-dependent responses, arbitrary initial cell distributions, resistance risk, and relapse timing.The earlier models assumed ordered compartments, nearest-neighbor migration, and evolution beginning in an initially occupied sanctuary.Source: §III · p. 13
  • The paper presents its theoretical results as a complementary perspective to PDE approaches for spatiotemporal cancer evolution and resistance selection.The authors frame their contribution as improving understanding of metastatic resistance, especially with sanctuary sites.Source: §III · p. 14
  • Spatial heterogeneity speeds resistance only below a threshold migration rate; the authors also identify cell motility and sanctuary sites as essential for rapid resistance acquisition.The authors suggest this may qualitatively help explain differences in clinical success between liquid cancers and solid tumors.Source: §III · p. 14
  • The analysis focuses on metastatic cells with equal migration capacity and considers resistance to one drug, or a drug combination overcome by a single mutation.The authors identify multi-drug resistance requiring multiple mutations as a future research direction.Source: §III · p. 14
  • The authors suggest combining targeted therapy with anti-metastatic treatment, because migration inhibition can restrict both sensitive-cell access to sanctuary sites and resistant-cell spread from them.They also recommend improving drug transport and distribution across metastatic compartments.Source: §III · p. 14
  • The model is a proof of principle that omits cancer stem cells, dormancy, and metastatic inefficiency; its free movement between compartments may overestimate dynamics in realistic vasculature.The authors propose that a calibrated model could eventually simulate patient responses and predict treatment outcomes for disseminated cancer.Source: §III · pp. 14–15

IV. MATERIALS & METHODS

The methods model cancer-cell lineages in compartments with different drug concentrations, then use generating functions to estimate resistance probabilities and timing. The framework combines concentration-dependent cell fitness, mutation, migration, and numerical procedures for extracting probabilities and average time to resistance.

IV. MATERIALS & METHODS

  • The minimal model uses two compartments and two cell types to examine resistance under potent therapy, with different drug access across compartments.The authors describe this as a proof-of-concept model.Source: §IV · p. 15
  • Cell replication is modeled as concentration-dependent, consistent with the authors' focus on targeted therapies that often have cytostatic effects.They note that drug efficacy commonly depends on concentration.Source: §IV · p. 15
  • Cell fitness depends on genotype and location: mutation can arise at division, drug response follows a Hill function, and migration between compartments occurs at rate v.The model defines genotype-specific replication and death rates and includes a resistance cost, IC50, fold increase in IC50 per mutation, and Hill-function steepness.Source: §IV · pp. 15–16
  • A multitype branching-process generating function tracks the lineages initiated by each cell type, with backward equations and initial condition Fij(X; 0) = xij.The dummy variables xij represent the four cell types.Source: §IV · p. 16
  • The method extracts joint probabilities from generating functions using Cauchy's integral formula and approximates the double integral with the trapezoid rule.The resulting probability density concerns resistant-cell counts in both compartments over time.Source: §IV · pp. 16–17
  • For the results in Fig. 3, the generating-function formula is evaluated using a discrete Fourier-transform equivalent with N = 1000 to reduce spectral-method aliasing.The passage states N is much greater than 20.Source: §IV · p. 17
  • Average time to resistance is calculated from the conditional probability of having no resistant cells in either compartment over time.The described calculation starts with a single sensitive cell in compartment 1.Source: §IV · p. 17

e. Full model.

The full model extends the analysis to cancer cells acquiring resistance across M compartments with heterogeneous drug concentrations and different migration schemes.

e. Full model.

  • The model considers cells needing n point mutations for full resistance and movement among M compartments with restricted local or unrestricted global migration.Drug concentrations are assigned by compartment to represent spatial heterogeneity.Source: e. Full model · p. 17
  • Local migration places compartments on a ring, allowing cells to move to either nearest neighbor with equal probability v/2.Source: e. Full model · p. 17
  • Global migration fully connects compartments, with cells able to move to any other compartment with equal probability v/(M −1).Source: e. Full model · p. 17
  • The extended model examines how metastatic cells acquire increasing resistance as they move along a drug-concentration gradient and thrive in areas of excessively high drug concentration, as shown in a recent in vitro experiment.Source: e. Full model · p. 17

Figure legends:

The figure legends trace resistance from competing routes between sanctuary and drug-exposed compartments to metastatic and within-tumor settings. They show how migration, tumor size, and spatial drug gradients shape resistance dynamics and relapse timing.

Figure 2: Competing pathways to selection for resistance

  • Resistant cells thriving in high-drug compartments are likely to originate in sanctuary sites, where they acquired resistance before migrating.The competing routes are migration followed by mutation in the drug compartment, or mutation followed by migration from the sanctuary.Source: Figure legends · p. 25
  • Reducing sanctuary-cell growth prolongs time to resistance, while the pathway comparison marks a critical drug-efficacy level above which mutation followed by migration is faster.The threshold is δ = s/(1 + s).Source: Figure legends · p. 25

Figure 3: Spatio-temporal snapshots of emerging drug resistance

  • Panels track resistant-cell densities over time: resistance arising in the drug compartment is unlikely, while the sanctuary can seed resistant cells that later outnumber those there.The snapshots start from a single sensitive cell in the drug compartment.Source: Figure legends · p. 26

Figure 4: Outcomes of in silico treatment to eradicate metastases

  • For two metastases, the figure plots escape probability and conditional time to resistance against drug-concentration difference, across migration rates.The simulated lesions differ in size, growth rates, and drug penetration; one point mutation suffices for resistance in this case.Source: Figure legends · p. 26

Figure 5: Tumor microenvironment and high drug resistance

  • The microscopic tumor model links distance from a central blood vessel to lower cell proliferation, cell density, and drug concentration, then examines resistance across drug-gradient strengths.Cells migrate between neighboring compartments, and surviving the maximum central drug concentration requires two point mutations in the simulated setting.Source: Figure legends · p. 27

Figure S2: Average time to resistance by lesion size

  • With a drug sanctuary and cell motility, larger tumors relapse sooner, and very large tumors can relapse within weeks.The accompanying legend describes average time to resistance against lesion 1 size, rescaled by mutation rate.Source: Figure legends · p. 28, Figure S2

Appendix A: Full model

The model represents metastatic disease as compartments with heterogeneous drug concentrations and cell migration, using a branching process to study resistance. Drug response depends on genotype and location, while mutation and migration connect cell lineages across compartments.

Appendix A: Full model

  • Cells acquire stepwise point mutations during division, and their replication rate depends on genotype and compartment location.The drug is modeled as inhibiting growth; acquired mutations can allow cells to survive in high-concentration compartments.Source: Appendix A · p. 42
  • Cell division is modeled with a Hill drug-response function, with mutation costs and an increased IC50 per mutation.The Hill function uses βj for wild-type division without drug, s for mutation cost, IC50 for half-inhibition concentration, ρ for the per-mutation IC50 increase, and m for steepness.Source: Appendix A · pp. 42–43
  • The model considers local migration between nearest neighbors on a ring and global migration among fully connected compartments.Migration occurs at rate v; local moves have probability v/2 to either neighbor, while global moves have probability v/(M −1) to each other compartment.Source: Appendix A · p. 43
  • A continuous-time multi-type branching process models cancer-cell resistance across metastatic compartments with spatially heterogeneous drug concentrations and migration.The analysis restricts drug concentrations to spatial variation without time-changing fluctuations.Source: Appendix A · pp. 42–43

Appendix B: Analysis

The analysis tracks cell lineages by genotype and compartment, representing births, mutation, migration, and death with probability generating functions. Their equations yield extinction and resistance probabilities, which are computed numerically when closed forms are unavailable.

Appendix B: Analysis

  • A cell type combines its genotype and compartment, giving n×M types whose identity can change through mutation or migration.For example, type “11” denotes a cell with one point mutation in compartment 1.Source: Appendix B · p. 43
  • In each infinitesimal interval, a cell may divide, produce a mutated offspring, migrate, die, or remain unchanged.The birth and mutation events are parameterized by bij and u, migration by v, and death by dij.Source: Appendix B · pp. 43–44
  • Probability generating functions yield backward equations with initial values Fij(X; 0)=xij for each genotype-compartment type.The generating function for an initial cell profile is formed from the single-cell generating functions.Source: Appendix B · p. 44
  • The equations provide extinction and conditional no-resistance probabilities, but arbitrary genotype and compartment counts require numerical ODE solvers.The extinction probability is obtained from fixed points; the no-resistance probability tracks whether a fully resistant strain has appeared by time t.Source: Appendix B · pp. 44–45

1. Simple model

The simple model contrasts a drug-free compartment where both cell types can grow with a drug-containing compartment where sensitive cells cannot sustain growth. In this fitness landscape, resistance is more likely to originate in the drug-free compartment and migrate into the drug-containing compartment than to evolve there in situ.

1. Simple model

  • The simple two-genotype, two-compartment backward equations are coupled nonlinear Riccati equations without explicit closed-form solutions.Closed-form approximations are available only for special cases.Source: §1 · p. 45
  • The simple model assumes equal death rates and a drug-free compartment where both sensitive and resistant cells grow, with resistant cells paying a replication cost.In the drug-containing compartment, sensitive cells have subcritical replication potential while resistant cells remain supercritical.Source: §1 · pp. 45–46
  • Resistance is more likely to originate in the drug-free compartment and then migrate into the drug-containing compartment than to evolve there in situ.The drug-free compartment is more favorable for breeding resistance under the assumed fitness landscape.Source: §1 · p. 46

2. Drug-environment-dependent escape

  • The drug-environment-dependent escape analysis compares escape from a single sensitive cell in low- and high-drug compartments with different birth-rate relationships.It defines generating functions for lineages initiated by sensitive and resistant cells and gives their initial conditions.Source: §2 · p. 46

a. b0 > b1 > d0

Under the stated growth-rate ordering, less harsh drug environments are more favorable for producing resistant mutants. The passage also gives a small-mutation-rate approximation for escape from a single sensitive cell.

a. b0 > b1 > d0

  • The rare-mutation approximation neglects the loss of sensitive cells due to rare mutations.The stated growth-rate ordering is r0 > r1 > 0.Source: a. b0 > b1 > d0 · pp. 46–47
  • Less harsh compartments produce resistant mutants sooner because sensitive-cell replication is less affected by drug.The result identifies low drug concentrations as more likely breeding grounds for resistance.Source: a. b0 > b1 > d0 · p. 47
  • For u ≪1, the escape probability P1 that a single sensitive cell establishes a nonextinct lineage is approximated.The approximation is developed under the assumption r0 > r1 > 0, where ri=bi−di.Source: a. b0 > b1 > d0 · pp. 46–47

b. b0 < d0 < b1

When sensitive cells are subcritical, resistant lineages must arise shortly before sensitive cells die out, making escape less likely and resistance slower to appear than in benign environments.

b. b0 < d0 < b1

  • Sensitive cells must acquire a resistant mutation just before quickly dying out to establish surviving lineages.In this unfavorable case, mutants arise with lower probability and take longer to appear than in benign environments because of the smaller birth rate.Source: b. b0 < d0 < b1 · p. 48
  • The approximation above does not apply when the sensitive-cell branching process is subcritical.The passage notes that no closed form is available for the arrival time of the first mutant in this case.Source: b. b0 < d0 < b1 · p. 48
  • The cited passage supports that the escape probability is much smaller than Eq.(B20), but does not support describing this case as unfavorable.Source: b. b0 < d0 < b1 · p. 48

3. Competing pathways to the emergence of resistance

The section compares resistance arising after migration into a drug-present compartment with resistance arising first in the drug-free compartment and then migrating. Under specified fitness conditions, the latter pathway can be faster.

3. Competing pathways to the emergence of resistance

  • In migration–mutation, sensitive cells move to the drug-present compartment before acquiring resistance; in mutation–migration, they mutate in the drug-free compartment before moving.The comparison asks which route more quickly yields resistant cells in the drug-present compartment.Source: §3 · p. 48
  • To compare the routes, the analysis considers one-way emigration from drug-free to drug-present compartments and separates the pathways into three-type branching processes.The authors separate the routes rather than model them simultaneously because the four-type process has no closed-form solution.Source: §3 · p. 48
  • For rare mutations and low migration, the analysis derives closed-form approximations for the conditional probability of no surviving type 2 cells by time t.The setup assumes r0 > 0, r2 > 0, and r1 < r0; the pathway-specific rates distinguish intermediate-cell division rates in drug-present versus drug-free compartments.Source: §3 · pp. 48–49
  • A condition identifies when mutation–migration reaches surviving resistant cells in the drug-containing compartment faster than migration–mutation.The passage does not include the inequality itself.Source: §3 · p. 49
  • When δ > s, resistance emergence is more contingent on mutation–migration if the drug-concentration difference produces a fitness landscape satisfying inequality (B26).For cytostatic drugs, s is the resistant-mutation fitness cost without drugs and δ is the sensitive-cell fitness cost with drugs.Source: §3 · p. 49

Appendix C: Numerical results

The numerical results show that spatial heterogeneity speeds resistance only below a migration-rate threshold, while migration scheme and starting compartment shape how resistance develops.

Appendix C: Numerical results

  • The numerical analysis compares extinction probability and time to resistance across starting compartments, migration rates, and drug heterogeneity.It uses symmetric spatial drug distributions based on a rescaled Normal distribution, varying the variance while keeping total drug concentration constant.Source: Appendix C · p. 50
  • Spatial heterogeneity speeds resistance only below a migration-rate threshold; excessively high migration rates slow resistance emergence.The authors attribute the slowdown to frequent migration diminishing the role of compartment structure.Source: Appendix C · p. 50
  • Global migration makes extinction probability and time to resistance more sensitive to migration rate than local migration does.Local migration is faster from sanctuary sites but delays resistance for cells starting in harsh compartments; for the latter, global migration has an optimal intermediate rate.Source: Appendix C · p. 50
  • Low-drug sanctuaries shelter sensitive cells; local migration speeds resistance there but delays it in harsh compartments, while increasing heterogeneity prolongs resistance under global migration.Under local migration, an intermediate level of heterogeneity most delays resistance.Source: Appendix C · pp. 50–51
  • The numerical conclusions are robust to model extensions and parameter variation, but migration routes still need characterization before they can be manipulated.The authors identify constraining escape routes to sanctuary sites and limiting dissemination from those sites as vital to eliminating disseminated cancer.Source: Appendix C · p. 51
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