Master of Science in Mathematical Engineering
Master’s thesis in Computational Science and Engineering
Computational models
for nanoparticle transport
in the vascular system
Advisor:
Prof. Paolo ZUNINO
Co-Advisors:
Prof. Paolo DECUZZI
Dr. Silvia LORENZANI
Dr. Alessandro COCLITE
Dr. Federica LAURINO
Annagiulia TIOZZO
Matr. 836483
Academic Year 2016-2017
of nanomedicine, in particular in the field of the treatment of complex dis-eases, for example cancer. A great contribution to the understanding of such therapies is given by mathematical models and numerical simulations, since they can provide complementary sophisticated and multiscale tools to experiments.
We build a mathematical model for the transport of nanoparticles in a microvascular network, for their adhesion to the vessel walls and for the cor-responding release or extravasation of therapeutic agents in the surrounding interstitial tissue.
All the biological systems share a multiscale structure, since several phe-nomena take place at different time- and space-scales. In order to take into account these effects in the mathematical models, we use the results of sev-eral numerical or microfluidic experiments as values for the parameters in the model.
Thanks to dimensional model reduction techniques, the blood flow and related transport phenomena can be described as a one-dimensional (1D) source within the 3D domain in order to reduce the computational cost of the simulations. From the mathematical standpoint, we notice that the high-dimensionality gap (3D/1D) causes an ill-posed formulation. We overcome this problem by directly exploiting the coupling in the variational finite ele-ment formulation, thanks to suitable restriction operators.
promettenti della nanomedicina, in particolare nel campo del trattamento di malattie complesse, ad esempio il cancro. I modelli matematici e le si-mulazioni numeriche possono dare un grande contributo alla comprensione delle terapie, perch´e sono in grado di fornire sofisticati strumenti di indagine complementari agli esperimenti.
Definiamo un modello matematico che descriva il trasporto di nanopar-ticelle in una rete microvascolare, la loro adesione alle pareti dei vasi e il corrispondente rilascio o extravasazione degli agenti terapeutici nel tessuto interstiziale circostante.
Tutti i sistemi biologici condividono una struttura multiscala, poich´e di-versi fenomeni avvengono a scale spaziali e temporali differenti. Per tene-re in considerazione questi effetti anche nel modello matematico, utilizzia-mo i risultati di vari esperimenti numerici o microfluidici come valori per i corrispondenti parametri nel modello.
Grazie a opportune tecniche di riduzione dimensionale di modelli, il flusso sanguigno e i relativi fenomeni di trasporto possono essere descritti come una sorgente unidimensionale (1D) all’interno del dominio 3D, al fine di ridurre il costo computazionale delle simulazioni. Dal punto di vista matematico, notiamo che il forte salto dimensionale (3D/1D) genera un problema mal po-sto. `E possibile superare queste difficolt`a esprimendo l’accoppiamento tra le equazioni direttamente a livello della formulazione variazionale agli elementi finiti, grazie ad un opportuno operatore di restrizione.
1 Introduction and motivations 1
1.1 Mathematical and computational models in nanomedicine . . 2
1.1.1 Nanomedicine . . . 2
1.1.2 Mathematical models for nanomedicine . . . 7
1.1.3 Discussions . . . 11
1.2 Multiscale models in biology . . . 12
1.2.1 Multiscale nature of biological systems . . . 12
1.2.2 Single-scale models and upscaling techniques . . . 12
1.2.3 Examples of application . . . 15
1.2.4 Nanoparticle delivery in a capillary network . . . 16
2 Mathematical model for particle transport in the microvas-culature 22 2.1 Three-dimensional model for microvasculature within a tissue 23 2.2 Model reduction: coupled 3D-1D problem . . . 25
2.2.1 Coupling term for the interstitial volume . . . 26
2.2.2 Model reduction for microvascular flow . . . 27
2.2.3 Governing equations for the coupled problem . . . 28
2.3 Dimensional analysis . . . 28
2.4 Boundary and initial conditions . . . 29
2.5 Junction treatment . . . 30
2.6 Weak formulation . . . 31
2.6.1 Weak formulation for the tissue problem . . . 31
2.6.2 Weak formulation for the vessel problem . . . 32
2.6.3 Coupled weak formulation . . . 34
2.6.4 Well posedness . . . 34
2.7 Alternative well-posed weak formulation . . . 35
2.7.1 Alternative 3D-1D coupling . . . 36
2.7.2 Alternative weak formulation for the tissue problem . . 36
2.7.3 Alternative weak formulation for the vessel problem . . 38
2.7.4 Alternative weak formulation for the coupled transport problem . . . 39 2.7.5 Well posedness . . . 39 2.8 Numerical approximation . . . 39 2.9 Algebraic counterpart . . . 41 3 Well-posedness analysis 44 3.1 Simplified problem setting . . . 44
3.1.1 Geometric setting . . . 44
3.1.2 Model equation . . . 45
3.1.3 Restriction operator . . . 46
3.1.4 Weak formulation . . . 46
3.2 Well-posedness analysis . . . 47
3.3 Conclusions and further developments . . . 49
4 Characterization of the model parameters 51 4.1 A model for particle adhesion to the vascular wall . . . 51
4.1.1 Vascular adhesion parameter . . . 52
4.1.2 Effective vascular adhesion parameter with saturation . 52 4.1.3 Explicit formula for Pa . . . 54
4.2 Multiscale model for particle adhesion . . . 57
4.2.1 Lattice Boltzmann approach for nanoparticle transport 57 4.2.2 Subscale model for particle adhesion . . . 59
4.3 Discussion . . . 61
4.4 Optimal control approach for the prediction of the diffusivity coefficient . . . 62 4.4.1 Experimental setup . . . 63 4.4.2 Numerical model . . . 64 4.4.3 Minimization problem . . . 65 4.4.4 Iterative method . . . 68 4.4.5 Numerical results . . . 68
4.4.6 Discussion and conclusions . . . 75
5 Numerical results 78 5.1 Nanoparticle transport and adhesion . . . 79
5.1.1 Fluid dynamics effects . . . 81
5.1.2 Explicit formula for Pa . . . 84
5.1.3 Pa from LB approach . . . 87
5.1.4 Effective vascular adhesion parameter and saturation . 91 5.1.5 Conclusions . . . 92
5.2.1 Available data and results . . . 94
5.2.2 Conclusions . . . 99
5.3 Drug delivery: combined nanoparticles and Dextran . . . 99
5.3.1 Model . . . 99
5.3.2 Parameters and results . . . 102
5.3.3 Conclusions . . . 107
6 Conclusions and future perspectives 109 6.1 Future developments . . . 112
A Fluid dynamics model for the vascularized tissue 115 A.1 Model set up . . . 115
A.2 Coupling microcirculation with interstitial flow . . . 117
A.2.1 A reduced model for microvascular flow . . . 118
A.2.2 Governing equations for the coupled problem . . . 118
A.3 Dimensional analysis . . . 119
A.4 Boundary conditions . . . 120
A.5 Junction treatment . . . 121
A.6 Variational formulation . . . 122
A.6.1 Weak formulation of the tissue problem . . . 122
A.6.2 Weak formulation of the vessel problem . . . 123
A.6.3 Coupled weak formulation . . . 126
A.7 Numerical approximation . . . 127
A.7.1 Discretization of the tissue problem . . . 127
A.7.2 Discretization of the vessel problem . . . 128
A.7.3 Discrete coupled weak formulation . . . 128
A.8 Algebraic formulation . . . 129
B Lattice Boltzmann method 132 B.1 Introduction . . . 132
B.2 The Boltzmann equation . . . 133
B.2.1 BGK kinetic model . . . 133
B.2.2 Time discretization . . . 134
B.2.3 Choice of the quadrature rule . . . 134
B.3 The lattice Boltzmann equation . . . 135
B.4 Boundary conditions . . . 137
B.4.1 Standard bounce-back . . . 137
B.4.2 Higher-order boundary conditions . . . 138
Bibliography 140
1.1 Effect of 3S parameters on the biodistribution of nanoparticles in organs. . . 3 1.2 The spectrum of a body-on-chip developed for drug screening
purposes. . . 4 1.3 Integrated microfluidic system for single-cell studies. . . 6 1.4 Technical challenges for theoretical and computational
scien-tists in nanomedicine. . . 8 1.5 Spatial distribution of different sized particles in an inflamed
arterial tree. . . 9 1.6 The multi-phase system within a Representative Elementary
Volume for tumor growth and nutrient evolution. . . 10 1.7 FSI simulation of a VAD in action. . . 10 1.8 Multiscale nature of a biological system: length and time scales. 12 1.9 A classification of methods for biological systems in terms of
their characteristic length- and time-scales. . . 13 1.10 Different scales within the human circulatory system. . . 15 1.11 Multiscale modules for vascular and extravascular transport
of nanoparticles. . . 17 2.1 Microvasculature within a tissue interstitium and reduction
from 3D to 1D description. . . 23 2.2 Y-shaped bifurcation for the description of the junction
treat-ment. . . 31 2.3 Sketch of the alternative weak formulation. . . 35 2.4 Vessel discretization for interpolation and average operators. . 43 3.1 Simplified geometric setting for well posedness analysis. . . 45 4.1 Particles trajectory within a Couette flow, solved via LB-IB
method. . . 58 4.2 Geometry for the Couette flow and particle transport . . . 59
4.3 Probability of adhesion computed using the Lattice
Boltzmann-Immersed Boundary method. . . 61
4.4 Geometric model of the Dextran diffusion experiment . . . 63
4.5 Diffusion experiments using 4 kDa Dextran molecules at the initial and final time. . . 64
4.6 Initial configuration for the code validation . . . 69
4.7 Reference and optimal concentrations for the code validation . 70 4.8 Intensity maps of the diffusion experiment using 4 kDa Dex-tran molecules . . . 72
4.9 Reference concentrations for 4 kDa Dextran molecules: possi-ble smooth and non smooth configurations. . . 74
4.10 Concentration with the optimal diffusivity coefficient: test4A. 74 5.1 Sketch of the three main classes of simulations. . . 80
5.2 rat93 geometry . . . 81
5.3 Pressure field in the vessel network: test1 and test2. . . 82
5.4 Velocity field in the vessel network: test1 and test2. . . 82
5.5 Wall shear rate in the vessel network: test1 and test2. . . . 83
5.6 Probability of adhesion in the vessel network with explicit for-mula: test1 and test2. . . 85
5.7 Vascular adhesion parameter in the vessel network without saturation model: test1 and test2. . . 85
5.8 Concentration of nanoparticles at the final time in the vessel network: test1 and test2. . . 85
5.9 Density of nanoparticle adhering per unit surface to the vascu-lar wall at the final time, without the saturation model: test1 and test2. . . 86
5.10 Reynolds number in the vessel network: test3. . . 88
5.11 Adhesive variables in the vessel network: test3. . . 88
5.12 Concentration of nanoparticle in the vessel network: test3 and test4. . . 90
5.13 Density of nanoparticle adhering per unit surface to the vas-cular wall at the final time: test3 and test4. . . 90
5.14 Pressure and velocity fields in the vessel network and in the tissue interstitium: test5. . . 95
5.15 Concentration in the vessel network and in the tissue intersti-tium at different simulation time: test5 and test6. . . 97
5.16 Mean concentration in the tissue region over time for 40 kDa and 250 kDa Dextran molecules release: test5 and test6. . . 98
5.17 Dextran release profile in time: test7, test8 and test9. . . . 104
5.18 Dextran concentration and density of adhering nanoparticle on the vessel wall in the cases of various molecular weight and
simulation time: test7, test8 and test9. . . 105
5.19 Total amount of Dextran in the tissue region over time for 4 kDa, 40 kDa and 250 kDa Dextran molecules release: test7, test8 and test9. . . 106
A.1 Y-shaped bifurcation for the description of the junction treat-ment. . . 121
B.1 D2Q9 lattice configuration . . . 136
B.2 D2Q9 on-grid bounce-back . . . 138
4.1 Probability of adhesion computed using the Lattice Boltzmann-Immersed Boundary method for strong ligand-receptor bond. . 60 4.2 Probability of adhesion computed using the Lattice
Boltzmann-Immersed Boundary method for mild ligand-receptor bond. . . 60 4.3 Model parameters for code validation . . . 69 4.4 Diffusivity coefficient computed solving the minimization
prob-lem with NLCG for code validation. . . 70 4.5 Sensitivity analysis of D∗ with respect to λ for code validation. 71 4.6 Numerical tests on the Dextran 4 kDa diffusion experiments . 75 4.7 Numerical tests at different final time T for test4E. . . 76 4.8 Numerical tests on the Dextran 40 kDa diffusion experiments . 76 4.9 Numerical tests on the Dextran 250 kDa diffusion experiments 76 5.1 Sketch of the tests for nanoparticle transport and adhesion. . . 81 5.2 Physical parameters characterizing the fluid dynamics
prob-lem in test3 . . . 89 5.3 Physical parameters characterizing the nanoparticle transport
problem in test3 . . . 89 5.4 Characteristic values for the non dimensional analysis in test3 89 5.5 Sketch of the tests for Dextran transport and extravasation. . 93 5.6 Physical parameters characterizing the fluid dynamics
prob-lem in test5 . . . 94 5.7 Physical parameters characterizing the 40 kDa Dextran molecules
transport problem in test5 . . . 96 5.8 Physical parameters characterizing the 250 kDa Dextran molecules
transport problem in test6 . . . 97 5.9 Sketch of the tests for nanoparticles transport, adhesion and
Dextran delivery. . . 100 5.10 Parameters characterizing the release of Dextran molecules in
test7, test8 and test9 . . . 103 5.11 Tissue diffusivity coefficient for Dextran delivery experiments . 104
Introduction and motivations
The final goal of the thesis is to perform simulations of the transport of nanoparticles in the capillary network and of the drug delivery in the inter-stitial tissue. The use of nanoparticles loaded by therapeutic agents is one of the most promising innovation in the field of the treatment of complex diseases, tipically cancer. Moreover, the microvasculature is where the ex-change of nutrients and drugs takes place. Therefore, the need to combine the two aspects is growing. A great help in the effort for creating a power-ful interconnection among medicine, technology and quantitative sciences is given by mathematical and numerical models. In order to provide a precise formulation of the microcirculation problem, it is necessary to exploit the multiscale nature that characterizes all the biological systems and the inter-actions among results obtained at different time- and space-scales. Within this general framework, the specific objective of this work is to encode into the macroscopic model for nanoparticle transport in microcirculation and drug delivery in the interstitial tissue some detailed information of the complex microscopic structure, in particular in terms of diffusion of the extracellular matrix with respect to nanoparticles and adhesive properties of the nanopar-ticles.
The plan of the thesis is as follows. Chapter 1 is an introductory chapter in which an overview of what nanomedicine is and what the role of mathematical and computational models in medical fields is. This is due to the fact that nanoparticle-based drug delivery systems fall within the general framework of nanomedicine. This chapter also contains an explanation on how the multiscale approach works in biological systems. Chapter 2 proposes the macroscopic model for microcirculation and drug delivery, by means of a network of one-dimensional channels immersed in a three-dimensional space. The well posedness of the weak formulation of the macroscopic model is
addressed in Chapter 3. At the microscale, the estimate of the adhesive parameters of the nanoparticles as function of the particle properties via a Lattice Boltzmann-Immersed Boundary method and the prediction of the diffusivity coefficient of the extracellular tissue by numerically reproducing physical experiments with the help of an optimization problem are presented in Chapter 4. Finally, in Chapter 5 we provide some results of the large scale simulation integrated with the information derived from the analysis at the microscale.
1.1
Mathematical and computational models
in nanomedicine
1.1.1
Nanomedicine
Nanomedicine is an emerging research branch which combines nanotechno-logical tools and biomedical studies. This research field aims at enhancing clinical diagnosis and providing opportunities for more effective therapeutic treatments. The use of nanotechnology for medical purposes leads to the miniaturization of engineered devices and nanostructures up to the molecu-lar level in order to improve the disease detection and the efficiency of the health care [45].
From its very beginning, nanotechnology applied to medicine has been a very fertile field, as many tools with a good therapeutic response have been developed. Applications can be found in many different fields of medicine, for example drug-delivery systems, nanosensors, imaging agents and implants.
In [5], Bao et al. draw up a list of issues in some biomedical fields that can be tackled with mechanical and technological tools, especially considering devices at the nanoscale. In particular, the three major challenges that they outline deal with (i) the drug delivery by means of injectable nanoparticles, (ii) the design of biomedical devices and (iii) the study of the mechanics of cellular processes.
(i) Functionalized nanoparticles have been developed for different pur-poses, mainly for cancer treatment [42] and diagnostic imaging [6], but also for atherosclerosis and neurodegenerative diseases, as vector for targeted drug or imaging contrast agent delivery or for hypertermia therapy. Nanoconstructs are widely used as drug carriers, since they can navigate through the human circulatory system and they can easily recognize tumor neovasculature. Nanoparticles are loaded with ther-apeutic or imaging agents that can be released as needed, improving
Figure 1.1: Nanoparticle size, shape and surface charge are responsible for the biodistribution among the different organs, such as lungs, liver, spleen and kidneys. Figure from [8].
the specificity and the personalization of the treatment. Moreover, it is well known that the tumor vasculature and the healthy blood vessel network exhibit several differences. In healthy vessels, blood circulates at higher mean velocity with respect to the blood that is in the dis-eased vasculature and there is lower interstitial fluid pressure. In the diseased case, the vessels are leakier and more tortuous and the vascu-lature is hyper-permeable [28]. Therefore, many types of nanoparticles with different sizes, shapes and surface properties have been designed in order to maximize the effectiveness of the treatments, the quantity of loaded agents, the circulation time and to minimize the sequestration by other organs. Maximizing the efficacy of the therapy means, in prac-tice, maximizing the accumulation of the agents in the targeted region. As shown in Figure 1.1, the properties of the nanoparticles also dictate the biodistribution among different organs. By manipulating the phys-ical characteristics of the nanoparticles, it is possible to observe the behaviour of their dynamics and their adhesion at the diseased walls. The aim of the optimization process is to find which nanoparticle can efficiently marginate and adhere to the diseased region, pushed by the hydrodynamic forces and the adhesive interactions [17].
(ii) Nanoscale technology turns out to be very useful also in the case of the design of many biomedical devices. For example, the develop-ment of microfluidic chips that are able to realistically reproduce the functionalities of some organs in order to perform the screening of new pharmaceuticals and to recreate biological processes outside the body is currently under investigation [30]. Many organs have been designed
Figure 1.2: The spectrum of a body-on-chip is being developed for drug screening purposes. System have been developed for (a) lung, (b) the blood-brain barrier, (c) heart tissue, (d) liver, (e) the gastrointestinal tract, (f) muscle and the (g) microcirculation. Figure from [5].
on a microfluidic platform, for example the lung [27], the liver [19] and the blood brain barrier. In each of these organs, there is a realistic exchange of signals and interaction. The most challenging issue in the framework of the organs-on-chip system is the realistic interaction of all the single organs in order to reproduce a global body-on-chip. This contribution could be extremely important in the perspective of study-ing possible side effects of therapeutic treatments on organs different from the targeted one. In Figure 1.2, we show a variety of organs on chip and functionalities.
(iii) The third issue proposed by Bao et al. in [5] is about cell mechanics and the use of microfluidic systems for the study of the behaviour of the cells. Traditional experiments on population of cells that are carried out by researchers with standard bulk methods usually show hetero-geneity in the results, even though the tests have been performed in identical environments [3]. This effect is due to the fact that the be-haviour of the cells is different in the case of bulk techniques and single-cell studies. Using the latter type of technique, some specific behaviours of the cells may not be captured, because the experimental ensemble average across the population can obscure an important subset within the data [31]. Nanoscale devices can be exploited for single-cell studies as they provide the opportunity of performing in vitro analysis, such as low-volume sampling, even with the integration of other tools, if needed. An example of an integrated microfluidic system for single cell studies is shown in Figure 1.3.
However, due to the complexity of the human biological system and of nanomedicines, many nanomedical devices that have been realized in the last decade are not yet employed in daily clinical use and their clinical integra-tion is still very challenging. For example, in order to design the process of a nanomolecule for targeted delivery, it is necessary to study the complex interactions that take place in the specific biological system. For the correct characterization of many tissue models, it is required to include the exchange of metabolites and the vascular perfusion, which still represent a limitation for the development. Indeed, some of these steps are currently based on trial and error. Using microfluidic systems for single-cell studies, it is still not possible to perform time-dependent analysis and the sensitivity of the detection scheme needs for improvement.
Therefore, there is the pressing need to examine in depth the intriguing possibilities given by the nanomedical technology. By studying them in a different light, it may lead to significant guidelines for the solution of the current problems.
Figure 1.3: Integrated microfluidic system for single-cell studies. The sistem would be made of several modules for multi-functions including single-cell manipulation, isolation, sampling and analysis in an automated and high-throughput manner. Figure from [5].
1.1.2
Mathematical models for nanomedicine
All the tools and techniques illustrated so far derive from experimental analy-ses and results. The experiments supply a wide understanding of the physics of the problems and the potential of the materials used, however in vitro stud-ies are expensive in terms of availability, time and cost. Moreover, as already seen, they not always provide any complete and feasible results. Therefore, mathematical and computational models can address the need and they can be powerful tools for the future progresses.
Given a physical problem, theoretical scientists may develop a mathe-matical model which describes the problem in a rigorous way and gives a quantitative understanding of its physical behaviour. Some reasonable hy-potesis should be considered and verified in the modelling process. They may elaborate some numerical techniques for the discretization of the prob-lem and, togheter with some computational scientists, they may provide some results as output of the numerical simulations.
Numerical tools should be used in order to lighten the costs associated with the physical experiments, especially in terms of availability of resources and repeatability of the experiments. Using numerical simulations, theoreti-cal and computational scientists aim at supplying some results which should be useful for the physical experiments. Indeed, the numerical results should limit the range of needed conditions and observations. This is the standard method for code validation: simulations are carried out using data for which a prior knowledge of the result is available as input. The same routine is performed using all the possible data for which the outputs are accessible. If the numerical results and the physical ones are in some sense similar, then it is possible to state with certainty that the numerical tool is robust. There-fore, if required, in silico experiments can substitute in vitro tests in order to reduce the number of different biological conditions that need to be analysed. Thanks to the combination of scientific and engineering knowledge, the use of computational models allows to simulate the effect of a medical treat-ment on an individual patient, leading to the promising field of personalized medical therapy, based on their own anatomy and history. Many interest-ing examples that deal with numerical tools for patient-specific treatments are available in literature as reported in [5] and [16], in the framework of both drug-encapsulated injectable nanoparticles and biomedical devices. Numerical tools for drug-encapsulated injectable nanoparticles The delivery of anticancer molecules using nanoparticles is one of the most widespread application of nanomedicine, however many challenges that
re-Figure 1.4: Technical challenges for theoretical and computational scientists in nanomedicine. Figure from [16].
quire the intervention of computational studies still need to be addressed (Figure 1.4). Decuzzi in [16] asserts that the first issue that is necessary to tackle is the maximization of the loading efficiency, namely the ratio between the mass of encapsulated drug and the total mass, while controlling the re-lease. Simulations for the analysis and optimization are made at molecular level by means of Molecular Dynamics, Monte Carlo or other simulations (see for example [29] and [50]).
Moreover, it has been studied that the formation of a protein corona around a nanoparticle affects its terapeutic performance [39]. Again, molecular simula-tions are used to model the interaction of blood proteins with the nanoparti-cles in order to predict and control their adsorption and to design the optimal surface features [18].
Lastly, there is the need to maximize the deposit at diseased sites and to avoid non-specific accumulation in organs different from the targeted ones. For this purpose, the description of the journey of nanoparticles within the vascular network can be done by means of different type of analysis, ac-cording to the specific aim, such as continuum mechanics approach [22] and molecular-based method [52].
An interesting application is about cardiovascular disease, which is the most important cause of death in the USA. Hearth attacks are caused by the blockage of the coronary artery due to the rupture of the so-called vulnerable
Figure 1.5: Spatial distribution (cm−2) of different sized particles in an in-flamed arterial tree: (A) dp = 0.1 µm, (B) dp = 0.5 µm and (C) dp = 2.0 µm
in terms of nadh/ (ninj× A), where nadh is the number of adhered particles,
ninj is the total number of injected particles and A (cm2) is the surfae area.
Figure from [26].
plaques that are not always detected by standard imaging techniques. It has been studied that plaque instability is mainly determined by the presence of plaque inflammation [20] and targeted drug-encapsulated nanoparticles can be used in the treatment in order to encourage rapid plaque stabilization [9]. For this purpose, in [26] Hossain et al. developed a numerical tool that was able to capture essential 3D aspects of the transport and the vascular deposition of nanoparticles in an inflamed arterial tree (Figure 1.5).
As previously underlined, computational modelling is particularly suit-able for the description and the analysis of cancer treatments. In order to monitor and predict the abnormal proliferation of mass caused by cancer, numerical tools have been developed for the prediction of the growth of the tumor. Indeed, the most evident consequences of the disease are invasion, metastasis and angiogenesis. Invasion is the capability of the tumor to get into the tissues, metastasis is its ability to spread to tissues without being di-rectly connected with them and angiogenesis is the unregulated blood vessels growth [21]. The precise description of tumor growth and nutrient evolution can be carried out in terms of an extracellular matrix, modelled as a solid, healty cells and tumor cells permeated by several fluids [48]. From the nu-merical point of view, a multi-phase approach (Figure 1.6), as the one just described, does not require some computationally expensive interfaces con-ditions and it is very flexible. Therefore, it is possible to predict the tumor growth as function of the nutrient concentration and the initial characteri-zation of the system and it allows to study and monitor the response of a
Figure 1.6: The multi-phase system within a Representative Elementary Vol-ume for tumor growth and nutrient evolution. Figure from [5].
Figure 1.7: FSI simulation of a VAD in action. Blood flow velocity during the fill (a) and the eject (b) stages; deformed configuration of the thin structural membrane during the fill (c) and the eject (d) stages. Figure from [5].
particular type of tumor to therapies.
Numerical tools for biomedical devices
In the field of design and optimization of biomedical devices, the ventricular assist device (VAD) is an example of biomedical tool which is already clin-ically used. VADs are implanted on a patient in case of heart failure and they give mechanical support to the heart, in particular to ventricles, for their pulsatile aim. There is the need for an improvement in the peadriatric field, as the devices available for adults are not suitable for children. To this end, numerical modelling and fluid-structure interaction simulations (Figure 1.7) are needed to describe and visualize the behaviour of VAD [7] and some shape-optimization studies for the design of the paediatric device can be per-formed. Other than the fluid-structure interaction models of the device, it is
necessary to model the blood coagulation process, since thrombus formation is one of the most important problem in VADs. In this way, it is possible to understand the source of the problem and to suggest some modifications in the design. An alternative to the direct modelization of the coagulation process that has been exploited in literature, see for example [34] and [35], is the analysis of the regions of blood recirculation and long residence times. Moreover, in order to study the effect of the physiological conditions on the device and viceversa, the model should be able to include information and parameters from the physiology of the patient.
1.1.3
Discussions
It is now clear that nanomedicine is a research field where innovative scien-tific discoveries can take place. It is also apparent that nanomedicine needs mathematical modeling, theoretical and computational analyses and simu-lations to rapidly flourish. Indeed, computational modelling is employed in nanomedicine for the optimization of the performances of nanostructures, that is equivalent to the optimization of the efficacy of nanomedicines. More-over, during a series of experiments, mathematical modelling and simulations allow to limit the required expensive parametric studies which are usually strictly dependent on the characteristics of the system. Numerical simula-tions are also less expensive than in vivo tests. Furthermore, computational studies provide some hints in the understanding of the physics and mechanics that regulate the biological systems. Thanks to the advantages provided by the modelling, useful guidelines for future improvements in the design pro-cess of devices and in the development of novel experimental techniques are now available. Theoretical and computational analyses should also help the effective clinical fruition of nanomedicines, since until now the use of models in nanomedicine is not yet widespread.
The most interesting, but also challenging aspect of this new frontier of the scientific knowledge is the need of an interdisciplinary environ-ment. The required coupling among medical, nanotechnological and numer-ical knowledge leads to the essential and stimulating cooperation of scientists coming from different disciplines. Biologists, chemists, clinicians, pharmacol-ogists, physicists should work side by side with engineers, mathematicians, theoretical and computational scientists, while up to now their knowledge has been developed in isolation.
Figure 1.8: Multiscale nature of a biological system: length and time scales. Figure from [55].
1.2
Multiscale models in biology
1.2.1
Multiscale nature of biological systems
All the biological systems are characterized by a myriad of small elements, typically molecules, that represent the elementary units of any organism. Molecules are assembled in bigger and bigger elements, up to supramulecular structures, then cells, tissues and organs. The integrated components that form a living organism interact and mutually depend on each other in order to play their own role [38]. In Figure 1.8, a sketch of the multiscale structure of the biological system is reported. Biological systems are organized in multiple scales and each one cannot be considered fully isolate from the others. Each level of this hierarchical structure has its own time-scale and length-scale and they are spread over a wide range. Most of the processes that take place at the largest scale cannot be observed at a smaller scale and viceversa. The events that happen at a given scale can be studied only if the investigation scale is the adequate one, nevertheless the effects can be seen at a larger and smaller scale, but under different forms. Moving from the smallest scale, typically the molecular level, where the elementary blocks are identifiable, with a larger and larger point of view it is possible to identify several intermediate scales, the so-called mesoscopic levels, up to the largest one, the macroscopic scale that typically is the organism or the organ level.
1.2.2
Single-scale models and upscaling techniques
As previously pointed out, the most effective strategy to understand the be-haviour of the biological systems and to control and predict the effects of nanotechnologies applied to medicine is the mathematical modelization and successively the computational simulation of the system. Even the mathe-matical model that describes a biological system has to take into account the
Figure 1.9: A classification of methods for biological systems in terms of their characteristic length- and time-scales. Figure from [55].
intrinsic multiscale structure of the organism. Therefore, to this end, it is necessary to use several types of methods for the study of the system at dif-ferent time-scales and length-scales. As reported in [55], two main classes of approaches can be identified: phenomenological and mechanistic. The former approach describes a process using laws based on empirical obser-vations and not from theoretical studies. It also uses lumped parameters, which are usually difficult to understand. In the latter one, a phenomenon is depicted by a set of microstructural models that are integrated exploiting some functional interdependence rules. A sketch of the two classes of method is depicted in Figure 1.9 and some examples are listed below:
• Ab initio methods (ABM) work at the atomistic scale. The properties of the atoms and the interactions with the surrounding environment do not accept any approximations, therefore they have to be considered from the quantum mechanics point of view [41]. Clearly, this method is computationally very expensive when it is applied to problems bigger than few hundred of atoms.
• The molecular dynamics (MD) approach is based on the generation of the atomic trajectories of a system composed by several particles [2]. The trajectory is obtained by numerical integrating the Newton’s law. Moreover, the method requires some hypothesis on the interatomic interactions that need to be given as input, preventing the possibility to analyse them.
• Coarse grained (CG) technique [56] represents a necessary link between the microscopic and the macroscopic scale. It is similar to the MD
method, as it exploits the interactions between single units, but the structural units are bigger than in MD.
• The discrete models (DM) are based on the description of a single cell and its interaction with the surrounding environment, which can be governed by deterministic or probabilistic rules [36]. The position, velocity and internal state of each cell are given as variables and the possible positions of a cell can be on a regular mesh for the lattice-based model or can be unrestricted for the lattice-free model.
• The continuous models (CM) are characterized by partial differential equations that describe the evolution in space and time of the variables. A CM is suitable in the case of large scale, since it is based on the assumption of the continuity of the matter [58].
Each one of the single-scale models that have just been detailed is conve-nient for a specific scale. However, the description of a process at different levels is not enough, since it is necessary to link the information that de-rives from each single-scale model. This means that a multiscale approach is needed. Indeed, a multiscale approach aims at gathering the models and the results from different levels and to combine them. Usually, the link-ing between different time-scales and length-scales is performed studylink-ing the problem at a characteristic scale and exporting the results at another scale. Depending on the “direction of study”, the multiscale analysis can be carried out in a bottom-up way, starting from the lowest scale, where the outputs are used as input conditions for a higher level, or viceversa in a top-down way, where the outputs of a higher scale represent some boundary conditions for a lower level [12]. In order to move from one level to another, several techniques are available in literature and some examples are summarized in [55]:
• Homogenization techniques are based on the average of the macroscale laws over the microscale. The method needs some linearity or periodic-ity properties as assumption. For example, the homogenization method has been used to determine the macroscopic transport properties of tu-mors in [51] and [43].
• Mixture theory is based on a weighted averaging of the single con-stituents of the matter. An example of application can be found in [4], where the mixture theory has been applied to consider the resid-ual stress caused by the growth and the remodeling of soft biological tissues.
Figure 1.10: Different scales within the human circulatory system: (a) whole cardiovascular system at the macroscale, (b) artery at the mesoscale and (c) red blood cells at the microscale. Figure from [55].
• Asymptotic expansion method can be used if there is a certain regularity at the microscopic level. It allows to perform an expansion of the macroscopic fields over the microscale. For example, in [46] the effective diffusivity of the outermost layer of the skin, the stratum corneum, has been obtained through the method of asymptotic expansion.
Therefore, in every biological system, each process at any scale can be anal-ysed provided that a suitable model is considered and that a good approxi-mation technique is used in order to link the scales among them.
1.2.3
Examples of application
In nature, many organisms and many biological systems share an intrinsic multiscale structure, here a couple of examples taken from [55] are reported. The cardiovascular system is composed by the heart, blood and blood vessels and it is divided into the systemic circulation and the pulmunary cir-culation. In the heart, there are four chambers, which are the left atrium, the left ventricle, the right atrium and the right ventricle, each of them occu-pies a volume of a few cubic centimeters (cm3). In the systemic circulation, which provides oxigenated blood to the body, the heart pumps blood from the left ventricle to the aorta, the biggest artery (diameter of about 2-3 cm), then the aorta branches in several blood vessels of decreasing size: arteries (cm), arterioles (mm) and arterial capillaries (µm). Since the capillaries are
very small, they are able to reach all the peripheral parts of the body and through their walls oxigen and the nutrients can extravasate and diffuse in the intracellular space (nm). At this stage, the deoxigenated blood enters the venous system in bigger and bigger vessels (venous capillaries, venules and veins) in order to go back to the heart. The deoxigenated blood enters the right atrium flowing in the superior and inferior vena cava. From the right ventricle, the deoxigenated blood is carried by the pulmonary circulation through the pulmonary artery (diameter of about 3 cm) to the lungs, where it is oxigenated and it goes back to the left atrium flowing in the pulmonary vein. The blood itself shares a multiscale structure, since it contains plasma, red blood cells, proteins, small molecules and ions that are of dimensions of the order of few µm. Therefore, it is clear that the human circulatory system involves different characteristic length-scales and time-scales (Figure 1.10), so that a multiscale analysis is needed for a complete description.
A second example is about solid tumors, which are a population of cells that abnormally grows and metastasizes in a distant region. The tumor mass exploits different properties during its developments, which are related to several characteristic scales in length and time. Tissue invasion, migration of the cells and creation of metastasis are some peculiar effects of tumor disease and they are related to the largest length scale of the human body, tipically the organs or the organism level (m). Angiogenesis, which is the unregulated blood vessels growth, usually occurs at lengths of cm and times of s, while the interactions among tumor cells and molecules take place at typical scales of µm and µs [24]. Consequently, also the study and the analysis of the growth of tumor masses require investigations at different time and length scales.
Furthermore, not only the multiscale nature of many biological systems needs to be exploited, but it is also necessary to take into account the hierar-chical structure of the system during the investigation of nanotechnological devices or nanoscale medicine. In this sense, a general overview of a multi-scale analysis applied to nanomedicine will be exploited in the next section. In the following chapters, a detailed and complete description of this case will be reported.
1.2.4
Nanoparticle delivery in a capillary network
In the framework of nanoparticle-based drug delivery system for the treat-ment and early-detection of cancer, an extended multiscale computational model is currently under investigation and in this work we will move in this direction. In this regard, the aim of the hierarchical computational model is the prediction of the vascular and extravascular transport of molecules,
Figure 1.11: Multiscale modules for vascular and extravascular transport of nanoparticles.
nanoconstructs and cells in neoplastic tissues. The complete model (Figure 1.11) can be split into several modules at different scales (the tissue module, the vascular module and the extravascular module) in order to better specify and understand the processes that are involved.
• At a macroscopic level, the tissue module focuses on the temporal and spatial distribution of injected agents within vascular and extravascu-lar compartments of a capilextravascu-lary network. Transport phenomena at the level of the microcirculation play a key role in the propagation of the characteristic effect of cancer and mass transport is also at the basis of most of the cancer pharmacological treatments. The model ends up to be a system of partial differential equations integrated by means of a finite element method. The system takes into account a two-way coupling between the capillary network and the surrounding environ-ment. This connection is exploited because the circulatory system is represented by means of a network of one-dimensional channels that act as a concentrated source of flow immersed into the interstitial vol-ume. Therefore, as a result of the 3D/1D coupling between the network and the external volume, the tissue module itself shares a multiscale nature. However, a multiscale approach needs to be set up among the
modules, as the characterization of some parameters of the model, such as the vascular adhesion parameter of the nanoparticles to the vessel wall, their diffusivity in the extravascular space or the permeability of the vessel walls with respect to some solutes, is still problematic at the tissue level.
• Moving to a smaller scale, in the vascular module the margination dynamics and vascular adhesion of injected agents are predicted con-sidering the actual blood rheology, transporting a dense suspension of deformable cells and particles by means of a Lattice Boltzmann-Immersed Boundary method. The adhesion of transported carriers to the targeted vessel site strongly depends on the particle geometry, on other particle properties and on the flow-related parameters, such as the red blood cells dynamics. In order to study the near wall dynam-ics of circulating agents and to take into account all these effects, a Lattice-Boltzmann methods is used for the description of the fluid dy-namics on a fixed lattice and an Immersed Boundary method is used for the presence of moving and deforming bodies.
• At the microscale, the extravascular module deals with the extrava-sation of the nanoconstructs and the following migration toward the malignant tissue. From a numerical point of view, these issues can be approached with a suitable hybrid version of the Cellular Potts model [47]. The extravascular dynamics is determined by a stochastic mini-mization of a discrete effective energy that contains several terms. The model is said to be hybrid in the sense that the nanostructure migra-tion is determined by the kinetics of some environmental chemical vari-ables, by their possible absorption and by the resulting directional cues. Therefore, it is possible to determine the influence in the nanoparticle extravasation and following migration of both external and internal de-terminants: it is possible to analyze dimensions, elasticity and remod-eling ability of the nanoconstructs in order to pass in between capillary walls, without disruptive consequences for the vasculature. Moreover, with this model it is possible to link the biophysical mesoscopic prop-erties of tumor masses to the kinetics of some biochemical variables, specifically the kinetics of the drugs carried by the nanoparticles. Some interesting studies for the extravascular module can be also per-formed by considering some experiments. More precisely, it is possible to determine the interstitial properties by means of inverse problems: by knowing part of the solution of a physical problem, we recover which parameters have generated that particular solution. In the specific case,
the diffusion of solutes and nanoparticles in an interstitial matrix is experimentally observed and some images of the phenomenon are cap-tured. From the numerical point of view, the diffusion is simulated and the diffusive properties of the solutes or nanoparticles in the matrix are derived by looking for the parameters that minimize the discrepancy between the experimental and numerical observations.
As can be seen in Figure 1.11, the two-way connections among the mod-ules are very efficient towards the achievement of the final aim, that is the development of a multiscale integrated approach for a new class of nanocon-structs for more effective cancer detection and treatment. In this respect, the coupling between the tissue and the extravascular module can be exploited in one way (from extravascular to tissue) as looking for a tissue equivalent diffusion. Indeed, the CPM or the control approach can provide a complete description of the nanoparticle behaviour in the extracellular matrix and it can be summarized in a single lumped parameter, such as the diffusion of the nanoconstruct in the tissue, that can be easily included in the macroscopic module. On the other way, the tissue module can supply some values that can be used as initial/boundary conditions in the study of the extravascular space, such as the value of concentration of chemoattractant. Also the rela-tion between the tissue and the vascular module has two direcrela-tions: from the tissue to the vascular module and viceversa. On one hand, some concentra-tion values or other biophysical condiconcentra-tions are given as boundary condiconcentra-tion for the vascular module. On the other hand, the vascular model can provide information in the form of lumped parameter to the tissue model, such as the vascular adhesion parameter.
In this work, we will focus on some particular aspects of the multiscale structure that has just been detailed. In particular, the work will be based on three different levels of the multiscale system and the final aim is to specify the general model of microcirculation in the case of transport of nanoparticle, exploiting the hierarchical structure. Here a general overview of the model at the adequate scales is proposed, then their complete description will be provided in the following chapters.
Macroscale: microcirculation and drug delivery
At the macroscopic scale, the mathematical model for microcirculation and drug delivery is analysed. The model has the ability to capture many phe-nomena that take place in the delivery of drugs and that represent a possible barrier to the final aim of reaching a target site, such as blood perfusion, particle transport and interaction with the tissue. The mathematical model
is able to combine many important aspects and mechanisms for nano-based treatments, such as realistic vasculature, coupled capillary and interstitial flow, coupled capillary and interstitial nanoparticle transport and coupled capillary and interstitial drug transport. Blood flow and mass transport is described by means of advection-diffusion equations in one dimensional chan-nels, immersed in a 3D space that represents the interstitial tissue. The main advantage of this approach is that the computational grids needed for the ap-proximation of the equations on the microvasculature and on the interstitial volume are completely independent. Therefore, the geometry of the capillary network can be arbitrarily complex and this effect will not affect too much the global computational cost. This work will be specifically addressed in Chapter 2.
Microscale: adhesion of a nanoparticle in microvessels
At the microscale, in the analysis of the vascular module, the study of the vascular adhesion parameter of nanoparticles is exploited. The model is based on a Lattice Boltzmann-Immersed Boundary method, through which it is possible to analyse the effect of size, shape, surface properties and mechanical stiffness of a nanoparticle on some parameters related to the margination dynamics, such as the probability of adhesion to the vessel wall. The particles are treated as Lagrangian solid domains immersed in the fluid. Fluid and structures interaction is considered, imposing no-slip boundary condition on the surface of the solids and evaluating the hydrodynamic and interaction forces on the moving bodies. The idea is to build a map for the vascular adhesion parameter as function of physical properties of the nanoparticles and then use it in the macroscopic model. The results of this work will be shown in Sections 4.1, 4.2 and 4.3.
Microscale: diffusion of drug in the extracellular matrix
Again at the microscopic scale, considering the extravascular computational module, a characterization of the interaction of delivered drugs with the in-terstitial tissue is performed by means of a predictive numerical model for some parameters. Many parameters can be estimated with this model, in par-ticular we will focus on the diffusivity coefficient of the extracellular matrix with respect to the chemical species. For the computation of the parameters, it is possible to take advantage of some available experiments in which the transport of molecules and nanoparticles is studied in a collagen matrix. The prediction of the parameters is made by means of a control problem in which the discrepancy between the particle concentration in the numerical and the
physical experiment is minimized, provided that suitable diffusion equations are satisfied in the numerical device. The idea is to provide the optimal value of the diffusivity coefficient of the extracellular matrix with respect to the chemical species from specific numerical experiments and, subsequently, include these results as lumped parameters in the macroscopic model. The result of this work will be presented in Section 4.4.
Mathematical model for
particle transport in the
microvasculature
The aim of this work is to provide tools for the investigation of a drug delivery system in the microcirculation based on injectable nanoparticles. The intent of the forthcoming study is to develop a model able to capture the distri-bution of nanoparticles injected in the microcirculation, accounting for their transport, their adhesion to the capillary wall and material extravasation from the capillary walls.
In Section 2.1 we state the problem in the microvasculature immersed in the interstitial tissue as a system of PDEs. Then, in Section 2.2 we exploit the embedded multiscale approach to couple the two problems leaving on separate scales, while the dimensionless formulation of the coupled problem is derived in Section 2.3. In Section 2.4 a set of suitable boundary and initial conditions for both the tissue and vessel variables is specified, while an overview of compatibility conditions at junctions between multiple vessels is given in Section 2.5. In order to approximate the coupled differential problem with the mixed finite element method, firstly, a dual mixed weak formulation of both the vessel and tissue problem is provided in Section 2.6 and successively, an alternative primal mixed weak formulation is derived in Section 2.7. The latter formulation is the foundation for the upcoming analysis as well as for the numerical approximation strategy presented in Section 2.8 and the corresponding algebraic formulation in Section 2.9.
Figure 2.1: On the left, microvasculature within a tissue interstitium; in the center, the interstitial tissue slab with one embedded capillary; on the right, the reduction from 3D to 1D description of the capillary vessel.
2.1
Three-dimensional model for
microvasculature within a tissue
We define a mathematical model for mass transport in a permeable biological tissue perfused by a vessel network. The domain Ω ∈ R3 where the model
is defined is composed by two parts, Ωt and Ωv, representing the interstitial
volume (tissue) and the capillary bed (vessel ), respectively. Assuming that the capillary vessels can be described as a set of cylinders, we denote with Γ the outer surface of Ωv and with Λ the one-dimensional line that describes the
centerline of the vessel. In general, the vessel radius R can change along the arc length of Λ (see Figure 2.1). In particular, the forthcoming model aims at describing the transport of nanoparticles that are injected in the capillary network and their behaviour in the blood stream and in the interstitial tis-sue. The physical quantities of interest are the concentration of transported nanoparticles in the capillary network and in the tissue interstitium, cv and
ct, respectively. The variables have to be intended as function of space, being
x ∈ Ω the spatial coordinates, and time t.
Concerning the interstitial volume Ωt, it can be considered as an isotropic
porous medium with velocity field ut and pressure pt. We assume that the
particles are advected by the fluid and diffuse in all Ωt. An important effect
for tissue perfusion is the lymphatic drainage. Excess of fluid extravasated from the blood circulation is drained by lymphatic vessels and returns to the blood stream. Since a geometrical description of the lymphatic vessels is not available, following [53], the lymphatic drainage is modeled as a sink term for the interstitial flow. Therefore, the distribution of particles in the interstitial tissue is also affected by the lymphatic drainage. Accounting for these effects, the equation for mass transport in the interstitium is
∂ct ∂t + ∇ · (ctut− Dt∇ct) + L LF p S V (pt− pL) ct= 0 in Ωt× (0, T ), (2.1)
where Dt is the particle diffusivity in the interstitium, assumed to be
con-stant, LLFp is the hydraulic conductivity of the lymphatic wall, VS is the surface area of lymphatic vessels per unit volume of tissue and pL is the hydrostatic
pressure within the lymphatic channels.
For the blood capillary, given the flow velocity vector uv and the pressure
field pv, the particles are advected by the fluid and diffuse in Ωv, similarly
to the previous case. Moreover, a peculiarity of the injected particles is their ability to adhere to the vessel wall. This effect is due to the presence of ligand molecules on the surface of the particles and of receptor molecules on the endothelial layer. The adhesion of particles to the wall is described as a sink term along the capillary network. The model accounting for the particle transport in the blood stream and their adhesion to the wall results in the following equation:
∂cv
∂t + ∇ · (cvuv− Dv∇cv) + Π
ef fc
v = 0 in Ωv× (0, T ), (2.2)
where Dv is the particle diffusivity in the capillary network, assumed to be
constant in Ωv, Πef f is the effective vascular adhesion parameter which could
depend on time.
In order to couple the two problems, it is necessary to impose some condi-tions at the interface Γ = ∂Ωv∩ ∂Ωt. In particular, we describe the capillary
wall as a semipermeable membrane allowing for the leakage of the fluid and for selective filtration of particles. A good model for mass transport across semi-permeable membranes is the Kedem-Katchalsky equation. According to this equation, the flux of particles per unit surface across the capillary walls is
(cvuv − Dv∇cv) · n = (ctut− Dt∇ct) · n = (2.3)
= (1 − σ) Lp[(pv− pt) − σ (πv − πt)] cavg + P (cv − ct) on Γ × (0, T ) .
In particular, Lp is the hydraulic conductivity of the vessel wall and πv− πt
is the difference in osmotic pressure, where π = RgT c is the osmotic pressure
given by a concentration c of a given solvent, Rg is the universal gas constant
and T is the absolute temperature. Indeed, because of osmosis, the pressure drop across the capillary wall is affected by the difference in concentration of the substances dissolved in blood. However, only the large molecules can induce a significant effect, for this reason we only consider the presence of proteins, therefore from now on πv and πt will represent the concentration of
proteins in the vessel network and in the tissue, respectively. The reflection coefficient σ quantifies how different a semi-permeable membrane is from the ideal permeability. The term accountig for the pressure difference is
multiplied by the average concentration within the capillary vessel cavg := wct+ (1 − w) cv
and it is defined as a suitable combination of cv and ct. In particular, 0 < w <
1 is a weight that depends on the P`eclet number of the particle transport through the wall, however, for the sake of simplicity, we will use w = 1
2.
Moreover, the flux depends on the concentration gradient across the capillary walls in terms of P , the permeability of the vessel wall with respect to the particle.
Therefore, these modelling assumptions lead to the following mass trans-port problem in the entire domain Ω:
∂ct ∂t + ∇ · (ctut− Dt∇ct) + L LF p S V (pt− pL) ct = 0 in Ωt× (0, T ), ∂cv ∂t + ∇ · (cvuv− Dv∇cv) + Π ef fc v = 0 in Ωv× (0, T ), (cvuv − Dv∇cv) · n = (ctut− Dt∇ct) · n = = (1 − σ) Lp[(pv− pt) − σ (πv − πt)] cavg + P (cv − ct) on Γ × (0, T ). (2.4) Denoting with f the flux per unit area released by the surface Γ, the flux continuity between the capillary network and the tissue is guaranteed by means of
(ctut− Dt∇ct) · n = f (ct, cv) on Γ × (0, T ).
In our case, f (ct, cv) = (1 − σ) Lp[(pv− pt) − σ (πv − πt)] cavg + P (cv − ct).
To ensure the uniqueness of the solution of problem (2.4), it is necessary to provide some initial conditions and some suitable boundary conditions on ∂Ωv and ∂Ωt. The prescription of these conditions significantly depends
on the particular features of the problem, as well as on the available data, therefore the discussion will be postponed to Section 2.4.
We observe that the expressions for the velocity and pressure fields ut,
uv, pt and pv have not been detailed yet. Indeed, a complete description of
the governing equations for the fluid-dynamical variables of problem (2.4) is given in Appendix A.
2.2
Model reduction: coupled 3D-1D
problem
The fully three-dimensional model (2.4) is able to capture the phenomena we are interested in. However, in order to face some technical difficulties
that arise in the numerical approximation of the coupling between a complex network with the surrounding volume, a multiscale approach [15, 14, 13] based on the Immersed Boundary Method (IBM) [59, 33] can be exploited.
To avoid resolving the complex 3D geometry of the vascular network, the combination of the IBM and the assumption of large aspect ratio between vessel radius and capillary axial length can be convenient. Precisely, with this approach, a suitable rescaling of the equations is applied and the cap-illary radius is let going to zero (R → 0) (Figure 2.1). In this way, the 3D description of the vessels is reduced to a simplified 1D representation and the immersed interface and the related interface conditions are replaced by an equivalent mass source.
2.2.1
Coupling term for the interstitial volume
Thanks to the IBM, the action of f on Γ can be represented as an equivalent source term F distributed on the entire domain Ω:
F (ct, cv) = f (ct, cv) δΓ.
More precisely, F is the Dirac measure concentrated on Γ, having density f , defined by: Z Ωt F (ct, cv) v dΩ = Z Γ f (ct, cv) v dσ ∀v ∈ C∞(Ωt) .
As specified in [10, 15], when R → 0 the mass flux per unit area can be replaced by an equivalent mass flux per unit length, distributed on the centerline Λ. Let γ (s) be the intersection of Γ with a plane orthogonal to Λ, located at s and denote by (s, θ) the local axial and angular coordinates on the cylindrical surface generated by Γ with radius R. It is possible to apply the mean value theorem in order to represent the action of F on v by means of an integral with respect to the arc length on Λ. To be precise, there exists
˜ θ ∈ [0, 2π] such that Z Ωt F (ct, cv) v dΩ = Z Λ Z γ(s) f (ct(s, θ) , cv(s, θ)) v (s, θ) R dθ ds = (2.5) = Z Λ |γ (s)| fct s, ˜θ, cv s, ˜θvs, ˜θds, ∀v ∈ C∞(Ωt).
Then, assuming that the capillaries are narrow with respect to the character-istic dimension of the surrounding volume, namely, R |Ωt|1/d, with d = 3
the space dimension of the model and assuming that f is a linear function, it is possible to have lim R→0f ct s, ˜θ γ(s), cv s, ˜θ γ(s) = f (¯ct(s) , ¯cv(s)) , (2.6) ¯ h (s) := 1 |γ (s)| Z γ(s) h (s, θ) R dθ,
where the bar operator is defined as the averaging operator on the circle of radius R laying on the cylindrical surface Γ and normal to the line Λ. In conclusion, combining (2.5) with (2.6), we recover the following expression for the distributed source term:
Z Ωt F (ct, cv) v dΩ ' Z Λ |γ (s)| f (¯ct(s) , ¯cv(s)) v (s) ds, ∀v ∈ C∞(Ωt). (2.7)
2.2.2
Model reduction for microvascular flow
In order to exploit the one-dimensional representation of the vessel, the gen-eral cylindrical coordinate system (R, θ, s) aligned with the centerline Λ can be reduced using uniquely the arc length s and the tangent unit vector λ that accounts for an arbitrary orientation. To this purpose, it is possible to assume the concentration cv to be constant on a section orthogonal to Λ
located at s. Therefore, the derivative with respect to the variables r and θ are negligible and an average of the quantities of interest on the section can be performed. For this reason, the pointwise and average concentrations in the channel coincide, namely cv(s, θ, r) = ¯cv(s). Thanks to the tangent unit
vector, differentiation is defined as ∂s := ∇ · λ.
Thanks to (2.7), the one dimensional representation of equation (2.4)(b) with the coupling forcing term f reads:
∂cv ∂t + ∂ ∂s cvuv· λ − Dv ∂cv ∂s +2πR πR2Π ef f cv = − 2πR πR2f (cv, ¯ct) , on Λ × (0, T ). (2.8) Notice that we have introduced the adhesion term as a flux per unit length
2πR Πef fc
v and the forcing term is again multiplied by 2πR. Successively,
we had to scale the obtained flux per unit length with the cross section, πR2, since the nanoparticle concentration inside the vessel c
v is measured as
number of particles per unit volume [#/m3].
Moreover, as a consequence of the geometric assumptions, the vessel ve-locity has a fixed direction, i.e. uv = uvλ, so the vessel problem could be
2.2.3
Governing equations for the coupled problem
At this stage, thanks to (2.7), equation (2.4)(a) can be expressed with the equivalent mass source term f . It is now possible to reformulate the flow problem (2.4) in differential form in terms of coupled equations in a three-dimensional space for the interstitial tissue and in a one-three-dimensional space for the capillary network.
The coupled problem for the transport of particles from the microvascu-lature to the interstitium consists to find the concentrations ct and cv such
that ( ∂ct ∂t + ∇ · (ctut− Dt∇ct) + L LF p VS (pt− pL) ct= 2πRf (cv, ¯ct) δΛ on Ω × (0, T ) ∂cv ∂t + ∂ ∂s cvuv− Dv ∂cv ∂s + 2πR πR2Π ef fc v = −2πRπR2f (cv, ¯ct) on Λ × (0, T ) (2.9) where f (cv, ¯ct) = (1 − σ) Lp[(pv − ¯pt) − σ (πv− ¯πt)] 1 2cv+ 1 2¯ct + P (cv − ¯ct)
and we also recall the average operator ¯ gt(s) = 1 2πR Z 2π 0 gt(s, θ) R dθ. (2.10)
Notice that the distinction between the subregion Ωt and the entire domain
Ω is no longer meaningful, since the one-dimensional Λ has zero measure in R3, therefore, for notational convienience, from now on Ωt will be identified
with Ω and Ωv with Λ.
2.3
Dimensional analysis
The first step toward the application of the model is to perform a dimensional analysis of equations (2.9), in order to highlight the relative magnitude of all the phenomena that affect mass transport, such as diffusion, convection, ligand-receptor interactions and lymphatic effects. For this purpose, the pri-mary variables that have been chosen for the analysis are length, velocity, pressure and concentration. We choose the average spacing between capil-lary vessels, d, as characteristic value for the length, the average velocity in the capillary bed, U , for the velocity, the typical pressure drop along the extrema of the vessel network, δP , for the pressure and the maximal admis-sible value of concentration at the systemic level, C, for the concentration. Correspondingly, the dimensionless groups that characterize the equations are:
• R0 = R
d the non dimensional radius,
• Av = DU dv the ratio of diffusion and transport in the vessel network,
• At = U dDt the ratio of diffusion and transport in the tissue interstitium,
• M = Πef f
U the magnitude of vascular deposition,
• Q = LpδP
U the hydraulic conductivity of the capillary walls,
• QLF = LLF p VS
dδP
U the non dimensional lymphatic drainage,
• Υ = P
U the magnitude of leakage from the capillary bed.
Therefore, the dimensionless form of (2.9) is ( ∂ct ∂t + ∇ · (ctut− At∇ct) + Q LF(p t− pL) ct = 2πR0fadim(cv, ¯ct) δΛ on Ω × (0, T ) ∂cv ∂t + ∂ ∂s cvuv− Av ∂cv ∂s + 2πR0 πR02M cv = −2πR 0 πR02fadim(cv, ¯ct) on Λ × (0, T ), (2.11) where fadim(cv, ¯ct) = {(1 − σ) Q [(pv− ¯pt) − σ (πv − ¯πt)] 1 2c¯t+ 1 2cv + Υ(cv− ¯ct)}.
For the sake of simplicity, the same symbols for the standard and dimension-less variables and for the standard and dimensiondimension-less operators have been used. Notice that the dimensionality of δΛ is [length]
−2
and that the dimen-sionless Dirac distribution is again called δΛ.
2.4
Boundary and initial conditions
As previously mentioned, in order to guarantee the uniqueness of the solu-tion of problem (2.11), it is necessary to specify some boundary condisolu-tions (BCs) on both the tissue and vessel boundary, that is ∂Ω and ∂Λ, respec-tively. Generally speaking, the prescription of adequate boundary conditions is a consequence of the variational formulation of the problem. The same occurs for the BCs for problem (2.11), indeed the choice will be driven by the particular integration by parts that will be performed or, alternatively, by the need to model some special behaviour of the problem unknowns on the boundary. However, the choice of the BCs also depends on the available data.
Concerning the capillary flow, we refer to the collection of inflow and outflow tips of the vessel network as ∂Λ ≡ Λin∪ Λout, i.e. non junction points
where the tangent unit vector is inward-pointing and outward-pointing. On the inflow boundary of the network, a given nanoparticle concentration cinj is
injected in the blood stream, while on the outflow boundary a homogeneous Neumann boundary condition is enforced, letting the particles to freely leave the system, namely:
cv = cinj on Λin× (0, T ), (2.12)
∂cv
∂s = 0 on Λ
out× (0, T ). (2.13)
We require cinj ∈ H1/2(Λ) .
For the interstitial volume Ω, we enforce on all the artificial interfaces of the tissue, ∂Ω, boundary conditions that mimic the resistance of the sur-rounding material. In particular, a fixed value for the normal diffusive flux will be imposed, namely:
− At∇ct· n = βcct on ∂Ω × (0, T ), (2.14)
where βc quantifies the conductivity of the outer tissue with respect to the
particle transport.
We observe that the transport equations depend on time, therefore some initial conditions are needed, in particular the system is assumed to be free of particles at time zero, namely
ct(t = 0) = 0 in Ω,
cv(t = 0) = 0 in Λ.
2.5
Junction treatment
The enforcement of boundary conditions is necessary, but not sufficient to close the problem (2.11). Indeed, the domain splitting approach requires the imposition of suitable compatibility conditions at the branching points (junctions) of the vessel tree. The aim of this section is to propose a suitable set of compatibility conditions at junction points.
Firstly, we assume that the concentration in the vessel is continuous in all ¯Λ and automatically at the junction points. Typically, the continu-ity of cv is guaranteed thanks to the choice of the functional framework
cv ∈ H1(Λ) ⊂ C0 Λ. In order to guarantee the conservation of mass, we¯
