Mathematical Modeling of Tumor Organoids: Toward Personalized Medicine

Abstract

Three-dimensional organoid and organoidal cell cultures can recreate certain aspects of in vivo tumors and tumor microenvironments, and thus can be used to test intratumoral interactions and tumor response to treatments. In silico organoid models, when based on biological or clinical data, are an invaluable tool for hypothesis testing, and provide an opportunity to explore experimental conditions beyond what is feasible experimentally. In this chapter, three different approaches to building in silico organoids are described together with methods for integration with experimental or clinical data. The first model will be used to determine the mechanisms of development of breast tumor acini, based on their in vitro morphology. The second model will be used to predict conditions for the most effective cellular uptake of therapies targeting pancreatic cancers that incorporate intravital microscopy data. The third model will provide a procedure for assessing patients’ response to chemotherapeutic treatments, based on the biopsy data. For each of the models, a protocol will be proposed indicating how it can be used to generate testable hypotheses or predictions. These models can help biologists in determining what experiments should be performed in the laboratory. They can also assist clinicians in assessing cancer patients’ response to a given therapy and their risk of tumor recurrence.

Appendices

Appendix A: Mathematical Framework of the IBCell Model

The IBCell model belongs to the class of fluid-structure interaction models, and utilizes the immersed boundary method framework [33, 38]. The boundaries of all cells Γ are discretized, and every material point X(l,t) represents a cell pseudo-receptor (l is a position along the boundary, t denotes time). The forces F(l,t) defined at each boundary point (Eq. A1) arise from combining the elastic properties of cell boundaries, from cell-to-cell adhesion, and from contractile forces splitting a cell during its division. In this equation, G denotes spring stiffness, and L denotes spring resting length. These forces are applied to the surrounding fluid, as described in Eq. A2. The source points Y _k and sink points Z _m are placed in the cell local microenvironment, and the source and sink values S ⁺(Y _k ,t) and S ⁻(Z _m ,t) are chosen such that they balance around each cell separately (Eq. A3). They assume the non-zero values only during cell growth (proliferation) or cell shrinkage (apoptosis). The transitions between the material points on cell boundaries and the Cartesian grid x = (x ₁,x ₂) in the domain Ω (Eqs. A3 and A7) are defined using the two dimensional Dirac delta function δ (Eq. A4). The fluid flow is described using the incompressible Navier-Stokes equation (Eq. A5), where p is the fluid pressure, μ is the fluid viscosity, ρ is the fluid density, s is the local fluid expansion, and f is the external force density. Eq. A6 is the law of mass balance. All material boundary points are carried along with the fluid (Eq. A7). The kinetics of ECM proteins γ(x,t) is defined along the cell boundaries and includes: constant secretion of ECM (at a rate κ₁) along the cells’ basal domains and ECM decay (at a rate κ₂) around all the cells’ boundaries (Eq. A8). More details on the mathematical formulation of IBCell and the implementation of cell life processes can be found in [38, 40, 44].

$$ \kern0.50em F\left(l,t\right)=G\frac{\parallel X\left(k,t\right)-X\left(l,t\right)\parallel -L}{\parallel X\left(k,t\right)-X\left(l,t\right)\parallel}\left(X\left(k,t\right)-X\left(l,t\right)\right), $$

(A1)

$$ f\left(x,t\right)=\underset{\varGamma }{\int }F\left(l,t\right)\delta \left(x-X\left(l,t\right)\right) dl, $$

(A2)

$$ \kern0.50em s\left(x,t\right)=\sum_{k\in \varXi +}{S}_{+}\left({Y}_k,t\right)\delta \left(x-{Y}_k(t)\right)+\sum_{m\in \varXi -}{S}_{-}\left({Z}_m,t\right)\delta \left(x-{Z}_m(t)\right), $$

(A3)

$$ {\delta}_h(r)=\Big\{{\displaystyle \begin{array}{l}\frac{1}{4h}\left(1+\cos \left(\frac{\pi r}{2h}\right)\right)\kern1.25em if \mid r\mid <2h\\ {}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\kern4em if\ \kern0.125em \mid r\mid \ge 2h\end{array}}\ \ \ r=\parallel x-X\left(l,t\right)\parallel, $$

(A4)

$$ {\displaystyle \begin{array}{l}\rho \left(\frac{\partial u\left(x,t\right)}{\partial t}+\left(u\left(x,t\right)\cdot \nabla \right)u\left(x,t\right)\right)=-\nabla p\left(x,t\right)+\mu \varDelta \mathbf{u}\left(x,t\right)\\ {}+\frac{\mu }{3\rho}\nabla s\left(x,t\right)+f\left(x,t\right),\end{array}} $$

(A5)

$$ \rho \nabla \cdot u\left(x,t\right)=s\left(x,t\right), $$

(A6)

$$ \frac{\partial X}{\partial t}=u\left(X(t),t\right)=\underset{\varOmega }{\int }u\left(x,t\right)\delta \left(x-X(t)\right) dx, $$

(A7)

$$ \frac{\partial \gamma \left(X\left(l,t\right)\right)}{\partial t}={\kappa}_1X\left(l,t\right)-{\kappa}_2\gamma \left(X\left(l,t\right)\right). $$

(A8)

Appendix B: Mathematical Framework of the microPK/PD Model

The microPK/PD model is a discrete Brownian diffusion model, defined on an irregular domain and coupled with binding kinetics equations. In this model, the explicitly defined tumor cells {C^l} _l=1,…,N (l is a cell index) are assumed to be non-motile and non-proliferative. The individual cell C^l is identified by a set of membrane virtual receptors C ^l = {(X _i , Y _i ) ^l , A _i ^l , B _i ^l} _i=1…M ^l, where (X _i ,Y _i ) ^l are the coordinates of the ith receptor, A _i ^l is affinity to the receptor, and B _i ^l is a receptor saturation level. Initially, the model was calibrated to the experiment-based values for moderate diffusion coefficient (D = 2.5 x 10⁻⁵ mm²/s), high binding affinity (K _A  = 100), and a slow release scheme (as if during intravenous injection). Subsequently, it was used to explore the parameter space beyond the experimental boundaries. The transport of particles is modeled as Brownian motion with an effective diffusion coefficient D that was varied between 10⁻⁴ and 10⁻⁶ mm/s². Receptor binding affinity is defined as the probability with which a ligand molecule binds to the receptor after successful recognition. Therefore, three conditions for the binding probability are used: strong (100%), moderate (10%), or weak (1%). This results in three values of the pseudo-association constant (K _A ): 100, 10, and 1, respectively.

The motion of a ligand particle (x,y) at the n + 1 time point is defined by Eq. (B1). The cell membranes are non-penetrable for the ligand particles, unless they are successfully recognized by receptors (Eq. B1a). The successful ligand–receptor binding condition (BC) requires that the agent particle is in close proximity to the receptor, i.e., ||(x _j ,y _j ) _n  – (X _i ,Y _i )^l|| < r _min , where (x _j ,y _j ) _n are coordinates of a j ^th ligand particle at n ^th simulation step, (X _i ,Y _i )^l are coordinates of the i ^th receptor of the l ^th cell, and r _min is the criterion for a minimum distance. In addition, the receptor may not be saturated (number of already bound particles is below B _i ^l), and the probability of binding meets the affinity criterion. Otherwise, if the new position will result in a particle crossing the cell boundary without satisfying the BC, the particle’s position will remain unmodified (Eq. B1b) or continue to move through the tissue space with Brownian motion, as in Eq. (B1c), where Δt is a time step, ϖ a randomly chosen direction of motion. The effective diffusion coefficient D is defined in Eq. (B2), with k _B being the Boltzmann constant, R the ligand molecule radius, and η the tissue viscosity. The receptor–ligand binding is quantified by fitting the association kinetics to simulated data that represent the averaged saturation per tissue area as a function of time (Eq. B3), where B corresponds to the receptor saturation parameter describing ligand–receptor complex formation [RL], with values between initial saturation and the maximum saturation B ₀ and B _max, respectively; k is a reaction rate constant; t is time. Equation (B4) gives the logarithmic formula used to generate specific binding curves from multiple ligand concentrations, where B, B ₀, and B _max are saturation parameters, K _D is a dissociation constant, h is the Hill slope defining the steepness of the fitting curve, and [L] is ligand concentration.

$$ \kern0.125em {\left(x,y\right)}_{n+1}=\Big\{{\displaystyle \begin{array}{cc}{\left({X}_i,{Y}_i\right)}^l\ & {}^{(a)}\ if the binding condition is satisfied\\ {}{\left(x,y\right)}_n& {}^{(b)}\ if the particle crosses cell boundary\\ {}{\left(x,y\right)}_n+\sqrt{2 D\varDelta t}{\overrightarrow{\omega}}_n& {}^{(c)}\ otherwise,\kern10em \end{array}} $$

(B1)

$$ \kern0.50em D=\frac{k_BT}{6\pi R\eta} $$

(B2)

$$ B={B}_0+\left({B}_{\mathrm{max}}-{B}_0\right)/\left(1-{e}^{- kT}\right) $$

(B3)

$$ B={B}_0+\left({B}_{\mathrm{max}}-{B}_0\right)/\left(1+{10}^{\left(\left[L\right]-\mathit{\log}{K}_D\right)}\right) $$

(B4)