The quest for balance between capturing data and model complexity: A quantitative clinical pharmacology approach applied to monoclonal antibodies
A completely mechanistic description of the processes involved in the distribution, metabolism and elimination of monoclonal antibodies (mAbs) is nontrivial because of the range in physiological processes that govern mAb pharmacokinetics (PKs), each with the potential to be rate‐limiting. As such, the PKs of mAbs can quickly become highly nonlinear and requires a systemic approach when attempting to understand these processes qualitatively, let alone quantitatively. The complexity is further compounded when the mAbs are not located in plasma but in tissue. Therefore, the objective of this tutorial is to develop a step‐wise and quantitative modeling framework for mAbs in order to better understand the relative contribution of each added layer of complexity to the overall dynamics. We will start out with the conceptual Target‐Mediated Drug Disposition (TMDD) model by Mager and Jusko, which assumes that the plasma compartment is the primary site for the interaction between ligand and receptor, whereas tissue simply serves as a homogeneous distribution compartment which is in rapid exchange with plasma. If the objective of the analysis is to fit a model to the plasma concentration‐time profile of an mAb, the use of this model or approximations thereof may be sufficient. However, if the objective is to understand the impact of, for example, different tissue layers as potential distribution barriers, binding to FcRn as a means to prolong the mAb's mean residence time in the body, target expression in tissue, or the impact of target shedding on the overall dynamics of the system, then more complex models are required. In order to highlight the relative contribution of each of these model expansions to the overall model dynamics, we will expand on the basic TMDD model by Mager and Jusko to the completely mechanistic model by Li et al. using trastuzumab as a paradigm compound. We chose trastuzumab because it is a well‐studied mAb whose PKs are governed by all of the previously mentioned processes (i.e., tissue distribution, FcRn‐recycling, and target shedding with membrane‐bound, as well as soluble receptors). At the same time, respective processes have been well‐characterized for this mAb resulting in the availability of respective model parameters in the literature. We will carry these parameters throughout the initial model‐building step of this tutorial in order to be able to compare the dynamics of the different model expansions. A summary of the assumptions and the parameters associated with each model are provided in Supplementary Material and , respectively. Once established, we will compare parameter adjustments as a part of a trastuzumab case example to guide the modeler's choice of parameters well‐suited for optimization in the second part of this tutorial.   Model 1: Basic TMDD In the conceptual model by Mager and Jusko, which we refer to as the basic TMDD model, the formation of the mAb‐receptor complex in plasma ( LR p ) is characterized by the interaction between free ligand (i.e., mAb) in plasma ( L p ) and free target (i.e., receptor) in plasma ( R p ) as shown in Figure . For didactic purposes, L p is supplied to plasma via an intravenous (i.v.) bolus injection, recognizing that more complex input functions, such as subcutaneous or intramuscular administrations, may be needed to accurately reflect clinically relevant dosing scenarios. Once in plasma, ligand can either bind to target at a rate k on,p , be eliminated from plasma at a rate k el,p , or further distribute into tissue at a rate k pt . Ligand can also be redistributed from tissue into plasma at a rate k tp or be released from the ligand‐receptor complex at a rate k off . Equations and initial conditions for the basic TMDD model Mathematically, the basic TMDD model described in Figure translates into a set of four ordinary differential Equations , , , , which describe the kinetics of ligand in plasma ( L p ), ligand in tissue ( L t ), receptor in plasma ( R p ), and ligand‐receptor complex in plasma ( LR p ): (1) d L p d t = − k on , p L p ⋅ R p + k off , p LR p − k el , p + k pt L p + k tp ⋅ V t V p ⋅ L t (2) d L t d t = k pt ⋅ V p V t ⋅ L p − k tp L t (3) d R p d t = k syn , p − k on , p L p ⋅ R p + k off , p LR p − k deg , p R p (4) d LR p d t = k on , p L p ⋅ R p − k int , p + k off , p LR p In Equations and , V p and V t denote the volumes of distribution of, respectively, the plasma and the tissue compartment and k pt = Cl dist / V p and k tp = Cl dist / V t , where Cl dist denotes the intercompartmental clearance between plasma and tissue. Note that in the absence of ligand, the baseline receptor concentration R 0 is given by: (5) R 0 = def k syn , p k deg , p At time zero, the initial conditions for the four differential equations are as follows: (6) where D is the dose and L 0 is the ligand concentration in plasma right after the i.v. bolus dose. At this point, we would also like to note that the following implicit assumptions have been made for model 1: (i) There is no endogenous generation of ligand. (ii) The target is located in plasma (and not in tissue). (iii) Binding of ligand to the receptor has a 1:1 stoichiometry. Discussion of the properties of the basic TMDD model In general, nonlinearity can be caused by multiple processes and is typically dose‐dependent. Nonlinearity as a result of TMDD was first described by Levy and later expanded upon by Mager and Jusko. A mechanistic description of the underlying TMDD processes can provide valuable insight into the dynamics of the systems as a whole as opposed to the behavior of individual constituents. A dynamic systems analysis by Peletier and Gabrielsson , sheds light on the understanding of the different parts of the ligand concentration‐time profiles in plasma. This analysis focused on the primary drivers of the dynamics of the overall TMDD system. Hence, the original two‐compartment model by Mager and Jusko was simplified to a one‐compartment model because distribution between plasma and tissue is generally very fast and does not contribute significantly to the overall model behavior as the formation and dissociation of the ligand‐target complex takes place in plasma. We recognize that this is an oversimplification of physiological reality because most targets are situated in tissue rather than in plasma. Therefore, we will explore the impact of the target being located in tissue versus plasma in subsequent model expansions, especially in model 4. Model 1 captures a broad range of complex dynamics and provides a versatile and mechanistic framework for the systematic evaluation of capacity‐limited nonlinear drug disposition. The typical concentration‐time profile for a drug showing TMDD along with its characteristic parameters was described by Peletier and Gabrielsson in 2012 and is summarized in Figure . In panel (i), it is shown that the nonlinear elimination of mAbs is generally observed within a certain concentration range which can further be categorized via four different phases, A, B, C, and D (Figure ). In the first phase (phase A), ligand rapidly binds to its target. This binding process occurs at a high affinity and at high doses it saturates the target completely. Once the target is completely saturated, ligand is slowly disposed via a first‐order process (phase B), which is characteristic of the ligand's clearance from plasma. The third phase (phase C) is characterized by a Michaelis–Menten type kinetics, which is the result of the target no longer being completely saturated by the ligand. As concentrations continue to drop, the ligand starts to enter a fourth phase, the terminal phase (phase D), where it exhibits linear kinetics again. Each of these phases provides valuable insight into the kinetics of ligand, target, and ligand‐target complex, as well as into model parameters that are characteristic for each of these phases. Panel (ii) of Figure shows a typical graph of total clearance (CL TOT ) versus ligand concentration in plasma ( L p ) at steady‐state. For small values of L p , total clearance is the sum of two processes: (1) nonspecific clearance (CL L ) and (2) internalization of the target (CL I or CL int ). It should be noted that if the L p is large, the target is completely saturated and elimination only occurs via nonspecific, linear clearance (CL L ). This also means that a further increase in dose will yield no additional effect because all target receptors are already occupied. Approximations of the basic TMDD model In practice, it is often difficult to estimate each parameter of the basic TMDD model due to the lack of clinical information on receptor turnover and complex internalization. This motivates simplifications , also referred to as approximations, of the basic TMDD model. In this tutorial, we briefly mention three of them. The Quasi‐Equilibrium (QE) model. In 2005, Mager and Krzyzanski introduced this model, also referred to as the rapid binding model, in which it is assumed that equilibrium between complex formation and complex dissociation is rapidly established. This simplification is based on the assumption that the ligand‐target complex dissociates much faster than it is internalized. This assumption does not always hold true. The Quasi‐Steady‐State (QSS) model by Gibiansky et al. which assumes that complex formation and complex dissociation are at QSS. In other words, it assumes that complex formation is equal to the sum of complex dissociation and internalization. The newly defined constant ( K ss ) represents a generalization of K D . It has improved predictive properties over K D for extensive, high affinity binding scenarios characterized in phase B of the general TMDD profile (Figure ). The Michaelis–Menten (MM) model represents a special case of the QE and QSS models because it assumes that targets are initially fully saturated. Therefore, the MM model is good approximation of the basic TMDD model at high ligand concentrations. It also simplifies the basic TMDD model by relating the reaction rate to ligand concentration. The resulting model has improved predictive properties for both phases B and C, but typically fails to appropriately capture phase D. At this point, we have covered three simplifications of the basic TMDD model. However, it should be noted that the basic TMDD model, itself, is an abstraction of the physiological reality because relevant nonlinear processes are assumed to occur in plasma and membranes do not represent significant distribution barriers. To overcome these limitations, several expansions of the Jusko and Mager model have been proposed which account for a more detailed description of the underlying (patho)physiology. , Some of the most pertinent features of these model expansions include consideration of tissue distribution, , the neonatal Fc receptor (FcRn), target shedding, immune response, receptor‐mediated endocytosis, multiple targets, dimerization, and competitive binding. The model by Li et al. integrates all of these features into an overarching modeling framework and has consequently become the default model which is widely used across physiologically‐based pharmacokinetic (PBPK) software platforms, such as Simcyp Simulator, GastroPlus, and PK‐Sim. Model 2: Expansion of the basic TMDD model to account for tissue distribution Tissue distribution of drugs is characterized in model 1 via a peripheral compartment into which ligand distributes at a rate k pt and out of at a rate k pt , respectively. However, this characterization is purely descriptive in nature and neglects the underlying physiological complexity of the tissue compartment which impacts the distribution of mAbs throughout the body. Model 2 structure To account for ligand distribution in tissue in a more mechanistic fashion, we expand upon the complexity of model 1 in model 2 (Figure ). In model 2, the plasma compartment is restricted to the central blood and plasma, whereas the peripheral blood, closer to the site of action, is designated as the vascular space compartment. Tissue is now subdivided into the endothelium and interstitial space to account for the underlying physiological structure. Target is still confined to plasma and associated TMDD processes retain their rate constants. The distribution of mAbs is primarily governed by convection and fluid moving from plasma to the “distal vascular space,” where they cross the endothelial layer into tissue (interstitial space). Movement to the interstitial compartment is driven by an osmotic pressure gradient and paracellular pores (i.e., cells between the endothelial cells). The nature of these pores is critical for tissue distribution because a decrease in pore size and/or increase in tortuosity of the paracellular path may create a greater impedance to drug movement. The resulting “sieving effect” is represented as a reflection coefficient ( σ ) in the mathematical model by Baxter et al. It captures differences between respective tissues. These concepts were formalized by Garg and Balthasar, who assumed that binding between drug and target only occurs in plasma, and that only the free drug can distribute into the tissue compartments. Equations and initial conditions of model 2 The mathematical formulation of model 2 comprises the three differential equations of model 1, which involve the receptor, and four turnover equations for the ligand concentration in each of the tissue compartments. Following Li et al., this results in a set of seven differential equations given by 7–11. The plasma compartment: (7) Exchange of ligand among the four tissue compartments adds convection terms to the ligand Equation involving the vascular and the lymph system. The vascular system (8) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v . where k up denotes the rate with which ligand is transferred to the endothelium. As observed, the flow rate Q is large compared to the other parameters, so that after a brief adjustment period, L p and L v move in lockstep, as might be expected in this physiological system. Specifically compared to model 1, the vascular compartment is the periphery of the plasma compartment, and these compartments readily equilibrate in the body. The endothelium (9) d L e d t = k up L v V v + δ k up L i V i − Cl int L e V e . Ligand concentration increases in the endothelium due to influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance. The expected concentration of ligand in this compartment will be determined by which of these two is more dominant. High clearance or low uptake rates might keep the concentration lower in this compartment. Low clearance, high uptake, and elevated vascular and interstitial concentrations would drive this concentration upward. The interstitial space (10) d L i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i . Loss of ligand to the endothelial space is accounted for by the first term ( δ k up L i V i ). Gain from the vascular system, governed by the term that drives flow to the lymphatic system, is modeled by the second term and loss to the lymphatic system by the third term. The lymphatic system (11) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Finally, the lymphatic system receives ligand from the interstitial space and loses ligand to the vascular system. Prior to the i.v. bolus dose, the system is assumed to be free of ligand. Therefore, the initial conditions are (12) where L 0 = D / V p , and R 0 is the baseline target concentration in the absence of ligand (5). Note that in model 2, besides the assumptions made in model 1, it is assumed that only free ligand can spread into the four peripheral compartments. Discussion of the properties of model 2 In order to acquire an impression of the disposition of the drug in the different compartments, as they evolve over time, we carry out a simulation for the parameters outlined in Supplementary Material . This simulation is shown in Figure . Here, we depict how the ligand concentration in each of the five tissue compartments develops over time after an i.v. bolus dose, D , is administered to the plasma compartment. The initial conditions are included in Equations and . We make the following observations about the graphs shown in Figure : Within a very short time, the ligand concentrations in plasma ( L p in red) and in the vascular space ( L v in dashed blue) coalesce and from then on stay together for all remaining time. More detailed simulations show that in this initial process L p and L v converge very rapidly with a half‐life of about 1 min or about 10 −3 days. Over this short period of time, effectively no ligand is lost and the complete dose D is divided between the plasma compartment and the vascular space. This results in a ligand concentration L ¯ equal to L ¯ = V p V p + V v ⋅ L 0 = 423 nM , where L 0 denotes the initial ligand concentration in plasma and is 650 nM. This is further discussed in Supplementary Material . Although the ligand concentration in the endothelium is very high, it contains only a relatively small amount of the initial dose, because the volume of the endothelium ( V e = 0.08 L) derived in Supplementary Material , is small compared to that of the plasma ( V p = 3 L). Over a larger time span the ligand concentration in the endothelium climbs to levels that are about twice the initial ligand concentration L ¯ , whereas the ligand concentrations in the endothelium, the interstitial space, and the lymph space, rise to no more than about one tenth of the initial concentration in plasma. Here, the endothelium serves as a protective sheath around the vascular system. The ligand versus time graphs shown in Figure on a semilogarithmic scale exhibit the typical ligand versus time graphs in plasma and in the vascular space, familiar from TMDD with a steep descent at about 7.5 days (Figure ). Simulations of the free target concentration show at the start that target rapidly binds the ligand ( t 1 / 2 ≈ 0.25 × 10 − 4 days; i.e., within 0.036 min). Subsequently, target stays fully saturated until L p becomes sufficiently small (i.e., drops steeply as shown in Figure ). From that time onward, the free target concentration settles on its baseline value R 0 and the ligand‐target complex eventually vanishes mono‐exponentially from the system. For the total target concentration R tot = R p + LR p , using Equation yields (13) d d t R p + LR p = k syn , p − k int , p LR p . By a short‐time analysis, it can be seen that ligand rapidly binds the target so that R p t ≈ 0 for t > δ , where δ = O 10 − 4 , until the ligand concentration had dropped enough. This means that for this period, Equation can be reduced to the equation (14) d L R p d t = k syn , p − k int , p LR p which can be solved explicitly, and yields, because LR p δ ≈ R 0 , the formula (15) In Figure it can be seen that as long as R p remains small, the graph of the ligand‐target complex, LR p , follows the graph of the function given by Equation 15, and climbs toward a maximum of R * . For our parameter values, we obtain R * = 83 nM . Model 3: Expanding model 2 to account for FcRn recycling The PKs of an mAb can be significantly affected by various physiological processes, such as receptor binding, binding to FcRn, immunogenicity, degree/nature of the glycosylation of the mAb, its susceptibility to proteolysis, and presence of multiple competing targets. In this section, we focus on the impact of FcRn binding on the disposition of ligand. The FcRn receptor is expressed in a wide variety of vascular endothelial cells in adult tissues and dendritic cells. Although the main physiological function of FcRn is to maintain ligand homeostasis by protecting the mAb from degradation, it also significantly reduces the clearance of structurally similar antibodies. As a consequence, FcRn‐mediated endothelial recycling of mAbs has been included in many PBPK models. , Model 3 structure Model 3 is obtained by adding the FcRn mechanism to the endothelium in model 2 (Figure ). Ligand in the endothelium may bind to the FcRn receptor depending on the ligand's dissociation constant ( K D , F = k off , F / k on , F ). Some of the FcRn‐bound ligand is recycled back to the vascular space and some is cleared into the interstitial space. These processes are characterized by the recycling rate of FcRn‐bound ligand ( k rc ) and the fraction of the ligand‐FcRn complex that is recycled into the vascular system (FR). Equations and initial conditions of model 3 The mathematical formulation of model 3 consists of the Equations , , , , , that are extended from model 2. Model 2 is comprised of equations for the plasma compartment, the vascular system, the endothelium, the interstitial space, and the lymph system. Here, we simply add the FcRn mechanism in the endothelium and the impact it has on the distribution of ligand in the vascular and the interstitial compartment. The plasma compartment (16) The vascular system (17) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . The last term in Equation results from the ligand in the endothelium that is recycled by the ligand‐FcRn complex into the vascular space. It is proportional to the concentration of bound ligand in the endothelium, f b L e = 1 − f u L e , the rate constant k rc , the fraction of this ligand (FR) which goes into the vascular space, and, finally, the ratio of the two volumes involved. Similarly, we add a term involving the fraction (1 − FR) to the ligand turnover equation for the interstitial space. In model 3, it is assumed that the QSS hypothesis holds for the FcRn receptor. If so, the fraction f u of ligand in the endothelium which is not bound to the FcRn‐receptor can be expressed as a function of the ligand concentration in the endothelium L e , the concentration R FcRn of the FcRn‐receptor, and the dissociation constant K D,F : (18) f u = 1 − 1 2 L e L e + R FcRn + K D , F − L e + R FcRn + K D , F 2 − 4 L e ⋅ R FcRn . This formula for the fraction of free ligand, and its role in the modeling of the impact of the FcRn receptor on the kinetics in the vascular space, the endothelium and the interstitial space, are explained in Supplementary Material . The endothelium (19) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . The ligand concentration increases here due to the influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance of the ligand that is not bound to the FcRn receptor, there is an additional term accounting for the decrease in free ligand due to the binding to FcRn, as well. The interstitial space (20) d L i d t = − δ k up L i V i + 1 − σ v L ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR ⋅ k rc 1 − f u L e V e V i . Here, the impact of the ligand recycling by the FcRn‐receptor is modeled by the last term. Note that if FR = 0, no ligand is recycled to the vascular space. If FR = 1, no ligand is recycled to the interstitial space but to the vascular space instead. The lymphatic system (21) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor does not recycle into the lymphatic system, Equation is the same as in model 2. For model 3, the initial concentrations are the same as for model 2: (22) where R 0 is given by Equation 5. Note that in this model the following assumptions have implicitly been made: (i) There is no endogenous generation of ligand. (ii) The receptor is located in the plasma (and not in the tissue). (iii) Binding of ligand to the receptor has a 1:1 stoichiometry. (iv) Only free ligand can move between the peripheral compartments. (v) The FcRn receptor is located in the endothelium. (vi) Ligand‐binding to the FcRn receptor is assumed to be in QSS. Discussion of the properties of model 3 The FcRn receptor which has been incorporated in model 3 is known to serve as a protective mechanism for the ligand in the tissue and increases its half‐life. This is seen in Figure where the concentration in the compartments stays elevated over a more extended period of time. Figures and , show the ligand and receptor concentrations, respectively, on a linear scale, whereas Figures and show the ligand and receptor concentrations, respectively, on a semilogarithmic scale. In these plots, the impact of the FcRn receptor is clearly visible. We make the following observations about the graphs in Figure : Compared to model 2, the ligand concentration (Figure ) has a lower peak in the endothelium compartment, is similar in the plasma and vascular compartments, and is higher in the interstitial and lymph compartments. In phase B, the slope is smaller in model 3 than in model 2 and the onset of nonlinearity is at a comparable ligand concentration in both the plasma and vascular spaces. This results in an increase of the time the ligand concentration in plasma remains larger than at the onset of nonlinearity by about 50%. After phase B, there is a return to the same terminal slopes seen in model 2 as shown after approximately day 30 in Figures and . The concentration of the ligand‐receptor complex remains higher over a longer period of time, lasting out until day 30 as opposed to peaking around day 7 in model 2. Model 4: Expansion of model 3 to account for target being located in tissue and target shedding In the models discussed thus far, it was assumed that target receptors are only located in plasma. This assumption does not hold for the majority of mAbs because their pharmacological targets are primarily located in tissue. In addition, membrane‐bound receptors can shed their ectodomains . This results in the coexistence of membrane‐bound and soluble (i.e., shed) targets in different physiological compartments of the body. For example, cluster differentiation (CD) 20 (target of rituximab, ibritumomab, and tositumomab), CD25 (target of basiliximab and daclizumab), and CD52 (target of alemtuzumab), the epidermal growth factor receptor (target of cetuximab and panitumumab), and the human epidermal growth factor receptor 2 (i.e., HER2, target of trastuzumab) all have membrane‐bound and soluble forms. It should also be noted that the corresponding mAb competes for binding to membrane‐bound and soluble receptors. Given that for trastuzumab only binding to membrane‐bound receptors can trigger pharmacological effects, this competition for target ultimately results in reduced or even no efficacy. Therefore, the impact of target shedding on target site receptor occupancy has to be considered when selecting an optimal dose for mAbs targeting membrane‐bound receptors. , , It should be noted, however, that there are also mAbs that specifically target soluble receptors. Therefore, the relationship between binding to membrane‐bound or soluble receptors and corresponding pharmacological effect needs to be carefully evaluated on a case‐by‐case basis. Model 4 structure Model 4 is expanded from model 3 by adding membrane bound receptors ( R m,i ) and soluble receptors ( R sh,i ) to the interstitial space. In addition, soluble receptors arrive in plasma via the lymph system ( R p , now resembling the shed receptors in plasma), where they can bind to ligand ( L p ) and form ligand‐shed receptor complexes ( LR sh,p ). Equations and initial conditions of model 4 Model 4 includes target shedding, as shown in Figure , and can be used to characterize the dynamic interplay among mAbs, their membrane‐bound target, and shed antigen. Synthesis and degradation of membrane bound target are characterized in this model by the respective zero‐order ( k syn,m ) and first‐order ( k deg,m ) rate constants. Shed antigen is formed in the interstitial space fluid at a rate, k sh , by cleavage of membrane‐bound receptor. From here, it can enter the systemic circulation either via diffusion across capillaries or via the lymph system ( λ ). Shed receptor degrades in both the interstitial space and plasma at rates ( k deg,sh , k deg,p ), respectively. Receptors (shed or membrane bound) compete for binding to drug ( K D,sh , K D,m , and k off,p / k on,p ). Elimination of drug‐receptor (membrane‐bound or soluble) complex is described by first‐order rate constants ( k int,sh , k int,p , and k int,m ). Effect of drug on target shedding was also incorporated in the model using αk sh , where α is the fraction of ligand‐bound, soluble receptor complexes added by shedding, and k sh is the first‐order shedding rate of the receptor. Whereas in model 4, we assume the drug does not have an effect on shedding (i.e., through a feedback mechanism) and set alpha equal to zero, changing α may account for the tumor's dynamic response to the drug, should that prove an observed phenomena via drug resistance. Additionally, the same mechanism of FcRn and ligand are also present in this model. The resulting mass balance equations describing the dynamics of drug and receptors are given by the following Equations 23–36. The plasma compartment (23) Note that R sh,p is sourced only through shedding in the interstitial compartment and movement through the lymph space into the plasma via λ . As compared to previous models, the target input into the plasma compartment comes exclusively from the shed receptors in the interstitial space, as the ligand is transported to the plasma (governed by λ ). This is why k syn,p has been removed from Figure . The vascular system (24) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . with the same definition of f u as model 3 (Equation ). The endothelium (25) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . This endothelial equation is also the same as in model 3 (Equation ). The interstitial space now contains two types of target: those bound to the membrane R m,i and those that have been “shed” (subscript: sh ) and circulate freely in the interstitial space, R sh,i . For these targets, we also add more explicit definitions of K D,sh and K D,m : (26) K D , sh = k off , sh k on , sh (27) K D , m = k off , m k on , m , where the off‐ and on‐rates for the ligand binding to the shed and the membrane bound receptors. For this tutorial, k off,sh = k off,m and k on,sh = k on,m . (28) In this system, k sh R m,i refers to shedding from free membrane‐bound target and αk sh LR m,i refers to drug‐induced partial return to free membrane‐bound target. At steady‐state, in the absence of ligand, the equilibrium concentrations of the three types of receptors, R m,i , R sh,i and R sh,p , are given by (29) R m , i = R 0 = def k syn , m k deg , m + k sh and R sh , i = k sh k deg , sh + λ ⋅ R 0 and by (30) R sh , p = λ k deg , p ⋅ k sh k deg , sh + λ ⋅ R 0 ⋅ V i V p . For the total ligand concentration in the interstitial space (31) L tot , i = L i + LR sh , i + LR m , i , we obtain, by adding the equations for L i , LR sh , i , and LR m , i , the following balance equation: (32) d L tot , i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR k rc 1 − f u V e ⋅ L e V i − k int , sh LR sh , i − k int , m + α k sh LR m , i . The total receptor concentration in the interstitial space is defined as: (33) R tot = R tot , m + R tot , sh , where (34) In combining, we obtain the following balance equation: (35) The lymphatic system (36) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor is membrane bound and thus does not recycle into the lymphatic system, this equation is the same as in models 2 and 3. Discussion of the properties of model 4 Model 4 shows a significant change in the long‐time behavior of the ligand because low levels of mAb remain in the system over an extended period of time. We make the following observations about the graphs in Figure for this model structure and parameterization: Compared to models 3 and 2, the ligand concentration (Figure ) has similar peaks in plasma and vascular compartments, compared to lower ones in the endothelial, interstitial, and lymph compartments. Compared to model 2, receptor concentrations are significantly lower and peak much sooner in model 4 (day 3 vs. day 7). Compared to model 3, the receptor concentrations are significantly lower in model 4, as they are being distributed across more states (membrane and shed) and there are added mechanisms of movement between tissues. Receptor concentrations also peak sooner (day 3 vs. day 18). The ligand concentration profiles are extended in time compared to model 2. But are dynamically different compared to model 3, resulting in less total exposure in model 4. For this parameterization, we have lost the characteristic TMDD profiles of the ligand and the target concentration versus time curves which we see for models 2 and 3 in Figures and (see also Supplementary Material ). The receptor is distributing across multiple compartments, making comparison to other models more difficult. However, it is clear that with this value of λ there is nearly no free receptor circulating in plasma and what arrives in plasma readily binds to ligand.   TMDD In the conceptual model by Mager and Jusko, which we refer to as the basic TMDD model, the formation of the mAb‐receptor complex in plasma ( LR p ) is characterized by the interaction between free ligand (i.e., mAb) in plasma ( L p ) and free target (i.e., receptor) in plasma ( R p ) as shown in Figure . For didactic purposes, L p is supplied to plasma via an intravenous (i.v.) bolus injection, recognizing that more complex input functions, such as subcutaneous or intramuscular administrations, may be needed to accurately reflect clinically relevant dosing scenarios. Once in plasma, ligand can either bind to target at a rate k on,p , be eliminated from plasma at a rate k el,p , or further distribute into tissue at a rate k pt . Ligand can also be redistributed from tissue into plasma at a rate k tp or be released from the ligand‐receptor complex at a rate k off . Equations and initial conditions for the basic TMDD model Mathematically, the basic TMDD model described in Figure translates into a set of four ordinary differential Equations , , , , which describe the kinetics of ligand in plasma ( L p ), ligand in tissue ( L t ), receptor in plasma ( R p ), and ligand‐receptor complex in plasma ( LR p ): (1) d L p d t = − k on , p L p ⋅ R p + k off , p LR p − k el , p + k pt L p + k tp ⋅ V t V p ⋅ L t (2) d L t d t = k pt ⋅ V p V t ⋅ L p − k tp L t (3) d R p d t = k syn , p − k on , p L p ⋅ R p + k off , p LR p − k deg , p R p (4) d LR p d t = k on , p L p ⋅ R p − k int , p + k off , p LR p In Equations and , V p and V t denote the volumes of distribution of, respectively, the plasma and the tissue compartment and k pt = Cl dist / V p and k tp = Cl dist / V t , where Cl dist denotes the intercompartmental clearance between plasma and tissue. Note that in the absence of ligand, the baseline receptor concentration R 0 is given by: (5) R 0 = def k syn , p k deg , p At time zero, the initial conditions for the four differential equations are as follows: (6) where D is the dose and L 0 is the ligand concentration in plasma right after the i.v. bolus dose. At this point, we would also like to note that the following implicit assumptions have been made for model 1: (i) There is no endogenous generation of ligand. (ii) The target is located in plasma (and not in tissue). (iii) Binding of ligand to the receptor has a 1:1 stoichiometry. Discussion of the properties of the basic TMDD model In general, nonlinearity can be caused by multiple processes and is typically dose‐dependent. Nonlinearity as a result of TMDD was first described by Levy and later expanded upon by Mager and Jusko. A mechanistic description of the underlying TMDD processes can provide valuable insight into the dynamics of the systems as a whole as opposed to the behavior of individual constituents. A dynamic systems analysis by Peletier and Gabrielsson , sheds light on the understanding of the different parts of the ligand concentration‐time profiles in plasma. This analysis focused on the primary drivers of the dynamics of the overall TMDD system. Hence, the original two‐compartment model by Mager and Jusko was simplified to a one‐compartment model because distribution between plasma and tissue is generally very fast and does not contribute significantly to the overall model behavior as the formation and dissociation of the ligand‐target complex takes place in plasma. We recognize that this is an oversimplification of physiological reality because most targets are situated in tissue rather than in plasma. Therefore, we will explore the impact of the target being located in tissue versus plasma in subsequent model expansions, especially in model 4. Model 1 captures a broad range of complex dynamics and provides a versatile and mechanistic framework for the systematic evaluation of capacity‐limited nonlinear drug disposition. The typical concentration‐time profile for a drug showing TMDD along with its characteristic parameters was described by Peletier and Gabrielsson in 2012 and is summarized in Figure . In panel (i), it is shown that the nonlinear elimination of mAbs is generally observed within a certain concentration range which can further be categorized via four different phases, A, B, C, and D (Figure ). In the first phase (phase A), ligand rapidly binds to its target. This binding process occurs at a high affinity and at high doses it saturates the target completely. Once the target is completely saturated, ligand is slowly disposed via a first‐order process (phase B), which is characteristic of the ligand's clearance from plasma. The third phase (phase C) is characterized by a Michaelis–Menten type kinetics, which is the result of the target no longer being completely saturated by the ligand. As concentrations continue to drop, the ligand starts to enter a fourth phase, the terminal phase (phase D), where it exhibits linear kinetics again. Each of these phases provides valuable insight into the kinetics of ligand, target, and ligand‐target complex, as well as into model parameters that are characteristic for each of these phases. Panel (ii) of Figure shows a typical graph of total clearance (CL TOT ) versus ligand concentration in plasma ( L p ) at steady‐state. For small values of L p , total clearance is the sum of two processes: (1) nonspecific clearance (CL L ) and (2) internalization of the target (CL I or CL int ). It should be noted that if the L p is large, the target is completely saturated and elimination only occurs via nonspecific, linear clearance (CL L ). This also means that a further increase in dose will yield no additional effect because all target receptors are already occupied. Approximations of the basic TMDD model In practice, it is often difficult to estimate each parameter of the basic TMDD model due to the lack of clinical information on receptor turnover and complex internalization. This motivates simplifications , also referred to as approximations, of the basic TMDD model. In this tutorial, we briefly mention three of them. The Quasi‐Equilibrium (QE) model. In 2005, Mager and Krzyzanski introduced this model, also referred to as the rapid binding model, in which it is assumed that equilibrium between complex formation and complex dissociation is rapidly established. This simplification is based on the assumption that the ligand‐target complex dissociates much faster than it is internalized. This assumption does not always hold true. The Quasi‐Steady‐State (QSS) model by Gibiansky et al. which assumes that complex formation and complex dissociation are at QSS. In other words, it assumes that complex formation is equal to the sum of complex dissociation and internalization. The newly defined constant ( K ss ) represents a generalization of K D . It has improved predictive properties over K D for extensive, high affinity binding scenarios characterized in phase B of the general TMDD profile (Figure ). The Michaelis–Menten (MM) model represents a special case of the QE and QSS models because it assumes that targets are initially fully saturated. Therefore, the MM model is good approximation of the basic TMDD model at high ligand concentrations. It also simplifies the basic TMDD model by relating the reaction rate to ligand concentration. The resulting model has improved predictive properties for both phases B and C, but typically fails to appropriately capture phase D. At this point, we have covered three simplifications of the basic TMDD model. However, it should be noted that the basic TMDD model, itself, is an abstraction of the physiological reality because relevant nonlinear processes are assumed to occur in plasma and membranes do not represent significant distribution barriers. To overcome these limitations, several expansions of the Jusko and Mager model have been proposed which account for a more detailed description of the underlying (patho)physiology. , Some of the most pertinent features of these model expansions include consideration of tissue distribution, , the neonatal Fc receptor (FcRn), target shedding, immune response, receptor‐mediated endocytosis, multiple targets, dimerization, and competitive binding. The model by Li et al. integrates all of these features into an overarching modeling framework and has consequently become the default model which is widely used across physiologically‐based pharmacokinetic (PBPK) software platforms, such as Simcyp Simulator, GastroPlus, and PK‐Sim.   TMDD model Mathematically, the basic TMDD model described in Figure translates into a set of four ordinary differential Equations , , , , which describe the kinetics of ligand in plasma ( L p ), ligand in tissue ( L t ), receptor in plasma ( R p ), and ligand‐receptor complex in plasma ( LR p ): (1) d L p d t = − k on , p L p ⋅ R p + k off , p LR p − k el , p + k pt L p + k tp ⋅ V t V p ⋅ L t (2) d L t d t = k pt ⋅ V p V t ⋅ L p − k tp L t (3) d R p d t = k syn , p − k on , p L p ⋅ R p + k off , p LR p − k deg , p R p (4) d LR p d t = k on , p L p ⋅ R p − k int , p + k off , p LR p In Equations and , V p and V t denote the volumes of distribution of, respectively, the plasma and the tissue compartment and k pt = Cl dist / V p and k tp = Cl dist / V t , where Cl dist denotes the intercompartmental clearance between plasma and tissue. Note that in the absence of ligand, the baseline receptor concentration R 0 is given by: (5) R 0 = def k syn , p k deg , p At time zero, the initial conditions for the four differential equations are as follows: (6) where D is the dose and L 0 is the ligand concentration in plasma right after the i.v. bolus dose. At this point, we would also like to note that the following implicit assumptions have been made for model 1: (i) There is no endogenous generation of ligand. (ii) The target is located in plasma (and not in tissue). (iii) Binding of ligand to the receptor has a 1:1 stoichiometry.   TMDD model In general, nonlinearity can be caused by multiple processes and is typically dose‐dependent. Nonlinearity as a result of TMDD was first described by Levy and later expanded upon by Mager and Jusko. A mechanistic description of the underlying TMDD processes can provide valuable insight into the dynamics of the systems as a whole as opposed to the behavior of individual constituents. A dynamic systems analysis by Peletier and Gabrielsson , sheds light on the understanding of the different parts of the ligand concentration‐time profiles in plasma. This analysis focused on the primary drivers of the dynamics of the overall TMDD system. Hence, the original two‐compartment model by Mager and Jusko was simplified to a one‐compartment model because distribution between plasma and tissue is generally very fast and does not contribute significantly to the overall model behavior as the formation and dissociation of the ligand‐target complex takes place in plasma. We recognize that this is an oversimplification of physiological reality because most targets are situated in tissue rather than in plasma. Therefore, we will explore the impact of the target being located in tissue versus plasma in subsequent model expansions, especially in model 4. Model 1 captures a broad range of complex dynamics and provides a versatile and mechanistic framework for the systematic evaluation of capacity‐limited nonlinear drug disposition. The typical concentration‐time profile for a drug showing TMDD along with its characteristic parameters was described by Peletier and Gabrielsson in 2012 and is summarized in Figure . In panel (i), it is shown that the nonlinear elimination of mAbs is generally observed within a certain concentration range which can further be categorized via four different phases, A, B, C, and D (Figure ). In the first phase (phase A), ligand rapidly binds to its target. This binding process occurs at a high affinity and at high doses it saturates the target completely. Once the target is completely saturated, ligand is slowly disposed via a first‐order process (phase B), which is characteristic of the ligand's clearance from plasma. The third phase (phase C) is characterized by a Michaelis–Menten type kinetics, which is the result of the target no longer being completely saturated by the ligand. As concentrations continue to drop, the ligand starts to enter a fourth phase, the terminal phase (phase D), where it exhibits linear kinetics again. Each of these phases provides valuable insight into the kinetics of ligand, target, and ligand‐target complex, as well as into model parameters that are characteristic for each of these phases. Panel (ii) of Figure shows a typical graph of total clearance (CL TOT ) versus ligand concentration in plasma ( L p ) at steady‐state. For small values of L p , total clearance is the sum of two processes: (1) nonspecific clearance (CL L ) and (2) internalization of the target (CL I or CL int ). It should be noted that if the L p is large, the target is completely saturated and elimination only occurs via nonspecific, linear clearance (CL L ). This also means that a further increase in dose will yield no additional effect because all target receptors are already occupied.   TMDD model In practice, it is often difficult to estimate each parameter of the basic TMDD model due to the lack of clinical information on receptor turnover and complex internalization. This motivates simplifications , also referred to as approximations, of the basic TMDD model. In this tutorial, we briefly mention three of them. The Quasi‐Equilibrium (QE) model. In 2005, Mager and Krzyzanski introduced this model, also referred to as the rapid binding model, in which it is assumed that equilibrium between complex formation and complex dissociation is rapidly established. This simplification is based on the assumption that the ligand‐target complex dissociates much faster than it is internalized. This assumption does not always hold true. The Quasi‐Steady‐State (QSS) model by Gibiansky et al. which assumes that complex formation and complex dissociation are at QSS. In other words, it assumes that complex formation is equal to the sum of complex dissociation and internalization. The newly defined constant ( K ss ) represents a generalization of K D . It has improved predictive properties over K D for extensive, high affinity binding scenarios characterized in phase B of the general TMDD profile (Figure ). The Michaelis–Menten (MM) model represents a special case of the QE and QSS models because it assumes that targets are initially fully saturated. Therefore, the MM model is good approximation of the basic TMDD model at high ligand concentrations. It also simplifies the basic TMDD model by relating the reaction rate to ligand concentration. The resulting model has improved predictive properties for both phases B and C, but typically fails to appropriately capture phase D. At this point, we have covered three simplifications of the basic TMDD model. However, it should be noted that the basic TMDD model, itself, is an abstraction of the physiological reality because relevant nonlinear processes are assumed to occur in plasma and membranes do not represent significant distribution barriers. To overcome these limitations, several expansions of the Jusko and Mager model have been proposed which account for a more detailed description of the underlying (patho)physiology. , Some of the most pertinent features of these model expansions include consideration of tissue distribution, , the neonatal Fc receptor (FcRn), target shedding, immune response, receptor‐mediated endocytosis, multiple targets, dimerization, and competitive binding. The model by Li et al. integrates all of these features into an overarching modeling framework and has consequently become the default model which is widely used across physiologically‐based pharmacokinetic (PBPK) software platforms, such as Simcyp Simulator, GastroPlus, and PK‐Sim.   TMDD model to account for tissue distribution Tissue distribution of drugs is characterized in model 1 via a peripheral compartment into which ligand distributes at a rate k pt and out of at a rate k pt , respectively. However, this characterization is purely descriptive in nature and neglects the underlying physiological complexity of the tissue compartment which impacts the distribution of mAbs throughout the body. Model 2 structure To account for ligand distribution in tissue in a more mechanistic fashion, we expand upon the complexity of model 1 in model 2 (Figure ). In model 2, the plasma compartment is restricted to the central blood and plasma, whereas the peripheral blood, closer to the site of action, is designated as the vascular space compartment. Tissue is now subdivided into the endothelium and interstitial space to account for the underlying physiological structure. Target is still confined to plasma and associated TMDD processes retain their rate constants. The distribution of mAbs is primarily governed by convection and fluid moving from plasma to the “distal vascular space,” where they cross the endothelial layer into tissue (interstitial space). Movement to the interstitial compartment is driven by an osmotic pressure gradient and paracellular pores (i.e., cells between the endothelial cells). The nature of these pores is critical for tissue distribution because a decrease in pore size and/or increase in tortuosity of the paracellular path may create a greater impedance to drug movement. The resulting “sieving effect” is represented as a reflection coefficient ( σ ) in the mathematical model by Baxter et al. It captures differences between respective tissues. These concepts were formalized by Garg and Balthasar, who assumed that binding between drug and target only occurs in plasma, and that only the free drug can distribute into the tissue compartments. Equations and initial conditions of model 2 The mathematical formulation of model 2 comprises the three differential equations of model 1, which involve the receptor, and four turnover equations for the ligand concentration in each of the tissue compartments. Following Li et al., this results in a set of seven differential equations given by 7–11. The plasma compartment: (7) Exchange of ligand among the four tissue compartments adds convection terms to the ligand Equation involving the vascular and the lymph system. The vascular system (8) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v . where k up denotes the rate with which ligand is transferred to the endothelium. As observed, the flow rate Q is large compared to the other parameters, so that after a brief adjustment period, L p and L v move in lockstep, as might be expected in this physiological system. Specifically compared to model 1, the vascular compartment is the periphery of the plasma compartment, and these compartments readily equilibrate in the body. The endothelium (9) d L e d t = k up L v V v + δ k up L i V i − Cl int L e V e . Ligand concentration increases in the endothelium due to influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance. The expected concentration of ligand in this compartment will be determined by which of these two is more dominant. High clearance or low uptake rates might keep the concentration lower in this compartment. Low clearance, high uptake, and elevated vascular and interstitial concentrations would drive this concentration upward. The interstitial space (10) d L i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i . Loss of ligand to the endothelial space is accounted for by the first term ( δ k up L i V i ). Gain from the vascular system, governed by the term that drives flow to the lymphatic system, is modeled by the second term and loss to the lymphatic system by the third term. The lymphatic system (11) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Finally, the lymphatic system receives ligand from the interstitial space and loses ligand to the vascular system. Prior to the i.v. bolus dose, the system is assumed to be free of ligand. Therefore, the initial conditions are (12) where L 0 = D / V p , and R 0 is the baseline target concentration in the absence of ligand (5). Note that in model 2, besides the assumptions made in model 1, it is assumed that only free ligand can spread into the four peripheral compartments. Discussion of the properties of model 2 In order to acquire an impression of the disposition of the drug in the different compartments, as they evolve over time, we carry out a simulation for the parameters outlined in Supplementary Material . This simulation is shown in Figure . Here, we depict how the ligand concentration in each of the five tissue compartments develops over time after an i.v. bolus dose, D , is administered to the plasma compartment. The initial conditions are included in Equations and . We make the following observations about the graphs shown in Figure : Within a very short time, the ligand concentrations in plasma ( L p in red) and in the vascular space ( L v in dashed blue) coalesce and from then on stay together for all remaining time. More detailed simulations show that in this initial process L p and L v converge very rapidly with a half‐life of about 1 min or about 10 −3 days. Over this short period of time, effectively no ligand is lost and the complete dose D is divided between the plasma compartment and the vascular space. This results in a ligand concentration L ¯ equal to L ¯ = V p V p + V v ⋅ L 0 = 423 nM , where L 0 denotes the initial ligand concentration in plasma and is 650 nM. This is further discussed in Supplementary Material . Although the ligand concentration in the endothelium is very high, it contains only a relatively small amount of the initial dose, because the volume of the endothelium ( V e = 0.08 L) derived in Supplementary Material , is small compared to that of the plasma ( V p = 3 L). Over a larger time span the ligand concentration in the endothelium climbs to levels that are about twice the initial ligand concentration L ¯ , whereas the ligand concentrations in the endothelium, the interstitial space, and the lymph space, rise to no more than about one tenth of the initial concentration in plasma. Here, the endothelium serves as a protective sheath around the vascular system. The ligand versus time graphs shown in Figure on a semilogarithmic scale exhibit the typical ligand versus time graphs in plasma and in the vascular space, familiar from TMDD with a steep descent at about 7.5 days (Figure ). Simulations of the free target concentration show at the start that target rapidly binds the ligand ( t 1 / 2 ≈ 0.25 × 10 − 4 days; i.e., within 0.036 min). Subsequently, target stays fully saturated until L p becomes sufficiently small (i.e., drops steeply as shown in Figure ). From that time onward, the free target concentration settles on its baseline value R 0 and the ligand‐target complex eventually vanishes mono‐exponentially from the system. For the total target concentration R tot = R p + LR p , using Equation yields (13) d d t R p + LR p = k syn , p − k int , p LR p . By a short‐time analysis, it can be seen that ligand rapidly binds the target so that R p t ≈ 0 for t > δ , where δ = O 10 − 4 , until the ligand concentration had dropped enough. This means that for this period, Equation can be reduced to the equation (14) d L R p d t = k syn , p − k int , p LR p which can be solved explicitly, and yields, because LR p δ ≈ R 0 , the formula (15) In Figure it can be seen that as long as R p remains small, the graph of the ligand‐target complex, LR p , follows the graph of the function given by Equation 15, and climbs toward a maximum of R * . For our parameter values, we obtain R * = 83 nM .   To account for ligand distribution in tissue in a more mechanistic fashion, we expand upon the complexity of model 1 in model 2 (Figure ). In model 2, the plasma compartment is restricted to the central blood and plasma, whereas the peripheral blood, closer to the site of action, is designated as the vascular space compartment. Tissue is now subdivided into the endothelium and interstitial space to account for the underlying physiological structure. Target is still confined to plasma and associated TMDD processes retain their rate constants. The distribution of mAbs is primarily governed by convection and fluid moving from plasma to the “distal vascular space,” where they cross the endothelial layer into tissue (interstitial space). Movement to the interstitial compartment is driven by an osmotic pressure gradient and paracellular pores (i.e., cells between the endothelial cells). The nature of these pores is critical for tissue distribution because a decrease in pore size and/or increase in tortuosity of the paracellular path may create a greater impedance to drug movement. The resulting “sieving effect” is represented as a reflection coefficient ( σ ) in the mathematical model by Baxter et al. It captures differences between respective tissues. These concepts were formalized by Garg and Balthasar, who assumed that binding between drug and target only occurs in plasma, and that only the free drug can distribute into the tissue compartments.   The mathematical formulation of model 2 comprises the three differential equations of model 1, which involve the receptor, and four turnover equations for the ligand concentration in each of the tissue compartments. Following Li et al., this results in a set of seven differential equations given by 7–11. The plasma compartment: (7) Exchange of ligand among the four tissue compartments adds convection terms to the ligand Equation involving the vascular and the lymph system. The vascular system (8) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v . where k up denotes the rate with which ligand is transferred to the endothelium. As observed, the flow rate Q is large compared to the other parameters, so that after a brief adjustment period, L p and L v move in lockstep, as might be expected in this physiological system. Specifically compared to model 1, the vascular compartment is the periphery of the plasma compartment, and these compartments readily equilibrate in the body. The endothelium (9) d L e d t = k up L v V v + δ k up L i V i − Cl int L e V e . Ligand concentration increases in the endothelium due to influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance. The expected concentration of ligand in this compartment will be determined by which of these two is more dominant. High clearance or low uptake rates might keep the concentration lower in this compartment. Low clearance, high uptake, and elevated vascular and interstitial concentrations would drive this concentration upward. The interstitial space (10) d L i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i . Loss of ligand to the endothelial space is accounted for by the first term ( δ k up L i V i ). Gain from the vascular system, governed by the term that drives flow to the lymphatic system, is modeled by the second term and loss to the lymphatic system by the third term. The lymphatic system (11) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Finally, the lymphatic system receives ligand from the interstitial space and loses ligand to the vascular system. Prior to the i.v. bolus dose, the system is assumed to be free of ligand. Therefore, the initial conditions are (12) where L 0 = D / V p , and R 0 is the baseline target concentration in the absence of ligand (5). Note that in model 2, besides the assumptions made in model 1, it is assumed that only free ligand can spread into the four peripheral compartments.   (7) Exchange of ligand among the four tissue compartments adds convection terms to the ligand Equation involving the vascular and the lymph system.   (8) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v . where k up denotes the rate with which ligand is transferred to the endothelium. As observed, the flow rate Q is large compared to the other parameters, so that after a brief adjustment period, L p and L v move in lockstep, as might be expected in this physiological system. Specifically compared to model 1, the vascular compartment is the periphery of the plasma compartment, and these compartments readily equilibrate in the body.   (9) d L e d t = k up L v V v + δ k up L i V i − Cl int L e V e . Ligand concentration increases in the endothelium due to influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance. The expected concentration of ligand in this compartment will be determined by which of these two is more dominant. High clearance or low uptake rates might keep the concentration lower in this compartment. Low clearance, high uptake, and elevated vascular and interstitial concentrations would drive this concentration upward.   (10) d L i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i . Loss of ligand to the endothelial space is accounted for by the first term ( δ k up L i V i ). Gain from the vascular system, governed by the term that drives flow to the lymphatic system, is modeled by the second term and loss to the lymphatic system by the third term.   (11) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Finally, the lymphatic system receives ligand from the interstitial space and loses ligand to the vascular system. Prior to the i.v. bolus dose, the system is assumed to be free of ligand. Therefore, the initial conditions are (12) where L 0 = D / V p , and R 0 is the baseline target concentration in the absence of ligand (5). Note that in model 2, besides the assumptions made in model 1, it is assumed that only free ligand can spread into the four peripheral compartments.   In order to acquire an impression of the disposition of the drug in the different compartments, as they evolve over time, we carry out a simulation for the parameters outlined in Supplementary Material . This simulation is shown in Figure . Here, we depict how the ligand concentration in each of the five tissue compartments develops over time after an i.v. bolus dose, D , is administered to the plasma compartment. The initial conditions are included in Equations and . We make the following observations about the graphs shown in Figure : Within a very short time, the ligand concentrations in plasma ( L p in red) and in the vascular space ( L v in dashed blue) coalesce and from then on stay together for all remaining time. More detailed simulations show that in this initial process L p and L v converge very rapidly with a half‐life of about 1 min or about 10 −3 days. Over this short period of time, effectively no ligand is lost and the complete dose D is divided between the plasma compartment and the vascular space. This results in a ligand concentration L ¯ equal to L ¯ = V p V p + V v ⋅ L 0 = 423 nM , where L 0 denotes the initial ligand concentration in plasma and is 650 nM. This is further discussed in Supplementary Material . Although the ligand concentration in the endothelium is very high, it contains only a relatively small amount of the initial dose, because the volume of the endothelium ( V e = 0.08 L) derived in Supplementary Material , is small compared to that of the plasma ( V p = 3 L). Over a larger time span the ligand concentration in the endothelium climbs to levels that are about twice the initial ligand concentration L ¯ , whereas the ligand concentrations in the endothelium, the interstitial space, and the lymph space, rise to no more than about one tenth of the initial concentration in plasma. Here, the endothelium serves as a protective sheath around the vascular system. The ligand versus time graphs shown in Figure on a semilogarithmic scale exhibit the typical ligand versus time graphs in plasma and in the vascular space, familiar from TMDD with a steep descent at about 7.5 days (Figure ). Simulations of the free target concentration show at the start that target rapidly binds the ligand ( t 1 / 2 ≈ 0.25 × 10 − 4 days; i.e., within 0.036 min). Subsequently, target stays fully saturated until L p becomes sufficiently small (i.e., drops steeply as shown in Figure ). From that time onward, the free target concentration settles on its baseline value R 0 and the ligand‐target complex eventually vanishes mono‐exponentially from the system. For the total target concentration R tot = R p + LR p , using Equation yields (13) d d t R p + LR p = k syn , p − k int , p LR p . By a short‐time analysis, it can be seen that ligand rapidly binds the target so that R p t ≈ 0 for t > δ , where δ = O 10 − 4 , until the ligand concentration had dropped enough. This means that for this period, Equation can be reduced to the equation (14) d L R p d t = k syn , p − k int , p LR p which can be solved explicitly, and yields, because LR p δ ≈ R 0 , the formula (15) In Figure it can be seen that as long as R p remains small, the graph of the ligand‐target complex, LR p , follows the graph of the function given by Equation 15, and climbs toward a maximum of R * . For our parameter values, we obtain R * = 83 nM .   FcRn recycling The PKs of an mAb can be significantly affected by various physiological processes, such as receptor binding, binding to FcRn, immunogenicity, degree/nature of the glycosylation of the mAb, its susceptibility to proteolysis, and presence of multiple competing targets. In this section, we focus on the impact of FcRn binding on the disposition of ligand. The FcRn receptor is expressed in a wide variety of vascular endothelial cells in adult tissues and dendritic cells. Although the main physiological function of FcRn is to maintain ligand homeostasis by protecting the mAb from degradation, it also significantly reduces the clearance of structurally similar antibodies. As a consequence, FcRn‐mediated endothelial recycling of mAbs has been included in many PBPK models. , Model 3 structure Model 3 is obtained by adding the FcRn mechanism to the endothelium in model 2 (Figure ). Ligand in the endothelium may bind to the FcRn receptor depending on the ligand's dissociation constant ( K D , F = k off , F / k on , F ). Some of the FcRn‐bound ligand is recycled back to the vascular space and some is cleared into the interstitial space. These processes are characterized by the recycling rate of FcRn‐bound ligand ( k rc ) and the fraction of the ligand‐FcRn complex that is recycled into the vascular system (FR). Equations and initial conditions of model 3 The mathematical formulation of model 3 consists of the Equations , , , , , that are extended from model 2. Model 2 is comprised of equations for the plasma compartment, the vascular system, the endothelium, the interstitial space, and the lymph system. Here, we simply add the FcRn mechanism in the endothelium and the impact it has on the distribution of ligand in the vascular and the interstitial compartment. The plasma compartment (16) The vascular system (17) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . The last term in Equation results from the ligand in the endothelium that is recycled by the ligand‐FcRn complex into the vascular space. It is proportional to the concentration of bound ligand in the endothelium, f b L e = 1 − f u L e , the rate constant k rc , the fraction of this ligand (FR) which goes into the vascular space, and, finally, the ratio of the two volumes involved. Similarly, we add a term involving the fraction (1 − FR) to the ligand turnover equation for the interstitial space. In model 3, it is assumed that the QSS hypothesis holds for the FcRn receptor. If so, the fraction f u of ligand in the endothelium which is not bound to the FcRn‐receptor can be expressed as a function of the ligand concentration in the endothelium L e , the concentration R FcRn of the FcRn‐receptor, and the dissociation constant K D,F : (18) f u = 1 − 1 2 L e L e + R FcRn + K D , F − L e + R FcRn + K D , F 2 − 4 L e ⋅ R FcRn . This formula for the fraction of free ligand, and its role in the modeling of the impact of the FcRn receptor on the kinetics in the vascular space, the endothelium and the interstitial space, are explained in Supplementary Material . The endothelium (19) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . The ligand concentration increases here due to the influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance of the ligand that is not bound to the FcRn receptor, there is an additional term accounting for the decrease in free ligand due to the binding to FcRn, as well. The interstitial space (20) d L i d t = − δ k up L i V i + 1 − σ v L ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR ⋅ k rc 1 − f u L e V e V i . Here, the impact of the ligand recycling by the FcRn‐receptor is modeled by the last term. Note that if FR = 0, no ligand is recycled to the vascular space. If FR = 1, no ligand is recycled to the interstitial space but to the vascular space instead. The lymphatic system (21) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor does not recycle into the lymphatic system, Equation is the same as in model 2. For model 3, the initial concentrations are the same as for model 2: (22) where R 0 is given by Equation 5. Note that in this model the following assumptions have implicitly been made: (i) There is no endogenous generation of ligand. (ii) The receptor is located in the plasma (and not in the tissue). (iii) Binding of ligand to the receptor has a 1:1 stoichiometry. (iv) Only free ligand can move between the peripheral compartments. (v) The FcRn receptor is located in the endothelium. (vi) Ligand‐binding to the FcRn receptor is assumed to be in QSS. Discussion of the properties of model 3 The FcRn receptor which has been incorporated in model 3 is known to serve as a protective mechanism for the ligand in the tissue and increases its half‐life. This is seen in Figure where the concentration in the compartments stays elevated over a more extended period of time. Figures and , show the ligand and receptor concentrations, respectively, on a linear scale, whereas Figures and show the ligand and receptor concentrations, respectively, on a semilogarithmic scale. In these plots, the impact of the FcRn receptor is clearly visible. We make the following observations about the graphs in Figure : Compared to model 2, the ligand concentration (Figure ) has a lower peak in the endothelium compartment, is similar in the plasma and vascular compartments, and is higher in the interstitial and lymph compartments. In phase B, the slope is smaller in model 3 than in model 2 and the onset of nonlinearity is at a comparable ligand concentration in both the plasma and vascular spaces. This results in an increase of the time the ligand concentration in plasma remains larger than at the onset of nonlinearity by about 50%. After phase B, there is a return to the same terminal slopes seen in model 2 as shown after approximately day 30 in Figures and . The concentration of the ligand‐receptor complex remains higher over a longer period of time, lasting out until day 30 as opposed to peaking around day 7 in model 2.   Model 3 is obtained by adding the FcRn mechanism to the endothelium in model 2 (Figure ). Ligand in the endothelium may bind to the FcRn receptor depending on the ligand's dissociation constant ( K D , F = k off , F / k on , F ). Some of the FcRn‐bound ligand is recycled back to the vascular space and some is cleared into the interstitial space. These processes are characterized by the recycling rate of FcRn‐bound ligand ( k rc ) and the fraction of the ligand‐FcRn complex that is recycled into the vascular system (FR).   The mathematical formulation of model 3 consists of the Equations , , , , , that are extended from model 2. Model 2 is comprised of equations for the plasma compartment, the vascular system, the endothelium, the interstitial space, and the lymph system. Here, we simply add the FcRn mechanism in the endothelium and the impact it has on the distribution of ligand in the vascular and the interstitial compartment. The plasma compartment (16) The vascular system (17) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . The last term in Equation results from the ligand in the endothelium that is recycled by the ligand‐FcRn complex into the vascular space. It is proportional to the concentration of bound ligand in the endothelium, f b L e = 1 − f u L e , the rate constant k rc , the fraction of this ligand (FR) which goes into the vascular space, and, finally, the ratio of the two volumes involved. Similarly, we add a term involving the fraction (1 − FR) to the ligand turnover equation for the interstitial space. In model 3, it is assumed that the QSS hypothesis holds for the FcRn receptor. If so, the fraction f u of ligand in the endothelium which is not bound to the FcRn‐receptor can be expressed as a function of the ligand concentration in the endothelium L e , the concentration R FcRn of the FcRn‐receptor, and the dissociation constant K D,F : (18) f u = 1 − 1 2 L e L e + R FcRn + K D , F − L e + R FcRn + K D , F 2 − 4 L e ⋅ R FcRn . This formula for the fraction of free ligand, and its role in the modeling of the impact of the FcRn receptor on the kinetics in the vascular space, the endothelium and the interstitial space, are explained in Supplementary Material . The endothelium (19) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . The ligand concentration increases here due to the influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance of the ligand that is not bound to the FcRn receptor, there is an additional term accounting for the decrease in free ligand due to the binding to FcRn, as well. The interstitial space (20) d L i d t = − δ k up L i V i + 1 − σ v L ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR ⋅ k rc 1 − f u L e V e V i . Here, the impact of the ligand recycling by the FcRn‐receptor is modeled by the last term. Note that if FR = 0, no ligand is recycled to the vascular space. If FR = 1, no ligand is recycled to the interstitial space but to the vascular space instead. The lymphatic system (21) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor does not recycle into the lymphatic system, Equation is the same as in model 2. For model 3, the initial concentrations are the same as for model 2: (22) where R 0 is given by Equation 5. Note that in this model the following assumptions have implicitly been made: (i) There is no endogenous generation of ligand. (ii) The receptor is located in the plasma (and not in the tissue). (iii) Binding of ligand to the receptor has a 1:1 stoichiometry. (iv) Only free ligand can move between the peripheral compartments. (v) The FcRn receptor is located in the endothelium. (vi) Ligand‐binding to the FcRn receptor is assumed to be in QSS.   (16)   (17) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . The last term in Equation results from the ligand in the endothelium that is recycled by the ligand‐FcRn complex into the vascular space. It is proportional to the concentration of bound ligand in the endothelium, f b L e = 1 − f u L e , the rate constant k rc , the fraction of this ligand (FR) which goes into the vascular space, and, finally, the ratio of the two volumes involved. Similarly, we add a term involving the fraction (1 − FR) to the ligand turnover equation for the interstitial space. In model 3, it is assumed that the QSS hypothesis holds for the FcRn receptor. If so, the fraction f u of ligand in the endothelium which is not bound to the FcRn‐receptor can be expressed as a function of the ligand concentration in the endothelium L e , the concentration R FcRn of the FcRn‐receptor, and the dissociation constant K D,F : (18) f u = 1 − 1 2 L e L e + R FcRn + K D , F − L e + R FcRn + K D , F 2 − 4 L e ⋅ R FcRn . This formula for the fraction of free ligand, and its role in the modeling of the impact of the FcRn receptor on the kinetics in the vascular space, the endothelium and the interstitial space, are explained in Supplementary Material .   (19) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . The ligand concentration increases here due to the influx from the adjacent compartments, the vascular and the interstitial compartments, and decreases due to systemic clearance of the ligand that is not bound to the FcRn receptor, there is an additional term accounting for the decrease in free ligand due to the binding to FcRn, as well.   (20) d L i d t = − δ k up L i V i + 1 − σ v L ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR ⋅ k rc 1 − f u L e V e V i . Here, the impact of the ligand recycling by the FcRn‐receptor is modeled by the last term. Note that if FR = 0, no ligand is recycled to the vascular space. If FR = 1, no ligand is recycled to the interstitial space but to the vascular space instead.   (21) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor does not recycle into the lymphatic system, Equation is the same as in model 2. For model 3, the initial concentrations are the same as for model 2: (22) where R 0 is given by Equation 5. Note that in this model the following assumptions have implicitly been made: (i) There is no endogenous generation of ligand. (ii) The receptor is located in the plasma (and not in the tissue). (iii) Binding of ligand to the receptor has a 1:1 stoichiometry. (iv) Only free ligand can move between the peripheral compartments. (v) The FcRn receptor is located in the endothelium. (vi) Ligand‐binding to the FcRn receptor is assumed to be in QSS.   The FcRn receptor which has been incorporated in model 3 is known to serve as a protective mechanism for the ligand in the tissue and increases its half‐life. This is seen in Figure where the concentration in the compartments stays elevated over a more extended period of time. Figures and , show the ligand and receptor concentrations, respectively, on a linear scale, whereas Figures and show the ligand and receptor concentrations, respectively, on a semilogarithmic scale. In these plots, the impact of the FcRn receptor is clearly visible. We make the following observations about the graphs in Figure : Compared to model 2, the ligand concentration (Figure ) has a lower peak in the endothelium compartment, is similar in the plasma and vascular compartments, and is higher in the interstitial and lymph compartments. In phase B, the slope is smaller in model 3 than in model 2 and the onset of nonlinearity is at a comparable ligand concentration in both the plasma and vascular spaces. This results in an increase of the time the ligand concentration in plasma remains larger than at the onset of nonlinearity by about 50%. After phase B, there is a return to the same terminal slopes seen in model 2 as shown after approximately day 30 in Figures and . The concentration of the ligand‐receptor complex remains higher over a longer period of time, lasting out until day 30 as opposed to peaking around day 7 in model 2.   In the models discussed thus far, it was assumed that target receptors are only located in plasma. This assumption does not hold for the majority of mAbs because their pharmacological targets are primarily located in tissue. In addition, membrane‐bound receptors can shed their ectodomains . This results in the coexistence of membrane‐bound and soluble (i.e., shed) targets in different physiological compartments of the body. For example, cluster differentiation (CD) 20 (target of rituximab, ibritumomab, and tositumomab), CD25 (target of basiliximab and daclizumab), and CD52 (target of alemtuzumab), the epidermal growth factor receptor (target of cetuximab and panitumumab), and the human epidermal growth factor receptor 2 (i.e., HER2, target of trastuzumab) all have membrane‐bound and soluble forms. It should also be noted that the corresponding mAb competes for binding to membrane‐bound and soluble receptors. Given that for trastuzumab only binding to membrane‐bound receptors can trigger pharmacological effects, this competition for target ultimately results in reduced or even no efficacy. Therefore, the impact of target shedding on target site receptor occupancy has to be considered when selecting an optimal dose for mAbs targeting membrane‐bound receptors. , , It should be noted, however, that there are also mAbs that specifically target soluble receptors. Therefore, the relationship between binding to membrane‐bound or soluble receptors and corresponding pharmacological effect needs to be carefully evaluated on a case‐by‐case basis. Model 4 structure Model 4 is expanded from model 3 by adding membrane bound receptors ( R m,i ) and soluble receptors ( R sh,i ) to the interstitial space. In addition, soluble receptors arrive in plasma via the lymph system ( R p , now resembling the shed receptors in plasma), where they can bind to ligand ( L p ) and form ligand‐shed receptor complexes ( LR sh,p ). Equations and initial conditions of model 4 Model 4 includes target shedding, as shown in Figure , and can be used to characterize the dynamic interplay among mAbs, their membrane‐bound target, and shed antigen. Synthesis and degradation of membrane bound target are characterized in this model by the respective zero‐order ( k syn,m ) and first‐order ( k deg,m ) rate constants. Shed antigen is formed in the interstitial space fluid at a rate, k sh , by cleavage of membrane‐bound receptor. From here, it can enter the systemic circulation either via diffusion across capillaries or via the lymph system ( λ ). Shed receptor degrades in both the interstitial space and plasma at rates ( k deg,sh , k deg,p ), respectively. Receptors (shed or membrane bound) compete for binding to drug ( K D,sh , K D,m , and k off,p / k on,p ). Elimination of drug‐receptor (membrane‐bound or soluble) complex is described by first‐order rate constants ( k int,sh , k int,p , and k int,m ). Effect of drug on target shedding was also incorporated in the model using αk sh , where α is the fraction of ligand‐bound, soluble receptor complexes added by shedding, and k sh is the first‐order shedding rate of the receptor. Whereas in model 4, we assume the drug does not have an effect on shedding (i.e., through a feedback mechanism) and set alpha equal to zero, changing α may account for the tumor's dynamic response to the drug, should that prove an observed phenomena via drug resistance. Additionally, the same mechanism of FcRn and ligand are also present in this model. The resulting mass balance equations describing the dynamics of drug and receptors are given by the following Equations 23–36. The plasma compartment (23) Note that R sh,p is sourced only through shedding in the interstitial compartment and movement through the lymph space into the plasma via λ . As compared to previous models, the target input into the plasma compartment comes exclusively from the shed receptors in the interstitial space, as the ligand is transported to the plasma (governed by λ ). This is why k syn,p has been removed from Figure . The vascular system (24) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . with the same definition of f u as model 3 (Equation ). The endothelium (25) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . This endothelial equation is also the same as in model 3 (Equation ). The interstitial space now contains two types of target: those bound to the membrane R m,i and those that have been “shed” (subscript: sh ) and circulate freely in the interstitial space, R sh,i . For these targets, we also add more explicit definitions of K D,sh and K D,m : (26) K D , sh = k off , sh k on , sh (27) K D , m = k off , m k on , m , where the off‐ and on‐rates for the ligand binding to the shed and the membrane bound receptors. For this tutorial, k off,sh = k off,m and k on,sh = k on,m . (28) In this system, k sh R m,i refers to shedding from free membrane‐bound target and αk sh LR m,i refers to drug‐induced partial return to free membrane‐bound target. At steady‐state, in the absence of ligand, the equilibrium concentrations of the three types of receptors, R m,i , R sh,i and R sh,p , are given by (29) R m , i = R 0 = def k syn , m k deg , m + k sh and R sh , i = k sh k deg , sh + λ ⋅ R 0 and by (30) R sh , p = λ k deg , p ⋅ k sh k deg , sh + λ ⋅ R 0 ⋅ V i V p . For the total ligand concentration in the interstitial space (31) L tot , i = L i + LR sh , i + LR m , i , we obtain, by adding the equations for L i , LR sh , i , and LR m , i , the following balance equation: (32) d L tot , i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR k rc 1 − f u V e ⋅ L e V i − k int , sh LR sh , i − k int , m + α k sh LR m , i . The total receptor concentration in the interstitial space is defined as: (33) R tot = R tot , m + R tot , sh , where (34) In combining, we obtain the following balance equation: (35) The lymphatic system (36) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor is membrane bound and thus does not recycle into the lymphatic system, this equation is the same as in models 2 and 3. Discussion of the properties of model 4 Model 4 shows a significant change in the long‐time behavior of the ligand because low levels of mAb remain in the system over an extended period of time. We make the following observations about the graphs in Figure for this model structure and parameterization: Compared to models 3 and 2, the ligand concentration (Figure ) has similar peaks in plasma and vascular compartments, compared to lower ones in the endothelial, interstitial, and lymph compartments. Compared to model 2, receptor concentrations are significantly lower and peak much sooner in model 4 (day 3 vs. day 7). Compared to model 3, the receptor concentrations are significantly lower in model 4, as they are being distributed across more states (membrane and shed) and there are added mechanisms of movement between tissues. Receptor concentrations also peak sooner (day 3 vs. day 18). The ligand concentration profiles are extended in time compared to model 2. But are dynamically different compared to model 3, resulting in less total exposure in model 4. For this parameterization, we have lost the characteristic TMDD profiles of the ligand and the target concentration versus time curves which we see for models 2 and 3 in Figures and (see also Supplementary Material ). The receptor is distributing across multiple compartments, making comparison to other models more difficult. However, it is clear that with this value of λ there is nearly no free receptor circulating in plasma and what arrives in plasma readily binds to ligand.   Model 4 is expanded from model 3 by adding membrane bound receptors ( R m,i ) and soluble receptors ( R sh,i ) to the interstitial space. In addition, soluble receptors arrive in plasma via the lymph system ( R p , now resembling the shed receptors in plasma), where they can bind to ligand ( L p ) and form ligand‐shed receptor complexes ( LR sh,p ).   Model 4 includes target shedding, as shown in Figure , and can be used to characterize the dynamic interplay among mAbs, their membrane‐bound target, and shed antigen. Synthesis and degradation of membrane bound target are characterized in this model by the respective zero‐order ( k syn,m ) and first‐order ( k deg,m ) rate constants. Shed antigen is formed in the interstitial space fluid at a rate, k sh , by cleavage of membrane‐bound receptor. From here, it can enter the systemic circulation either via diffusion across capillaries or via the lymph system ( λ ). Shed receptor degrades in both the interstitial space and plasma at rates ( k deg,sh , k deg,p ), respectively. Receptors (shed or membrane bound) compete for binding to drug ( K D,sh , K D,m , and k off,p / k on,p ). Elimination of drug‐receptor (membrane‐bound or soluble) complex is described by first‐order rate constants ( k int,sh , k int,p , and k int,m ). Effect of drug on target shedding was also incorporated in the model using αk sh , where α is the fraction of ligand‐bound, soluble receptor complexes added by shedding, and k sh is the first‐order shedding rate of the receptor. Whereas in model 4, we assume the drug does not have an effect on shedding (i.e., through a feedback mechanism) and set alpha equal to zero, changing α may account for the tumor's dynamic response to the drug, should that prove an observed phenomena via drug resistance. Additionally, the same mechanism of FcRn and ligand are also present in this model. The resulting mass balance equations describing the dynamics of drug and receptors are given by the following Equations 23–36. The plasma compartment (23) Note that R sh,p is sourced only through shedding in the interstitial compartment and movement through the lymph space into the plasma via λ . As compared to previous models, the target input into the plasma compartment comes exclusively from the shed receptors in the interstitial space, as the ligand is transported to the plasma (governed by λ ). This is why k syn,p has been removed from Figure . The vascular system (24) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . with the same definition of f u as model 3 (Equation ). The endothelium (25) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . This endothelial equation is also the same as in model 3 (Equation ). The interstitial space now contains two types of target: those bound to the membrane R m,i and those that have been “shed” (subscript: sh ) and circulate freely in the interstitial space, R sh,i . For these targets, we also add more explicit definitions of K D,sh and K D,m : (26) K D , sh = k off , sh k on , sh (27) K D , m = k off , m k on , m , where the off‐ and on‐rates for the ligand binding to the shed and the membrane bound receptors. For this tutorial, k off,sh = k off,m and k on,sh = k on,m . (28) In this system, k sh R m,i refers to shedding from free membrane‐bound target and αk sh LR m,i refers to drug‐induced partial return to free membrane‐bound target. At steady‐state, in the absence of ligand, the equilibrium concentrations of the three types of receptors, R m,i , R sh,i and R sh,p , are given by (29) R m , i = R 0 = def k syn , m k deg , m + k sh and R sh , i = k sh k deg , sh + λ ⋅ R 0 and by (30) R sh , p = λ k deg , p ⋅ k sh k deg , sh + λ ⋅ R 0 ⋅ V i V p . For the total ligand concentration in the interstitial space (31) L tot , i = L i + LR sh , i + LR m , i , we obtain, by adding the equations for L i , LR sh , i , and LR m , i , the following balance equation: (32) d L tot , i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR k rc 1 − f u V e ⋅ L e V i − k int , sh LR sh , i − k int , m + α k sh LR m , i . The total receptor concentration in the interstitial space is defined as: (33) R tot = R tot , m + R tot , sh , where (34) In combining, we obtain the following balance equation: (35) The lymphatic system (36) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor is membrane bound and thus does not recycle into the lymphatic system, this equation is the same as in models 2 and 3.   (23) Note that R sh,p is sourced only through shedding in the interstitial compartment and movement through the lymph space into the plasma via λ . As compared to previous models, the target input into the plasma compartment comes exclusively from the shed receptors in the interstitial space, as the ligand is transported to the plasma (governed by λ ). This is why k syn,p has been removed from Figure .   (24) d L v d t = Q ⋅ L p − Q − S L v − 1 − σ v S ⋅ L v V v − k up L v + FR ⋅ k rc 1 − f u V e ⋅ L e V v . with the same definition of f u as model 3 (Equation ).   (25) d L e d t = k up L v V v + δ k up L i V i − f u Cl int L e V e − k rc 1 − f u L e . This endothelial equation is also the same as in model 3 (Equation ). The interstitial space now contains two types of target: those bound to the membrane R m,i and those that have been “shed” (subscript: sh ) and circulate freely in the interstitial space, R sh,i . For these targets, we also add more explicit definitions of K D,sh and K D,m : (26) K D , sh = k off , sh k on , sh (27) K D , m = k off , m k on , m , where the off‐ and on‐rates for the ligand binding to the shed and the membrane bound receptors. For this tutorial, k off,sh = k off,m and k on,sh = k on,m . (28) In this system, k sh R m,i refers to shedding from free membrane‐bound target and αk sh LR m,i refers to drug‐induced partial return to free membrane‐bound target. At steady‐state, in the absence of ligand, the equilibrium concentrations of the three types of receptors, R m,i , R sh,i and R sh,p , are given by (29) R m , i = R 0 = def k syn , m k deg , m + k sh and R sh , i = k sh k deg , sh + λ ⋅ R 0 and by (30) R sh , p = λ k deg , p ⋅ k sh k deg , sh + λ ⋅ R 0 ⋅ V i V p . For the total ligand concentration in the interstitial space (31) L tot , i = L i + LR sh , i + LR m , i , we obtain, by adding the equations for L i , LR sh , i , and LR m , i , the following balance equation: (32) d L tot , i d t = − δ k up L i V i + 1 − σ v S ⋅ L v − 1 − σ i S ⋅ L i V i + 1 − FR k rc 1 − f u V e ⋅ L e V i − k int , sh LR sh , i − k int , m + α k sh LR m , i . The total receptor concentration in the interstitial space is defined as: (33) R tot = R tot , m + R tot , sh , where (34) In combining, we obtain the following balance equation: (35)   (36) d L s d t = 1 − σ i S ⋅ L i − S ⋅ L s V s . Because the FcRn receptor is membrane bound and thus does not recycle into the lymphatic system, this equation is the same as in models 2 and 3.   Model 4 shows a significant change in the long‐time behavior of the ligand because low levels of mAb remain in the system over an extended period of time. We make the following observations about the graphs in Figure for this model structure and parameterization: Compared to models 3 and 2, the ligand concentration (Figure ) has similar peaks in plasma and vascular compartments, compared to lower ones in the endothelial, interstitial, and lymph compartments. Compared to model 2, receptor concentrations are significantly lower and peak much sooner in model 4 (day 3 vs. day 7). Compared to model 3, the receptor concentrations are significantly lower in model 4, as they are being distributed across more states (membrane and shed) and there are added mechanisms of movement between tissues. Receptor concentrations also peak sooner (day 3 vs. day 18). The ligand concentration profiles are extended in time compared to model 2. But are dynamically different compared to model 3, resulting in less total exposure in model 4. For this parameterization, we have lost the characteristic TMDD profiles of the ligand and the target concentration versus time curves which we see for models 2 and 3 in Figures and (see also Supplementary Material ). The receptor is distributing across multiple compartments, making comparison to other models more difficult. However, it is clear that with this value of λ there is nearly no free receptor circulating in plasma and what arrives in plasma readily binds to ligand.   Trastuzumab is an mAb targeting the HER2 receptor, which plays an important role in cell growth and division. HER2 overexpression leads to strong proliferation signaling and tumor formation. Trastuzumab binds to the extracellular domain of the HER2 receptor resulting in reduced downstream signaling and tumor growth. , Tokuda et al. conducted a phase‐I PK dose escalation study of trastuzumab in 18 patients with metastatic breast cancer. Patients received doses of 1, 2, 4, and 8 mg/kg as single 90‐min i.v. infusions. The data suggests that trastuzumab clearance decreases from 14.1 (at a dose of 1 mg/kg) to 5.6 ml/kg/day (at a dose of 8 mg/kg). This dose‐dependent nonlinearity has been previously characterized by Li et al. using a PBPK modeling approach. We will now use the data from Tokuda et al. to evaluate the performance of models 1–4 via simulations. For the sake of model comparison, we used the same set of parameter values across all four models. Respective parameter values originated from the most complex model (i.e., model 4 which is based on Li et al., and can be found in Supplementary Material ). We also updated the i.v. bolus administration used during modeling framework development to a 90‐min i.v. infusion to account for the clinically used dosing regimen. When overlaying the simulations made by models 1–4 with the Tokuda et al. data, we can see that the basic TMDD model (model 1) captures the clinical data very well (Figure ) and performs better than models 2–4. Specifically, the drug is cleared faster from the system than suggested by the clinical data for model 2 (Figure ), whereas the opposite holds true for the three highest dose levels of model 3 (Figure ). Model 4 captures the higher dose levels very well, whereas simulations for the lower doses approach a steady‐state in plasma that is not supported by the clinical data (Figure ). These mispredictions warrant parameter adjustment or optimization. To guide the modeler's choice of which parameter or which set of parameters to change, we conducted a local sensitivity analysis, included in Supplementary Material . Of all the parameters evaluated, k up , k rc , λ , and k sh were most sensitive. When we now use this set of sensitive parameters to adjust the parameterization of models 2–4, the following results are obtained. For model 2, the faster than expected clearance of drug from the system could be addressed by a reduction in k up (Figure : 0.0298 to 0.00536 h −1 ). This is not a surprising finding because the uptake into the tissue has been very rapid with the original parameterization of k up , thus leaving more drug in the plasma compartment. The added benefit of using model 2 compared to model 1 is that we can now capture the presence of ligand in additional tissue compartments, where we might expect the ligand to reside. For model 3, the overprediction of the three highest drug concentrations as the result of too extensive FcRn recycling could be addressed by reducing the FcRn recycling rate, k rc (Figure : 16.7 to 1.7 h −1 ). This is also not surprising because FcRn serves as a protective mechanism allowing the antibody to reside for a longer period of time in tissue. Reducing the time the antibody spends recycling in tissue will increase its clearance. The added benefit of using model 3 compared to model 1 is that we can capture both the location of the ligand while still accounting for the well‐established effect of FcRn recycling on overall drug kinetics.For model 4, a number of adjustments had to be made, as shown in Figure . First, lowering the FcRn recycling rate ( k rc 16.7 to 3.13 h −1 ) helped to fit the data. This finding is consistent with the parameter optimization results of model 3 and suggests that an increased residence time of drug in the endothelium is important for the overall model dynamics. This finding is also supported by the fact that an increase in k up (0.0298 to 0.0507 h −1 ) results in more free ligand moving to the endothelium improves the model fit. Finally, an increase in λ (0.01 to 0.5) suggests that the transfer of shed antigen from tissue to plasma may play an important role for the overall kinetics in plasma. We acknowledge, however, that the required change in λ was quite drastic and may require further investigation because physiological flowrates are usually confined to a much tighter parameter range. Specifically, this investigation may require additional experimental evidence that allows dissection of this hybrid parameter into physiologically‐plausible lymph flow and drug‐specific antibody transfer rate constants. The overall advantage of using model 4 compared to model 1 is that one can now also estimate the concentration of the membrane‐bound ligand‐receptor complex in tissue ( LR m,i ) which may enable a better understanding of the target site PK/pharmacodynamic (PD) relationships, particularly when highly nonlinear dynamics are governing the rate and extent of mAb binding to its target.   From this tutorial, it is evident that characterizing the dose‐exposure‐response relationship of mAbs is nontrivial and that the complexity of the approach needed is highly dependent on the research question to be answered. Typical questions in drug development and regulatory evaluation include, but are not limited to: What is a reasonable starting dose? What is an optimal dose, considering both efficacy and safety, although safety is oftentimes less of a concern for mAbs? What happens if we change the dose? Are we still in the therapeutic range? Are we still in the linear range? What if we modulate the FcRn binding affinity of our new drug candidate? Is there variability in target receptor expression? If so, what are the consequences? Does target shedding change over time and can shed antigen in plasma be used to make inferences about target site mAb kinetics? Which experimental design should we use for a particular question? What should be measured, when, and where? Addressing these questions in isolation is neither time‐ nor cost‐efficient and can be in part misleading because they are inter‐related. Integration of information through the use of quantitative models is consequently key when attempting to streamline the drug development and evaluation process. At the same time, the limited availability of data during initial stages of the development program may force us to start simple and step‐wise increase the complexity of the model as the development program matures. In this tutorial, we provide a roadmap of how to approach this challenge by outlining the setup of an overarching modeling framework that compares and contrasts the relative importance of key pharmacological processes governing mAbs PKs at various level of physiological complexity. The framework not only provides the conceptual basis for understanding basic TMDD behavior (model 1), tissue distribution (model 2), the impact of FcRn recycling (model 3), and tissue target sites (model 4) but also sets stage for further model expansion to evaluate the impact of, for example, endogenous immunoglobulin G (IgG), immunogenicity, glycosylation patterns, competing targets, and downstream PD effects. Prior to allocating time and resources to a full‐fleshed model expansion, conducting sensitivity analyses can provide valuable insights into the relative importance of a particular mechanism for the overall dynamics of the system. To this end, we conducted initial sensitivity analyses for our trastuzumab example to evaluate the impact of different parameters, such as the concentration and affinity of the FcRn receptor in models 3 and 4 as well as the shedding rate in model 4, on the overall mAb kinetics. The findings of this sensitivity analysis show that the sensitivity of these parameters can vary significantly and should, thus, be interpreted in a physiological context. In this tutorial, we provide a roadmap of how to strike a balance between physiological and mathematical complexity in light of the research question asked. We also provide the reader with the tools to make educated choices in this regard and set the stage for further model parameter refinement as well as potential future model expansions. Coding resources This work was developed in MATLAB version 2021b and R. Data were digitized with open‐source software: Plot Digitizer 2.6.9 ( http://plotdigitizer.sourceforge.net/ ).   This work was developed in MATLAB version 2021b and R. Data were digitized with open‐source software: Plot Digitizer 2.6.9 ( http://plotdigitizer.sourceforge.net/ ).   The authors acknowledge support funding one post‐doctoral associate working on the project from Janssen Pharmaceuticals of Johnson & Johnson.   The authors declared no competing interests for this work.   Supplementary Material S1. List of model assumptions. Supplementary Material S2. Parameter values for Models 1‐4. Supplementary Material S3. Mathematical flowrate analysis. Supplementary Material S4. Derivation of equation (3.18). Supplementary Material S5. Sensitivity analysis and comparison. Click here for additional data file. Data S1 Click here for additional data file.   