A deterministic mathematical model of HL-60 cell kinetics in presence of bispecific antibody and T cellWe designed and validated a deterministic mathematical model of HL-60 cell kinetics in presence of bispecific antibody and T cell (shown in Fig. 1A,B). Model equations are shown in System 3–6. Proposed mathematical models describe interaction kinetics among tumour HL-60 cell, T cell and bispecific antibody in two scenarios, including (A) bispecific antibody concentration is constant (shown in Fig. 1A, and System 3–6) and and (B) bispecific antibody concentration rapidly decays after infusion due to antibody clearance (shown in Fig. 1B, and System 7). First, in absence of bispecific antibody and T cell, proliferation of HL-60 cell subjects to logistic growth with birth rate \(\rho\) and maximum capacity \(\beta\) (Fig. S1, Sect. 1.1, supplimenary material). Next, because T cell regenerates naturally and die naturally at homeostasis, CD8 T cell concentration maintains constant and range 1.5 ~ 10 × 105 cells/ml35. Bispecific antibody binds to both T cell and HL-60 cell, induces T-cell activation, and then lyses HL-60 cells. In presence of antibody clearance, bispecific antibody is rapidly cleared and regularly infused to maintain its high levels (shown in Fig. 1B).Fig. 1A mechanistic mathematical model of HL-60 cell kinetics and bispecific antibody induced tumour-lysing. (A) Model schematic: HL-60 growth follows logistic growth and T cell concentration maintain constant at homeostasis. Bispecific antibodies bind to both HL-60 cell and T cell, induced T-cell activation, and initiate tumour-lysing activity. (B) For In vivo experiment, bispecific antibody is constantly cleared and supplied by antibody infusion.No existed mathematical models were proposed to understand effeciency of bispecific antibody induced tumour lysing, and here we proprose three categories First, the effeciency is assumed to linearly increase with increase of T cell, bispecific antibody and HL-60 cell concentration, which is modelled by law of mass action (given by System 3). Law of mass action assumes that the rate of a chemical reaction is directly proportional to the product of the activities or concentrations of the reactants36, which is widely used in theoretical immunology37. Next, the effeciency is assumed to increase but converge to maximun with increase of bispecific antibody, T cell or/ and HL-60 cell concentration, which is modelled by Hill equation (given by System 4 and 5). Hill function describes rate of a chemical reaction with saturation effect, which is commonly used in biochemistry and pharmacology36,38. Finally, it is modelled by Hill equation and exponential function (given by System 6).In this following context, we will select proper mathematical model (System 3–6) to delineate efficiency bispecific antibody induced tumour lysing by fitting in vitro cytotoxicity assay data and comparing sum of squared error (SSE), Akaike information criterion (AIC) and modifed AIC for small sample size (AICc). SSE measures goodness-of-fitting of proposed model against data. Given a collection of models for the data, AIC estimates the quality of each model, relative to each of the other models. AICc is AIC with punishment term of sample size to balance goodness-of-fitting, quality of model and small sample size. Next, in absence of antibody clearance, we will qualitatively understand how bispecific antibody and tumour burden determine HL-60 cell kinetics. Finally, we will investigate whether oberved bispecific induced bistable HL-60 kinetics still exist in presence of bispecific antibody clearance.Efficiency of bispecific antibody induced tumour lysing is increasing and saturated with increase of bispecific antibody, T cell and HL-60 concentrationEfficiency of bispecific antibody induced tumour lysing were quantified for HL-60 cell tested against bispecific antibody (ABL602 2 + 1) with purified human T cell, using in vitro assay allowing estimation of total HL-60 cell and remaining viable HL-60 cell concentration before and during 48-h incubation (see Methods). HL-60 cells were cocultured at different effector to target cell ratio (E:T ratio) of 5:1 with bispecific antibody (ABL602 2 + 1) with tenfold serial dilutions from 50 nM in 200uL medium (Panel HL-60 cell lysis percentage, Fig. 2C, Reference18). Figure 2C in Reference18 analysed mean of in vitro cytotoxicity data performed T cell harvested from eight healthy donors, and here we use three of them to analyse lysing efficiency.Fig. 2Quantitative relationship between HL-60 cell concentration and ABL602 2 + 1 bispecific antibody with different antibody concentration through cytotoxicity assay.To quantitatively understand efficiency of bispecific antibody induced tumour lysing, we developed four mathematical models to describe the rate of change of HL-60 cell, bispecific antibody and T cell concentration (See System 3 – System 6, Method). Surprisingly, the model allowing saturation with increase of HL-60 cell, T cell and bispecific antibody concentration (System 5, Method) fits cytotoxicity assay data (Fig. 2A), but unsaturated models (System 3), semi-saturated models (System 4) and saturated model with exponential function (System 6) cannot (Table S1-S4, Sect. 1.3. Supplementary Material). System 5 describes the rate of change of HL-60 cell, bispecific antibody and T cell concentration as \(\left\{\begin{array}{c}\frac{dT}{dt}=rT\left(1-\frac{T}{\beta }\right)-\frac{\kappa }{1+aA}\frac{1}{1+\gamma E+\eta T}ATE\\ \frac{dA}{dt}=-\frac{\varphi }{1+aA}\frac{1}{1+\gamma E+\eta T}ATE\\ \frac{dE}{dt}=-dE\end{array}\right.\). System 5 uses \(\frac{\kappa }{1+aA}\frac{1}{1+\gamma E+\eta T}ATE\) to understand lysing efficiency. \(\frac{\kappa }{1+aA}\) describes saturation in lysing efficiency with increase of antibody concentration for 48-h incubation when HL-60 (2 × 104 cells) and T cell number (105 cells) are fixed (shown in Panel HL-60 lysis percentage, Fig. 2C, Reference18); namely, HL-60 cell lysis percentage increase less rapidly on high antibody concentration interval (> 10–3 nM/ml) and very rapidly on antibody concentration interval (< 10–3 nM/ml, Fig. 2C, Reference18). \(\frac{1}{1+\gamma E+\eta T}\) describes saturation in lysing efficiency with increase of HL-60 and T cell concentration when antibody concentration is fixed. \(ATE\) describes rate of antibody binding and tumour lysing among HL-60 cell, T cell and bispecific antibody.System 5 fits well for cytotoxicity assay data performed T cell harvested from Donor 1–3 (shown in Fig. 2A–C) and geometric mean of cytotoxicity assay data of Donor 1–3 (shown in Fig. 2D) when bispecific antibody concentration is higher than 10–3 nM/ml (10–3 − 102 nM/ml) but does not fit well when antibody concentration is very low (10–5 − 10–3 nM/ml). This may be induced by small denominator dilemma (1 nano mole = 10–9 mol) and accumulated calculated error (for example, converting gram to mole dividing by molecular weight 1.5 × 105). However, current clinical trial administered 21 patients with a median age of 69.0 years have received bispecific antibody treatment dose levels of 0.1–7.5 mg/kg39. Because typical human weight is 80 kg and blood volume is 5000 ml, 21 patients received dose levels of 1.07 × 10–2—8 × 10–1 nm/ml, which corresponds to good fitting antibody concentration interval (green bar in Fig. 2A–D).Human T cells were isolated from the PBMC of three healthy donors using MACS beads18. In vitro cytotoxicity assay data performed with three healthy donors can be considered as three biological replicates. Thus, we can use maximum likelihood estimation or minimum squared error to estimate efficiency of bispecific antibody induced tumour lysing. To prevent individual differences of donors leading to estimation error, we selected minimum squared error. For example, the degree of T cell proliferation is highly variable from donors to donors, for example 2–3 folds increase for donors but 20 folds increase for others over the course of a seven-day culture40. If three biological replicates are performed by T cell from same donor, maximum likelihood estimation is a better candidate to select mathematical model and estimate parameters.Cytotoxicity assays data indicates that remaining viable HL-60 cell concentration decreases with increase of antibody concentration (2.5 × 10–5-2.5 × 102 nm/ml) at 48 h for Donor 1 (A), Donor 2 (B), Donor 3 (C) and geometric mean of Donor 1–3 (D) for 48 h. Red curve represents cytotoxicity assay data. Blue, purple, cyan and black curve represents estimated valued provided by saturated model (System 5), unsaturated model (System 3), semi-saturated model (System 4) and saturated model with experiment function (System 6). Green bar represents antibody concentration interval proposed by current clinical trial.Bispecific antibody induced bistable HL-60 cell kineticsIn above section, we found that bispecific antibody induced tumour lysing kinetics follows a saturated model, \(-\frac{\kappa }{1+aA}\frac{1}{1+\gamma E+\eta T}ATE\). In this section, using bifurcation analysis and numerical simulation, we found that bispecific antibody induced bistable HL-60 cell kinetics. Bistability means that for a given bispecific antibody concentration and T cell concentration, HL-60 cell kinetics with high tumour burden persists and that with low high tumour burden is supressed, whereas monostability means that persistence or inhibition of HL-60 cell kinetics is only determined by magnitude/intensity of bispecific antibody and human T cell concentration.T cell concentration of a healthy human maintains constant and range 2.62 ± 0.8 × 105 cells/ml, and here we chose T cell concentration 105 cells/ml, 2 × 105 cells/ml, 3 × 105 cells/ml and 4 × 105 cells/ml as examples. First, in simulations of HL-60 cell kinetics with T cell concentration 105 cells/ml, there existed of one threshold C* = 27.4 nm/ml that divided bispecific antibody concentration interval into two regimes, shown as a bifurcation diagram (Fig. 3C). At different antibody concentration intervals, HL-60 cell kinetics exhibited different dynamics, including monostability to HL-60 cell persistence and bistability between HL-60 cell persistence and inhibition (shown in Fig. 3C–G). HL-60 cell kinetics exhibited bistability at the antibody cell concentration interval between 0 and C*, where HL-60 cell kinetics with low tumour burdens were inhibited, and that with high tumour burdens persists at same bispecific antibody concentration (Fig. 3A,B,C). HL-60 cell concentration threshold (above which HL-60 cell concentration is not affected) increased with increase of antibody concentration (the red dashed curve in Fig. 3C). For example, at low antibody concentration C = 20 nm/mL, only HL-60 cell kinetics with high tumour burden 5 × 106 cells /mL and 7 × 106 cells /mL persists (Fig. 3A,C). At antibody concentration higher than the threshold C* = 27.4 nm/ml, HL-60 cell kinetics was inhibited independent of tumour burden (Fig. 3B,C).Fig. 3Simulated kinetics of HL-60 cell with different combinations of tumour burden, T cell and antibody concentration. (A) When T cell concentration was 105 cell/ml and antibody concentration was between 0 and C*, HL-60 cell kinetics with high tumour burden persists and that with low tumour burden is inhibited. (B) HL-60 kinetics with any tumour burden was inhibited when antibody concentration was greater than C*. Bifurcation diagram (C) showing HL-60 concentration as a function of antibody concentration. Tumour burden threshold increases with increase of antibody concentration (red dashed curve). The maximal capacity of HL-60 cell concentration decreases with increases of antibody concentration (black solid curve). Purple arrows represent any tumour burden. (D) Bifurcation diagram at T cell concentration 2 × 105 cells/ml provides threshold value C* = 13.7 nm/ml corresponding to good treatment outcomes. (E) Bifurcation diagram at T cell concentration 3 × 105 cells/ml provides threshold value C* = 9.1 nm/ml corresponding to good treatment outcomes. (F) Bifurcation diagram at T cell concentration 4 × 105 cells/ml provides threshold value C* = 6.8 nm/ml corresponding to good treatment outcomes. Blue, red, green, and purple lines and circles represent HL-60 cell kinetics with pro-infusion tumour burden 1 × 105 cells/mL, 3 × 105 cells /mL, 5 × 105 cells /mL and 7 × 105 cells /mL in (A–F). (G) 2D bifurcation diagram; x-axis refers ABL602 bispecific antibody concentration, y-axis refers T cell concentration and z-axis refer HL-60 cell concentration corresponding ABL602 bispecific antibody and T cell concentration. Cyen region and surface represent HL-60 cell persistence due to high per-infusion tumour burden. Yellow region and surface represent HL-60 cell inhibition due to high per-infusion tumour burden. Purple region and surface represent HL-60 cell inhibition due to high antibody and T cell concentration. Red dashed curve and surface represent critical value separating HL-60 cell persistence and inhibition due to per-infusion tumour burden.Next, the threshold value C* of bispecific antibody concentration corresponding to good treatment outcomes decreases with increase of T cell concentration (shown in Fig. 3D,E,F,G). Specifically, 2 × 105 cells/ml T cell concentration provide threshold value C* = 13.7 nm/ml (shown in Fig. 3D), 3 × 105 cells/ml T cell concentration provide threshold value C* = 9.1 nm/ml (shown in Fig. 3E), 4 × 105 cells/ml T cell concentration provide threshold value C* = 6.8 nm/ml (shown in Fig. 3F). 2D bifurcation diagram is given in Fig. 3G and describes HL-60 concentration corresponding to different bispecific antibody and T cell concentration.Bispecific antibody induced bistable HL-60 cell kinetics also exist in presence of antibody clearance.In above section, we found that bispecific antibody induced bistable HL-60 cell kinetics when antibody concentration is constant. For In vivo experiment, bispecific antibody rapidly decreases after bispecific antibody infusion, and its concentration cannot be considered as constant. In this section, we will understand whether observed bispecific antibody induced bistable HL-60 cell kinetics still exists in presence of bispecific antibody clearance.Reference18 administered bispecific antibody to HL-60 orthotopic mouse model with human T cell transfer twice a week. Reference39,41,42 administered bispecific antibody to human volunteer twice, fourth time and seven time a week. current clinical trial administered 21 patients with a median age of 69.0 years have received bispecific antibody treatment dose levels of 0.1–7.5 mg/kg, which approximately equal to 1.07 × 10–2—8 × 10–1 nm/ml39. Here, we propose four treatment scenarios, Group 1 will receive 1 nm/ml bispecific antibody seven times a week. Group 2 will receive 1 nm/ml bispecific antibody four times a week. Group 3 will receive 3 nm/ml bispecific antibody and seven times a week. Group 4 will receive 3 nm/ml bispecific antibody and four times a week. We assume that T cell concentration is 106 cell/ml.First, due to bispecific antibody clearance, Group 1 suppresses HL-60 kinetics with low tumour burden but not high tumour burden, thereby explaining observed phenomenon that bispecific antibody was less efficacious at high tumour burden even with enough activated cytotoxic CD8 + T cells (shown in Fig. 4A,C,D)21. In absence of antibody clearance, 1 nm/ml and 3 nm/ml bispecific antibody and 106 cells/ml T cell concentration can suppress HL-60 growth independent of any tumour burden (Fig. 3G). This indicate that antibody clearance can change antibody induced monostable HL-60 cell kinetics to antibody induced bistable HL-60 cell kinetics.Fig. 4Simulated HL-60 cell kinetics with different infusion dosage, infusion frequency and T cell concentration. (A) and (B) HL-60 cell and bispecific antibody kinetics with infusion dosage, infusion frequency and T cell concentration kinetics under Group 1. (C) and (D) HL-60 cell and bispecific antibody kinetics with infusion dosage, infusion frequency and T cell concentration kinetics under Group 2. (E) and (F) HL-60 cell and bispecific antibody kinetics under Group 3. (G) and (H) HL-60 cell and bispecific antibody kinetics under Group 4. Blue, red, green, and purple curve represent HL-60 cell kinetics with pro-infusion tumour burden 1 × 105 cells/mL, 3 × 105 cells /mL, 5 × 105 cells /mL and 7 × 105 cells /mL in (A–H). In Panel (B), (D), (F) and (H), blue, red, green, and purple curve overlaps, because initial infusion dosage and frequency are same. BsAB refers bispecific antibody concentration.Next, increased infusion frequency (Group 2, four 1 nm/ml antibody infusion in a week) cannot suppress HL-60 cell kinetics with any tumour burden (shown in Fig. 4C), because accumulated bispecific antibody concentration rapidly decreases. In addition, comparing with Group 1, 3 nm/ml antibody infusion (Group 3) can efficiently reduce maximum HL-60 cell concentration and per-infusion tumour burden corresponding to HL-60 cell persistence (shown in Fig. 4E). Similarly, comparing Group 3 and Group 4, enhanced infusion frequency can efficiently suppress maximum HL-60 cell concentration and per-infusion tumour burden corresponding to HL-60 cell persistence (shown Fig. 4G). Accumulated antibody concentration of Group 1–4 are given in Fig. 4B,D,F,H. Thus, reducing tumour burden, increasing infusion frequency and dosage, maintaining high T cell concentration can efficiently promote long-term AML control.For human volunteers, Reference39,41,42 administered bispecific antibody to twice, fourth time and seven time a week. Reference18 administered bispecific antibody to HL-60 orthotopic mouse model with human T cell transfer twice a week. We found that antibody clearance, antibody infusion dosage (1 nm/ml or 3 nm/ml), infusion frequency (four or seven infusion a week), and T cell concentration are key factors leading to different treatment outcomes. In absence of antibody clearance, 1 nm/mL antibody concentration and 106 cells/ml T cell concentration can suppress HL-60 growth independent any tumour burden. However, in presence of antibody clearance, seven 1 nm/ml antibody infusion can suppress HL-60 growth with low pro-infusion burden but not high tumour burden, because accumulated antibody concentration rapidly decay. Comparing seven 1 nm/ml antibody infusion, seven 3 nm/ml antibody infusion can reduce maximum HL-60 cell concentration and pro-infusion tumour burden corresponding HL-60 cell persistence, because accumulated antibody concentration is relatedly high. Moreover, T cell concentration is strongly correlated with HL-60 cell concentration (shown Fig. 3C–G). Thus, maintaining high T cell concentration and accumulated antibody concentration is equivalently important in bispecific antibody treatment success.
