T-DM1 ADC Example

[Ravua1992]

Hi everyone,

I'm currently working on a project where I aim to link my pop-PK model (in mrgsolve) with tumor growth inhibition (TGI) and other pharmacodynamics (PD) aspects. My ultimate goal is to calibrate drug-specific parameters to explore TGI based on drug-antibody ratio (DAR) or target expression to understand the sink effect etc.

As a first step, I'm trying to implement an already available model in mrgsolve
(https://github.com/metrumresearchgroup/PMSimulatorSimulate.jl/blob/main/test/ADCExample.jl)

While I've successfully implemented the PK part (full code below), I'm encountering issues with the TGI model. The predicted tumor diameter changes are significantly higher than expected, despite using the exact model and parameters provided.

I've reviewed the initial conditions and the drug's kill rate, but these haven't resolved the issue. I'm unsure where to go from here and would appreciate any suggestions on how to start debugging this problem.

Any insights or advice would be greatly appreciated!

Thanks in advance for your help.

Expected output ;

code<-'

$PROB

$PARAM @annotated
BW              : 70          : Body Weight (kg)
k_on_ADC        : 0.37        : 2nd-order association rate constant between T-DM1 and HER2 antigen ((nmol*L^-1)^-1*hr^-1)
k_off_ADC       : 0.097       : Dissociation rate constant between T-DM1 and HER2 antigen (hr^-1)
k_int_ADC       : 0.09        : Internalization rate of the HER2-T-DM1 complex inside the cell (hr^-1)
k_deg_ADC       : 0.03        : Proteasomal degradation rate of T-DM1 in endosomal/lysosomal space (hr^-1)
k_on_Tub        : 0.03        : 2nd-order association rate constant between DM1 catabolites and intracellular tubulin protein ((nmol*L^-1)^-1*hr^-1)
k_off_Tub       : 0.0285      : Dissociation rate constant between tubulin catabolites and tubulin protein (hr^-1)
Tub_total       : 65          : Total concentration of intracellular tubulin protein (nmol*L^-1)
k_dec_ADC       : 0.0223      : Non-specific deconjugation rate of T-DM1 from the extracellular space (hr^-1)
k_diff_Drug     : 0.092       : Bidirectional diffusion rate constant for DM1 catabolites (hr^-1)
k_out_Drug      : 0           : Active efflux rate constant of DM1 catabolites (hr^-1)
Ag_total        : 1660        : Total antigen expression levels (nmol*L^-1)
R_Cap           : 8E-4        : Radius of the tumor blood capillary (cm)
R_Krogh         : 7.5E-3      : Average distance between two capillaries (cm)
P_ADC           : 0.001391667    : Permeability rate of T-DM1 across the blood vessels (cm*hr^-1)
P_Drug          : 0.0875      : Permeability rate of DM1 catabolites across the blood vessels (cm*hr^-1)
D_ADC           : 9.167E-4    : Diffusion rate of T-DM1 (mm^2*hr^-1)
D_Drug          :  0.0104166   : Diffusion rate of DM1 catabolites (mm^2*hr^-1)
varepsilon_ADC  : 0.24        : Tumor void volume for T-DM1
varepsilon_Drug : 0.44        : Tumor void volume for DM1 catabolites
CL_ADC          : 1.792E-4    : Central clearance of ADC (L*hr^-1*kg^-1)
CLD_ADC         : 5.833E-4    : Distributional clearance of ADC (L*hr^-1*kg^-1)
V1_ADC          : 0.034       : Central volume of distribution of ADC (L*kg^-1)
V2_ADC          : 0.04        : Peripheral volume of distribution of ADC (L*kg^-1)
CL_Drug         : 0.09291667    : Central clearance of DM1 catabolites (L*hr^-1*kg^-1)
CLD_Drug        : 0.04166667   : Distributional clearance of DM1 catabolites (L*hr^-1*kg^-1)
V1_Drug         : 0.034       : Central volume of distribution of DM1 catabolites (L*kg^-1)
V2_Drug         : 5.0         : Peripheral volume of distribution of DM1 catabolites (L*kg^-1)
k_dec_ADC_plasma: 0.241*1/24     : Non-specific deconjugation rate constant for T-DM1 in the systemic circulation (hr^-1)
k_growth_exponential    : 0.2    : Exponential growth phase of the tumor (hr^-1)
k_growth_linear : 25.875      : Linear growth phase of the tumor (mm^3*hr^-1)
Psi             : 20.0        : Switch between exponential and linear growth phases
V_max           : 530*1000     : Maximum achievable tumor volume (mm^3)
k_kill_max      :0.139/24    : Linear killing constant (hr^-1)
KC_50           : 23.8        : Concentration of drug (payload) corresponding to half-maximal killing (nM)
BASE            :  2500 : Baseline  
DAR0:3.5: 
$CMT @annotated
X1_ADC_nmol      : Amount of T-DM1 in plasma central compartment (nmol)
X2_ADC_nmol      : Amount of T-DM1 in peripheral compartment (nmol)
C_ADC_f_ex_nM    : Concentration of T-DM1 in tumor extracellular space (nmol*L^-1)
C_ADC_b_ex_nM    : Concentration of T-DM1 bound to tumor cell surface (nmol*L^-1)
C_ADC_endolyso   : Concentration of internalized ADC in lysosomal/endosomal space (nmol*L^-1)
C_Drug_f_cell_nM : Concentration of unconjugated intracellular free DM1 (nmol*L^-1)
C_Drug_b_cell_nM : Concentration of unconjugated intracellular tubulin-bound DM1 (nmol*L^-1)
C_Drug_f_ex_nM   : Concentration of free DM1 in tumor extracellular space (nmol*L^-1)
C1_Drug_nM       : Concentration of DM1 in plasma central compartment (nmol*L^-1)
C2_Drug_nM       : Concentration of DM1 in peripheral compartment (nmol*L^-1)
DAR              : Average number of DM1 molecules conjugated to Trastuzumab (unitless)
TV_mm3           : Tumor volume (mm^3)

$MAIN

capture TAD = self.tad();
if (NEWIND<=1){
DAR_0 =0;
TV_mm3_0 = BASE ;
}

double PI = 3.141592653589793238462643383279502884197; 


$ODE

double R_tumor = pow(3.0 * TV_mm3 / (4.0 * PI), 1.0 / 3.0);

dxdt_DAR =-k_dec_ADC_plasma*DAR;

dxdt_X1_ADC_nmol = 
    -(CL_ADC/V1_ADC)*X1_ADC_nmol - (CLD_ADC/V1_ADC)*X1_ADC_nmol
    + (CLD_ADC/V2_ADC)*X2_ADC_nmol - k_dec_ADC_plasma*X1_ADC_nmol 
    - ((2*P_ADC*R_Cap)/(R_Krogh*R_Krogh))*((X1_ADC_nmol/(V1_ADC*BW))*varepsilon_ADC - C_ADC_f_ex_nM)*(TV_mm3*1E-6)
    - ((6*D_ADC)/(R_tumor*R_tumor))*((X1_ADC_nmol/(V1_ADC*BW))*varepsilon_ADC - C_ADC_f_ex_nM)*(TV_mm3*1E-6);

dxdt_X2_ADC_nmol = (CLD_ADC/V1_ADC)*X1_ADC_nmol - (CLD_ADC/V2_ADC)*X2_ADC_nmol;

dxdt_C_ADC_f_ex_nM = ((2*P_ADC*R_Cap)/(R_Krogh*R_Krogh))*((X1_ADC_nmol/(V1_ADC*BW))*varepsilon_ADC - C_ADC_f_ex_nM)
    + ((6*D_ADC)/(R_tumor*R_tumor))*((X1_ADC_nmol/(V1_ADC*BW))*varepsilon_ADC - C_ADC_f_ex_nM) 
    - k_on_ADC*C_ADC_f_ex_nM*((Ag_total - C_ADC_b_ex_nM)/varepsilon_ADC) + k_off_ADC*C_ADC_b_ex_nM - k_dec_ADC*C_ADC_f_ex_nM;

dxdt_C_ADC_b_ex_nM = k_on_ADC*C_ADC_f_ex_nM*((Ag_total - C_ADC_b_ex_nM)/varepsilon_ADC)
    - (k_off_ADC + k_int_ADC + k_dec_ADC)*C_ADC_b_ex_nM;

dxdt_C_ADC_endolyso = k_int_ADC*C_ADC_b_ex_nM - k_deg_ADC*C_ADC_endolyso;

dxdt_C_Drug_f_cell_nM = k_deg_ADC*DAR*C_ADC_endolyso - k_on_Tub*C_Drug_f_cell_nM*(Tub_total - C_Drug_b_cell_nM)
    + k_off_Tub*C_Drug_b_cell_nM - k_out_Drug*C_Drug_f_cell_nM + k_diff_Drug*(C_Drug_f_ex_nM - C_Drug_f_cell_nM);

dxdt_C_Drug_b_cell_nM = k_on_Tub*C_Drug_f_cell_nM*(Tub_total - C_Drug_b_cell_nM) - k_off_Tub*C_Drug_b_cell_nM;

dxdt_C_Drug_f_ex_nM = ((2*P_Drug*R_Cap)/(R_Krogh*R_Krogh))*(C1_Drug_nM*varepsilon_Drug - C_Drug_f_ex_nM)
    + ((6*D_Drug)/(R_tumor*R_tumor))*(C1_Drug_nM*varepsilon_Drug - C_Drug_f_ex_nM)
    + k_out_Drug*C_Drug_f_cell_nM + k_dec_ADC*DAR*(C_ADC_f_ex_nM + C_ADC_b_ex_nM) - k_diff_Drug*(C_Drug_f_ex_nM - C_Drug_f_cell_nM);

dxdt_C1_Drug_nM = -(CL_Drug/V1_Drug)*C1_Drug_nM - (CLD_Drug/V1_Drug)*C1_Drug_nM + (CLD_Drug/V2_Drug)*C2_Drug_nM
    + (X1_ADC_nmol*DAR*k_dec_ADC_plasma)/(V1_Drug*BW) + (CL_ADC*DAR*(X1_ADC_nmol/V1_ADC))/(V1_Drug*BW)
    - ((2*P_Drug*R_Cap)/(R_Krogh*R_Krogh))*(C1_Drug_nM*varepsilon_Drug - C_Drug_f_ex_nM)
    - ((6*D_Drug)/(R_tumor*R_tumor))*(C1_Drug_nM*varepsilon_Drug - C_Drug_f_ex_nM);

dxdt_C2_Drug_nM = (CLD_Drug/V2_Drug)*C1_Drug_nM - (CLD_Drug/V2_Drug)*C2_Drug_nM;



double A = pow(k_growth_exponential * (TV_mm3 / k_growth_linear), 20);
double Growth_TV = (k_growth_exponential * (1 - (TV_mm3 / V_max))) ;
double Growth_TV1=Growth_TV/ pow(1+A, 1.0 / 20);

double DrugEffect = (k_kill_max*(C_Drug_f_cell_nM + C_Drug_b_cell_nM) ) / 
                    (KC_50 + (C_Drug_f_cell_nM + C_Drug_b_cell_nM));

dxdt_TV_mm3 = (Growth_TV1 - DrugEffect) * TV_mm3;


double R_tumor1 = pow(3.0 * (TV_mm3/1000) / (4.0 * PI), 1.0 / 3.0);


double TumorDiameter =2 * R_tumor1;


$TABLE
double C1_ADC_nM = X1_ADC_nmol / (V1_ADC * BW);
double C2_ADC_nM = X2_ADC_nmol / (V2_ADC * BW);

double IntracellularPayload = C_Drug_f_cell_nM + C_Drug_b_cell_nM;
double ExtracellularFreePayload = C_Drug_f_ex_nM;

$CAPTURE C1_ADC_nM C2_ADC_nM TumorDiameter Growth_TV Growth_TV1 A

$CAPTURE C1_ADC_nM IntracellularPayload ExtracellularFreePayload TumorDiameter

'


library(mrgsolve)
library(ggplot2)

# Define model
mod <- mcode("example",code)
(init(mod))
#
#Initial conditions
dose <- 3.6 * 1E-3 * 70 / (148781 / 1E9) # Converted dose to nmol
DAR_0 <- 3.5
TV_0 <- (19 * 16^2) / 2 # Initial tumor volume in mm3

init <- list(
  X1_ADC_nmol = 0,
  X2_ADC_nmol = 0,
  DAR = DAR_0,
  TV_mm3_0 = TV_0
)

# Dosing events
evs <- ev(amt = c(dose,DAR_0), cmt = c(1,11), time = (0), ii = 168*3, addl = 2,evid=1)# %>% as.data.frame()


out <- mod %>%
  ev(evs) %>%
  mrgsim(start = 0, end = 65*24, delta = 0.1) #%>% as.data.frame() %>% mutate(TUM_DIA=2*((3*(TV_mm3/1000)/4*3.14)^1/3))



plot(out,A+Growth_TV+Growth_TV1+TV_mm3+TumorDiameter~time)
# Convert time from hours to days
out$time <- out$time / 24

# Plot 1: Plasma concentration
p1 <- ggplot(out, aes(x=time)) +
  geom_line(aes(y=C1_ADC_nM, color="ADC (T-DM1)")) +
  geom_line(aes(y=IntracellularPayload, color="Drug (Payload, DM-1)")) +
  scale_y_log10() +
  labs(title="Plasma Human PK (3.6 mg/kg Q3W)", x="Time (days)", y="Concentration (nM)") +
  theme_minimal() +
  scale_color_manual(values = c("blue", "green"))

# Plot 2: Tumor Payload
p2 <- ggplot(out, aes(x=time)) +
  geom_line(aes(y=IntracellularPayload, color="Intracellular payload")) +
  geom_line(aes(y=C_Drug_f_ex_nM, color="Extracellular free payload"), linetype="dashed") +
  geom_hline(yintercept = 23.8, color="gray", linetype="dotted") +
  labs(title="Tumor Payload (3.6 mg/kg Q3W)", x="Time (days)", y="Concentration (nM)") +
  theme_minimal() +
  geom_hline(yintercept = 25)+
  scale_color_manual(values = c("blue", "green"))

# Plot 3: Tumor Diameter
p3 <- 
  ggplot(out, aes(x=time, y=TumorDiameter)) +
  #geom_line(color="blue") +
 geom_line(aes(x=time, y=TUM_DIA)) +
  labs(title="Tumor diameter (cm)", x="Time (days)", y="Tumor Diameter (cm)") +
  theme_minimal()

# Display the plots
print(p1)
print(p2)
print(p3)

grid.arrange(p1,p2,p3)

[smouksassi]

you need to initialize the compartments within your code see here
https://mrgsolve.org/docs/articles/extra/init.html

[Ravua1992]

Thank you for the suggestion. I am setting the initial conditions in the compartment
I think the models initial conditions are comparable to the example, but the tumor volume seems to be increasing at a much faster rate. The PK looks comparable.

The TGI simulations also looks comparable when I adjust the vmax parameter which corresponds to the Maximum achievable tumor volume (mm^3)

I am trying to reproduce this

https://www.metrumrg.com/wp-content/uploads/2023/11/ACoP14-workshop-ADC-example.pdf

Any suggestions on how to proceed?

Thankyou,

; if (NEWIND<=1){
DAR_0 =0;
TV_mm3_0 = BASE ;
} 

/// Tumor Code 

double A = pow(k_growth_exponential * (TV_mm3 / k_growth_linear), 20);
double Growth_TV = (k_growth_exponential * (1 - (TV_mm3 *1/ V_max))) ;
double Growth_TV1=Growth_TV/ pow(1+A, 1.0 / 20);

double DrugEffect = (k_kill_max*(C_Drug_f_cell_nM + C_Drug_b_cell_nM) ) / 
                    (KC_50 + (C_Drug_f_cell_nM + C_Drug_b_cell_nM));

dxdt_TV_mm3 = ((k_growth_exponential * (1 - (TV_mm3 *1/ V_max)))/pow(1+ pow(k_growth_exponential * (TV_mm3 / k_growth_linear), 20), 1.0 / 20) - DrugEffect) * TV_mm3;

double R_tumor1 = pow(3.0 * (TV_mm3/1000) / (4.0 * PI), 1.0 / 3.0);

double TumorDiameter =2 * R_tumor1;

When I change the Vmax from 523.81000 to 5.231000 it seems to be working.

Original:

Its closer to this output which I am trying to reproduce;

[kylebaron]

Hi @Ravua1992 -

I'm just coming back from a week out of office. Will try to take a look this afternoon. Will update if I can find anything.

Kyle

[Ravua1992]

Thank you ! I think it’s the Tumor growth model. I am not sure if I am implementing the equation correctly. The radius at baseline is comparable between the outputs when running it in Julia and r , but the growth is different. Thanks again for your help! Get Outlook for iOS<https://aka.ms/o0ukef>

…

________________________________ From: Kyle Baron ***@***.***> Sent: Monday, August 26, 2024 9:22:33 AM To: metrumresearchgroup/mrgsolve ***@***.***> Cc: Ravua1992 ***@***.***>; Mention ***@***.***> Subject: Re: [metrumresearchgroup/mrgsolve] T-DM1 ADC Example (Discussion #1224) Hi @Ravua1992<https://github.com/Ravua1992> - I'm just coming back from a week out of office. Will try to take a look this afternoon. Will update if I can find anything. Kyle — Reply to this email directly, view it on GitHub<#1224 (comment)>, or unsubscribe<https://github.com/notifications/unsubscribe-auth/ANLFXQNW24CG52URWJG3K5TZTMTZTAVCNFSM6AAAAABM2P6B3SVHI2DSMVQWIX3LMV43URDJONRXK43TNFXW4Q3PNVWWK3TUHMYTANBVGE4TIMI>. You are receiving this because you were mentioned.Message ID: ***@***.***>

[kylebaron]

@Ravua1992 - I re-coded some of the math from the original Julia code just as a check; tumor growth is still looking more like the "original" profile posted above. Also confirmed your observation around V_max ... tweaking that seems to replicate more of the behavior seen in the pdf. I'll have to put it down for now, but will keep looking when I get a chance.

[Ravua1992]

Thank you!